2013 03 01 Maurizio Di Noia Presentation

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Entanglement Concentration and

Distillation in Continuous Variable Systems

Maurizio Di Noia

University of Physics of Milan

01 March 2013
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix

1 Introduction

EntanglementDefinitionExample

Gaussian Systems

2 Entanglement Concentration

System definitionSeparabilityEntanglement Measure

3 Entanglement DistillationInitial System

IPSQuantum TeleportationRelative Improvement of Quantum Teleportation Fidelity

4 Conclusions
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1 Introduction

EntanglementDefinitionExample

Gaussian Systems

2 Entanglement Concentration

System definitionSeparabilityEntanglement Measure



4 Conclusions
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Introduction

Entanglement

Mathematical Definition

Let a bipartite system consists of two subsystems A and B, describedby the density operator

HA HB,with density operators j HA and j HB then pj 0,

j pj = 1,

entangled =

j pjj j separable =j pjj j

Since Entanglement permits to implement operations not achievableonly with LOCC it represents a resource to manipulate information in

an effective way.

I t d ti E t l t C t ti E t l t Di till ti C l i A di
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Introduction

Entanglement



HA HB,with density operators j HA and j HB then pj 0,

j pj = 1,

entangled =


Since Entanglement permits to implement operations not achievableonly with LOCC it represents a resource to manipulate information in

an effective way.

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Introduction

Entanglement



HA HB,

with density operators j HA and j HB then pj 0,

j pj = 1,

entangled =


Since Entanglement permits to implement operations not achievableonly with LOCC it represents a resource to manipulate information inan effective way.

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Introduction

Entanglement



HA HB,

with density operators j HA and j HB then pj 0,

j pj = 1,

entangled =


Since Entanglement permits to implement operations not achievableonly with LOCC it represents a resource to manipulate information inan effective way.
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Introduction

Entanglement

Example

Is entangled the state :

| =

n=0cn|n |n = ||

while is separable the state :

=

n=0cn|

n

n

| |n

n

|Both and show correlations in photon numbers, but it does existan observable for which those of disappear while those of do not.

also pure quantum correlations

only classical correlation achievable through LOCC
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Introduction

Gaussian Systems

The systems studied in quantum information can be classified indiscrete Variable Systems(DV) and Continuous Variable Systems(CV).

Definition of Gaussian System

A CV System G

with n free parameters is Gaussian if its Wignerfunction W[ G](X) is Gaussian:

W[ G](X) =e

12(XX)T1(XX)

(2)n

Det[].

The Gaussian state G is completely defined by its first two moments:

the mean square vector X

the covariance matrix which contains the second moments


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Introduction

Gaussian Systems - Separability Criterion

For two mode bipartite Gaussian Systems exists a necessaryand sufficient criterion to distinguish separable states from theentangled one.Let be a separable bipartite state, we define the partialtranspose state (over the second subsystems) :

=

i

pii iT.

The Peres-Horodecki Condition1, or PPT Condition, establishes

that the state is separable if and only if2

0.

1Asher Peres, Phys. Rev. Lett. 77 1413 (1996), Horodecki, P.Horodecki, and

R.Horodecki Phys. Lett. A 223 18 (1996)2R.Simon, Phys. Rev. Lett. 84 2726, (2000)
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1 IntroductionEntanglement

DefinitionExample

Gaussian Systems

2 Entanglement ConcentrationSystem definitionSeparabilityEntanglement Measure



4 Conclusions
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Entanglement Concentration

System definition

S() is the squeezing operator given by

S() = e12

(a)2(a)2

C

i, i = 1,2 the thermal state (noise)

i =ea

iai

Tr[ea

iai]

=1

1 + ni

s=0

ni

1 + ni

s|si is|

with average number of photons ni, where = 1kBT .The state 0 interacts with a two mode mixer, a linear devicedescribed by the unitary operator U(),

U() = e(abab) = ei

C
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System definition

S() is the squeezing operator given by

S() = e12

(a)2(a)2

C

i, i = 1,2 the thermal state (noise)

i =ea

iai

Tr[ea

iai]

=1

1 + ni

s=0

ni

1 + ni

s|si is|

with average number of photons ni, where =1

kBT .The state 0 interacts with a two mode mixer, a linear devicedescribed by the unitary operator U(),

U() = e(abab) = ei

C

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System definition

The operator U() describes a complex rotations of the two modes.

