2005 q 0031 Density Matrices 2

8/11/2019 2005 q 0031 Density Matrices 2

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1

Richard Cleve

Lectures 10 ,11 and 12

DENSITY

MATRICES, traces,

Operators andMeasurements

Michael A. Nielsen

Michele Mosca

Sources:


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Review: Density matrices

of pure statesWe have represented quantum states as vectors(e.g. ,and all such states are called pu re states)

An alternative way of representing quantum states is in terms

of densi ty m atr ices(a.k.a. densi ty operators)

The density matrix of a pure state is the matrix =

Example:the density matrix of 0+1is

2

2


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Example:Notation of Density

Matrices and traces

Notice that 0=0|, and 1=1|.

So the probability of getting 0 when measuring |is:220 0)0( p

0000

0000

0000

TrTr

Tr

where = || is called

the density matrixfor thestate |

10 10


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Review:Mixture of pure states

A state described by a state vector |is called apure state.

What if we have a qubit which is known to be in thepure state |1with probabilityp1, and in |2withprobabilityp2?

More generally, consider probabilistic mixturesofpure states (called mixed states):

...,,,, 2211 pp


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Density matrices of mixed states

A probability distribution on pure states is called a mixed state:( (1, p1), (2, p2), , (d, pd))

The densi ty matr ixassociated with such a mixed state is:

d

kkkkp

1

Example: the density matrix for ((0, ), (1, ))is:

10

01

2

1

10

00

2

1

00

01

2

1

Question:what is the density matrix of

((0+1,

),(0 1,

)) ?


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Density matrix of a mixed

state (use of trace)then the probability of measuring 0 is given byconditional probability:

i

iipp statepuregiven0measuringofprob.)0(

00

00

00

Tr

pTr

Trp

i

iii

i

iii

where i

iiip is the density matrixfor the mixedstate

Density matrices contain all the useful information about anarbitrary quantum state.


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Recap: operationally

indistinguishable states

Since these are expressible in

terms of density matrices alone(independent of any specific

probabilistic mixtures), states with

identical density matrices areoperat ional ly ind ist inguishable


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Applying Unitary Operator to a

Density Matrix of a pure state

If we apply the unitary operation U tothe resulting state is

with density matrix

U

tt UUUU


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Density Matrix

If we apply the unitary operation U to

the resulting state is

with density matrix

kkq , kk Uq ,

t

t

t

UU

UqU

UUq

k

k

kk

k

k

kk

Applying Unitary Operator to a

Density Matrix of a mixedstate

How do quantum operations work for these mixedstates?


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Operators on Density matrices of

mixed states.

Effect of a unitary operation on a density matrix:

applying U to st i l lyields UU

Effect of a measurement on a density matrix:

measuring statewith respect to the basis1, 2,..., d,st i l lyields the kthoutcome with probability kk

Why?

Thus this

is true

always


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Effect of a measurement on a density matrix:

measuring state with respect to the basis1,2,..., d, yields the kthoutcome with probabilitykk

How do quantum operations

work using density matrices?

(this is because kk=kk=k2)

and thestate collapses tok

k


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More examples of density matrices

The densi ty matr ixof the mixed state

((1, p1), (2,p2), ,(d,pd)) is:

d

k

kkk p1

1. & 2. 0+1and 01both have

3. 0with prob. 1with prob.

4. 0+1 with prob.

01 with prob.

6. 0 with prob. 1 with prob. 0+1 with prob.

01 with prob.

Examples (from previous lecture):

11

11

2

1

1001

21


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5. 0 with prob. 0+1 with prob.

7. The first qubit of 0110

Examples (continued):

4/12/1

2/14/3

2/12/1

2/12/1

2

1

00

01

2

1has:

...?(later)

More examples of density matrices


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To Remember:Three Properties of Density

Matrices

Three properties of :

Tr=1(TrM =M11+M22 + ... +Mdd )

=(i.e.is Hermitian)

0, for all states

d

kkkk p

1

Moreover, for anymatrix satisfying the above properties,

there exists a probabilistic mixturewhose density matrix is

Exercise:show this


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Use of Density Matrix and Trace to

Calculate the probability of obtaining

state in measurement

If we perform a Von Neumann measurementof the state wrt a basiscontaining , the probability ofobtaining is

Tr2 This is for a pure

state.

