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Quantum Computers, Algorithms and Chaos - Varenna 2005

ENTANGLEMENT IN QUANTUM OPTICS

Paolo Tombesi

Department of PhysicsUniversity of Camerino

Quantum Computers, Algorithms and Chaos - Varenna 2005

OUTLINE

• Introduction-definition of entangled states• The Peres’ criterion for separability of bipartite states• Experimental realization•A general bipartite entanglement criterion• Continuous variable case•The Simon’s criterion for Gaussian bipartite states• One example of continuous variable bipartite Gaussian states• Tripartite continuous variable Gaussian states• Example of tripartite Gaussian states

Quantum Computers, Algorithms and Chaos - Varenna 2005

entanglement: polarization of two photons

(H1 H2 ± V1 V2)/√2 (H1 V2 ± H2 V1 )/√2 or

In general, for a bipartite system, it is separable

= i wii1i2 wi ≥ 0 i wi =1

i.e. it can be prepared by means of local operations and classical communicationsacting on two uncorrelated subsystems 1 and 2

Simple criterion for inseparability or entanglement was derived by Peres (PRL 77, 1413 (1996)

These are the so-called Bell states () and ()

Quantum Computers, Algorithms and Chaos - Varenna 2005

Given an orthonormal basis in H12 = H1 H2 the arbitrary state of the bipartite state 1+2 is describedby the density matrix ( 12 )m,n (Latin indices for the first system and Greek indices for the second one). To have the transpose operation it means to invert row indices with column indices ( 12 )n ,m

The partial transpose operation (PT) is given by the the inversion of Latin indices (Greek) PT : ( 12 )m,n ( 12 )n,m ( T1

12 )m,n

We ask if the operator T1 12 is yet a density operator i.e.

Tr (T1 12 ) = 1 and T1

12 ≥ 0

It easy to prove this because the transposition does not change the diagonal elements, Thus the Trace remains invariant, and the positivity is connected with the positivity of the eigenvalues of the matrix, which do not change under transposition.

Then the violation of the positivity of the partial transpose is a sufficient criterion forentanglement

It easy to prove that the positivity of partial transpose of the state is a necessary condition forseparability. i.e. 12 separable T1

12 ≥ 0

T1 12 < 0 12 entangled

In 2x2 and 2x3 dimension for the Hilbert space 12 separable T1

12 ≥ 0 Horodecki3 Phys Lett A 223, 8 (1996)

Ox

Quantum Computers, Algorithms and Chaos - Varenna 2005

Non-linear crystal

Pump laser

sω

iωik

sk

sθ

iθpω

pk

Type I Type II

Quantum Computers, Algorithms and Chaos - Varenna 2005

Quantum Computers, Algorithms and Chaos - Varenna 2005

Dichroic Mirror

Aperture

((2))NLC

Laser beam

Phase Matching TYPE II

0.0

0.05

-0.050.050.0-0.05

810nmExtraordinary

Ordinary810nm

θy

θx

Pump Laser @405nm

| > = | H V > + e i| V H >

Quantum Computers, Algorithms and Chaos - Varenna 2005

Quantum Computers, Algorithms and Chaos - Varenna 2005

det T212 = 0.54

Quantum Computers, Algorithms and Chaos - Varenna 2005

We shall derive a general separability criterion valid for any state of any bipartite system.

Let us consider a bipartite system whose subsystems, not necessarily identical, are labeled as 1 and 2, and a sep separable state on the Hilbert space Htot = H1 H2.Ox

sep = i wii1i2 wi ≥ 0 i wi =1

Let us now choose a generic couple of observables for each subsystem, say rj , sj

on Hj (j = 1, 2),

Cj = i [rj , sj ] , j = 1, 2

Cj is typically nontrivial Hermitian operator on the Hilbert subspaces.Let’s define two Hermitian operators on Htot

u = a1r1 + a2r2 , v = b1s1 + b2s2 where aj , bj are real parameters

From the standard form of the uncertainty principle, it follows that

every state on Htot must satisfy

< ( u )2> < ( v )2> ≥| a1b1< C1> + a2b2< C2>|2

4

Quantum Computers, Algorithms and Chaos - Varenna 2005

For separable states the following Theorem holds

sep < ( u )2> < ( v )2> ≥ O2

With O = ( | a1b1| < O1> + | a2b2| < O2> ) /2 < Oj> = kwk < Cj >k

< Cj >k = Tr [Cj k j] i.e. the expectation value of operator Cj onto k j

Proof :From the definition of < ( u )2> and sep it is easy to see that

< ( u )2> = kwk [ a12 < ( r1

(k) )2>k + a22 < ( r2

(k) )2>k ] + kwk < uk>2 - (kwk < uk>)2

With rj(k) = rj - < rj>k the variance of rj onto the state kj

The same for < ( v )2>

For the Cauchy-Schwartz inequality kwk < uk>2 ≥(kwk < uk>)2 the last two terms in

