Impossible Operations in the Quantum World-I and II

Debasis SarkarDebasis Sarkardsappmathcalunivacindsappmathcalunivacin

Department of Applied Mathematics Department of Applied Mathematics

University of CalcuttaUniversity of Calcutta

Impossible Impossible Operations in the Operations in the Quantum World-I Quantum World-I

and IIand II

A List of Impossible A List of Impossible Operations (No Go Operations (No Go

Principles)Principles) 1 Impossibility of Exact Cloning (No 1 Impossibility of Exact Cloning (No

Cloning)Cloning) 2 Impossibility of Exact Deletion (No 2 Impossibility of Exact Deletion (No

Deleting)Deleting) 3 Stronger No-Cloning 3 Stronger No-Cloning 4 Non-Existence of Universal Exact 4 Non-Existence of Universal Exact

Flipper (No Flipping)Flipper (No Flipping) 5 No Partial Cloning5 No Partial Cloning 6 No Partial Erasure6 No Partial Erasure

7 Non-existence of Universal 7 Non-existence of Universal Hadamard GateHadamard Gate

8 Impossibility of Probabilistic 8 Impossibility of Probabilistic CloningCloning

9 Impossibility of Broadcasting 9 Impossibility of Broadcasting mixed statesmixed states

10 No Splitting10 No Splitting 11 No Hiding 11 No Hiding EtchellipEtchellip

Physical System-Physical System- associated with a associated with a separable complex Hilbert spaceseparable complex Hilbert space

ObservablesObservables are linear self-adjoint are linear self-adjoint operators acting on the Hilbert spaceoperators acting on the Hilbert space

StatesStates are represented by density are represented by density operators acting on the Hilbert spaceoperators acting on the Hilbert space

Some Basic Notions about Some Basic Notions about Quantum SystemsQuantum Systems

Measurements are governed by two rulesMeasurements are governed by two rules 1 1 Projection Postulate-Projection Postulate- After the After the

measurement of an observable A on a measurement of an observable A on a physical system represented by the state physical system represented by the state ρ the system jumps into one of the eigen ρ the system jumps into one of the eigen states of Astates of A

2 2 Born Rule-Born Rule- The probability of The probability of obtaining the system in an eigen state obtaining the system in an eigen state is given by is given by

Tr(ρP[ ])Tr(ρP[ ])

The evolution is governed by an unitary The evolution is governed by an unitary operator or in other words by operator or in other words by Schrodingerrsquos evolution equationSchrodingerrsquos evolution equation

States of a Physical States of a Physical SystemSystem

Suppose H be the Hilbert space Suppose H be the Hilbert space associated with the physical systemassociated with the physical system

Then by a state Then by a state ρ we mean a linear ρ we mean a linear Hermitian operator acting on the Hermitian operator acting on the Hilbert space H such that Hilbert space H such that

It is non-negative definite andIt is non-negative definite and Tr(ρ)= 1Tr(ρ)= 1 A state is pure iff ρA state is pure iff ρ22 = ρ and otherwise = ρ and otherwise

mixedmixed Pure state has the form ρ=|Pure state has the form ρ=| | |HH

Consider physical systems consist of two or Consider physical systems consist of two or more number of parties A B C D helliphellipmore number of parties A B C D helliphellip

The associated Hilbert space is The associated Hilbert space is HHAAHHB B HHC C

HHD D hellip hellip States are then classified in two waysStates are then classified in two ways (I) (I) Separable-Separable- have the form have the form

ρρABCDABCD = =wwii ρ ρiiAA ρρii

BB ρρiiCC ρρii

DD with 0 with 0 w wii 11

and and wwi i =1=1 (II) All other states are (II) All other states are entangledentangled

Composite SystemsComposite Systems

States are represented by States are represented by unique Schmidt formunique Schmidt form

herehere is the is the Schmidt Schmidt

rankrank

andand are the are the Schmidt coefficientsSchmidt coefficients such that such that

Schmidt Schmidt vector of this State is vector of this State is

Pure Bipartite SystemPure Bipartite System

B

A

AB

)( 21 d

Pure Bipartite Pure Bipartite EntanglementEntanglement

Entanglement of pure state is uniquely Entanglement of pure state is uniquely measured by von Nuemann entropy of its measured by von Nuemann entropy of its subsystemssubsystems

States are States are locally unitarily connectedlocally unitarily connected if and if and only if they have only if they have same Schmidt vectorsame Schmidt vector hence hence their entanglement must be equaltheir entanglement must be equal

Thermo-dynamical law of EntanglementThermo-dynamical law of Entanglement Amount of Entanglement of a state cannot be Amount of Entanglement of a state cannot be increased by any LOCC (local operations increased by any LOCC (local operations performed on subsystems together with performed on subsystems together with classical communications between the classical communications between the subsystems)subsystems)

Some Use of Quantum Some Use of Quantum EntanglementEntanglement

Quantum Teleportation(Bennett Quantum Teleportation(Bennett etal PRL 1993)etal PRL 1993)

Dense coding (Bennett etal PRL Dense coding (Bennett etal PRL 1992)1992)

Quantum cryptography (Ekert PRL Quantum cryptography (Ekert PRL 1991)1991)

Physical OperationPhysical Operation

Suppose a physical system is described by a Suppose a physical system is described by a statestate

Krause describe the notion of a physical Krause describe the notion of a physical operationoperation

defined on as a defined on as a completely positive mapcompletely positive map

acting on the system and described by acting on the system and described by

where each is a linear operator thatwhere each is a linear operator that

satisfies the relationsatisfies the relation

kA

If then the operation is trace preserving When the state is shared between a number of parties say A B C D and each has the form

with all of

are linear operators then the operation is said to be a separable super operator

kA

A B C Dk k k kL L L L

Separable Super Separable Super operatoroperator

kA

Local operations with Local operations with classical classical communications communications (LOCC)(LOCC)

Consider a physical system shared between a number of parties situated at distant laboratories Then the joint operation performed on this system is said to be a LOCC if it can be achieved by a set of some local operations over the subsystems at different labs together with the communications between them through some classical channel

A result A result Every LOCC is a Every LOCC is a separable superoperatorseparable superoperator But whether the converse But whether the converse isisalso true or not also true or not It is affirmed that It is affirmed that there are separable there are separable superoperators which could superoperators which could not be expressed by finite not be expressed by finite LOCCLOCC

Bennett et al Phys Rev A 59 1070 Bennett et al Phys Rev A 59 1070 (1999)(1999)

