Characterization of Bipartite Entanglement Werner Vogel and Jan Sperling University of Rostock Germany Paraty, September 2009 Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 1
Table of Contents Introduction Partial Transposition Continuous Variable Entanglement Entanglement Witnesses Quasidistributions for Entanglement Entanglement Measures Summary and Conclusions Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 2
Introduction and Definition Early investigations of entanglement EPR paradox [Einstein, Podolsky, Rosen, Phys. Rev. 44, 777 (1935)] Schrödinger s cat [Schrödinger, Naturwiss. 23, 807 (1935)] Today, key resource for new research fields: Quantum-information, -computation, and -technology Definition. [R. F. Werner, PRA 40, 4277 (1989)] A bipartite state ˆϱ is called entangled, if: ˆϱ ˆσ : ˆσ n=0 p n ˆϱ (n) 1 ˆϱ (n) 2, with p n 0, and n p n = 1 Mean values with ˆσ: analogy to classical correlations Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 3
Partial Transposition (PT) Peres condition for partial transposition [A. Peres, Phys. Rev. Lett. 77, 1413 (1996)] ˆσ PT = n p n ˆϱ (n) 1 ˆϱ (n)t 2 0 Violation of Peres condition: sufficient for entanglement Necessary and sufficient: for quantum systems 2 2 or 2 3 and bipartite Gaussian states General approach for bipartite oscillator system: [E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502 (2005)] ˆρ with ˆρ PT 0 Ŵ = (ˆf ˆf ) PT : tr(ˆϱŵ ) = ˆf ˆf PT < 0 General operator expansion: ˆf = n,m,k,l c nmklâ n k l âmˆb ˆb Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 4
Partial Transposition (PT) Quadratic form: ˆf ˆf PT = n,m,k,l p,q,r,s c pqrsc nmkl â q â p â n â mˆb l ˆbk ˆb r ˆbs }{{} D N Delivers matrix of moments D N For states with negative PT (at least one) negative principal minor: det D N < 0 Continuous variable entanglement criterion Characterization of non-gaussian states Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 5
Continuous Variable Entanglement In general, quantum systems given by continuous variables (CV) CV-entanglement completely characterized in finite dimensional Hilbert spaces [J. Sperling and W. Vogel, Phys. Rev. A 79, 052313 (2009)] Theory of Hilbert spaces delivers: ˆρ has a spectral decomposition with a countable number of eigenvectors ψ i ψ i has a countable Schmidt decomposition Existence of finite supspaces V 1, V 2, with a projected entangled state [ˆPV1 ˆP ] V2 ˆρ [ˆPV1 ˆP ] V2 Theorem for CV-entanglement. A quantum state ˆρ is entangled, if and only if it is entangled in a finite space. Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 6
Optimized general entanglement conditions Necessary and sufficient condition by entanglement witnesses: [M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223, 1 (1996)] Ŵ is a bounded Hermitian operator tr(ˆσŵ ) 0 ˆσ separable, ˆϱ with tr(ˆϱŵ ) < 0 Generalized and optimized entanglement conditions [J. Sperling and W. Vogel, Phys. Rev. A 79, 022318 (2009)] Maximal expectation value of an arbitrary observable  for all separable states: f AB (Â) = sup{tr(ˆσâ)} Theorem. ˆϱ is entangled Â: fab(â) < tr(ˆϱâ). Optimal entanglement conditions! General, optimal witnesses: Ŵ = f AB (Â)ˆ1 Â, no longer needed! Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 7
Separability Eigenvalue Problem For general entanglement condition: need to find f AB (Â) Optimization problem: g(a, b) = a, b  a, b max. Normalization condition h(a, b) = a, b a, b 1 0 Method of Lagrangian multipliers delivers algebraic expression Definition of separability eigenvalue equations.  b a = g a and  a b = g b With Âa = tr A [ ( a a ˆ1 ) ) B ] and Âb = tr B [ (ˆ1A b b ], reduced observables Result: f AB (Â) = sup {g} Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 8
Optimized general entanglement conditions Implementation of entanglement condition f AB (Â) < tr(ˆϱâ) Spherical grid of operators {Âi} i=1...n,: Âi = 1,  (with  = 1) i : Âi  < ɛ. Error ɛ of test: related to experimental precision! Figure: Grid of operators, with Ŵ i = f AB (Âi)ˆ1 Âi. Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 9
Identifying PPT bound entanglement SE g and SE vector a, b transform under PT to g and a, b State ˆϱ BE is PPT bound entangled (1) Ĉ = ˆf ˆf : 0 tr(ˆϱbe Ĉ PT ) (2) ˆf : inf{ a, b Ĉ a, b } > tr(ˆϱ BEĈPT ) Figure: BE states Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 10
Entanglement by Quasi-Probabilities Characterizing entanglement by quasi-probabilities: Representation of entangled states by separable ones [Sanpera, Tarrach, Vidal, PRA 58, 826 (1998); Vidal, Tarrach, PRA 59, 141 (1999)] ˆρ = (1 + µ)ˆσ µˆσ, µ 0 Representation of any state ˆρ: ˆρ = k p k a k, b k a k, b k, P Ent = (p k ) k In general P Ent 0: quasi-probability distribution Problem: P Ent ambiguous Optimization: determining P Ent with minimal negativity. Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 11