< ( u )2> are bound below by zero

Follows < ( u )2> ≥ kwk [ a12< ( r1

(k) )2>k + a22 < ( r2

(k) )2>k ]

and < ( v )2> ≥ kwk [ b12 < ( s1

(k) )2>k + b22 < ( s2

(k) )2>k ]

Quantum Computers, Algorithms and Chaos - Varenna 2005

Given any two real non negative numbers and

< ( u )2> + < ( v )2> ≥ kwk [ a1 2< ( r1(k) )2>k + b1

2 < ( s1(k) )2>k +

+ kwk [ a2 2< ( r2(k) )2>k ] + b2

2 < ( s2(k) )2>k ]

Furthermore, by applying the uncertainty principle to the operators rj and sj on the state

k j , it follows

aj 2< ( rj(k) )2>k+ bj

2 < ( sj(k) )2>k ≥

aj 2< ( rj

(k) )2>k + bj2

|< Cj>k |2

4 < ( rj(k) )2>k

≥

√ |ajbj| |< Cj>k |

(min f(x) = 1x + 2 /x fmin = 2√ 1 2)

Finally < ( u )2> + < ( v )2> ≥ 2 √ O

That has to be satisfied for any and positive, thus maximizing g(x) = 2 x O - x2 < ( v )2> for x > 0

< ( u )2> ≥ O2

< ( v )2>

Quantum Computers, Algorithms and Chaos - Varenna 2005

Connection with other criteria

The Duan et al. criterion (the sum criterion) Phys. Rev. Lett. 84, 2722 (2000)is a particular case of

sep < ( u )2> < ( v )2> ≥ O2

Indeed, if we pose = = 1 in < ( u )2> + < ( v )2> ≥ 2 √ O

We get < ( u )2> + < ( v )2> ≥ 2 O

< ( u )2> ≥ O2

< ( v )2>however

Then < ( u )2> + < ( v )2> ≥ + < ( v )2> ≥ 2 O O2

< ( v )2>

The last inequality holds because of the function (min f(x) = 1x + 2 /x fmin = 2√ 1 2)

Giovannetti et al. Phys. Rev. A 67, 022320 (2003)

Quantum Computers, Algorithms and Chaos - Varenna 2005

Continuous variables

A single qubit forms a 2-dim Hilbert space, a single quantum mechanicaloscillator (i.e. single mode of the electromagnetic field or an acoustic vibrationalmode) forms an ∞ - dim Hilbert space.

This system can be described by observables (position and momentum), which have a continuous spectrum of eigenvalues. We will refer to this system as acontinuous variable system (CV)

.One can introduce the so-called rotated quadratures for the CV system

X() = ( a e- i + a+e i )/√ 2 P() = ( a e- i - a+e i )/i √ 2 = X( +/2 )

With [X() , P() ] = i

A single mode of the e.m. field in free space can be written as

E (r,t) = E0[X cos (ωt - k r ) + P sin (ω t - k r ) ]

in phase out of phase

Quantum Computers, Algorithms and Chaos - Varenna 2005

An arbitrary single mode state can be associated, by a 1 to 1 correspondence, toa symmetrically ordered characteristic function

() = Tr [ D() ] = R +i I

D() = exp( a + - * a ) = exp i √2( I x - R p )

With x and p position and momentum, or quadratures, of the CV mode

The Wigner function W() of the state is related to () by an inverse FourierTransform

W() = - 2 ∫d2 exp( - * + * ) ()

() = ∫d2 exp( * - * ) W()

Where the complex amplitude = R + i I represents the coherent state |>in the phase space.

Passing to N modes the state H Ox N

The W-function becomes

W(1… N) = - 2N [k ∫d2 k exp( - k k* + k* k )] (1.. N)

Quantum Computers, Algorithms and Chaos - Varenna 2005

We shall consider N modes Gaussian states with the characteristic function (1,…., N) Gaussian as it is the W-function W(1,…,N)

To the k-th mode we associate the complex variable k = kR +i k

I which is

Represented by the vector ( )= k real R2 (k = 1,..,N)kI

-kR

In terms of the real variables ( T1,…..,T

N) T R2N the arbitrary N-mode

Gaussian characteristic function takes the form

() = exp ( - TV + i dT )

Where d R2N and V is a 2N x 2N real, symmetric strictly positive matrix V = V T V > 0

The corresponding N-mode Gaussian W-function by the inverse F-T is:

W() = exp( -1/4 dTV -1d)