The number of Schmidt coefficients of a pure The number of Schmidt coefficients of a pure bipartite entangled states cannot be increased bipartite entangled states cannot be increased by any LOCCby any LOCC

Measure of Entanglement-Measure of Entanglement- 1 For 1 For pure bipartite statepure bipartite state entanglement is entanglement is

measured by the measured by the Von Neumann entropyVon Neumann entropy of any of any of itrsquos subsystems This is the of itrsquos subsystems This is the unique measureunique measure for all pure bipartite statesfor all pure bipartite states

2 For 2 For mixed bipartitemixed bipartite entangled states there entangled states there is no unique way to define entanglement of a is no unique way to define entanglement of a state state

Two useful measure of entanglementTwo useful measure of entanglement Entanglement of Formation and Distillable Entanglement of Formation and Distillable

EntanglementEntanglement For For multipartitemultipartite case the situation is very case the situation is very

complexcomplex

CloningCloning

Basic tasks Is to copy quantum Basic tasks Is to copy quantum information exactly in (arbitrary) information exactly in (arbitrary) quantum statesquantum states

If there exists a machine which can If there exists a machine which can perform such tasks then we call it a perform such tasks then we call it a quantum cloning machinequantum cloning machine

The question is whether there exists The question is whether there exists quantum cloning machine which can quantum cloning machine which can copy exactly arbitrary quantum copy exactly arbitrary quantum information or notinformation or not

The No-Cloning TheoremThe No-Cloning Theorem

MMb

MMb

Arbitrary quantum information cannot Arbitrary quantum information cannot be copiedbe copied

To prove it suppose we have provided To prove it suppose we have provided with two states and assume with two states and assume that there is an exact cloning machine that there is an exact cloning machine Then we could write its action asThen we could write its action as

where |bgt|Mgt are the blank and where |bgt|Mgt are the blank and machine statesmachine states

Sketch of the proofSketch of the proof

Use the Unitarity of quantum operations ie Use the Unitarity of quantum operations ie consider U be the unitary operation consider U be the unitary operation responsible for cloning two above states responsible for cloning two above states Unitarity impliesUnitarity implies

==

If the states are different then the above If the states are different then the above relation immediately implies both must be relation immediately implies both must be orthogonalorthogonal

One may also prove the theorem using linearityOne may also prove the theorem using linearity

MM2

Some points to NoteSome points to Note Quantum mechanics prohibits exact Quantum mechanics prohibits exact

cloning of arbitrary information cloning of arbitrary information encoded in quantum states This does encoded in quantum states This does not imply inexact cloning is not not imply inexact cloning is not possiblepossible

For qubit system there exists optimal For qubit system there exists optimal universal isotropic quantum cloning universal isotropic quantum cloning machine with fidelity 56 If it is machine with fidelity 56 If it is restricted further to states in a great restricted further to states in a great circle of Bloch sphere then fidelity is circle of Bloch sphere then fidelity is increased further and is frac12+frac14(increased further and is frac12+frac14())

There is a nice relation between There is a nice relation between impossibility of discriminating non-impossibility of discriminating non-orthogonal states with certainty and no orthogonal states with certainty and no cloning theorem (exercise)cloning theorem (exercise)

There are lot of results already in There are lot of results already in literature to study the possibility of doing literature to study the possibility of doing inexact quantum cloning in different inexact quantum cloning in different quantum systems(see arXiv)quantum systems(see arXiv)

One can also prove no-cloning theorem One can also prove no-cloning theorem using some other physical constraints on using some other physical constraints on the system We will do some laterthe system We will do some later

Ref-Wootters amp Zurek Nature 299(1982)802 Diecks Phys Lett A Ref-Wootters amp Zurek Nature 299(1982)802 Diecks Phys Lett A 92(1982)27192(1982)271

Yuen Phys Lett A 113(1986)405 Scarani et al Rev Mod Phys Yuen Phys Lett A 113(1986)405 Scarani et al Rev Mod Phys 77(2005)122577(2005)1225

N Gisin amp S Massar PRL 79(1997) 2153 Bruss et al PRA 57(1998)2368N Gisin amp S Massar PRL 79(1997) 2153 Bruss et al PRA 57(1998)2368

Quantum DeletingQuantum Deleting

Here the task is Given two copies of Here the task is Given two copies of a quantum state (unknown) |a quantum state (unknown) | is it is it possible to delete the information of possible to delete the information of one of the part one of the part

In other words given two distinct In other words given two distinct states whether there is any states whether there is any quantum operation which could quantum operation which could perform the following operationperform the following operation

| | | | 0

| | | | 0

No Deleting theoremNo Deleting theorem

Given two copies of an unknown quantum Given two copies of an unknown quantum state it is impossible to delete the state it is impossible to delete the information encoded in one of the copyinformation encoded in one of the copy

Suppose we assume there is a quantum Suppose we assume there is a quantum operation which can perform the operation which can perform the following taskfollowing task

Ref-Pati amp Braunstein Nature 404(2000)164Ref-Pati amp Braunstein Nature 404(2000)164

1

0

111

000

MbM

MbM


Use linearity of quantum operations Use linearity of quantum operations ie as the operation is linear the ie as the operation is linear the above task is also possible for the above task is also possible for the states |states | where where 2 2 22

Then we have from Then we have from and using other two and using other two relationsrelations

Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM

Clearly the above expressions imply it is Clearly the above expressions imply it is impossible to delete exactly arbitrary impossible to delete exactly arbitrary quantum information encoded in a statequantum information encoded in a state

Exercise- What would be the status of Exercise- What would be the status of ancilla states|Mancilla states|M00 |M |M11 and |M and |M

One may prove the result with other One may prove the result with other ways also eg considering some ways also eg considering some constraints on the systemsconstraints on the systems

Observe the differences between cloning Observe the differences between cloning and deleting operations Have there any and deleting operations Have there any dual kind of relation between themdual kind of relation between them

Stronger No-Cloning Stronger No-Cloning TheoremTheorem

Here the question is Here the question is ldquoldquoHow much or what kind of additional How much or what kind of additional

quantum information is needed to quantum information is needed to supplement one copy of a quantum state supplement one copy of a quantum state in order to be able to produce two copies in order to be able to produce two copies of that state by a physical operationrdquoof that state by a physical operationrdquo

ie what information is needed initially ie what information is needed initially to cloneto clone

Ref-R Jozsa quant-ph0305114 quant-ph0204153Ref-R Jozsa quant-ph0305114 quant-ph0204153