The action of U() can be decomposed as in the picture

where U1 2 = ei2 aa are unitary operations which cannot modifythe entanglement of the system.= in order to study the entanglement behaviour we can reduce to atwo mode mixer with real parameter U(), R (BS).
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Separability

The PPT Condition for the state after the interaction with the BScan be written as

f(r1, r2, n1,n2,T, ) = K(n1,n2)+4T(1T)W(r1, r2,n1,n2, ) 0

Lets call f PPT functionIf we set the squeezing parameters r1 = r2 = r the minimumvalue rmin of r for which the state after the interaction with the BSwith T = 1

2is entangled

rmin = rminDet[] = 14 Arcosh 1 + 16 Det[]8Det[]=

1

4Arcosh

1 + 16

n1 +12

2 n2 +

12

28

n1 +12

n2 +

12


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g

Separability

In case of same thermal noise in both branches n1 = n2 = n,

rmin =1

4Arcosh

1 + 16(n+ 12

)4

8(n+ 12

)2.
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Entanglement Measure

In general the PPT f function can be used only to determine thethreshold between separable and entangled states through thecondition f 0, while it does not give a measure of the quantumcorrelation of the system

In the specific case of the variables from which the determinantof the covariant matrix Det[] is not dependent on, the function fhas a monotone behaviour with respect of the symplectic

eigenvalue d and hence with respect of the entanglement of thesystem

the PPT function f is easier to compute than the symplecticeigenvalue dSince Det[] = Det[](n1,n2) the f provides an entanglementmeasure as function of the phase and the BS transmissivity T


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Once we set the thermal noise n1 and n2, the squeezing parametersr1 and r2, the output state has always the maximum entanglement for

relative phase between the two squeezing =

BS transmissivity T = 12

(balanced BS)

Lets then set = and T =1

2 .

Once we set the total squeezing photons ns, and hence thesqueezing energy,

sinh2

r1 + sinh2

r2 = ns

the best squeezing photon distribution of the initial state to maximizethe entanglement of the output state from the BS is always

r1 = r2


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Once we set the thermal noise n1 and n2, the squeezing parametersr1 and r2, the output state has always the maximum entanglement for

relative phase between the two squeezing =

BS transmissivity T = 12

(balanced BS)

Lets then set = and T =1

2 .

Once we set the total squeezing photons ns, and hence thesqueezing energy,

sinh2

r1 + sinh2

r2 = ns

the best squeezing photon distribution of the initial state to maximizethe entanglement of the output state from the BS is always

r1 = r2
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Entanglement Distillation

IPS


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IPS

The outcome of the IPS are 4 conditional states ij, with i,j = 0,1,each given by

ij =Trcd

Uac(T) Ubd(T)0 |0cd dc0|Uac(T) Ubd(T)Ia Ib ij()

pij(r1, r2,n1,n2,T, )

where the pij are the relative probability.

State parameters after IPS process:

average number of thermal photons n1 and n2

squeezing parameters r1 and r2

BS transmissivity T and APD efficiency
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Quantum Teleportation


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Quantum Teleportation

Lets define the average fidelity F between the pure state = ||to be teleported and the state out after the Teleportation.

F

Tr[out] =

|out

|

,

as Wigner functions

F C

d2 W[]() W[out]().

The average fidelity measure the overlap between the two states.Shared state not entangled Classical Limit= max[Fcl] = 12
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Relative Improvement of Quantum Teleportation


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p p

Fidelity

For non Gaussian states lets consider only average fidelity valuesgreater than 1

2.

We studied the following average fidelities:

F0(r1, r2, n1,n2) obtained using as shared state between Alice

and Bob the Gaussian state 0 not processed by IPSFij(r1, r2,n1, n2, ,T), i,j = 0,1 obtained using as shared statesbetween Alice and Bob the states ij processed through IPS

Lets define the relative improvement Rij(r1, r1,n1,n2, ,T) of averagefidelity obtained with the states ij over the one obtained with the

state 0 as

Rij(r1, r2,n1,n2, ,T) = Fij(r1, r2,n1,n2, ,T) F0(r1, r2,n1,n2)F0(r1, r2,n1,n2)
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Relative Improvement of Quantum Teleportation


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p p

Fidelity

Selecting on an high number of input copies only thosecorresponding to a joint click by APD

Increase of Entanglement of the original Gaussian Systemthrough the non Gaussian IPS map
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Conclusions


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Optimal conditions for Entanglement Concentration through theinteraction with a linear passive device

