Bipartite Continuous-Variable Entanglement. Werner Vogel Universität Rostock, Germany

Transcription

1 Bipartite Continuous-Variable Entanglement Werner Vogel Universität Rostock, Germany

2 Contents Introduction NPT criterion for CV entanglement Negativity of partial transposition Criteria based on moments Relation to previously known conditions Optimized entanglement conditions Entanglement witnesses General optimization Implementing entanglement test Identifying bound entanglement CV entanglement in finite spaces

3 Contents Entanglement quasi-probabilities State representation with factorized pure states Unambiguous entanglement quasi-probabilities Quantifying entanglement Universal entanglement measures The Schmidt number Operational measure and pseudo-measures Summary and conclusions

4 Introduction Early studies of entanglement EPR paradoxon 1 Schrödinger s cat 2 Entangled State: Ψ = yes cat dead + no cat alive ψ cat Φ Ra 1 Einstein, Podolsky, Rosen (1935) 2 Schrödinger (1935)

5 Introduction Nowadays: entanglement as key resource for Quantum information Quantum computation Quantum technology Applications for Quantum key distribution 3 Quantum dense coding 4 Quantum teleportation 5 3 A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991) 4 C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992) 5 C. H. Bennett, G Brassard,..., Phys. Rev. Lett. 70, 1895 (1993)

6 Bipartite entanglement: Introduction A bipartite state ˆσ is called separable if it is a convex combination of factorizable ones: 6 ˆσ = p n ˆϱ (n) 1 ˆϱ (n) 2, where p n 0 and n=0 p n = 1. n=0 A bipartite state ˆϱ is called entangled if it cannot be represented in such a form: ˆϱ ˆσ For separable states classical correlations: Â1 ˆB 2 = n p n (n) Â1 ˆϱ ˆB 2 (n) 1 ˆϱ 2 6 R. F. Werner, Phys. Rev. A, 40, 4277 (1989)

7 Non-positivity of partial transposition The Peres criterion for entanglement: 7 Partial transposition: ˆϱ ˆϱ PT For any separable state partial transposition is non-negative ˆσ PT = p n ˆϱ (n) 1 ˆϱ (n)t 2 ˆσ PT 0 n=0 For ˆϱ PT 0 ˆϱ is entangled But: entangled states ˆϱ BE with positive partial transposition (PPT) exist: PPT (bound) entangled states NPT condition necessary and sufficient for entanglement: Hilbert spaces of dimension 2 2 and 2 3 Gaussian states (see later) 7 A. Peres, Phys. Rev. Lett. 77, 1413 (1996), P. Horodecki et. al, Phys. Lett. A 223, 1 (1996)

8 Non-positivity of partial transposition General form of NPT criterion for harmonic oscillators 8 Nonnegativity of an operator Â: ψ Â ψ 0, ψ Equivalent condition: ˆf ˆf = Tr( Â ˆf ˆf) 0 for all operators ˆf whose normally-ordered form exists. Normally ordered expansion: Quadratic form: ˆf = ˆf ˆf PT = + n,m,k,l=0 + n,m,k,l=0 p,q,r,s=0 c nmkl â n â mˆb kˆbl c pqrsc nmkl M pqrs,nmkl 0 where M pqrs,nmkl = â q â p â n â mˆb sˆbrˆb kˆbl PT PT moments: â q â p â n â mˆb sˆbrˆb kˆbl PT = â q â p â n â mˆb lˆbkˆb rˆbs 8 E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, (2005)

9 Criteria based on moments Introducing (appropriately ordered) matrix: M 11 M M 1N M 21 M M 2N M N1 M N2... M NN where M ij = â q â p â n â mˆb lˆbkˆb rˆbs Explicit form (graded antilexicographical order): 1 â â ˆb ˆb... â â â â 2 â ˆb â ˆb... â â 2 ââ âˆb âˆb... ˆb âˆb â ˆb ˆb ˆb ˆb2... ˆb âˆb â ˆb ˆb 2 ˆbˆb...