Consider a finite set of non-orthogonal Consider a finite set of non-orthogonal states |states |ii and a set of states and a set of statesii(generally mixed)(generally mixed)

Then stronger no-cloning theorem states Then stronger no-cloning theorem states thatthat

There is a physical operationThere is a physical operation

if and only if there is a physical operationif and only if there is a physical operation

ie full information is needed to cloneie full information is needed to clone

iiiiiii

iii

Outline of the proofOutline of the proof

Use the lemma that Given two sets Use the lemma that Given two sets of pure states |of pure states |ii | |ii if they have if they have equal matrices of inner products equal matrices of inner products iijjiijj for all i and j then there is for all i and j then there is a unitary operation U on the direct a unitary operation U on the direct sum of the state spaces with U|sum of the state spaces with U|ii=|=|ii for all i and vice-versafor all i and vice-versa

Now consider first Now consider first i i be a pure state be a pure state Then prove the result using lemma Then prove the result using lemma After that using After that using i i as mixture of pure as mixture of pure states prove the result in generalstates prove the result in general

Some consequencesSome consequences

For cloning assisted by classical For cloning assisted by classical information (information (i i are required to be mutually are required to be mutually commuting) supplementary data must commuting) supplementary data must contain full identity of the states as contain full identity of the states as classical informationclassical information

The proof of no-deleting theorem could be The proof of no-deleting theorem could be done using the lemma that used in done using the lemma that used in stronger no cloning theorem Both no-go stronger no cloning theorem Both no-go theorems together establish permanence theorems together establish permanence of quantum information of quantum information

Spin FlippingSpin Flipping

Here the task is Whether it is Here the task is Whether it is possible to flip spin directions of any possible to flip spin directions of any arbitrary qubit or notarbitrary qubit or not

In other words is it possible to In other words is it possible to construct a quantum device which construct a quantum device which could take an arbitrary(unknown) could take an arbitrary(unknown) qubit and transform it into the qubit and transform it into the orthogonal qubit or not ie orthogonal qubit or not ie possibility of following operationpossibility of following operation1010

No-Flipping theoremNo-Flipping theorem

It is not possible to flip arbitrary It is not possible to flip arbitrary (unknown) spin directions(unknown) spin directions

To prove it we must careful about the To prove it we must careful about the nature of the operation we are trying to nature of the operation we are trying to do Unlike cloning here it is not possible do Unlike cloning here it is not possible to prove the result by considering only to prove the result by considering only two non-orthogonal statestwo non-orthogonal states

WhyWhy Because flipping is possible for states Because flipping is possible for states

lying on a great circle of the Bloch lying on a great circle of the Bloch sphere (sphere (exerciseexercise))


One way is to show the operation we One way is to show the operation we have written is in general an anti-have written is in general an anti-unitary operation and proof is unitary operation and proof is complete as anti-unitary operations complete as anti-unitary operations are not Physicalare not Physical(Not a CP map)(Not a CP map)

Other way is to consider three states Other way is to consider three states not lying on a great circle and prove not lying on a great circle and prove the impossibilitythe impossibility

Consider states like the followingConsider states like the following

| 0

| | 0 |1

| | 0 |1i

a b

c de

Then consider there is a flipping Then consider there is a flipping machine which could act on those machine which could act on those three states three states

Now try to prove the operation is Now try to prove the operation is impossibleimpossible

One may also prove by considering One may also prove by considering some constraints on the quantum some constraints on the quantum systemssystems

Ref-Buzek et al PRA 60(1999) R2626Ref-Buzek et al PRA 60(1999) R2626 Martini et al Nature 419(2002)815Martini et al Nature 419(2002)815

Probabilistic Quantum Probabilistic Quantum CloningCloning

Here the task is copyingcloning exactly but Here the task is copyingcloning exactly but not with certainty ie the condition is not with certainty ie the condition is relaxed one relaxed one

Interesting fact is that if one try to copy a set Interesting fact is that if one try to copy a set of states exactly but probabilistically then of states exactly but probabilistically then the set should be a linearly independent one the set should be a linearly independent one

Ref-Duan amp Guo PRL 80 (1998)4999Ref-Duan amp Guo PRL 80 (1998)4999 The intuitive proof follows from a constraint The intuitive proof follows from a constraint

on the system(Pati PLA 270(2000)103 on the system(Pati PLA 270(2000)103 Hardy amp Song PLA 259(1999)331Hardy amp Song PLA 259(1999)331

Partial CloningPartial Cloning

Here we have to study the possibility Here we have to study the possibility of cloning partially ie whether there of cloning partially ie whether there is any quantum device which could is any quantum device which could perform the following operation or notperform the following operation or not

where where are are arbitrary and F is a function of the arbitrary and F is a function of the originaloriginal

Ref- Pati PRA 66(2002)062319Ref- Pati PRA 66(2002)062319

MFMb )(

10

Outline of the proofOutline of the proof Consider some cases of F first SupposeConsider some cases of F first Suppose |F(|F( where K is a unitary or anti- where K is a unitary or anti-

unitary operator Then using linearity unitary operator Then using linearity and anti- linearity of the operation it is and anti- linearity of the operation it is straight forward to provestraight forward to prove

Then consider K as a combination Then consider K as a combination (linear) unitary and anti-unitary operator (linear) unitary and anti-unitary operator and prove the result and prove the result

One may also prove in general wayOne may also prove in general way

Non-Existence of Non-Existence of Universal Hadamard Universal Hadamard

GateGate There are two ways of defining itThere are two ways of defining it Check the possibility of the following Check the possibility of the following

operationsoperations

OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i

Result There is no Hadamard gate Result There is no Hadamard gate of the above kind for arbitrary of the above kind for arbitrary unknown qubitsunknown qubits

The proof is straightforward if we The proof is straightforward if we consider any two distinct qubits and consider any two distinct qubits and then taking inner products of them then taking inner products of them and their orthogonals before and and their orthogonals before and after the operations (exercise)after the operations (exercise)


Partial ErasurePartial Erasure

Reversibility of a quantum Reversibility of a quantum operation(unitary) prohibits complete operation(unitary) prohibits complete erasure of state (say qubit) ie to erasure of state (say qubit) ie to transform any qubit to a standard state transform any qubit to a standard state by unitary evolution is not possibleby unitary evolution is not possible

However one may ask the followingHowever one may ask the following Whether it is possible to erase partially Whether it is possible to erase partially

quantum information even by using quantum information even by using irreversible onesirreversible ones

