quantum versus classical correlations in gaussian states

Quantum versus Classical Correlations in Gaussian States

Gerardo Adesso

joint work with Animesh Datta (Imperial College / Oxford)

School of Mathematical Sciences

Imperial College London 10/08/2010

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Outline

•Quantum versus classical correlations

•Quantum discord

•Gaussian quantum discord

•Structure of Gaussian correlations

•Open problems


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Correlations

Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010

A B

Classical correlations

Quantum correlations

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Correlations


•Pure global composite states: ▫entanglement = nonlocality

= nonclassicality (quantum correlations)

•Mixed global composite states:▫Werner 1989: separable = classically

correlated

A B

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Quantumness in separable statesNonorthogonal separable states cannot

be discriminated exactlyMeasuring a local observable on a

separable bipartite state will perturb the stateThe eigenvectors of a separable state

can be entangled superpositions

…

In general separable states have not a purely classical nature


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A new paradigm


M. Piani, P. Horodecki, R. Horodecki, PRL 2008

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Quantum discord• A measure that strives at capturing all quantum

correlations, beyond entanglement, which can be nonzero also in separable states

• Introduced a decade ago in two independent works (Ollivier/Zurek and Henderson/Vedral)

• Recently became very popular: stats from arXiv:quant-ph…


05101520253035

# pr

eprin

ts

year

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Quantum discord• Almost all bipartite states have nonzero quantum discord

(purely classically correlated states are of zero measure) A. Ferraro et al. PRA 2010

• Reduces to the entropy of entanglement on pure bipartite states

• Quantum discord without entanglement may allow for a computational speed-up in the DQC1 model of quantum computation A. Datta et al. 2008-2010; experiment: M. Barbieri et al. PRL 2008 discord

entanglement


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Mutual information: classical

measuring total

correlations…

( )H A ( )H B

( : ) ( ) ( ) ( , )I A B H A H B H A B= + -( : ) ( ) ( | )( : ) ( ) ( | )

J A B H A H A BJ B A H B H B A

= -= -

all equal (Bayes’

rule)



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what are these ??

Mutual information: quantum

( )AS ñ ( )BS ñ

( ) ( ) ( ) ( )AB A B ABI S S S= + -ñ ñ ñ ñ



( ) Tr[ log ]H S® = -ñ ñ ñ

( ) ( ) ( | )( ) ( ) ( | )

AB A

BA B

J S S A BJ S S B A

¬

®= -= -

ñ ññ ñ

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Conditional entropy

( )AS ñ ( )BS ñ



• Introduce POVM on B:

• Posterior state of A after B has been measured:

{ },B Bi i

iP P =å 1

|Tr [ ],

with r[ ]T

BB i AB

A ii

Bi i AB

pp P=

P= ññ

ñ

|( | ) inf ( )Bi

i A ii

S A B pSP

º å ñ• looking for the “least disturbing measurement”:

( )ABI ñ

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Bipartite correlations•Total correlation

•One-way classical correlation Henderson, Vedral, JPA 2001

•Quantum discord Ollivier, Zurek, PRL 2001


( ) ( ) ( ) ( )AB A B ABI S S S= + -ñ ñ ñ ñ

|( ) ( ) ( | ) ( ) inf ( )Bi

AB A A i A ii

S S A B S pS¬P

= - = - åJ ñ̂ ñ ñ ñ

|

( ) ( ) ( )( ) ( ) inf ( )

Bi

AB AB AB

B AB i A ii

IS S pS

¬ ¬

P

= -= - + å

D ñ̂ ñ J ñ̂ñ ñ ñ

A B

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Quantum discord• For classical states (classical probability distribution embedded into

density matrices) I=J hence the quantum discord vanishes

• Zurek introduced it in the context of environment-induced selection, identifying classical states with the pointer states

• The optimization involved in the conditional entropy is hard. Closed analytical formulas are available only for special families of two-qubit staes (X-shaped), not even for arbitrary states of two qubits

• Two recent independent works, including this one, defined a Gaussian version of the quantum discord for bipartite Gaussian states, where the optimization is restricted to Gaussian measurements P. Giorda & M.G.A. Paris PRL 2010; GA & A. Datta PRL 2010

• We have solved the optimization problem and obtained a simple formula for the Gaussian quantum discord of arbitrary two-mode Gaussian states


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Gaussian states


Very natural: ground and thermal states of all physical systems in the harmonic approximation regime (M.S.Kim: like orange juice and sunshine)

Relevant theoretical testbeds for the study of structural properties of entanglement and correlations, thanks to the symplectic formalism

Preferred resources for experimental unconditional implementations of continuous variable protocols

Crucial role and remarkable control in quantum optics- coherent states- squeezed states- thermal states