10 Criteria based on moments Necessary and sufficient condition for nonnegative PT: 9 Notations: d k 0 Principal minors with rows and columns k 1 < < k n : d k, where k = (k 1,..., k n ) Leading principal minors: d n Necessary and sufficient condition for nonpositivity of PT (NPT): k : d k < 0 sufficient conditions for entanglement 9 E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, (2005), A. Miranowicz and M. Piani, Phys. Rev. Lett 97, (2006)

11 Criteria based on moments Additional useful conditions: Identifying some of the coefficients c nmkl in the condition ˆf ˆf PT = + n,m,k,l=0 p,q,r,s=0 c pqrsc nmkl M pqrs,nmkl 0, Leads to determinants whose entries are linear combinations of original moments Example: for ˆf = c 1 +c 2 â+c 3ˆb one can put c2 = c 3, which yields ˆf = c 1 + c 2 (â + ˆb).

12 Criteria based on moments A simple condition in terms of higher-order moments: Special choice of operator ˆf: ˆf = c 1 â n â mˆb kˆbl + c 2 â p â qˆb rˆbs Separability condition: â m â n â n â mˆb lˆbkˆb kˆbl â q â p â p â qˆb sˆbrˆb rˆbs â m â n â p â qˆb sˆbrˆb kˆbl 2 ( ) Useful for characterizing non-gaussian states!

13 Relation to previous conditions Simon condition: 10 S 0, where ( ) 2 1 ) S = det A 1 det A det C Tr( A 1 JCJA 2 JC T J 1 ) (det A 1 + det A 2 0, 4 and A i = C = ( ) ( ˆxi ) 2 { ˆx i, ˆp i } { ˆx i, ˆp i } ( ˆp i 2, ( ) ˆx1 ˆx 2 ˆx 1 ˆp 2, J = ˆp 1 ˆx 2 ˆp 1 ˆp 2 ( ) Our approach: leading principal minor S d 5 10 R. Simon. Phys. Rev. Lett. 84, 2726 (2000)

14 Relation to previous conditions Duan et al. condition: 11 For any real parameter r (r 0) define operators û and ˆv: û = r ˆx 1 + r 1ˆx 2, ˆv = r ˆp 1 r 1 ˆp 2. For any separable state the inequality ( û) 2 + ( ˆv) 2 (r 2 + r 2 ) 0 holds true for all r 0. Equivalent (optimized) form â â ˆb ˆb Re 2 â ˆb 0. Our approach: principal minor 1 â ˆb d (1,2,4) = â â â â ˆb ˆb âˆb ˆb ˆb = â â ˆb ˆb â ˆb 2 11 L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000)

15 Relation to previous conditions Example: CV Bell-type state ( ) ψ = N (α, β) α, β α, β, Both d 5 (Simon) and d (1,2,4) (Duan et al.) fail! Principal minor: Explicit form: d (1,5,12) = 1 ˆb âˆb ˆb ˆb ˆb âˆb ˆb â ˆb â ˆb ˆb â âˆb ˆb. d (1,5,12) = α 2 β 4 coth( α 2 + β 2 ) sinh 2 ( α 2 + β 2 ), always negative (except for α = 0 or β = 0)! Experiment with non-gaussian entanglement P. H. Souto Ribeiro, S. P. Walborn, et al., in preparation

16 Relation to previous conditions Raymer et. al.: 13 ˆf = c 1 + c 2  1 + c 3 ˆB1 + c 4  2 + c 5 ˆB2 Mancini et. al.: 14 Principal minor d (1,2,4) together with: ˆf = c1 + c 2 (â + ˆb) + c 3 (â + ˆb ) Agarwal and Biswas: 15 ˆf = c 1 + c 2 âˆb + c 3 â ˆb Hillery and Zubairy: 16 Special cases of ( ) 13 M. G. Raymer, A.C. Funk, B.C. Sanders, H. de Guise, Phys. Rev. A 67, (2003) 14 S. Mancini, V. Giovannetti, D. Vitali, P. Tombesi, Phys. Rev. Lett. 88, (2002) 15 G. S. Agarwal and A. Biswas, New J. Phys. 7, 211 (2005) 16 M. Hillery and M. Zubairy, Phys. Rev. Lett. 96, (2006)