What do we mean by partial erasureWhat do we mean by partial erasure A partial erasure is a trace preserving A partial erasure is a trace preserving

completely positive map that maps all completely positive map that maps all pure states of a n-dimensional Hilbert pure states of a n-dimensional Hilbert space to pure states in a m-dimensional space to pure states in a m-dimensional subspace (mltn) via a constraintsubspace (mltn) via a constraint

In other words partial erasure reduces In other words partial erasure reduces the dimension of the parameter domain the dimension of the parameter domain and does not leave the state entangled and does not leave the state entangled with other systemwith other system

No-partial ErasureNo-partial Erasure Theorem 1 Given any pair of non-Theorem 1 Given any pair of non-

orthogonal qudits in general there is orthogonal qudits in general there is no physical operation that can partially no physical operation that can partially erase themerase them

Linearity of quantum theory gives us Linearity of quantum theory gives us even more than the aboveeven more than the above

Theorem 2 Any arbitrary qudit cannot Theorem 2 Any arbitrary qudit cannot be partially erased by an irreversible be partially erased by an irreversible operationoperation

Ref-AKPati amp BCSanders PLA 359(2006)31Ref-AKPati amp BCSanders PLA 359(2006)31

Outline of the proofsOutline of the proofs For theorem-1 Consider two statesFor theorem-1 Consider two states in a n-dim Hilbert space and suppose there in a n-dim Hilbert space and suppose there

is a partial erasure with a constraint K is a partial erasure with a constraint K For simplicity choose the constraint such For simplicity choose the constraint such

that it reduces the parameter space at that it reduces the parameter space at least 1 dim The reduced dim states then least 1 dim The reduced dim states then may not have in general equal inner may not have in general equal inner product with the initial states product with the initial states

Now attach ancilla to see the effect of the Now attach ancilla to see the effect of the quantum evolution and considering quantum evolution and considering unitarity take inner product of both sideunitarity take inner product of both side

For theorem-2 take an orthonormal For theorem-2 take an orthonormal basis |basis |ii i= 1hellipd Then partial i= 1hellipd Then partial erasure machine yieldserasure machine yields

Where|Agt is the ancilla stateWhere|Agt is the ancilla state Now consider an arbitrary state |Now consider an arbitrary state |

ThenThen

Clearly the resultant state of the Clearly the resultant state of the system is mixed that contradicts the system is mixed that contradicts the definition of erasure to be puredefinition of erasure to be pure

iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22

Some ConsequencesSome Consequences

Check the result of theorem-2 for d=2 not Check the result of theorem-2 for d=2 not using ancillausing ancilla

For d=2 in theorem-1 an immediate For d=2 in theorem-1 an immediate consequence is ldquoA universal NOT gate is consequence is ldquoA universal NOT gate is impossiblerdquo ie No-flipping theorem impossiblerdquo ie No-flipping theorem (Check it)(Check it)

Also from theorem-2 the no-splitting of Also from theorem-2 the no-splitting of quantum information quantum information followsfollows

There are also some other consequencesThere are also some other consequences

No-HidingNo-Hiding

Perfect Hiding is a process Perfect Hiding is a process where where

is arbitrary input state and be the is arbitrary input state and be the fixed output statefixed output state

The input states forms a subspace of a The input states forms a subspace of a larger Hilbert space and output state larger Hilbert space and output state resides in a subspace where it has no resides in a subspace where it has no dependence on inputdependence on input

No-Hiding theorem states ldquoQuantum No-Hiding theorem states ldquoQuantum information can run but it canrsquot hiderdquoinformation can run but it canrsquot hiderdquo

oi i

o


Use linearity and unitarity of quantum Use linearity and unitarity of quantum processprocess

Consider only pure input states using Consider only pure input states using linearitylinearity

Unitarity allows suitable choice of Unitarity allows suitable choice of ancilla making the space larger Thenancilla making the space larger Then

Examining physical nature of hiding we Examining physical nature of hiding we havehave

)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp

So perfect hiding is impossible as So perfect hiding is impossible as ancilla contains full information of ancilla contains full information of hiding state which by definition hiding state which by definition impossibleimpossible

One may also study the imperfect hiding One may also study the imperfect hiding processes and found the implication of processes and found the implication of the result of no-hiding theoremthe result of no-hiding theorem

Further it has severe implication in Further it has severe implication in Black-hole information paradoxBlack-hole information paradox

Ref-Braunstein amp Pati PRL 98(2007)080502Ref-Braunstein amp Pati PRL 98(2007)080502

Constraints over Physical Constraints over Physical Operations Operations

1) No-Signalling 1) No-Signalling 2) Non-increase of Entanglement 2) Non-increase of Entanglement

under LOCCunder LOCC 3) Impossibility of inter conversion 3) Impossibility of inter conversion

by deterministic LOCC of a pair of by deterministic LOCC of a pair of Incomparable statesIncomparable states

No-SignallingNo-Signalling

Status It is a very strong constraint over any physical system This is not only a quantum mechanical constraint

Power This is very powerful restriction and senses easily in most of the settings and for almost all kind of impossible operations Even not using quantum mechanical formalism eg linearity and unitary dynamics

Impossible Operations detected by it UNIVERSAL EXACT CLONING UNIVERSAL EXACT DELETING UNIVERSAL EXACT SPIN-FLIPPING etc

Non-Increase of Non-Increase of Entanglement under Entanglement under

LOCCLOCCStatusStatus It is quite similar to the thermo-dynamical It is quite similar to the thermo-dynamical constraint over physical systems This is entirely constraint over physical systems This is entirely a constraint of quantum information a constraint of quantum information processing processing The physical reason behind this restriction entirely The physical reason behind this restriction entirely depends on the existence of entangled states of adepends on the existence of entangled states of a physical system Entanglement is a measurable physical system Entanglement is a measurable physical resource that can be applied to performphysical resource that can be applied to perform various kinds of information and computationalvarious kinds of information and computational tasks As it may be viewed as an amount tasks As it may be viewed as an amount of non-locality of the system therefore we have of non-locality of the system therefore we have a principle that entanglement can not be increased a principle that entanglement can not be increased by local operations with classical communicationsby local operations with classical communications

Power This is also a very powerful restriction over any quantum mechanically allowed Evolution of any physical system and it detects many impossible operations of quantum Information processing As it is purely a quantum mechanical constraint so we can use it to detect any local operation to be physical or not and we have the facility of using the operation to act linearly on superposition level of quantum states

