Geometry on the set of quantum states and quantum correlations

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Geometry on the set of quantum states and quantum correlations


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FOURIER INSTITUT i f Geometry on the set of quantum states and quantum correlations Dominique Spehner Institut Fourier et Laboratoire de Physique et Mod´elisation des Milieux Condens´es, Grenoble Short course, GSI’2015, ´Ecole Polytechnique, Paris, 28/10/2015     Quantum Correlations & Quantum Information ™ Quantum Information Theory (QIT) studies quantum systems that can perform information-processing tasks more efficiently than one can do with classical systems: - computational tasks (e.g. factorizing into prime numbers) - quantum communication (e.g. quantum cryptography, ...)  A quantum computer works with qubits, i.e. two-level quantum systems in linear combinations of |0y and |1y.  Entanglement is a resource for quan- tum computation and communication [Bennett et al. ’96, Josza & Linden ’03] However, other kinds of “quantum correlations” differing from entanglement could also explain the quantum efficiencies.  
Outlines  Entangled and non-classical states  Contractive distances on the set of quantum states  Geometrical measures of quantum correlations     Basic mathematical objects in quantum mechanics (1) A Hilbert space H (in this talk: n  dim H   V). (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 € MatpC, nq finite Hermitian matrices) (4) An evolution is given by a linear map Φ : MatpC, nq Ñ MatpC, nq which is (TP) trace preserving (so that trpΦpρqq  trpρq  1) (CP) Completely Positive, i.e. for any integer d ¥ 1 and any d ¢d matrix pAijqd i,j1 ¥ 0 with elements Aij € MatpC, nq, one has pΦpAijqqd i,j1 ¥ 0. Special case: unitary evolution Φpρq  U ρ U¦ with U unitary.     Pure and mixed quantum states  A pure state is a rank-one projector ρψ  |ψyxψ| with |ψy € H, }ψ}  1 (actually, |ψy belongs to the projective space PH). 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  ¸ i pi|ψiyxψi|, with pi ¥ 0, ° i pi  1 and |ψiy € PH. Statistical interpretation: the pure states |ψiy have been prepared with probability pi.     Quantum-classical analogy Hilbert space H Ø finite sample space Ω state ρ Ø probability p on pΩ, PpΩqq observable Ø random variable on pΩ, PpΩqq set of quantum states Ø probability simplex EpHq Eclass  2 p € Rn  ; ° k pk  1 @ CPTP map Φ Ø stochastic matrices pΦklqk,l1,...,n (Φkl ¥ 0, ° k Φkl  1 d l)     Separable states A bipartite system AB is composed of two subsystems A and B with Hilbert spaces HA and HB. It has Hilbert space HAB  HA ˜HB. For instance, A and B can be the polarizations of two photons localized far from each other ñ HAB C2 ˜C2 (2 qubits):  A pure state |Ψy of AB is separable if it is a product state |Ψy  |ψy˜|φy with |ψy € PHA and |φy € PHB.  A mixed state ρ is separable if it admits a pure state decomposition ρ  ° i pi|ΨiyxΨi| with |Ψiy  |ψiy ˜ |φiy separable for all i.     Entangled states  Nonseparable states are called entangled. Entanglement is ãÑ the most specific feature of Quantum Mechanics. ãÑ used as a resource in Quantum Information (e.g. quantum cryptography, teleportation, high precision interferometry...).  Examples of entangled & separable states: let HA HB C2 (qubits) with canonical basis t|0y, |1yu. The pure states |Ψ¨ Belly  1c 2 ¡ |0 ˜0y¨|1 ˜1y © are maximally entangled. ãÑ lead to the maximal violation of the Bell inequalities observed experimentally [Aspect et al ’82] ñnonlocality of QM In contrast, the mixed state ρ  1 2 |Ψ  BellyxΨ  Bell| 1 2 |Ψ¡ BellyxΨ¡ Bell| is separable ! (indeed, ρ  1 2 |0 ˜0yx0 ˜0|  1 2 |1 ˜1yx1 ˜1|).  
Classical states  A state ρ of AB is classical if it has a spectral decomposition ρ  ° k pk|ΨkyxΨk| with product u states |Ψky  |αky˜|βky. Classicality is equivalent to separability for pure states only.  A state ρ is A-classical if ρ  ° i qi|αiyxαi|˜ρB|i with t|αiyu orthonormal basis of HA and ρB|i arbitrary states of B.  The set CAB (resp. CA) of all (A-)classical states is not convex. Its convex hull is the set of separable states SAB. ρ CA C ABS CAB B  Some tasks impossible to do clas- sically can be realized using sepa- rable non-classical mixed states.  Such states are easier to produce and presumably more robust to a coupling with an environment.     Quantum vs classical correlations  Central question in Quantum Information theory: identify (and try to protect) the Quantum Correlations responsible for the exponential speedup of quantum algorithms. classical correlations quantum correlations  For mixed states, two (at least) kinds of QCs Õ entanglement [Schr¨odinger ’36] × nonclassicality (quantum discord) [Ollivier, Zurek ’01, Henderson, Vedral ’01]  
Outlines Entangled and non-classical states  Contractive distances on the set of quantum states  
Contractive distances ρ φ(ρ) φ(σ) σ CONTRACTIVE DISTANCE  The set EAB of all quantum states of a bipartite system AB (i.e. , operators ρ ¥ 0 on HAB with tr ρ  1) can be equipped with many distances d.  From a QI point of view, interesting distances must be contractive under CPTP maps, i.e. for any such map Φ on EAB, d ρ, σ € EAB, dpΦpρq, Φpσqq ¤ dpρ, σq Physically: irreversible evolutions can only decrease the distance between two states.  A contractive distance is in particular unitarily invariant, i.e. dpUρU¦, UσU¦q  dpρ, σq for any unitary U on HAB  The Lp -distances dppρ, σq  }ρ ¡σ}p  ptr |ρ ¡σ|p q1{p are not contractive excepted for p  1 (trace distance) [Ruskai ’94].  