How it would be for a

mixed state?

U f D it M t i d T t C l l t th


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Density Matrix

If we perform a Von Neumann measurementof the state

wrt a basis containing the probabilityof obtaining is

kkq ,

Tr

qTr

Trqq

k

kkk

k

kkk

k

kk

2

Use of Density Matrix and Trace to Calculate the

probability of obtaining state in measurement (now

for measuring a mixed state)

The same

state


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Conclusion: Density Matrix Has

Complete Information

In other words, the density matrix containsall the information necessary to computethe probability of any outcome in anyfuture measurement.


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Spectral decomposition can be used to

represent a useful form of density matrix

Often it is convenient to rewritethedensity matrix as a mixture of itseigenvectors

Recall that eigenvectors with distinct

eigenvalues are orthogonal;for the subspace of eigenvectors with a

common eigenvalue(degeneracies), wecan select an orthonormal basis


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Continue - Spectral decomposition used

to diagonalize the density matrix

In other words, we can alwaysdiagonalizea density matrix so that it

is written ask

k

kkp

where is an eigenvectorwitheigenvalue and forms anorthonormal basis

kkp k


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Taxonomy ofvarious normal

matrices


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Normal matrices

Definition:A matrixMis normalifM

M =MM

Theorem:Mis normal iff there exists a unitary U such that

M =UDU, whereD is diagonal (i.e. unitarily diagonalizable)

Examples of abnormal matrices:

10

11 is not even

diagonalizable

20

11 is diagonalizable,but not unitarily

eigenvectors:

d

D

00

0000

2

1


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Unitary and Hermitian matrices

d

M

00

0000

2

1 with respect to someorthonormal basis

Normal:

Unitary:MM =I which implies |k |2=1, for all k

Hermitian:M =M which implies kR, for all k

Question:which matrices areboth unitary andHermitian?

Answer:reflections (k{+1,1}, for all k)


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Positive semidefinite matrices

Positive semidefinite:Hermitian and k0, for all k

Theorem:Mis positive semidefinite iffMis Hermitian and,

for all , M 0

(Positive defini te:k>0, for all k)


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Projectors and density matrices

Projector:Hermitian andM2 =M, which implies thatMispositive semidefinite and k{0,1}, for all k

Density matrix:positive semidefinite and TrM=1, so 11

d

k

k

Question:which matrices areboth projectors anddensity

matrices?

Answer:rank-one projectors(k=1if k = k0and k=0 ifk k0)


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Taxonomy of normal matrices

normal

unitary Hermitian

reflectionpositive

semidefinite

projector densitymatrix

rank one

projector

If Hermitian then

normal


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Review:Bloch sphere for qubits

Consider the set of all 2x2 density matrices

Note that the coefficient ofI is , sinceX,Y,Zhave trace zero

They have a nice representation in terms of the Paul i m atr ices:

01

10 Xx

0

0

i

iYy

10

01 Zz

Note that these matricescombined withIform a basisforthe vector space of all 2x2 matrices

We will express density matrices in this basis


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Bloch sphere for qubits: polar

coordinates

2

ZcYcXcI

zyx We will express

First consider the case of pure states , where, withoutloss of generality, =cos()0+e2isin()1 (, R)

2cos12sin

2sin2cos1

2

1

sinsincos

sincoscos

2

2

22

22

i

i

i

i

e

e

e

e

Therefore cz= cos(2),cx= cos(2)sin(2),cy= sin(2)sin(2)

These are po lar co ordinatesof a unit vector (cx ,cy ,cz)R3

Bloch sphere for qubits: location of


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Bloch sphere for qubits: location of

pure and mixed states

+

0

1

+i

i

+i=0+i1

i=0i1

=01+=0+1

Pure statesare on the surface, and mixed statesare inside

(being weighted averages of pure states)

Note that or thogonalcorresponds to ant ipodalhere


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General

quantum

operationsDecoherence, partial traces,

measurements.


