i f INSTITUT FOURIER Quantum correlations and Geometry Dominique Spehner Institut Fourier et Laboratoire de Physique et Modélisation des Milieux Condensés, Grenoble
Outlines Entangled and non-classical states Bures distance on the set of quantum states Geometrical measures of quantum correlations Link with Quantum State Discrimination Joint work with: M. Orszag, Pontificia Universidad Católica de Chile, Santiago
Outlines Entangled and non-classical states
Basic mathematical objects in quantum mechanics (1) A Hilbert space H (in this talk: n dim H 8). (2) States ρ are non-negative operators on H with trace one. (3) Observables A are self-adjoint operators on H (in this talk: A P MatpC, nq finite Hermitian matrices) (4) An evolution is given by a map M : MatpC, nq Ñ MatpC, nq which is (i) trace preserving (so that trpmpρqq trpρq 1) (ii) Completely Positive (CP), i.e. for any integer d 1 and any d d matrix pa ij q d i,j1 0 with elements A ij P MatpC, nq, one has pmpa ij qq d i,j1 0. Special case: unitary evolution Mpρq U ρ U with U unitary.
Pure and mixed quantum states A pure state is a rank-one projector ρ ψ ψyxψ with ψy P H, }ψ} 1 (actually, ψy belongs to the projective space P H). The set EpHq of all quantum states is a convex cone. Its extremal elements are the pure states. A mixed state is a non-pure state. It has infinitely many pure state decompositions ρ η i ψ i yxψ i, with η i 0, i η i 1 and ψ i y P P H. i Statistical interpretation: the pure states ψ i y have been prepared with probability η i.
Separable and entangled states A bipartite system AB is composed of two subsystems A & B with Hilbert spaces H A and H B. It has Hilbert space H AB H A b H B. A pure state Ψy of AB is separable if it is a product state Ψy ψy b φy with ψy P P H A and φy P P H B. A mixed state ρ is separable if it admits a pure state decomposition ρ i η i Ψ i yxψ i with Ψ i y ψ i y b φ i y separable for all i. Nonseparable states are called entangled. Entanglement is ãñ the most intringuing property of quantum theory. ãñ used as a resource in quantum information (e.g. quantum cryptography, teleportation, high precision interferometry...).
Examples of entangled and separable states Let H A H B C 2 Then the pure states Ψ Bell y 1? 2 0 b 0y 1 b 1y (qubits) with canonical basis t 0y, 1yu. are maximally entangled. ãñ states leading to the maximal violation of the Bell inequalities observed experimentally ñ nonlocality [Aspect et al 82,... ]. The mixed state ρ 1 2 Ψ Bell yxψ Bell 1 2 Ψ Bell yxψ Bell is separable! (indeed, ρ 1 2 0 b 0yx0 b 0 1 1 b 1yx1 b 1 ). 2
Classical states A state ρ of AB is classical if it has a spectral decomposition ρ k p k Ψ k yxψ k with separable states Ψ k y α k y b β k y. Classicality is equivalent to separability only for pure states. A state ρ is A-classical if ρ i q i α i yxα i b ρ B i with t α i yu orthonormal basis of H A and ρ B i arbitrary states of B. The set C (resp. C A ) of all (A-)classical states is not convex. Its convex hull is the set of separable states S AB. ρ S AB C A C B Some tasks impossible to do classically can be realized using separable non-classical mixed states. C AB
Quantum vs classical correlations Central question in quantum information theory: identify (and try to protect) the quantum correlations (QCs) responsible of the exponential speedup of quantum algorithms. classical correlations For mixed states, two (at least) measures of QCs quantum correlations Õ entanglement [Schrödinger, Werner 89] nonclassicality Ñ quantum discord [Ollivier & Zurek 01, Henderson & Vedral 01]
Outlines Entangled and non-classical states Bures distance on the set of quantum states
Distances between quantum states The set of quantum states EpHq can be equipped with several distances: L p -distance: d p pρ, σq }ρ σ} p trp ρ σ p q 1 p with p 1. ãñ not natural from a quantum information point of view (excepted maybe for p 1) Bures distance: d B pρ, σq 2 2 a F pρ, σq 1 2 with the fidelity F pρ, σq }? ρ? σ} 2 1 F pσ, ρq [Bures 69, Uhlmann 76] Kubo-Mori distance with metric ds 2 g KM pdρ, dρq ds 2 pρq with Spρq trpρ ln ρq von Neumann entropy [Balian, Alhassid and Reinhardt, 86].
