BONA FIDE MEASURES OF NON-CLASSICAL CORRELATIONS

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1 BON FIDE MESURES OF NON-CLSSICL CORRELTIONS New J. Phys. 16, (2014). De Pasquale in collaboration with. Farace, L. Rigovacca and V. Giovannetti

2 Outline MIN IDE: Introduction of measures of non-classical correlations classical and non-classical correlations discord, as difference between two definitions of mutual information brief review of some measures of discord, in the context of quantum metrology Discriminating Strength (DS)

3 Correlations: classical and quantum Two systems are correlated if together they contain more information than taken separately Classical Correlations B We measure correlations in terms of experimental observations on both subsystems Quantum Correlations Can we establish the nature of such correlations? inspired by G. desso s seminar at "46 Symposium on Mathematical Physics: Information Theory & Quantum Physics", Torun 2014

4 Classical correlations p a =, B random variables with joint probability distribution, p ab, b p ab p b = a p ab H() = a p a log p a Shannon entropy the amount of information we gain on average when we learn the value of MUTUL INFO I( : B) =H()+H(B) H(, B) = J(B ) =H(B) H(B ) H() I(:B) H(B ) H(B) we measure H(, B) H(B ) = a p a H(B a) Bayes rule p b a = p ab p a Nielsen & Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

5 Correlations and Entanglementent the best possible knowledge of a whole does not include the best possible knowledge of its parts, even though they may be entirely separated Entanglement, Schrodinger 1935 : ψ = i,j c ij ψ i ψ j B = ψ ψ B B INTERFERENCE among probability amplitudes pure states: entanglement quantum (non-classical) correlations. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935); E. Schrodinger, Natu!rwissenschaften 23, 807 (1935)

6 Correlations and Entanglementent the best possible knowledge of a whole does not include the best possible knowledge of its parts, even though they may be entirely separated Entanglement, Schrodinger 1935 : incomplete knowledge of the system (e.g. uncontrolled interactions with the environment) B ensemble of pure states {p i, ψ i } ρ = i p i ψ i ψ i = k p k ρ k ρ k B mixed states: entanglement? quantum correlations R. F. Werner, Phys. Rev. 40 (1989) 4277; R. Horodecki, M. Horodecki, Phys. Rev. 54, 1838 (1996)

7 Quantum correlations B a bipartite system described by the density matrix ρ ρ =Tr B [ρ] ρ B =Tr [ρ] Shannon entropy H() = a p a log p a von Neumann entropy S(ρ) = Tr[ρ log ρ] I( : B) =S(ρ )+S(ρ B ) S(ρ) = J(B ) =max S(ρB ) S(B {Π Π j }) j we measure MUTUL INFO

8 Discord: purely quantum correlations D(B ) =I( : B) J(B ) 0 all possible correlations between and B measures on : classical fraction of correlations For pure states (i.e. ρ = ψψ ) discord is a measure of entanglement There are separable mixed states (i.e. ρ = p i ρ i ρ B i ) such that D(B ) > 0 i ρ sep ρ cl ρ disc ρ ent D(B ) =0 ρ = j { j } p j j j ρ B j orthonormal basis CQ states L. Henderson and V. Vedral, J. Phys. : Math. Gen (2001); H. Ollivier and W. H. Zurek, Phys. Rev. Lett (2002)

9 How to measure discord? D(B ) =S(ρ ) S(ρ B )+min??? Π j S(ρB ) S(B {Π j }) difficult to compute? MESURE: well-defined, easy to compute, with a clear operative meaning geometric discord trace distance discord discord of response measurement-induced disturbance measurement-induced non-locality local quantum uncertainty interferometric power QUNTUM ILLUMINTION ND METROLOGY B? S. Lloyd, Science 321 (2008)

10 Measures of discord related to q. metrology Local Quantum Uncertainty Skew Info. 1 2 Tr{[ ρ,h Λ ]} U Λ B(ρ) = min {H Λ } I(ρ,H Λ ) D. Girolami, T. Tufarelli and G. desso, Phys. Rev. Lett. 110, (2013) (b) minimum quantum uncertainty associated to local observables B B zero q. uncertainty iff zero discord Bures distance Discord of Response 1 D R (ρ) =min U 2 D2 Bu(ρ,U ρu ) local unitary whose eigenvalues are the roots of the unity faithful measure of non-classical correlation, which generalizes the ENTNGLEMENT OF RESPONSE for pure states E R ( ψ) =1 max F ( ψ,u ψ) U U F ( ψ,u ψ) B U W. Roga, S.M. Gianpaolo and F. Illuminati, J. Phys. : Math. Theor. 47, (2014)