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General quantum operations (I)

Example 1(unitary op): applying U to yields UU

General quantum operationsare also calledcompletely positive trace preserving maps,or

admissible operations

IAA jm

j

j 1

t

Then the mapping m

jjj AA1

t

is a general quantumoperator

LetA1,A2, ,Ambe matrices satisfying

condition

General quantum operations:


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General quantum operations:

Decoherence Operations

Example 2(decoherence):letA0 =00andA1 =11This quantum op maps to 0000+1111

Corresponds to measuring without looking at the outcome

2

2

2

2

0

0

For =0+1,

After looking at the outcome, becomes 00 with prob. ||211 with prob. ||2

General quantum operations: measurement


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General quantum operations: measurementoperations

Example 3(trine statemeasurement):

Let 0=0, 1=1/20+ 3/21, 2=1/203/21

Then IAAAAAA 221100ttt

The probability that state kresults in outcome stateAk is 2/3.This can be adapted to actually yield the value of

kwith this success

probability

00

01

3

2DefineA0 =2/300

A1=

2/311 A2=

2/322

62

232

4

1

62

232

4

1

We apply the general quantum mapping operator

m

j

jj AA1

t

Condition satisfied

General quantum operations: Partial trace discards


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General quantum operations: Partial tracediscards

the second of two qubits

Example 4(discarding the second of two qubits):

LetA0=I0 andA1=I1

0100

0001

1000

0010

State becomes

State becomes110011002

1

2

1

2

1

2

1

Note 1:its the same density matrix as for ((0, ), (1, ))

10

01

2

1

Note 2:the operation is the partial traceTr2

We apply the general quantum mapping operator

m

j

jj AA1

t


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Distinguishing

mixed statesSeveral mixed states can have the same

density matrixwe cannot distinguish

between them.

How to distinguish by two different density

matrices?

Try to find an orthonormal basis 0,1in which both

density matrices are diagonal:


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Distinguishing mixed states (I)

10

01212

0 with prob. 0+1 with prob.

0 with prob. 1 with prob.

412121431

////

0 with prob. cos2(/8)1 with prob. sin2(/8)

0

+

0

1

0 with prob. 1 with prob.

Whats the best distinguishing strategybetween these two

mixed states?

1also arises from this

orthogonal mixture: as does 2from:

/8=180/8=22.5

Di ti i hi i d t t (II)


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Distinguishing mixed states (II)

8sin0

08cos2

2

2/

/

0

+

0

1

10

01

2

11

Weve effectively found an orthonormal basis 0,1inwhich both density matrices are diagonal:

Rotating 0,1to 0, 1the scenario can nowbe examined using classical probability theory:

Question:what do we do if we arent so lucky to get two

density matrices that are simultaneously diagonalizable?

Distinguish between two classicalcoins, whose probabilities

of headsare cos2(/8)and respectively (details: exercise)

1

Density matrices 1and 2are simultaneously diagonalizable

B i ti f


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Reminder:Basic properties of

the trace

d

k

k,kMM1

Tr

NMNM TrTrTr

NMNM TrTrTr

MNNM TrTr

adcbdcba Tr

d

k

kMUUM1

1 TrTr

The t raceof a square matrix is defined as

It is easy to check that

The second property implies

and

Calculation maneuvers worth remembering are:

aMMa bb Tr and

Also, keep in mind that, in general,


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Partial Trace How can we compute probabilities fora partial system?

E.g.

yx

p

p

yx

yx

y x y

xy

y

y xxy

yxxy

,

Partial

measurement


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Partial Trace

If the 2ndsystem is taken away and neveragain (directly or indirectly) interacts withthe 1stsystem, then we can treat the firstsystem as the following mixture

E.g.