Properties of the Bures distance (1) d B pρ, σq 2 2 a F pρ, 2 σq 1 with F pρ, σq }? ρ? σ} 2 1. (i) Monotonicity property: for any trace-preserving CP map M, d B pmpρq, Mpσqq d B pρ, σq (not true for d p for p 1). (ii) For pure states d B coincides with the Fubiny-Study metric d FS pρ ψ, ρ φ q inf } ψy φy} 2 2 2 xψ φy 1 on P H. ψy, φy (iii) If ρ k p k ψ k yxψ k and σ k q k ψ k yxψ k commute, then d B coincides with the Heilinger distance on the probability simplex E clas tpp k q; p k 0, k p k 1u R n,? d B pρ, σq d H pp, qq 2 2 pk q 2 k 1. k
Properties of the Bures distance (2) (iii ) d B pρ, σq sup d H pp, qq, the sup is over all measurements giving the outcome i with proba. p i (for state ρ) and q i (for state σ). (iv) d B pρ, σq 2 is jointly convex in pρ, σq. (v) d B is a Riemannian distance on EpHq with metric ds 2 pg B q ρ pdρ, dρq (not true for d p, p 2). g B is the smallest monotonous metric, and the only such metric coinciding with the Fubiny-Study metric on the projective space P H (not true for the monotonous metric g KM ) [Petz 96] (vi) The Bures metric g B (= quantum Fisher information) is physical: it allows to estimate the best phase precision in quantum interferometry. [Braunstein and Caves 94]
Outlines Entangled and non-classical states Bures distance on the set of quantum states Geometrical measures of quantum correlations
ρ Geometric approach of quantum correlations Geometric entanglement: E B pρq min d B pρ, σ sep q 2 σ sep PS AB Geometric quantum discord : S AB C A C AB C B D A pρq min d B pρ, σ A-cl q 2 σ A-cl PC A Properties: E B pρ Ψ q D A pρ Ψ q for pure states ρ Ψ E B is convex Monotonicity: E B pm A b M B pρqq E B pρq for any tracepreserving CP maps M A and M B for subsytems A and B (also true for D A but only for unitary evolutions on A).
E B as a measure of entanglement E B pρq d B pρ, S AB q 2 2 2 a F pρ, S AB q with F pρ, S AB q max σsep PS AB F pρ, σ sep q = maximal fidelity between ρ and a separable state. ÝÑ Main physical question: determine F pρ, S AB q explicitely. pb: it is not easy to find the geodesics for the Bures distance! The closest separable state to a pure state ρ Ψ is a pure product state, so that F pρ Ψ, S AB q max ϕy, χy xϕ b χ Ψy 2 Ñ easy! For mixed states ρ, F pρ, S AB q coincides with the convex roof F pρ, S AB q max t Ψi y,η i u [Streltsov, Kampermann and Bruß 10] i η if pρ Ψi, S AB q Ñ not easy! max. over all pure state decompositions ρ i η i Ψ i yxψ i of ρ.
The two-qubit case Assume that both subsystems A and B are qubits, H A H B C 2. Concurrence: [Wootters 98] Cpρq maxt0, λ 1 λ 2 λ 3 λ 4 u with λ 2 1 λ2 2 λ2 3 λ2 4 the eigenvalues of ρ σ y b σ y ρ σ y b σ y 0 i σ y = Pauli matrix i 0 ρ = complex conjugate of ρ in the canonical (product) basis. Then [Wei and Goldbart 03, Streltsov, Kampermann and Bruß 10] F pρ, S AB q 1 2 1 a 1 Cpρq 2
Outlines Entangled and non-classical states Bures distance on the set of quantum states Geometrical measures of quantum correlations Link with Quantum State Discrimination
ψ 2 > ψ 1 > ψ 3 > Quantum State Discrimination? "ancilla" 0> MEASUREMENT APPARATUS A receiver gets a state ρ i randomly chosen with probability η i among a known set of states tρ 1,, ρ m u. To determine the state he has in hands, he performs a measurement on it. ãñ Applications : quantum communication, cryptography,... If the ρ i are K, one can discriminate them unambiguously Otherwise one succeeds with probability P S i η i trpm i ρ i q M i = non-negative operators describing the measurement, i M i 1. Open pb (for n 2): find the optimal measurement tm opt i u and highest success probability P opt S. Π 2 ψ > 2 Π 1 ψ > 1
Geometric quantum discord The square distance D A pρq d B pρ, C A q 2 to the set C A of A- classical states is a geometrical analog of the quantum discord characterizing the quantumness of states (actually, the A-classical states are the states with zero discord) P opt S p α iyq optimal success proba. in discriminating the states ρ i η i 1? ρ αi yxα i b 1? ρ with proba η i xα i tr B pρq α i y, where t α i yu orthonormal basis of H A. The geometric quantum discord is given by solving a state discrimination problem [Spehner and Orszag 13] D A pρq 2 2 max t αi yu b P opt S p α iyq
Closest A-classical states to a state ρ The closest A-classical states to ρ are σ ρ 1 F pρ,c A q i αopt i yxα opt i b xα opt i? ρ Π opt i? ρ α opt i y where tπ opt i t α opt i [Spehner and Orszag 13] u is the optimal von Neumann measurement and yu the orthonormal basis of H A maximizing P opt S, i.e. F pρ, C A q n A i1 η opt i trpm opt i ρ opt i q. ρ can have either a unique or an infinity of closest A-classical states.
The qubit case If A is a qubit, H A C 2, and dim H B n B, then F pρ, C A q 1! 2 max 1 tr Λpuq 2 }u}1 n B l1 ) λ l puq [Spehner and Orszag 14] λ 1 puq λ 2nB puq eigenvalues of the 2n B 2n B matrix K c 3 M Λpuq? ρ σ u b 1? ρ with u P R 3, }u} 1, and σ u u 1 σ 1 u 2 σ 2 u 3 σ 3 with σ i Pauli matrices. L σ τ τ σ τ I c 1 σ ρ ρ J c 2 G Η + σ ρ Η ρ c 1 =c 2 N G +