Petz’s characterization of contractive distances  Classical setting: there exists a unique (up to a multiplicative factor) contractive Riemannian distance dclas on the probability simplex Eclas, with Fisher metric ds2  ° k dp2 k{pk [Cencov ’82]  Quantum generalization: any Riemannian contractive distance on the set of states EpHq with n  dim H   V has metric ds2  gρpdρ, dρq  n¸ k,l1 cppk, plq|xk|dρ|ly|2 where pk and |ky are the eigenvalues and eigenvectors of ρ, cpp, qq  pfpq{pq qfpp{qq 2pqfpp{qqfpq{pq and f : R  Ñ R  is an arbitary operator-monotone function such that fpxq  xfp1{xq [Morozova & Chentsov ’90, Petz ’96]     Distance associated to the von Neumann entropy ™ Quantum analog of the Shannon entropy: von Neumann entropy Spρq  ¡trpρ ln ρq ™ Since S is concave, the physically most natural metric is ds2  gSpdρ, dρq  ¡d2 Spρ tdρq dt2 § § § t0  d2 F pX sdXq ds2 § § § s0 [Bogoliubov; Kubo & Mori; Balian, Alhassid & Reinhardt, ’86, Balian ’14]. with FpXq  ln trpeX q and ρ  eX¡F pXq  eX {trpeX q. ™ ds2 has the Petz form with fpxq  x¡1 ln x ãÑ the corresponding distance is contractive. ™ Loss of information when mixing the neighboring equiprobable states ρ¨  ρ ¨ 1 2dρ: ds2 {8  Spρq¡ 1 2Spρ q¡ 1 2Spρ¡q     Bures distance and Uhlmann fidelity ™ Fidelity (generalizes F|xψ|φy|2 for mixed states) [Uhlmann ’76] Fpρ, σq  2 trrcσρ cσs1{2 @2  Fpσ, ρq ™ Bures distance: dBupρ, σq    2 ¡2 — Fpρ, σq ¨1 2 [Bures ’69] ãÑ has metric of the Petz form with fpxq  x 1 2 ãÑ smallest contractive Riemannian distance [Petz ’96] ãÑ coincides with the Fubiny-Study metric on PH for pure states ãÑ dBupρ, σq2 is jointly convex in pρ, σq ™ dBupρ, σq  sup dclaspp, qq with sup over all measurements giving outcome k with proba pk (for state ρ) and qk (for state σ) [Fuchs ’96]  
Bures distance and Fisher information In quantum metrology, the goal is to estimate an unknown parameter φ by measuring the output states ρoutpφq  e¡iφH ρ eiφH and using a statistical estimator depending on the measurement results (e.g. in quantum interferometry: estimate the phase shift φ1 ¡φ2) ãÑ precision ∆φ  e¡§ § § fxφestyφ fφ § § § ¡1 φest ¡φ ©2i1{2 φ The smallest precision is given by the quantum Cr´amer-Rao bound p∆φqbest  1c N cFpρ,Hq , Fpρ, Hq  4dBupρ, ρ  dρq2 , dρ  ¡irH, ρs N = number of measurements Fpρ, Hq= quantum Fisher information [Braunstein & Caves ’94]     Summary CONTRACTIVE RIEMANNIAN METRICS: Classical Quantum Interpretation Bures ds2 Bu Q. metrology unique: Õ ... (Fisher information) ds2 clas  ¸ k dp2 k pk Ñ ds2 S  ¡d2 S Loss of information (Fisher) × ... when merging 2 states Hellinger ds2 Hel Q. state discrimination ... with many copies  
Outlines Entangled and non-classical states Contractive distances on the set of quantum states  Geometrical measures of quantum correlations     Geometric approach of quantum correlations ρ CA C ABS CAB B Geometric entanglement: Epρq  min σsep€SAB dpρ, σsepq2 Geometric quantum discord : DApρq  min σA-cl€CA dpρ, σA-clq2 Properties: EpρΨq  DApρΨq for pure states ρΨ Ð for Bures distance E is convex Ð if d2 is jointly convex Entanglement monotonicity: EpΦA ˜ ΦBpρqq ¤ Epρq for any TPCP maps ΦA and ΦB acting on A and B (also true for DA but only when ΦApρAq  UA ρA U¦ A). Ð if d is contractive     Bures geometric measure of entanglement EBupρq  dBupρ, SABq2  2 ¡2 — Fpρ, SABq with Fpρ, SABq  maxσsep€SAB Fpρ, σsepq = maximal fidelity between ρ and a separable state. ÝÑ Main physical question: determine Fpρ, SABq explicitely. pb: it is not easy to find the geodesics for the Bures distance! ™ The closest separable state to a pure state ρΨ is a pure product state, so that FpρΨ, SABq  max|ϕy,|χy |xϕ ˜χ|Ψy|2 Ñ easy! ™ For mixed states ρ, Fpρ, SABq coincides with the convex roof [Streltsov, Kampermann and Bruß’10] Fpρ, SABq  maxt|Ψiy,ηiu ° i piFpρΨi , SABq Ñ not easy! max. over all pure state decompositions ρ  ° i pi|ΨiyxΨi| of ρ.     The two-qubit case Assume that both subsystems A and B are qubits, HA HB C2 .