11 Measures of discord related to q. metrology Local Quantum Uncertainty Skew Info. 1 2 Tr{[ ρ,h Λ ]} U Λ B(ρ) = min {H Λ } I(ρ,H Λ ) D. Girolami, T. Tufarelli and G. desso, Phys. Rev. Lett. 110, (2013) (b) minimum quantum uncertainty associated to local observables B B zero q. uncertainty iff zero discord B C? e iϕh D E!" Interferometric Power Quantum Fisher Information P B (ρ) = 1 4 min {H Λ } F(ρ,H Λ ) GOL:estimation of a continuous parameter ϕ D. Girolami et al, Phys. Rev. Lett. 112, (2014)

12 Measures of discord related to q. metrology D(B ) =S(ρ ) S(ρ B )+min Π j local measurements on S(ρB ) S(B {Π j }) difficult to compute Quantum METROLOGY Skew Info. LQU U Λ B(ρ) =min H Λ I(ρ,H Λ ) Bures distance min/max over the special unitary group on DR Quantum Fisher Info. IP P B (ρ) = 1 4 min H Λ F(ρ,H Λ )

13 The discriminating strength B ρ ρ }ρ n ρ (1) (2) (4) (3) S R R R I I R (I ρi ) n (p =1/2) (R? ρr? ) n (p =1/2) lice Robert (1) lice: n copies of ρ + Robert: a set of unitary rotations S ={R,R,R,..} (2) lice sends the probing subsystems to Robert while keeping the reference subsystems B R (3) Robert: apply or not apply? (4) Subsystems back to lice + Robert reveals R (5) lice via optimal POVM on the n copies of B: or ρ n (R ρr ) n?

14 The discriminating strength B ρ ρ }ρ n ρ (1) (2) (4) (3) S R R R I I R (I ρi ) n (p =1/2) (R? ρr? ) n (p =1/2) GOL: lice aims to discriminate, via optimal POVM, between ρ n and (R ρr ) n lice Robert P (n) err,min := 1 2 (1 ρ n (R ρr ) n 1 ) e nξ(ρ,r ρr ) =: Q(ρ,R ρr )n D B (ρ) :=1 max R S Q(ρ,R ρr ) n 1 QUNTUM CHERNOV BOUND = Q = min 0 s 1 Tr[ρs (R ρr )1 s ] worst case with respect to the choice of the initial state (1) We will only require that I / S : I S = D B (ρ) =0

15 The discriminating strength B ρ ρ }ρ n ρ (1) (2) (4) (3) S R R R I I R (I ρi ) n (p =1/2) (R? ρr? ) n (p =1/2) We set H Λ { R =exp[i(u ΛU )] Λ = diag{λ 1,,λ d }, λ i > λ i+1 lice Robert D Λ B(ρ) =1 max H Λ Q(ρ,e ihλ ρe ih Λ ) We have proved that: 1. DS is a bona-fide measure of non-classical correlations * (b) 2. for generic bipartite systems: D B(ρ) Λ =U B(ρ)+O(Λ Λ 3 ) 2.1 for qubit-qudit systems: D B(ρ) Λ =f(λ) U B(ρ) Λ LQU B B 3. DS is invariant under constant shifts of the spectrum: D Λ B(ρ) =D Λ+b B (ρ), b R 4. for qubit-qudit systems the maximum over the set of separable states is reached by pqc states 5. there exist simple closed expressions for the set of pure states * K. Modi et al, Rev. Mod. Phys. 84, 1665 (2012), F. Ciccarello et al, New J. Phys. 16, (2014)

16 Conclusions There exist non-classical correlations going beyond the definition of entanglement Such correlations can be seen as resources in the context of quantum illumination (local quantum uncertainty, discord of response, interferometric power) Discriminating strength: well-defined measure of discord, easy to compute, with a clear operative meaning