17 Optimized entanglement conditions Entanglement witnesses: 17 Definition: Ŵ is a bounded Hermitian operator with tr(ˆσŵ ) 0 ( ˆσ separable), ˆϱ : tr(ˆϱŵ ) < 0 General test, if general form of Ŵ is known! Lemma: 18 For any entanglement witness Ŵ exists a real number λ > 0 and a positive Hermitian operator Ĉ so that Ŵ can be written as Ŵ = λˆ1 Ĉ. Proof: λ sup S(Ŵ ), the largest eigenvalue ( ) Ŵ = w d ˆP (w) = λˆ1 λˆ1 w d ˆP (w) S(Ŵ ) S(Ŵ ) = λˆ1 (λ w) d S(Ŵ ) }{{} ˆP (w) = λˆ1 Ĉ 0 17 M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223, 1 (1996) 18 J. Sperling and W. Vogel, Phys. Rev. A 79, (2009)

18 Optimized entanglement conditions Definition: Let  be a general Hermitian operator and f AB (Â) = sup{tr(ˆσâ) : ˆσ separable} = sup{ a, b  a, b : a a = b b = 1} The condition tr(ˆσŵ ) 0 for a witness (Ŵ = λˆ1 Ĉ) implies The condition is optimal, iff λ f AB (Ĉ) λ = f AB (Ĉ) Theorem: A state ˆϱ is entangled, iff there exists a positive semidefintie Hermitian operator Ĉ: f AB (Ĉ) < tr(ˆϱĉ) Alternatively: With an arbitrary Hermitian operator,  = κˆ1 + Ĉ (κ R), the condition reads as f AB (Â) < tr(ˆϱâ)

19 General Optimization The separability eigenvalue (SE) equations Sought: Extremum of g(a, b) = a, b  a, b, constraints h 1 (a) = a a 1 = 0 and h 2 (b) = b b 1 = 0 Lagrange multipliers L 1, L 2 Functional calculus: 0 = g a L h 1 1 a L h 2 2 a, 0 = g b L h 1 1 b L h 2 2 b Defining:  a = tr A [ ( a a ˆ1 B ) ], Âb = tr B [ (ˆ1 A b b ) ] SE equations:  b a = g a,  a b = g b Solution: f AB (Â) = sup {g}

20 Implementing entanglement tests General test operators: General Hermitian operator Â: a b + ic  = b ic d...., where a, b, c, d,... are real numbers. Sperical grid of operators: Choosing {Âi} i=1...n, with  :  = 1, i : Âi  < ɛ. Error ɛ of test experimental precision! Optimal entanglement witnesses: Ŵ i = f AB (Âi)ˆ1 Âi

21 Implementing entanglement tests Approximation by 6 Hermitian operators:

22 Identifying PPT bound entanglement 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 )

23 CV entanglement in finite spaces Continuous Variable Entanglement in general, quantum systems given by continuous variables (CV) CV-entanglement: completely characterized in finite dimensional Hilbert spaces 19 from theory of Hilbert spaces: 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. 19 J. Sperling and W. Vogel, PRA 79, (2009)

24 Entanglement quasi-probabilities State representation with factorized pure states: Representation of entangled states by separable ones: 20 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: a k, b k non-orthogonal P Ent 0 quasi-probability Problem: P Ent ambiguous Optimization: determining P Ent with minimal negativity 20 Sanpera, Tarrach, Vidal, Phys. Rev A 58, 826 (1998); Vidal, Tarrach, Phys. Rev. A 59, 141 (1999)

25 Entanglement quasi-probabilities Construction/ reconstruction of quasi-probabilities: Tomographic or local quantum state reconstruction 2. Spectral decomposition of state ˆρ: ˆρ = i p φ i φ i φ i 3. Schmidt decomposition of all states φ i with Schmidt rank r(φ i ): r(φ) φ = λ k e k, f k, k=1 φ φ = k λ 2 k e k, f k e k, f k + k>l λ k λ l ( e k, f k e l, f l + e l, f l e k, f k ) 4. Representation with factorized states: with e k, f k e l, f l + e l, f l e k, f k = 3 n=0 ( 1) n s (k,l) n, s (k,l) s (k,l) n, s (k,l) n = 1 2 ( e k + i n e l ) ( f k + i n f l ) n s (k,l) n, s (k,l) n 21 J. Sperling and W. Vogel, Phys. Rev. A 79, (2009)