Concurrence: [Wootters ’98] Cpρq  maxt0, λ1 ¡λ2 ¡λ3 ¡λ4u with λ2 1 ¥ λ2 2 ¥ λ2 3 ¥ λ2 4 the eigenvalues of ρ σy ˜σy ρ σy ˜σy σy  ¢ 0 ¡i i 0  = Pauli matrix ρ = complex conjugate of ρ in the canonical (product) basis.
Then [Wei and Goldbart ’03, Streltsov, Kampermann and Bruß’10] Fpρ, SABq  1 2   1   — 1 ¡Cpρq2 ¨     Quantum State Discrimination 000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000 0000000000000000000000000000000000000000000000 111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111 1111111111111111111111111111111111111111111111 | |ψ ψ2 > 1 > >3 ψ| APPARATUS MEASUREMENT ? "ancilla" |0>  A receiver gets a state ρi randomly chosen with probability ηi among a known set of states tρ1, ¤¤¤ , ρmu.  To determine the state he has in hands, he performs a measurement on it. ãÑ Applications : quantum communication, cryptography,...  If the ρi are u, one can discriminate them unambiguously  Otherwise one succeeds with probability PS  ° i ηi trpMiρiq Mi = non-negative operators describing the measurement, ° i Mi  1. |ψ >2 |ψ >1 Π2 Π1 Open pb (for n ¡ 2): find the optimal measurement tMopt i u and highest success probability Popt S .     Bures geometric quantum discord The square Bures distance DApρq  dBupρ, CAq2 to the set CA of A-classical states is a geometric analog of the quantum discord characterizing the “quantumness” of states (actually, the A-classical states are the states with zero discord)
Popt S p|αiyq optimal success proba. in discriminating the states ρi  η¡1 i cρ |αiyxαi|˜1 cρ with proba ηi  xαi|trBpρq|αiy, where t|αiyu orthonormal basis of HA.
The geometric quantum discord is given by solving a state discrimination problem [Spehner and Orszag ’13] DApρq  2 ¡2 maxt|αiyu ˜ Popt S p|αiyq     Closest A-classical states to a state ρ
The closest A-classical states to ρ are σρ  1 F pρ,CAq ° i |αopt i yxαopt i |˜xαopt i |cρ Πopt i cρ |αopt i y [Spehner and Orszag ’13] where tΠopt i u is the optimal von Neumann measurement and t|αopt i yu the orthonormal basis of HA maximizing Popt S , i.e. Fpρ, CAq  nA¸ i1 ηopt i trpMopt i ρopt i q.
ρ can have either a unique or an infinity of closest A-classical states.     The qubit case
If A is a qubit, HA C2 , and dim HB  nB, then Fpρ, CAq  1 2 max }u}1 3 1 ¡tr Λpuq 2 nB¸ l1 λlpuq A [Spehner and Orszag ’14] λ1puq ¥ ¤¤¤ ¥ λ2nB puq eigenvalues of the 2nB ¢2nB matrix Λpuq  cρ σu ˜1 cρ with u € R3 , }u}  1, and σu  u1σ1  u2σ2  u3σ3 with σi Pauli matrices. c2 c1 ρ ρσ L J I K τ στ στ c3 ρ Η−+Η c1 2=c N G+−G M σρ     Conclusions & perspectives  Conclusions: ™ Contractive Riemannian distances on the set of quantum states provide useful tools for measuring quantum correlations in bipartite systems. ™ Major challenges are ãÑ compute the geometric measures for simple systems ãÑ compare the measures obtained from different distances and look for universal properties  References: - Review article: D. Spehner, J. Math. Phys. 55, 075211 (’14) - D. Spehner, M. Orszag, New J. Phys. 15, 103001 (’13) - D. Spehner, M. Orszag, J. Phys. A 47, 035302 (’14) - R. Roga, D. Spehner, F. Illuminati, arXiv:1510.06995