Impossible Operations detected by Impossible Operations detected by using the principle of non-increase using the principle of non-increase of entanglement by LOCC of entanglement by LOCC

UNIVERSAL EXACT CLONING - IUNIVERSAL EXACT CLONING - In a pure n a pure statestate setting by Horodecki etal In a much more setting by Horodecki etal In a much more relaxedrelaxed mixed state scenariomixed state scenario by I Chattopadhyay et by I Chattopadhyay et al (in al (in preparation) preparation) UNIVERSAL EXACT DELETING ndash UNIVERSAL EXACT DELETING ndash ByBy Horodecki etal Horodecki etal considering the constraint that considering the constraint that `In a closed system the entanglement remain `In a closed system the entanglement remain unchanged unchanged under LOCCrsquo under LOCCrsquo Further using Further using `Non-increase of `Non-increase of entanglement underentanglement underLOCCrsquoLOCCrsquo in a very in a very simple settingsimple setting with linear with linearityity assumption only on density matrix level and assumption only on density matrix level and applying the deletion applying the deletion over only twoover only two arbitrary arbitrary qubits by I Chattopadhyay et al qubits by I Chattopadhyay et al

UNIVERSAL EXACT SPIN-FLIPPING -UNIVERSAL EXACT SPIN-FLIPPING -

Assuming theAssuming the existence of exact flipping operation existence of exact flipping operation acted acted onon only three states not lying in a great circleonly three states not lying in a great circle of the of the Bloch sphere Bloch sphere and and applying this operation applying this operation locally locally on a joint systemon a joint system using a using a less restricted less restricted linearitylinearity assumption assumption the entanglement content of the entanglement content of the systemthe systemcan be increased can be increased which establishes thewhich establishes the impossibility impossibility of universal exact spin-flippingof universal exact spin-flipping

In all the settings we assume linearity of In all the settings we assume linearity of the operations only on density matrix levelthe operations only on density matrix level

Quantum Deleting and Quantum Deleting and SignallingSignalling

Alice-Bob share 2 copy of Singlet state Alice-Bob share 2 copy of Singlet state and Bob has 1-2 copy exact deletion and Bob has 1-2 copy exact deletion machine acting on the arbitrary qubit machine acting on the arbitrary qubit

asas

Bob apply the deleting machine on joint Bob apply the deleting machine on joint state state

1sin0cos

M

M

MbM

MbM

2

1

3412 M

If Alice measure her particles on either If Alice measure her particles on either 01 or in qubit basis and Bob traced out 01 or in qubit basis and Bob traced out Machine state then the remaining 2 qubit Machine state then the remaining 2 qubit mixed state depends on choice of basis(mixed state depends on choice of basis() ) Which can be distinguished exactly from Which can be distinguished exactly from =0=0

Thus assumption of perfect deletion Thus assumption of perfect deletion machine can lead to signallingmachine can lead to signalling

Ref- Pati amp Braunstein PLA 315(2003)208Ref- Pati amp Braunstein PLA 315(2003)208

2455245542

1234513524

4

1trtrbbI

tr outout

No Signalling and Probabilistic No Signalling and Probabilistic Quantum CloningQuantum Cloning

Alice Bob share a large no of entangled Alice Bob share a large no of entangled state state ABAB

BBnnB B are arbitrary (not necessarily orthogonal)are arbitrary (not necessarily orthogonal)

If Bob have a 1-2 copy PQCM then he have a 1 If Bob have a 1-2 copy PQCM then he have a 1 to to copy PQCM for a large copy PQCM for a large and for a less and for a less (but gt 0) probability as(but gt 0) probability as

for some values n=12for some values n=12middotmiddotmiddotN and N+1middotmiddotmiddotN and N+1

N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(

Alice has 1 cbit information Alice has 1 cbit information corresponding to it she make a corresponding to it she make a measurement on each of her particle but measurement on each of her particle but do not communicate to Bob BOB apply do not communicate to Bob BOB apply PQCM on his particlePQCM on his particle

For general inputFor general input BBkk Bobrsquos output Bobrsquos output

CbitCbit

valuevalue

Measurement Measurement Basis of AliceBasis of Alice

Input set for Bobrsquos PQCMInput set for Bobrsquos PQCM

00 AA11 11 22middotmiddotmiddotmiddot middotmiddotmiddotmiddot NN

BB11==BB11 BB22middotmiddotmiddotmiddot middotmiddotmiddotmiddot BBNN

11 AA22 11acute 22acutemiddotmiddotmiddotmiddot middotmiddotmiddotmiddot NNacute

BB22==BBN+1N+1 BBN+2N+2middotmiddotmiddotmiddot middotmiddotmiddotmiddot BB2N2N

121 0 where1

1

NlBdBc lkkk

N

il

kl

Bob succeeds to clone exactly Bob succeeds to clone exactly copies copies of particle B if input is one of of particle B if input is one of BBnn n=12hellipN+1n=12hellipN+1

PP00(A(A11)) = = n=1n=1NN Prob(n| A Prob(n| A11) ) = 1= 1 PP11(A(A11))

= P(N+1| A= P(N+1| A11) ) = 0= 0

PP00(A(A22))= = NN P(n| A P(n| A22) ) 1 1- P(N+1| A- P(N+1| A22) ) PP11(A(A22))= P(N+1| A= P(N+1| A22) ) 0 0

After many repetition PAfter many repetition P00(A(A22))rarrrarr1 Bob 1 Bob can correctly know the bit without any can correctly know the bit without any actual communication imply actual communication imply superluminal signalingsuperluminal signaling

Ref- L Hardy amp D D Song PLA Ref- L Hardy amp D D Song PLA 259(1999)259(1999) 331 331 See also Pati PLA 270(2000)103See also Pati PLA 270(2000)103

Impossibility of Exact CloningImpossibility of Exact Cloning Consider a mixed state Consider a mixed state ρρABC ABC with A in one with A in one

location and B C are in another location so that location and B C are in another location so that the system may be considered as a bipartite the system may be considered as a bipartite systemsystem

where where = a= a00+b+b11 and and = c= c00+de+deii 11 are are two non-orthogonal qubit state on which we two non-orthogonal qubit state on which we assume exact cloning is possibleassume exact cloning is possible

The logarithmic negativity (an upper bound of The logarithmic negativity (an upper bound of distillable entanglement) of the state isdistillable entanglement) of the state is

If exact cloning is possible then we can extract If exact cloning is possible then we can extract one ebit of entanglement however in ABC cut one ebit of entanglement however in ABC cut LN(LN() ) 11