Threshold separability on r as function of the noises n1 and n2 of

the state evolved through a balanced BS


Conditions for entanglement increase for a two mode bipartiteGaussian state through the de-Gaussification IPS process


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Entanglement


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Examples of applications

Quantum Information

dense coding

quantum teleportation

Quantum ComputingImplementation of algorithms more effective than the classical one

Groover searching algorithm

Shor factorization algorithm

Quantum Cryptography

BB84 protocol

E91 protocol

Back


Entanglement
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Examples of applications

Quantum Information

dense coding

quantum teleportation

Quantum ComputingImplementation of algorithms more effective than the classical one

Groover searching algorithm

Shor factorization algorithm

Quantum Cryptography

BB84 protocol

E91 protocol

Back


DV and CV Systems
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DV Systems

The DV Systems are those systems whose degrees of freedom aredescribed in Hilbert spaces of finite dimensions.ExampleThe two levels system (qubit)

|+

= 1

2(

|0

+

|1

) described in the

Hilbert space C2 physically implemented by the two polarizations of aphoton or from the base and excited states of an electron in an atom.

CV Systems

The CV Systems are those systems whose degrees of freedom aredescribed in Hilbert spaces of infinite dimensions..ExampleA radiation mode described in the Hilbert space H = L2(R)


DV and CV Systems
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DV Systems

The DV Systems are those systems whose degrees of freedom aredescribed in Hilbert spaces of finite dimensions.ExampleThe two levels system (qubit)

|+

= 1

2(

|0

+

|1

) described in the

Hilbert space C2 physically implemented by the two polarizations of aphoton or from the base and excited states of an electron in an atom.

CV Systems

The CV Systems are those systems whose degrees of freedom aredescribed in Hilbert spaces of infinite dimensions..ExampleA radiation mode described in the Hilbert space H = L2(R)
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DC and CV Systems


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DV Systems

unitarian operations are difficult to

implementstates are easily destroyed

they are probabilistic (e.g.:success rate of the teleportationprotocol of 1

4)

CV Systems

they can leverage experimentalapparatuses of quantum optics

they are states with low race ofdecoherence, i.e. resistant tonoise

they are deterministic

thanks to the size of the Hilbertspace they potentially provide agreater bandwidth (e.g. arbitrary

systems of d-levels can be storedin one CV mode)

they can be implemented inoptical cables

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Wigner Function


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Let O be a generic Hilbert-Schmidt operator, D() the displacement

operator, we define the characteristic function [O]() as

[O]() = Tr[OD()], = (1,...,n)T.

We define the Wigner function W[O]() as the complex Fouriertransformation of the characteristic function:

W[O]() =

Cn

d2n

2ne(

)[O](), = (1,...,n).

Back


Quantum Teleportation Fidelity

Quantum Teleportation Fidelity
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Minimum value of average fidelity Fmin(n1,n2) for two mode bipartiteGaussian States with shared state entangled

Fmin(n1,n2) =1

2n1+12n2+1

+

2n2+12n1+1

+ 2

Fmin = 12

n1 = n2.
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Probability of IPS States



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Due to the symmetry of IPS process the probabilities p01 and p10 arethe same. Probability behaviour shown for r1 = r2 = r and for thermalphoton number n1 = n2 = n= 0.5, quantum efficiency = 0.9, BSstransmissivity T = 0.9.

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Fidelity of the state 11


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The Fidelity F11 of the state 11 is shown as function of tanh r.

thermal noise fixed n1 = n2 = n from

the top to the bottom values n= 0,n = 0.25, n = 0.75, n= 1.25, n= 2,quantum efficiency = 0.9,transmissivity T = 0.99. Dashed linefidelity obtained with TWB.

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Relative Improvement and Probability as a function of T

Relative Improvement and Probability as a

function of T
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function of T

The relative improvement R11 of the average fidelity obtained as theIPS state 11 proportionally increases with the transmissivity T.

squeezing parameters fixedr1 = r2 = 0.3 thermal noises fixedn1 = n2 = 0.001.

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Relative Improvement and Probability as a function of T

Relative Improvement and Probability as a

function of T
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function of T

The probability p11 decreases as the transmissivity T increases.

thermal noises fixed n1 = n2 = 0.1.


Experimental Values

Experimental Values
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Best Experimental Values achievable with the current technology:

maximum squeezing parameter: r

2

thermal noise: order of decimal of thermal photon

BS transmissivity: T 0.99usual nominal quantum photodetector efficiency: 0.9
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2013 03 01 Maurizio Di Noia Presentation

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