22,2 Trx

p

p

yx

p

p

x y

xy

y

Trace

y x y

xy

y

From previous

slide

Partial Trace: we derived an important


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Partial Trace:we derived an important

formula to use partial trace

22,2 Trx

pp

yxp

p

x y

xy

y

Trace

y x y

xy

y

yy

y

ypTr 2 x y

xy

y xp

Derived in

previous

slide


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Why?

the probability of measuring e.g. inthe first register depends only on

2

2

2

TrwwTr

pwwTr

wwTrp

pp

yy

y

y

yy

y

y

y y y

wy

ywy

w2Tr

Partial Trace can be calculated in


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Partial Trace can be calculated in

arbitrary basis

Notice that it doesnt matter in whichorthonormal basis we trace outthe2ndsystem, e.g.

11001100 222 Tr

In a different basis

12

1

02

1

102

1

1100

1

2

10

2

110

2

1

(cont) Partial Trace can be calculated in


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Partial Trace

1100

10102

1

10102

1

22

**

**2

Tr

1

2

10

2

110

2

1

12

1

02

1

102

1

(cont) Partial Trace can be calculated in

arbitrary basis

Which is the same as in

previous slide for otherbase


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Methods to calculate the Partial Trace

Partial Traceis a linear mapthat takesbipartite statesto single system states.

We can also trace out the first system

We can compute the partial trace directlyfrom the density matrixdescription

kijljlki

ljTrkiljkiTr

2


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Partial Trace using matrices

Tracing out the 2ndsystem

33223120

13021100

3332

2322

3130

2120

1312

0302

1110

0100

33323130

23222120

13121110

03020100

2

aaaa

aaaa

aa

aaTr

aa

aaTr

aa

aaTr

aa

aaTr

aaaa

aaaaaaaa

aaaa

Tr

Tr 2

E l P ti l t (I)


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Examples: Partial trace (I)

In such circumstances, if the second register (say) is discardedthen the

state of the first register remains

Two quantum registers (e.g. two qubits) in states and (respectively) are independentif then the combined system

is in state =

In general, the state of a two-register system may not be of the

form (it may contain entanglementor correlat ions)

We can define the part ial trace, Tr2,as the unique linearoperator satisfying the identity Tr2()=

For example, it turns out that

110011002

1

2

1

2

1

2

1

10

01

2

1Tr2( )=

index means

2ndsystem

traced out

E l P ti l t (II)


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Examples:Partial trace (II)

Weve already seen this defined in the case of 2-qubit systems:discarding the second of two qubits

LetA0=I0 andA1=I1

0100

0001

1000

0010

For the resulting quantum operation, state becomes

For d-dimensional registers, the operators areAk =Ik,

where0, 1, , d1are an orthonormal basisAs we see in last slide, partial trace is a

matrix.

How to calculate this matrix of partial trace?


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Unitary transformations dont


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Unitarytransformations don t

change the local density matrix

A unitary transformation on the systemthat is traced outdoes not affect theresult of the partial trace

I.e.

22,2

Trp

UIyUp

yy

Trace

y

yy

Distant transformations dont


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Distant transformationsdon t

change the local density matrix

In fact, any legal quantum transformationon the traced out system, includingmeasurement (without communicatingback the answer) does not affect thepartial trace

I.e.

22,2 ,

Trp

yp

yy

Trace

yy


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Why??

Operations on the 2ndsystem should notaffect the statistics of any outcomes ofmeasurements on the first system

Otherwise a party in control of the 2ndsystem could instantaneouslycommunicate information to a partycontrolling the 1stsystem.