00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N

Partial CloningPartial Cloning We define the partial cloning machine on We define the partial cloning machine on

two qubit states two qubit states as follows as follows

where where F(F()) F(F()) are some functions of are some functions of resp resp

Suppose two distant parties A B share a stateSuppose two distant parties A B share a state

where the first particle is with A and other two where the first particle is with A and other two with B and with B and bbBB is the blank state is the blank state

Applying partial cloning machine the final state Applying partial cloning machine the final state has the formhas the form

| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb

Tracing out the last two qubits of initial and Tracing out the last two qubits of initial and final final

states the reduced state between A and B arestates the reduced state between A and B are

AndAnd

So whenever So whenever 0 ie two states remain 0 ie two states remain non-orthogonal the entanglement content of non-orthogonal the entanglement content of the final reduced state is always greater than the final reduced state is always greater than the initial onethe initial one

Comment violation of principle of non-Comment violation of principle of non-increase of entanglementincrease of entanglement

2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F

Quantum DeletionQuantum Deletion Consider a pure entangled state shared Consider a pure entangled state shared

between three parties A B Cbetween three parties A B C

Assume A has a exact deleting machine for Assume A has a exact deleting machine for two non-orthogonal states two non-orthogonal states acting as acting as

Final state is thenFinal state is then

1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P

Tracing out first two qubits of A the Tracing out first two qubits of A the reduced state between B and C arereduced state between B and C are

andand

The concurrence (used to measure The concurrence (used to measure entanglement of formation) of initial and entanglement of formation) of initial and final states are final states are

C(C() =) =22 and and C(C() = ) = therefore therefore which implieswhich implies

| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E

Spin FlippingSpin Flipping Consider three arbitrary qubit states Consider three arbitrary qubit states

not lying in one great circle of the not lying in one great circle of the Bloch sphereBloch sphere

where awhere a0 c0 c0 00 0 bd are real nos bd are real nos such that asuch that a22 +b +b22 =1 c =1 c22 +d +d22 =1 and =1 and 00 11 are orthogonal to each other are orthogonal to each other

Assume the most general flipping operation Assume the most general flipping operation for those three states asfor those three states as 00 MM 11 MM00

MMeeii MM MM eeii MM

| 0

| | 0 |1

| | 0 |1i

a b

c de

Consider a bipartite pure entangled state Consider a bipartite pure entangled state shared between two spatially separated parties shared between two spatially separated parties A BA B

||ii AB AB =(1=(13)[3)[|0|0A A |0|0BB+|1+|1AA||BB+|2+|2AA||B B ] |M] |MBB

where A holds 3-dim system with basis where A holds 3-dim system with basis |0|0A A | |11AA|2|2AA

Applying exact flipper on the side of B the state Applying exact flipper on the side of B the state takes the formtakes the form

||ff AB AB =(1=(13)[3)[|0|0A A |1|1MM00BB+ + eeii |1 |1AA||MMBB++

eeii|2|2AA||MMB B ] ] Now if we calculate the reduced density Now if we calculate the reduced density

matrices of Arsquos side then it is easy to check matrices of Arsquos side then it is easy to check that the initial and final density matrices are that the initial and final density matrices are different unless different unless when b=0 or d=0 or sin(when b=0 or d=0 or sin()=0)=0

Conclusion No-signalling implies no-Conclusion No-signalling implies no-flippingflipping

Note- the above setting does not show any Note- the above setting does not show any change in entanglementchange in entanglement

Next consider a bipartite state shared Next consider a bipartite state shared between A B with A has a 2-dim system between A B with A has a 2-dim system and B has four qubit system(say 1234)and B has four qubit system(say 1234)

||ii AB AB =(1=(18)[(8)[(|000|000++ |111|111))A12A12 |10 |103434 ndash ndash

(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB

Assume B has a exact flipping machine that Assume B has a exact flipping machine that acts on last qubit(4-th) Then the final joint acts on last qubit(4-th) Then the final joint system takes the formsystem takes the form

||ff AB AB =(1=(18)[(8)[(|000|000++ |111|111))A12A12 |11 |11MM00 34B34B ndash ndash

eeii(|010(|010+|100+|100+|101+|101))A12A12||MM34B34B ndash ndash

eeii(|011(|011+|110+|110+|001+|001))A12A12||MM34B34B]] Now if we calculate the reduced density Now if we calculate the reduced density

matrices of Arsquos side we find that the largest matrices of Arsquos side we find that the largest eigen values of initial and final density eigen values of initial and final density matrices are related bymatrices are related by

f f ii which implies E(|which implies E(|ffAB AB )) E(|E(|iiABAB)) ie entanglement has increased by local ie entanglement has increased by local

operations in AB cutoperations in AB cut

Incomparability ~ As a New Constraint over Pure

Bipartite System Incomparability of two pure bipartiteIncomparability of two pure bipartite entangled states imply that either of the entangled states imply that either of the two states can not be converted to the other two states can not be converted to the other by LOCC with certainty We propose by LOCC with certainty We propose this as a constraint over the joint system throughthis as a constraint over the joint system through LOCC Thus if a local operation can inter-convert LOCC Thus if a local operation can inter-convert an incomparable pair of states then the an incomparable pair of states then the

operation operation is certainly an unphysical operation is certainly an unphysical operation

Nielsenrsquos criteriaNielsenrsquos criteriaLet and be any two bipartite pure Let and be any two bipartite pure

states of Schmidt rank states of Schmidt rank d d withwith Schmidt Schmidt vectors vectors

and and

respectively where respectively where

and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1

Then the state can be Then the state can be deterministicallydeterministically

transformed to the state by LOCC transformed to the state by LOCC (denoted (denoted

by ) if and only if by ) if and only if majorized bymajorized by

(denoted by ) ie if(denoted by ) ie if

AB

If this criteria is not satisfied for a pair of If this criteria is not satisfied for a pair of statesstates

then we denote it as then we denote it as Though it may happen that Though it may happen that The pair of states are said to be The pair of states are said to be

comparable if either or comparable if either or

From thermo-dynamical law of entanglement From thermo-dynamical law of entanglement one may concludeone may conclude

For any pair of states implies For any pair of states implies rank of is greater or equals to the rank of rank of is greater or equals to the rank of

ABAB

AB

AB

ABAB

ABAB

Incomparability of two bipartite Incomparability of two bipartite pure states -pure states -