N √ det V

exp ( - T V-1 + dT V-1 ) 2 √2

T (x1,p1,….,xN,pN) R2N

Define a 2N-dim phase space

Quantum Computers, Algorithms and Chaos - Varenna 2005

V is the correlation matrix (CM) of the N-mode state

Vnm = ( n m + m n) /2 (n,m = 1,2,….,2N)

m= m - < m >

d = < > is the displacement of the state

Both d and V are measurable quantities defined for every N-mode state

When the state is Gaussian it is fully characterized by these two quantities

Gaussian ( V, d )

The CM expresses the covariance between the position and momentum quadratures(in-phase and out-of-phase) of the state . It must respect the uncertainty principle < ( xk ) 2 > < ( pk’ ) 2 > 1/4 k k’ (k,k’ = 1,..,N)

The uncertainty principle reads V + i I(N)/2 o

(-1 0)0 1Introducing the N-mode symplectic matrix I(N)= Ii Ii =

1

N

Quantum Computers, Algorithms and Chaos - Varenna 2005

CV entanglement

Simon (Phys. Rev. Lett. 84, 2726 (2000) showed how to extend the Peres criterion for separability to the CV case.

Consider a bipartite CV state AB and introduce its phase space representationthrough the W-function W() T ( xA, pA, xB, pB) the PT operation on the stateAB in HAB is equivalent to a partial mirror reflection of W() in the phase space

PT : AB PT(AB ) W() W()

with diag (1,1,1,-1)

PT is a local time-reversal which inverts the momentum of only one subsystem (B in our case)

The extension to CV of the Peres criterion is: AB separable W() is a genuine W-function

genuine means corresponding to a physical stateA genuine W-function implies a genuine V CM i.e. AB separable V is a genuine CM

where V is the CM of W() Then the PT implies V V

AB separable V + i I(2)/2 o V > 0

Quantum Computers, Algorithms and Chaos - Varenna 2005

The condition AB separable V + i I(2)/2 o

if V is of the form ( )A CCT B

implies

In the case of two-mode Gaussian states, the positivity of PT represents a necessary andsufficient condition for separability, i.e.

AB(Gaussian) separable PT(AB) o

This is the Simon’s criterion

Where J = ( ) 0 1-1 0 is the one-mode symplectic matrix

det A det B + ( 1/4 - |det C|)2 - Tr (AJCJBJCTJ) - det A + det B

4 o

Quantum Computers, Algorithms and Chaos - Varenna 2005

Generation of CV bipartite entangled state

a1

a2

b1

b2Consider a beam splitter described by theoperator B(θ,) = expθ/2(a+

1a2 e i - a1a+2 e-i )

where the transmission and reflection coefficients are t = cos θ/2 r = sin θ/2while is the phase difference between the reflected and transmitted fields

If the input fields are coherent states we have B(θ,)| 1 >1)| 2 >2 = B(θ,)D1(1)D2(2)|0 >1)|0 >2 = D() = exp( a+- *a ) D1(1 cos θ/2 + 2 e i sin θ/2) D2(2 cos θ/2 - 1 e -i sin θ/2)| 0 >1|0 >2=| 1 t + 2 r e i >1| 2 t - 1 r e -i >2

And are not entangled.When the input states are squeezed states we get: (with j= rj e -2i)

B(θ,) S1 (1)S2(2)|0 >1)|0 >2 = B(θ,) S1(1) B+(θ,) B(θ,) S2(2) B+(θ,) B(θ,) | 0 >1)|0 >2 = S1(r1 + r2e2i ) S2(r1 e-2i + r2) S12(r1 e-i - r2 ei ) | 0 >1)|0 >2

S12()=exp( - a1a2+ * a+1a+

2) is the two-mode squeezing operator

Quantum Computers, Algorithms and Chaos - Varenna 2005

Applying the two-mode squeeze operator S() = exp - r( a+1 a+

2- a1 a2)( = r e 2i with = 0 is the squeezing parameter) to two vacuum modes | 0 >1| 0 >2

And using the disentangling theorem of Collet (Phys. Rev. A 38, 2233 (1988)).

S() | 0 >1| 0 >2 = exp(a+1 a+

2 tanh r )

( )1

cosh r

a+1 a1+ a+

2 a2 + 1

X exp - (a1 a2 tanh r ) | 0 >1| 0 >2 = √ 1 - ∑n n/2 | n >1| n >2

= tanh2 r

The two-mode squeezed vacuum state is the quantum optical representative for bipartitecontinuous-variable entanglement.

Quantum Computers, Algorithms and Chaos - Varenna 2005

An arbitrary two-mode Gaussian state AB can be associated to its displacement d and CM V (such association is a 1-1 correspondence in the case of a Gaussian state). Since its separability properties do not vary under LOCCs, we may, first, cancel its displacement d via local displacement operators Dk ( k = A,B), and then reduce its CM V to the normal form

For the two-mode vacuum squeezed state we have

Quantum Computers, Algorithms and Chaos - Varenna 2005

The best way to see that it is really entangled is to consider the Simon criterion.

