Quantum versus Classical
Correlations in Gaussian States
Gerardo Adesso
joint work with Animesh Datta (Imperial College / ...
Outline
• Quantum versus classical correlations
• Quantum discord
• Gaussian quantum discord
• Structure of Gaussian corre...
Correlations
3
Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
A B
Classical c...
Correlations
4
Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
• Pure global c...
Quantumness in separable states
Nonorthogonal separable states cannot be
discriminated exactly
Measuring a local observabl...
A new paradigm
6
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
M. Piani, P. H...
Quantum discord
• A measure that strives at capturing all quantum correlations,
beyond entanglement, which can be nonzero ...
Quantum discord
• Almost all bipartite states have nonzero quantum discord (purely
classically correlated states are of ze...
Mutual information: classical
9
measuring total
correlations…
( )H A ( )H B
( : ) ( ) ( ) ( , )I A B H A H B H A B= + -
( ...
what are these ??
Mutual information: quantum
10
( )A
S ð ( )B
S ð
( ) ( ) ( ) ( )AB A B AB
I S S S= + -ð ð ð ð
Imperial C...
Conditional entropy
11
( )A
S ð ( )B
S ð
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussi...
Bipartite correlations
• Total correlation
• One-way classical correlation Henderson, Vedral, JPA 2001
• Quantum discord O...
Quantum discord
• For classical states (classical probability distribution embedded into
density matrices) I=J hence the q...
Gaussian states
14
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
 Very natur...
Gaussian operations
Gaussian states can be
efficiently:
 displaced
(classical currents)
 squeezed
(nonlinear crystals)
...
Gaussian operations
Gaussian states can be
efficiently:
 displaced
(classical currents)
 squeezed
(nonlinear crystals)
...
Gaussian operations
Gaussian states can be
efficiently:
 displaced
(classical currents)
 squeezed
(nonlinear crystals)
...
Gaussian operations
Gaussian states can be
efficiently:
 displaced
(classical currents)
 squeezed
(nonlinear crystals)
...
Gaussian operations
Gaussian states can be
efficiently:
 displaced
(classical currents)
 squeezed
(nonlinear crystals)
...
Gaussian states: formalism
• Up to local unitaries, Gaussian states are completely
specified by the covariance matrix…
• …...
Gaussian POVMs
21
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
• All the mea...
Gaussian quantum discord
• The Gaussian quantum discord is the quantum discord
of a bipartite Gaussian state where the opt...
Gaussian quantum discord
• Optimal POVM: heterodyne for squeezed thermal states,
homodyne for another class of states, som...
Discord/separability/entanglement
• By relating the nullity of discord with saturation of
strong subadditivity of entropy,...
• Consider this class of states (box=two-mode squeezing)
▫ s: initial entanglement; r: entanglement degradation
25
Imperia...
26
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
max discord is limited
if 1 ...
27
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
Discord/separability/entangl...
Other results & comments
• Via the Koashi-Winter duality between entanglement and
one-way classical correlations we can de...
Summary
• The concept of quantum correlations goes beyond
entanglement
• Quantum discord is a bona fide measure of such ge...
Open problems
• Maximum discord for separable states in any
dimension.
▫ known for qubits,
numerically, to be 1/3
Al-Qasim...
Open problems
• Operational interpretation of discord
• Usefulness of quantum correlations in separable
states for quantum...
Thank you
Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
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Gaussian discord imperial

  1. 1. 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
  2. 2. Outline • Quantum versus classical correlations • Quantum discord • Gaussian quantum discord • Structure of Gaussian correlations • Open problems 2 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
  3. 3. Correlations 3 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 A B Classical correlations Quantum correlations
  4. 4. Correlations 4 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 • Pure global composite states: ▫ entanglement = nonlocality = nonclassicality (quantum correlations) • Mixed global composite states: ▫ Werner 1989: separable = classically correlated A B
  5. 5. Quantumness in separable states Nonorthogonal separable states cannot be discriminated exactly Measuring a local observable on a separable bipartite state will perturb the state The eigenvectors of a separable state can be entangled superpositions … In general separable states have not a purely classical nature 5 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  6. 6. A new paradigm 6 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States M. Piani, P. Horodecki, R. Horodecki, PRL 2008
  7. 7. 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… 7 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States 0 5 10 15 20 25 30 35 #preprints year
  8. 8. 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 8 discord entanglement Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  9. 9. Mutual information: classical 9 measuring total correlations… ( )H A ( )H B ( : ) ( ) ( ) ( , )I A B H A H B H A B= + - ( : ) ( ) ( | ) ( : ) ( ) ( | ) J A B H A H A B J B A H B H B A = - = - all equal (Bayes’ rule) Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  10. 10. what are these ?? Mutual information: quantum 10 ( )A S ð ( )B S ð ( ) ( ) ( ) ( )AB A B AB I S S S= + -ð ð ð ð Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States ( ) Tr[ log ]H S® = -ð ð ð ( ) ( ) ( | ) ( ) ( ) ( | ) AB A BA B J S S A B J S S B A ¬ ® = - = - ð ð ð ð