If for some pair of bipartite pure states we have both If for some pair of bipartite pure states we have both | |ΨΨ |Φ|Φ and and |Φ|Φ |Ψ|Ψ (ie either (ie either αα11ββ11 and and i=i=

kk1 1 ββi i

i=i=kk1 1 ααii for some for some kk23hellipn or β23hellipn or β11αα11 and and i=i=

ss1 1 ααi i i=i=

ss1 1

ββii for some s for some s23hellipn Then 23hellipn Then ((|Ψ|Ψ |Φ |Φ)) is said to be an is said to be an incomparable pair of states and this phenomenon is incomparable pair of states and this phenomenon is denoted as denoted as |Ψ|Ψ|Φ|Φ

It is to be noted that from the incomparability criteria It is to be noted that from the incomparability criteria ||ΨΨ|Φ|Φ will not indicate will not indicate either E(|Ψeither E(|Ψ) ) E(|Φ E(|Φ) or E(|) or E(|ΦΦ) ) E(|Ψ E(|Ψ))

If If ((|Ψ|Ψ |Φ |Φ) are m) are mn statesn states where where minminmn=3 then mn=3 then the criteria for the criteria for |Ψ|Ψ |Φ|Φ can be expressed ascan be expressed as either either α α1 1 ββ1 1 and and αα3 3 ββ33

or βor β1 1 αα1 1 and βand β33 αα33

Consider an operation ( ) defined on an operation ( ) defined on the set of input statesthe set of input states i = 12hellipk i = 12hellipk

in the following mannerin the following manner

Now consider the pure bipartite Now consider the pure bipartite statestate

shared between two distant parties shared between two distant parties Alice Alice and Boband Bob

(| ) | i iA AA

A

1| | |

k

AB i A i Bii

Applying locally on Bobrsquos system Applying locally on Bobrsquos system the joint state transforms to another the joint state transforms to another pure bipartite statepure bipartite state

Now the Schmidt coefficients Now the Schmidt coefficients corresponding to the two pure corresponding to the two pure bipartite states are bipartite states are

If the two states If the two states are incomparable then the are incomparable then the operation ( ) operation ( ) is an impossible oneis an impossible one

1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A

Universal exact FlippingUniversal exact Flippingvia Incomparabilityvia Incomparability

Special feature Flipping operation is defined only on a minimum number of three qubit states as in the earlier cases

The flipping operation is described as

We omit the machine part as we assume the quantum formalism

| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c

Now consider a pure bipartite state shared between A B

After applying the flipping operation on the second qubit of B the state takes the form

Now if we calculate the eigen values of the initial and final density matrices of A then we find that they satisfies the Incomparability criteria unless they are in a great circleSo incomparability implies no-flipping Also we find the reverse case We illustrating it by examples

1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e

Consider three qubits representing three axes of the Bloch sphere as

|Z =|0 |x = frac12 (|0 +|1) and |Y = frac12 (|0 + i|1)

We define our flipping operation on those three state as follows

|Z |Z

|x eiμ |x helliphellip(1)

|Y eiν |Y

The flipped state |i is orthogonal to the initial state |i i= zxy

and eiμ eiν are arbitrary phase factors

Let |χi = 1 3 |0A |ZZ B+|1A |xyB+|2A |yxB be a 34

( 322 ) state between Alice and Bob situated in two distant labs

Now if Bob applies the flipping operation defined in (1) on the

2nd qubit of his local system then the joint state between them

will transform as follows

|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB

+ eiν |2A|yxBSchmidt vector corresponding to |χi and |χf are

λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12

) lt 23 which imply that the initial and final joint state between Alice and Bob are incomparable Hence it is not possible to transform |χi to |χf by LOCC though |χf is being prepared from |χi by applying the flipping operation locally Thus the operation defined in (1) is not a valid (physical) operation So we have |χi |χf Universal Flipping is not possible

Incomparability from No-Flipping Incomparability from No-Flipping PrinciplePrinciple

Lets consider two pure state of 34 system shared between two spatially separated parties Alice and Bob

|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B

Bobrsquos local system has the following form |0B= |ΨB1 |ΨB2 |1B = |ΨB1 |ΨB2 |2B = |ΨB1 |ΨB2

where |Ψ be a arbitrary qubit state |ΨΨ|= frac12 [ I + n middot σ ]Local systems of Bobrsquos first qubit (B1) corresponding to the two joint state |1 and |2 between Alice and Bob are

1= Tr A B2 ( |1 1| ) = frac12 [ I + 02 n middot σ ] 2 = Tr A B2 ( |2 2| ) = frac12 [ I - 02 n middot σ ]

Hence local transformation of |1 to |2 will flip the spin direction of 1 associated with |Ψ

ConclusionConclusion If somehow we extend the LOCC If somehow we extend the LOCC transformation criteria so that either transformation criteria so that either ||11 | |22 or or

||22 | |11 then then consequently on a subsystem the direction consequently on a subsystem the direction

of spin-polarization of an arbitrary state of spin-polarization of an arbitrary state 1 1 will be reversedwill be reversed

That is equivalent of preparing an exact universal flipping That is equivalent of preparing an exact universal flipping machinemachine

SoSo ````No-Flipping principlersquorsquoNo-Flipping principlersquorsquo ||11 | |22

This two result together shows an equivalence between the This two result together shows an equivalence between the constraint on LOCC state transformation criteria and the constraint on LOCC state transformation criteria and the

No-Flipping principle The relation is drawn from the No-Flipping principle The relation is drawn from the unphysical nature of flipping operation as it is not an unitaryunphysical nature of flipping operation as it is not an unitary

operation but an anti-unitary operationoperation but an anti-unitary operation

Cloning and Deleting Cloning and Deleting via Incomparabilityvia Incomparability

This work connects the Cloning and Deleting with the concept of Incomparability as a constraint imposed over the system by LOCC We define the Cloning operation to act on just two arbitrary input qubits in the following manner

| 0 | | 0 | 0b

| 0 |

| | | |b

Now consider a pure bipartite state shared between Alice and Bob situated at distant labs in the form

Let Bob applies the Cloning operation on his last two qubits Then the joint state transforms to another pure bipartite state

1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B

Tracing out Bobrsquos subsystem we observe the initial and final form of Alicersquos local system and compute the Schmidt vector corresponding to the two joint states are and and find that

Thus the cloning operation defined over is unphysical in nature

| |AB AB

| AB | AB

| 0 |

If instead of Cloning we consider the Deleting operation defined on the two arbitrary inputstates in the following way that

