#### Transcript Quantum versus Classical Correlations in Gaussian States

Quantum versus Classical Correlations in Gaussian States Gerardo Adesso Imperial College London 10/08/2010 joint work with Animesh Datta (Imperial College / Oxford) School of Mathematical Sciences 2 Outline • Quantum versus classical correlations • Quantum discord • Gaussian quantum discord • Structure of Gaussian correlations • Open problems Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 3 Correlations Classical correlations A B Quantum correlations Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 4 Correlations A B • Pure global composite states: ▫ entanglement = nonlocality = nonclassicality (quantum correlations) • Mixed global composite states: ▫ Werner 1989: separable = classically correlated Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 6 A new paradigm M. Piani, P. Horodecki, R. Horodecki, PRL 2008 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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) # preprints • Recently became very popular: stats from arXiv:quant-ph… 35 30 25 20 15 10 5 0 year Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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 discord entanglement Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 9 Mutual information: classical H (A ) H (B ) measuring total correlations… I (A : B ) = H (A ) + H (B ) - H (A , B ) J (A : B ) = H (A ) - H (A | B ) all equal (Bayes’ rule) J (B : A ) = H (B ) - H (B | A ) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 10 Mutual information: quantum H ® S ( ñ ) = - T r[ñ log ñ ] S (ñA ) S (ñB ) I (ñA B ) = S (ñA ) + S (ñB ) - S (ñA B ) J J ¬ ® (ñA B ) = S (ñA ) - S ( A | B ) w h a t a r e t h ese ? ? (ñB A ) = S (ñB ) - S ( B | A ) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 11 Conditional entropy • Introduce POVM on B: B S (ñA ) {P i } , S (ñB ) å P B = 1 i i • Posterior state of A after B has been measured: B ñ A |i = I (ñA B ) • looking for the “least disturbing measurement”: Quantum versus Classical Correlations in Gaussian States T rB [ P i ñ A B ] , pi B w it h p i = T r[P i ñ A B ] S ( A | B ) º in f P B i å p i S ( ñ A |i ) i Imperial College London 10/08/2010 12 Bipartite correlations A • Total correlation B I (ñA B ) = S (ñA ) + S (ñB ) - S (ñA B ) • One-way classical correlation J ¬ Henderson, Vedral, JPA 2001 ^ñ ( A B ) = S ( ñ A ) - S ( A | B ) = S ( ñ A ) - in f P • Quantum discord D ¬ ^ñ ( A B ) = I (ñA B ) - J å p i S ( ñ A |i ) i Ollivier, Zurek, PRL 2001 ¬ (^ñ A B ) = S ( ñ B ) - S ( ñ A B ) + in f P Quantum versus Classical Correlations in Gaussian States B i B i å p i S ( ñ A |i ) i Imperial College London 10/08/2010 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 twoqubit 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 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 14 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 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 15 Gaussian operations Gaussian states can be efficiently: displaced (classical currents) squeezed (nonlinear crystals) rotated (phase plates, beam splitters) measured (homodyne detection) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 16 Gaussian operations Gaussian states can be efficiently: displaced (classical currents) squeezed (nonlinear crystals) rotated (phase plates, beam splitters) measured (homodyne detection) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 17 Gaussian operations Gaussian states can be efficiently: displaced (classical currents) squeezed (nonlinear crystals) rotated (phase plates, beam splitters) measured (homodyne detection) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 18 Gaussian operations Gaussian states can be efficiently: displaced (classical currents) squeezed (nonlinear crystals) rotated (phase plates, beam splitters) measured (homodyne detection) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 19 Gaussian operations Gaussian states can be efficiently: displaced (classical currents) squeezed (nonlinear crystals) rotated (phase plates, beam splitters) measured (homodyne detection) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 20 Gaussian states: formalism • Up to local unitaries, Gaussian states are completely specified by the covariance matrix… æa ö c ÷ çç ÷ ÷ ç æa ö g÷ a d÷ çç ÷ çç ÷ ÷ s = s AB = ç T = ç ÷ ÷ ç ÷ ÷ st a n d a r d ççè g c b b ø÷ ÷ ç for m ÷ çç ÷ ÷ çç d b ÷ è ø • … or equivalently by the four symplectic invariants A = d et a , B = d et b , C = d et g , D = d et s Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 21 Gaussian POVMs • All the measurements that can be done by linear optics (appending Gaussian ancillas, manipulating with symplectic transformations, plus homodyne detection): - 1 0 † † * P B ( h ) = p Wˆ B ( h ) P BWˆ B ( h ) , w h er e Wˆ B ( h ) = ex p ( hbˆ - h bˆ ) , p - 1 ò 2 d h P B ( h ) = 1, a n d P 0 B is t h e d en sit y m a t r ix of a sin gle-m od e G a u ssia n st a t e w it h cov a r ia n ce m a t r ix s 0 • The posterior state ñ A | h of A after measuring B has a covariance matrix e (independent of the measurement outcome) e = a - g(b + s 0 ) Quantum versus Classical Correlations in Gaussian States - 1 g T Imperial College London 10/08/2010 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 D ¬ ( ñ A B ) = S ( ñ B ) - S ( ñ A B ) + in f P B (h) ò dhp B ( h )S ( ñ A | h ) • and can be rewritten as D ¬ ( s A B ) = f ( B ) - f ( n - ) - f ( n + ) + in f f ( d et e ) s 0 ▫ where the symplectic eigenvalues are 2 2n ± = D ± Quantum versus Classical Correlations in Gaussian States 2 D - 4D , D = A + B + 2C Imperial College London 10/08/2010 23 Gaussian quantum discord • Optimal POVM: heterodyne for squeezed thermal states, homodyne for another class of states, something inbetween for the other two-mode Gaussian states ìï ï 2C ï ï ï ï in f d et ( e ) = ïí s0 ï ï ï ï ï ï ïî 2 + (- 1 + B )(- A + D )+ 2 |C | (AB - C 2 C 2 + (- 1 + B )(- A + D) 2 1+ B + D - Quantum versus Classical Correlations in Gaussian States , (D (A B + D) - AB 2 ) £ (1 + B )C 2 (A + D ); ) C 4 2 + (- A B + D ) - 2C 2 , o t h er w ise . 2B Imperial College London 10/08/2010 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… Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 25 Discord/separability/entanglement • Consider this class of states (box=two-mode squeezing) ▫ s: initial entanglement; r: entanglement degradation w h en s , r ® ¥ A s s B r * AB D D ¬ ® 0 ® ® 1 C Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 26 Discord/separability/entanglement m a x d iscor d is lim it ed t o 1 if D Quantum versus Classical Correlations in Gaussian States ¬ > 1 Þ en t a n gled Imperial College London 10/08/2010 27 Discord/separability/entanglement 1 D ® (s * AB ) ¬ D (s * AB ) EG : G a u ssia n E n t a n glem en t of F or m a t ion Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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.) Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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 Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 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…) •… Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010 Thank you Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010