26 Unambiguous quasi-probabilities Intermediate result: ˆρ in terms of factorized states ˆρ = q k x k, y k x k, y k dp Ent (a, b) a, b a, b k Possible conclusions: P Ent (a, b) 0 quantum state is separable P Ent (a, b) < 0 entangled Needed: optimization, so that P Ent (a, b) has minimal amount of negativity

27 Unambiguous quasi-probabilities Idea of optimization: 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 : d P Ent (a, b) + f(a, b) 2 min Optimization within the convex set of separable quantum states Optimization with respect to separability norm ˆρ ˆσ sep = sup{ a, b ˆρ ˆσ a, b } min separability eigenvalue problem for the state ˆρ

28 Unambiguous quasi-probabilities Idea of optimization: ˆρ bi a i = g i a i and ˆρ ai b i = g i b i

29 Unambiguous quasi-probabilities Procedure of optimization: Solution of SE equations: SE (g j ) j = g, SE vectors a i, b i deliver G = ( a i, b i x j, y j 2 ) i,j, ˆρ = i q i x i, y i x i, y i, q = (q i ) i Linear system: a j, b j ˆρ a j, b j Yields: p P Ent = i p iδ ai,b i Final result: based on optimized P Ent ˆρ = dp Ent (a, b) a, b a, b State is entangled a, b : P Ent (a, b) < 0 State is separable a, b : P Ent (a, b) 0

30 Quantifying entanglement Entanglement measures 22 entanglement amount of separable states vanishes LOCC paradigm: Entanglement cannot increase under certain operations. Definition: entanglement measure arising questions: (i) ˆσ separable E(ˆσ) = 0 ( ) Λ(ˆρ) (ii) E(ˆρ) E trλ( ˆρ) maximally entangled states? existence of superior measure? usefull amount of entanglement? 22 C. H. Bennett et al., Phys. Rev. A 54, 3824 (1996); V. Vedral et al., Phys. Rev. Lett. 78, 2275 (1997); G. Vidal, J. Mod. Opt. 47, 355 (2000)

31 Quantifying entanglement Fundamental entanglement measure: Schmidt number 23 Pure state ψ k : r(ψ k ) Schmidt rank Mixed quantum state ˆρ = k p k ψ k ψ k : r max (ˆρ) = max k r(ψ k ) Under all possible decompositions minimal r max (ˆρ) yield Schmidt number r S : r S (ˆρ) = inf{r max (ˆρ) : ˆρ = k p k ψ k ψ k } r S (ˆσ) = 1 ˆσ separable for all separable operations Λ: ( ) Λ(ˆρ) r S (ˆρ) r S trλ( ˆρ) 23 J. Sperling and W. Vogel, arxiv: [quant-ph]

32 Quantifying entanglement 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 Λ(ˆρ) = i (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

33 Quantifying entanglement 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 = r k=1 1 r e k, f k Local invertable transform, Λ(ˆρ) = (ˆ1 A ˆT )ˆρ(ˆ1 A ˆT ), with ˆT f k = rλ k f k Transformation φ 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. (ˆ1 A = E ˆT 1 )ˆρ(ˆ1 A ˆT ) 1 ) tr(ˆ1 A ˆT 1 )ˆρ(ˆ1 A ˆT 1 )

34 Universal entanglement measures 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 24 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. Example: the Schmidt number r S ; generalization of the Schmidt rank to mixed quantum states 24 G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, arxiv: [quant-ph]

35 The Schmidt number 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 ˆρ 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

36 Operational- and pseudo-measures Operational entanglement measures Not every procedure can use any kind of entanglement Example: PPT entangled state no destillable entanglement Leads to pseudo-measures: (i) (i) ˆσ separable E(ˆσ) = 0 Experimental setting: ˆM denotes measured quantity Λ 1,..., Λ n X denote the separable operations, which can be performed (all configurations lead to LOCCs, C X ) f( ˆM) denotes the maximal eigenvalue of ˆM

37 Operational- and pseudo-measures Operational entanglement measures Usable entanglement for experiment with given pseudomeasure 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) is value between one (perfect entanglement for a given experiment), and zero (no usable entanglement) For example, different types of entanglement are needed for quantum teleportation and quantum computation D. Gross, S. Flammia, J. Eisert, Phys. Rev. Lett. 102, (2009)

38 Summary and Conclusions 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