Consider now that Alice and Bob will initially share the joint state Then by assuming the existence of the exact Deleting operation and applying it on Bobrsquos system the final joint state will be The incomparability of the pair of states will imply that the exact deleting operation defined on the two arbitrary non-orthogonal states does not exists

Question- Is it possible to Question- Is it possible to extend the proof via extend the proof via incomparability for general incomparability for general anti-unitary operations and anti-unitary operations and inner-product preserving inner-product preserving operationsoperations

Answer- YesAnswer- Yes (ref- Quant Inf amp Comp 7 (2007) (ref- Quant Inf amp Comp 7 (2007)

392-400)392-400)

Thanks to the organizers for inviting me in IPQI-2010 January 4-30 2010 at Institute of

Physics Bhubaneswar

A List of Impossible A List of Impossible Operations (No Go Operations (No Go

Principles)Principles) 1 Impossibility of Exact Cloning (No 1 Impossibility of Exact Cloning (No

Cloning)Cloning) 2 Impossibility of Exact Deletion (No 2 Impossibility of Exact Deletion (No

Deleting)Deleting) 3 Stronger No-Cloning 3 Stronger No-Cloning 4 Non-Existence of Universal Exact 4 Non-Existence of Universal Exact

Flipper (No Flipping)Flipper (No Flipping) 5 No Partial Cloning5 No Partial Cloning 6 No Partial Erasure6 No Partial Erasure























BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar























BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



















BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar















BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar










BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





BB ρρiiCC ρρii






rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



rankrank




B

A

AB

)( 21 d
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar
















kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar












kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar








kA


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar


with all of


kA



kA











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar











complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar









complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar







complexcomplex

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar

CloningCloning





MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar


MMb

MMb






==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



==



MM2













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar













| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar










| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




| | | | 0

| | | | 0





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





1

0

111

000

MbM

MbM




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




Mbb

bMbM

M

)10(

2)10(

11)1001(00

12

02

22

MbM















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar















iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar











iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






iiiiiii

iii



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar
















| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar













| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar










| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





| 0

| | 0 |1

| | 0 |1i

a b

c de














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar














MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar










MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





MFMb )(

10








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar








OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




OrOr

)(2

1

)(2

1

)(2

1

)(2

1

i

i






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar















ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar












ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar







ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



ThenThen


iAA ii

n

nn ACoseACoseA nn

d

n

in

nd

n

i

22






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar

No-HidingNo-Hiding





oi i

o






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






)(1

ko

d

kki

Akp

)0(1

ko

d

kki

qkp























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar























asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



















asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar















asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar











asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar









asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar








asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



asas


1sin0cos

M

M

MbM

MbM

2

1

3412 M




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




2455245542

1234513524

4

1trtrbbI

tr outout






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






N

nBnNAN

N

nBnANAB

BnBn1

1

1

1

nprob

n BbB 0 )1(



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



CbitCbit

valuevalue







121 0 where1

1

NlBdBc lkkk

N

il

kl



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



= P(N+1| A= P(N+1| A11) ) = 0= 0









00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






00 11 00 11| | (1 ) | |

2 2ABC

BC

P P

11( ) ( ) 1

2 2

bc adLN N







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar







| | | | ( )

| | | | ( )

b F

b F

1| | 0 | 0 ( ) |1 |1 ( )

2f i

AB A B A BF e F

1| | 0 | 0 |1 |1 |

2AB A B A B Bb



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



AndAnd



2 2100 11 | | 00 11| | |11 00 |

2iAB P P

100 11 | | | 00 11|

2f

AB P P F F

| | |11 00 |F F





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





1| | |1 |1 | | 0 | 0

2ABC A B C A B C

| | | | 0

| | | | 0

1| | 0 |1 |1 | 0 | 0 | 0

2d

ABC A B C A B C

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar

2 2100 11 | | 00 11| | |11 00 |

2P P


andand



| |BC A ABCTr

| |d d dBC A ABCTr

100 11 | | 00 11| | |11 00 |

2P P

( ) ( )C C

( ) ( )f fE E






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






| 0

| | 0 |1

| | 0 |1i

a b

c de












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar












(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





(|010(|010+|100+|100+|101+|101))A12A12||3434 ndash ndash

(|011(|011+|110+|110+|001+|001))A12A12||3434] ] |M |MBB













and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar












and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



and and


and and

(PRL 83 436 (1999))(PRL 83 436 (1999))

AB

AB

jiji 01 ji 1





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





AB






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar






ABAB

AB

AB

ABAB

ABAB



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



kk1 1 ββi i


ss1 1 ααi i i=i=

ss1 1









(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





(| ) | i iA AA

A

1| | |

k

AB i A i Bii




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




1| | |

k

AB i A i Bii

1 2i i k

1 2i i k

| |AB AB A





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





| 0

| | 0 |1

| | 0 |1i

a b

c de

| 0 |1

| | | 0 |1

| | | 0 |1

i i

i i i

e e b a

e e de c




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




1| | 0 | 00 |1 | | 2 |

3i

AB A B A B A B

1| | 0 | 01 |1 | | 2 |

3f i i

AB A B A B A Be e




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar




|Z |Z


|Y eiν |Y








|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar

|χi |χf = 1 3 |0A |ZZ B + eiμ |1A

|xyB


λf = 1 3 ( 1 3 + 12

) 13 1 3 ( 1 3 - 12

)

and λi = 23 16 16

Obviously 1 3 ( 1 3 - 12

) lt 16 lt

13 lt1 3 ( 1 3 + 12




|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



|1 = 51 |0A |0B + 30 |1A |1B + 19 |2A |2B

|2 = 49 |0A |0B + 36 |1A |1B + 15 |2A |2B















| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar











| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



| 0 | | 0 | 0b

| 0 |

| | | |b



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar



1| |1 (| 0 0 | 0 0 )AB A B

N

| 2 (| 0 0 | 00 )A B

| 3 (| 00 | 00 ) |A B Bb

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar

1| |1 (| 0 0 | 0 00 )AB A B

N

| 2 (| 0 00 | 00 )A B

| 3 (| 00 | 000 )A B



| |AB AB

| AB | AB

| 0 |





392-400)392-400)


Physics Bhubaneswar





392-400)392-400)


Physics Bhubaneswar



392-400)392-400)


Physics Bhubaneswar


Physics Bhubaneswar

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Impossible Operations in the Quantum World-I and II

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