Ambiguity of quasi-probabilities Unambiguous characterization of entanglement [J. Sperling and W. Vogel, Phys. Rev. A 79, 042337 (2009)] ˆρ in terms of factorized states: ˆρ = dp Ent (a, b) a, b a, b Signed measures f generating the ˆ0 operator: ˆ0 = df (a, b) a, b a, b New quasi-probabilities, P Ent P Ent + f, for the given state ˆρ ˆρ + ˆ0 Minimal negativity d P Ent (a, b) + f (a, b) 2 min Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 12
Entanglement quasi-probabilities Leads to SE equations for the quantum state ρ itself Solution of SE equations: linear system of equations Solution yields p P ent = p i δ ai,b i i unambiguous representation of any quantum state ˆρ with optimized quasi-probability P Ent State ˆρ is entangled a, b : P Ent (a, b) < 0 State ˆρ is separable a, b : P Ent (a, b) 0 Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 13
Entanglement Measures Quantification of entanglement [C. H. Bennett et al., Phys. Rev. A 54, 3824 (1996); V. Vedral et al., PRL 78, 2275 (1997); G. Vidal, J. Mod. Opt. 47, 355 (2000).] Entanglement amount of separable states vanishes LOCC paradigm: Entanglement cannot increase under certain operations. Definition entanglement measure. (i) ˆσ separable E(ˆσ) = 0 ( ) Λ(ˆρ) (ii) E(ˆρ) E trλ( ˆρ) Arising questions: maximally entangled states?; existence of superior measure?; useful amount of entanglement? Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 14
Local operation and classical communication Definition of measure requires fundamental understanding of LOCCs Operations which only can make a state separable Λ via separable operations i Λ(ˆρ) = (Âi ˆB i )ˆρ(Âi ˆB i ) i tr(âi ˆB i )ˆρ(Âi ˆB i ) LOCC subset of all such operations Used devices or protocolls can perform operation Λ X All configurations of such operations, Λ 1 (Λ 2 (... Λ n (ˆρ))), define LOCCs C X Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 15
Maximally entangled states Maximal amount of entanglement for ˆρ max : E(ˆρ max ) E(ˆρ) Consider maximally entangled state ˆρ max = φ φ, with a Schmidt decomposition φ = k 1 r e k, f k Consider Λ(ˆρ) = (ˆ1 A ˆT )ˆρ(ˆ1 A ˆT ), with local invertable ˆT f k = rλ k f k Transformation to ˆ1 A ˆT φ = k λ k e k, f k ˆρ = (ˆ1 A ˆT ) φ φ (ˆ1 A ˆT ) maximally entangled for the measure E, ( ) E (ˆρ) def. = E (ˆ1 A ˆT 1 )ˆρ(ˆ1 A ˆT 1 ) tr(ˆ1 A ˆT 1 )ˆρ(ˆ1 A ˆT 1 ) Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 16
Maximally entangled states Observation. For any choice of Schmidt coefficients of ψ we can define measure E by local invertable operations, such that ψ is maximally entangled. Note: implementation of local invertable operation in noise-free amplification [G. Y. Xiang, T. C. Ralph, et al., arxiv:0907.3638] Which rule play Schmidt coefficients for the amount of entanglement? No importance for universal entanglement measures Definition. A measure, which is invariant under local invertable operations, is called universal. [J. Sperling and W. Vogel, arxiv:0908.3974] Example: the Schmidt number r S ; generalization of the Schmidt rank to mixed quantum states Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 17
Schmidt number monotones Local projections ˆP can decrease the Schmidt number, ˆP = r 1 k=1 e k e k ˆP ˆ1 B ψ r = r 1 k=1 λ k e k, f k = ψ r 1 Observation: Schmidt number monotones. Entanglement measures have a monotonic behavior with respect to the Schmidt number, E( ψ r ψ r ) E( ψ r 1 ψ r 1 ). Boundaries for the amount of entanglement of ρ can be given by the Schmidt number: r S (ˆρ 1 ) < r S (ˆρ) < r S (ˆρ 2 ) E(ˆρ 1 ) E(ˆρ) E(ˆρ 2 ) Schmidt coefficients are of minor, and Schmidt number of major importance for the quantification of entanglement Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 18
Operational entanglement measures Usable entanglement for experiment with given pseudo-measure Definition: operational entanglement. ( tr Λ(ˆρ) ˆM ) tr Λ(ˆρ) f 12 ( ˆM) E ˆM(ˆρ) = sup Λ C X f ( ˆM) f 12 ( ˆM) E ˆM(ˆρ): one (perfect entanglement for a given experiment) zero (no usable entanglement), f ( ˆM): maximal eigenvalue of ˆM For quantum teleportation and quantum computation other types of entanglement are needed [D. Gross, S. Flammia, J. Eisert, Phys. Rev. Lett. 102, 190501 (2009)] Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 19
Summary Identification of PT entanglement Reduction of CV entanglement to finite spaces Optimized, necessary and sufficient conditions in terms of arbitrary Hermitian operators Separability eigenvalue equations Optimized quasi-distributions Universal entanglement measure: Schmidt number Operational entanglement measures Paraty, September 2009 UNIVERSITÄT ROSTOCK INSTITUT FÜR PHYSIK 20