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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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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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 Concentration Entanglement Distillation Conclusions Appendix
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.
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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 inan effective way.
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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 inan effective way.
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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 Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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 Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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1 IntroductionEntanglement
DefinitionExample
Gaussian Systems
2 Entanglement ConcentrationSystem definitionSeparabilityEntanglement Measure
3 Entanglement DistillationInitial System
IPSQuantum TeleportationRelative Improvement of Quantum Teleportation Fidelity
4 Conclusions
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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 =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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Entanglement Concentration
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Concentration
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Entanglement Concentration
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
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Concentration
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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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Concentration
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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
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Concentration
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Entanglement Measure
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
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Concentration
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Entanglement Measure
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Distillation
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Distillation
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Entanglement Distillation
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Conclusions
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Entanglement Concentration
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
Entanglement Distillation
Conditions for entanglement increase for a two mode bipartiteGaussian state through the de-Gaussification IPS process
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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)
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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).
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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Probability of IPS States
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
Probability of IPS States
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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.
Introduction Entanglement Concentration Entanglement Distillation Conclusions Appendix
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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