Principle of implicit


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Principle of implicit

measurement

If some qubits in a computationarenever used again, you can assume (if

you like) that they have beenmeasured (and the result ignored)

The reduced density matrix of theremaining qubits is the same

POVMs (I)


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POVMs (I)Posit ive operator valued measurement(POVM):

LetA1,A2 , ,Ambe matrices satisfying IAA jm

j

j 1

t

Then the corresponding POVM is a stochastic operation on

that, with probability produces the outcome:

j (classicalinformation) t

jj AATr

t

t

jj

jj

AA

AA

Tr

(the collapsed quantum state)

Examp le 1:Aj = jj(orthogonal projectors)

This reduces to our previously definedmeasurements

POVMs (II): calculating the measurement


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POVMs (II):calculating the measurement

outcome and the collapsed quantum state

Moreover,

tjj AATr

jj

j

jjjj

jj

jj

AA

AA

2Tr t

t

WhenAj=

j

jare orthogonal projectors and= ,

= Trjjjj

= jjjj

= j2

(the collapsed quantum state)

probability

of the

outcome:

The measurement postulate


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The measurement postulateformulated

in terms of observablesA measurement is described by a complete set of

projectors onto orthogonal subspaces. Outcome occurs

with probability Pr( ) .

The corresponding post-measure

Our

ment state is

form

:

j

j

P j

j P

P

.j

jP

This is aprojector

matrix

The measurement postulate formulatedf l


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pin terms of observables

A measurement is described by a complete set ofprojectors onto orthogonal subspaces. Outcome occurs

with probability Pr( ) .

The corresponding post-measure

Our

ment state is

form

:

j

j

P j

j P

P

.j

jP

A measurement is described by an ,

a Hermitian operator , with spectral decomposiOld form: o

tion

bservable

.j jj

MM P

The possible measurement outcomes correspond to theeigenvalues , and the outcome occurs with probability Pr( ) .

j j

j jP

The corresponding post-measurement state is

.j

j

P

P

The same

An example of observables in action


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Suppose we "measE uxample: re ".Z

has spectral decomposition 0 0 - 1 1 , so

this is just like measuring in the computational basis,and calling the outcomes "1" and "-1", respectively, for0 and 1.

Z Z

Find the spectral decomposition of .

Show that measuring corresponds to measuringthe parity of two qubits, with the result +1 correspondingto even parity, and the result

Exercis

-1 correspon

:

i

e

d

Z Z

Z Z

ng to oddparity.

00 00 11Hint: 11 10 10 01 01Z Z



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Suppose we measure the observable for astate which is an eigenstate of that observable. Showthat, with certainty, the outcome of the measurement isthe corresponding eigenvalue

Exerci

of the ob

se: M

servable.

What can be measured in


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What can be measured inquantum mechanics?

Computer science can inspire fundamental questions aboutphysics.

We may take aninformatic approach to physics.(Compare thephysical approach to information.)

Problem:What measurements can be performed in

quantum mechanics?

What can be measured in quantum mechanics?


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Traditional approach to quantum measurements:A quantum measurement is described by an observable MM is a Hermitian operator acting on the state spaceof the system.

Measuring a system prepared in an eigenstate of

Mgives the corresponding eigenvalue of Mas themeasurement outcome.

The question now presents itself Can every observable

be measured? The answer theoretically is yes. In practiceit may be very awkward, or perhaps even beyond the ingenuityof the experimenter, to devise an apparatus which couldmeasure some particular observable, but the theory alwaysallows one to imagine that the measurement could be made.

- Paul A. M. Dirac

What can be measured in quantum mechanics?

V N t


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Von Neumann measurement

in the computational basis

Suppose we have a universal set of quantumgates, and the ability to measure each qubitin the basis

If we measure we getwith probability

}1,0{

2

bb

)10( 10


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In section 2.2.5, this is described as follows

00P0 11P1

We have the projection operatorsand satisfying

We consider the projection operatororobservable

Note that 0 and 1 are the eigenvalues

When we measure this observable M, theprobability of getting the eigenvalue isand we are in

that case left with the state

IPP 10

110 PP1P0M

b2

)Pr( bbPb bb

)b(p

P

b

bb

What is an Expected value


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What is an Expected valueof an observable

b If we associate with outcome theeigenvalue then the expected outcome is

)Pr(

MTrbPTr

bPPb

bb

b

b

b

b

b

b

b

b

Von Neumann measurement in


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Von Neumann measurement in

the computational basis

Suppose we have a universal set of quantumgates, and the ability to measure each qubitin the basis