Applied to the previous CM gives 1/2 (cosh 4r)/2 that is never satisfied for r 0.

With the V matrix given in normal form the Simon’s separability criterion V + iI (2)/2 0reads

4(ab - c2) (ab - c’2) (a2 + b2) + 2 |c c’| - 1/4

Quantum Computers, Algorithms and Chaos - Varenna 2005

CV TRIPARTITE entangled states

We’ll consider the scheme introduced by Dür et al. PRL 83, 3562 (1999)According with their classification one has five entanglement classes

A tripartite state is composed by three distinguishable parties A,B and C

Quantum Computers, Algorithms and Chaos - Varenna 2005

The extension to more dimensions of the Simon’s criterion was proved byWerner and Wolf PRL 86, 3658 (2001) and is based on the positivity of thepartial transpose

Let us consider the Gaussian state 1N , which is a bipartite state of 1 X N ( i.e. 1 mode at Alice site and N on Bob’s site) and V 1,N be its CM, the partial transposition is given applying 1 = I (the partial transpose at Alice).

CRITERIONA Gaussian state 1N of a 1 X N system is separable (with respect to the grouping A ={1} and B ={2,…..,N}) if and only if 1 V1N 1 + iI (N+1)/2 0

The classification is given then as: (V’K = K VABC K+ iI (3)/2 K = A,B,C ) class 1 V’A 0, V’B 0, V’C 0 class 2 V’A 0, V’B 0, V’C 0 (permutation of A,B,C) class 3 V’A 0, V’B 0, V’C 0 (permutation of A,B,C) class 4 or 5 V’A 0, V’B 0, V’C 0

Quantum Computers, Algorithms and Chaos - Varenna 2005

For small mirror displacements, in interaction picture

H = –∫d2r P(r,t) x(r,t)

P is the radiation pressure force and x is the mirror displacement(r is the coordinate on the mirror surface)

x(r,t) BOB

(Loudon et al. 1995)

Pirandola et al. J. Mod. Opt. 51, 901 (2004)

Quantum Computers, Algorithms and Chaos - Varenna 2005

Pinard et al. Eur.Phys. J. D 7,107 (1999)

Frequency and mass M

ω0

ω0-

ω0+

x(r,t)b e-i t + b+ei t)exp[-r2/w2]fundamental Gaussian mode

Quantum Computers, Algorithms and Chaos - Varenna 2005

Quantum Computers, Algorithms and Chaos - Varenna 2005

RWA (neglecting all terms oscillating faster than )

Heff = - i(a1b - a1+b+) - i θ(a2b+ - a2

+b)

a1 @ ω0- and a2 @ ω0+

parametric interactiongenerates EPR-like entangled states used in continuous variable teleportation

rotation (BS) it might degradeentanglement

quant-ph/02/07094 - JOSA B 2003

Quantum Computers, Algorithms and Chaos - Varenna 2005

Dynamics is studied with the normally orderedcharacteristic function

(,t=0) = e -nth

||2

corresponding to a1,b,a2

nth = average number of thermal excitations for mode b

i.e. initially vacuum states for a1 and a2 and b in a thermal state

Quantum Computers, Algorithms and Chaos - Varenna 2005

After an interaction time the state is

1 b 2 = ∫ d2 ∫ d2 ∫ d2(,) e -(||2+ ||2+ ||2)D1(-) Db(-) D2(-)

(,) is the evolution of (,) which is still Gaussian(,) = e- V T

)

Di = normally ordered displacement operators Di () = e-ci e*ci

We can now study the class of entanglement of the tripartite state 1 b 2

Werner & Wolf PRL 86, 3653 (2001)

Quantum Computers, Algorithms and Chaos - Varenna 2005

It turns out that 1 b 2 is bi-separable with respect to b at interaction times = ….

i.e. 1 b 2 is a one-mode bi-separable state (class 2 entangled)

In particular the tripartite state at these times can be written as the tensor product of the initial thermal state of the mirror and a pure EPR state for a1 and a2 with squeezing parameter depending on θ /

!!Radiation pressure could be a source of two-mode entangled states!!

Quantum Computers, Algorithms and Chaos - Varenna 2005

At all other times the tripartite state 1 b 2 is fully entangled (class 1) at any nth

By tracing out one mode of the three we study the entanglement of a bipartite subsystem

We find that mode a2 and b are never entangled

Modes a1 and a2 are entangled (extremely robust with respect to the mirror temperature nth)

Modes a1 and b are entangled even though the region of entanglement is small and depends on nth

a1

a2

b

Quantum Computers, Algorithms and Chaos - Varenna 2005

Braunstein and van Loock Quantum Information with Continuous Variablesto appear in Rev. Mod. Phys.quant-ph/0410100

REFERENCES

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