Gaussian operationsGaussian states can beefficiently: displaced (classical currents)

squeezed (nonlinear crystals)

rotated (phase plates, beam splitters)

measured (homodyne detection)

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Gaussian states: formalism• Up to local unitaries, Gaussian states are

completely specified by the covariance matrix…

• … or equivalently by thefour symplectic invariants


standardform

TAB

a ca d

c bd b

a gs s

g b

æ ö÷ç ÷ç ÷çæ ö ÷ç ÷÷ç ç ÷÷ç= = = ç ÷÷ç ç ÷÷ç ÷÷ç çè ø ÷ç ÷ç ÷ç ÷çè ø

det , det , det , detA B C Da b g s= = = =

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Gaussian POVMs


• All the measurements that can be done by linear optics (appending Gaussian ancillas, manipulating with symplectic transformations, plus homodyne detection):

• The posterior state of A after measuring B has a covariance matrix (independent of the measurement outcome)

1 0 † †

1 2 0

0

is the densityˆ

matrix of a single-mode Gaussian state with covariance ma

ˆ ˆ ˆ ˆ( ) ( ) ( ), where ( ) exp( )

trix

,( ) , a

n

d

B B B B B

B B

W W W b bd

h p h h h h hp h h

s

- *

-P = P = -

P = Pò 1

|A hñe

10( ) Te a g b s g-= - +

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Gaussian quantum discord• The Gaussian quantum discord is the quantum

discord of a bipartite Gaussian state where the optimization in the conditional entropy is restricted to Gaussian POVMs

• and can be rewritten as

▫ where the symplectic eigenvalues are


( |)( ) ( ) ( ) inf ( ) ( )

BAB B AB B AD S S d p S hh

h h¬P

= - + òñ ñ ñ ñ

0( ) ( ) ( ) ( ) inf ( det )ABD f B ff f

ss n n e¬

- += - - +

2 22 4 , 2D A B Cn± = D ± D - D = + +

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Gaussian quantum discord• Optimal POVM: heterodyne for squeezed thermal

states, homodyne for another class of states, something in-between for the other two-mode Gaussian states


( )( ) ( )( )( ) ( ) ( ) ( )

( ) ( )0

2 22 2

2

22 4 2

2 1 2| | 1, 1 ;

1inf det( )2

, .2

C B A D C C B A DD AB B C A D

B

AB C D C AB D C AB Dotherwise

B

se

ìï + - + - + + + - + - +ïïï - £ + +ïï - +ï= íïïï - + - + - + - +ïïïïî

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Discord/separability/entanglement• By relating the nullity of discord with saturation

of strong subadditivity of entropy, we demonstrated that (for finite mean energies) the only two-mode Gaussian states with zero Gaussian discord are product states

• All correlated Gaussian states (including all entangled states and all non-product separable mixed states) are quantumly correlated!

• This proves the truly quantum nature of Gaussian states despite their positive Wigner function…


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• Consider this class of states (box=two-mode squeezing)▫ s: initial entanglement; r: entanglement

degradation


Discord/separability/entanglement

sA

B

Cr

ABs *when ,

01

s r¬

®

® ¥®®

DD

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max discord is limited if 1 en

to tangled

1¬ > ÞD


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( )ABs® *D( )ABs¬ *D

pure

: Gaussian Entanglement of FormationGE

1

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Other results & comments• Via the Koashi-Winter duality between entanglement

and one-way classical correlations we can derive a closed formula for the Gaussian EoF of a family of three-mode Gaussian states

• Only in very special cases we can prove that the Gaussian quantum discord realizes the absolute minimum in the conditional entropy optimization not constrained to Gaussian POVMs (this is related to the problem of additivity of bosonic channel capacity etc…)

• It would be interesting to prove, or show counterexamples to it, that Gaussian POVMs are always optimal among all continuous variable measurements (including photodetection etc.)


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Summary• The concept of quantum correlations goes

beyond entanglement• Quantum discord is a bona fide measure of such

general quantum correlations• Quantum discord can be computed for Gaussian

states under Gaussian measurements• All correlated Gaussian states have quantum

correlations• They are limited for separable states• They admit upper and lower bounds as a

function of the entanglement, for entangled states


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Open problems•Maximum discord for separable states in

any dimension.

▫known for qubits,numerically, to be 1/3

Al-Qasimi & James, arXiv:1007.1814


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Open problems• Operational interpretation of discord• Usefulness of quantum correlations in separable

states for quantum information processing• Understanding connection with other

nonclassicality indicators in continuous variable systems (e.g. in terms of P function)

• Producing a theory of quantum correlations, with axioms to be satisfied by any valid measure of quantum correlations (e.g. nonincreasing under local operations and classical communication…)

• …



Thank you


quantum versus classical correlations in gaussian states

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