Say we have the state If we measure all n qubits, then we obtain

with probability

Notice that this means that probability ofmeasuring a in the first qubit equals

}1,0{x

n}1,0{xx

x2

x

0

1n}1,0{0x

2

x

Partial measurements


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Partial measurements

(This is similar to Bayes Theorem)

x

p1n}1,0{0x 0

x

0

1n}1,0{0x

2

x0p

If we only measure the first qubit and leavethe rest alone, then we still get with

probability The remaining n-1 qubits are then in the

renormalized state

M t l t


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Most general measurement

kk

000 U


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In section 2.2.5

This partial measurementcorresponds tomeasuring the observable

1n1n I111I000M

V N M t


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Von Neumann Measurements

A Von Neumann measurement is a type ofprojective measurement. Given anorthonormal basis , if we perform a

Von Neumann measurement with respect to of the state thenwe measure with probability

}{ k

kk}{ kk

kkkk

kk2

k2

k

TrTr


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E.x. Consider Von Neumann measurement ofthe state with respect tothe orthonormal basis

Note that

2

10,

2

10

2

10

22

10

2

)10(

We therefore get with probability

2

10

2

2


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Note that

22

10

22 10

**

22

10

2

10Tr

2

10

2

10

2

How do we implement


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How do we implement

Von Neumann measurements?

If we have access to a universal set ofgates and bit-wise measurements in the

computational basis, we can implement VonNeumann measurements with respect to anarbitrary orthonormal basis asfollows.

}{ k

How do we implement


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How do we implement

Von Neumann measurements?

Construct a quantum network thatimplements the unitary transformation

kU k Then conjugate the measurement

operation with the operation U

kk U k2

kprob

1U k


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Example: Bell basis change


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Example:Bell basis change

100101

Consider the orthonormal basisconsistingof the Bell states

110000

110010 100111 Note that

xyx

y

H

We discussed Bell basis in

lecture about superdense

coding and teleportation.

Bell measurements:destructiveand


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non-destruct ive

We can destructivelymeasure

Or non-destructivelyproject

xy

y,x

y,x x

y

H2

xyprob

xyy,x

y,x xyy,xH

2

xyprob 00

H



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000 U

0000002 Tr

Simulations among operations:


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Simulationsamong operations:general quantumoperations

Fact 1:any general quantum operat ioncan be simulatedby applying a unitary operation on a larger quantum system:

U000

Example:decoherence

0

0+1

2

2

0

0

output

discard

input

zeros discard

Simulations among operations:


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g p

simulations of POVM

Fact 2:any POVMcan also be simulated by applying aunitary operation on a larger quantum system and then

measuring:

U0

00

quantum outputinput

classical outputj

Separable states


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Separable states

m

j

jjjp1

product stateif =

separablestateif

A bipartite (i.e. two register) stateis a:

Question: which of the following states are separable?

110011001100110021

21

2

(i.e. a probabilistic mixture

of product states)

(p1 ,,pm 0)

1100110021

1

Continuous-time evolution


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Continuous-time evolution

Although weve expressed quantum operations in discreteterms, in real physical systems, the evolution is continuous

0

1LetHbe any Hermitianmatrix and tR

Then eiHtis uni tarywhy?

H =UDU, where

d

D

1

Therefore eiHt=UeiDtU= Ue

e

Uti

ti

d

1

t (unitary)

P ti ll d i 2007


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Partially covered in 2007:

Density matrices and indistinguishable states Taxonomy of normal operators

General Quantum Operations

Distinguishing states Partial trace

POVM

Simulations of operators

Separable states

Continuous time evolution

Date post:	02-Jun-2018
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2005 q 0031 Density Matrices 2

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