  11. 11. Conditional entropy 11 ( )A S ð ( )B S ð Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States • Introduce POVM on B: • Posterior state of A after B has been measured: { },B B i i i P P =å 1 | Tr [ ] , with r[ ]T B B i AB A i i B i i AB p p P= P = ð ð ð | ( | ) inf ( )B i i A i i S A B pS P º å ð• looking for the “least disturbing measurement”: ( )AB I ð
  12. 12. Bipartite correlations • Total correlation • One-way classical correlation Henderson, Vedral, JPA 2001 • Quantum discord Ollivier, Zurek, PRL 2001 12 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States ( ) ( ) ( ) ( )AB A B AB I S S S= + -ð ð ð ð | ( ) ( ) ( | ) ( ) inf ( )B i AB A A i A i i S S A B S pS¬ P = - = - åJ ^ð ð ð ð | ( ) ( ) ( ) ( ) ( ) inf ( )B i AB AB AB B AB i A i i I S S p S ¬ ¬ P = - = - + å D ^ð ð J ^ð ð ð ð A B
  13. 13. 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 13 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  14. 14. Gaussian states 14 Imperial College London 10/08/2010Quantum versus Classical Correlations in 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
  15. 15. Gaussian operations Gaussian states can be efficiently:  displaced (classical currents)  squeezed (nonlinear crystals)  rotated (phase plates, beam splitters)  measured (homodyne detection) 15 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  16. 16. Gaussian operations Gaussian states can be efficiently:  displaced (classical currents)  squeezed (nonlinear crystals)  rotated (phase plates, beam splitters)  measured (homodyne detection) 16 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  17. 17. Gaussian operations Gaussian states can be efficiently:  displaced (classical currents)  squeezed (nonlinear crystals)  rotated (phase plates, beam splitters)  measured (homodyne detection) 17 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  18. 18. Gaussian operations Gaussian states can be efficiently:  displaced (classical currents)  squeezed (nonlinear crystals)  rotated (phase plates, beam splitters)  measured (homodyne detection) 18 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  19. 19. Gaussian operations Gaussian states can be efficiently:  displaced (classical currents)  squeezed (nonlinear crystals)  rotated (phase plates, beam splitters)  measured (homodyne detection) 19 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  20. 20. Gaussian states: formalism • Up to local unitaries, Gaussian states are completely specified by the covariance matrix… • … or equivalently by the four symplectic invariants 20 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States standard form TAB a c a d c b d b a g s 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= = = =
  21. 21. Gaussian POVMs 21 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States • 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 density ˆ matrix of a single-mode Gaussian state with covariance ma ˆ ˆ ˆ ˆ( ) ( ) ( ), where ( ) exp( ) trix , ( ) , and B B B B B B B W W W b b d h p h h h h h p h h s - * - P = P = - P = Pò 1 |A h ð e 1 0 ( ) T e a g b s g- = - +
  22. 22. 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 22 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States ( |) ( ) ( ) ( ) inf ( ) ( ) B AB B AB B A D S S d p S hh h h¬ P = - + òð ð ð ð 0 ( ) ( ) ( ) ( ) inf ( det )AB D f B f f f s s n n e¬ - + = - - + 2 2 2 4 , 2D A B Cn± = D ± D - D = + +
  23. 23. 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 23 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 2 2 2 2 22 4 2 2 1 2 | | 1 , 1 ; 1inf det( ) 2 , . 2 C B A D C C B A D D AB B C A D B AB C D C AB D C AB D otherwise B s e ìï + - + - + + + - + - +ïïï - £ + +ïï - +ï= í ïïï - + - + - + - +ïïïïî
  24. 24. 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… 24 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  25. 25. • Consider this class of states (box=two-mode squeezing) ▫ s: initial entanglement; r: entanglement degradation 25 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States Discord/separability/entanglement s A B C r AB s * when , 0 1 s r ¬ ® ® ¥ ® ® D D
  26. 26. 26 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States max discord is limited if 1 en to tangled 1 ¬ > ÞD Discord/separability/entanglement
  27. 27. 27 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States Discord/separability/entanglement ( )AB s® * D ( )AB s¬ * D : Gaussian Entanglement of FormationG E 1
  28. 28. 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.) 28 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  29. 29. 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 29 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  30. 30. Open problems • Maximum discord for separable states in any dimension. ▫ known for qubits, numerically, to be 1/3 Al-Qasimi & James, arXiv:1007.1814 30 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  31. 31. 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…) • … 31 Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States
  32. 32. Thank you Imperial College London 10/08/2010Quantum versus Classical Correlations in Gaussian States

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