für Mathematik in den Naturwissenschaften Leipzig

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1 ŠܹÈÐ Ò ¹ÁÒ Ø ØÙØ für Mathematk n den Naturwssenschaften Lepzg Bounds for multpartte concurrence by Mng L, Shao-Mng Fe, and Zh-X Wang Preprnt no.:

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3 Bounds for multpartte concurrence Mng L 1, Shao-Mng Fe,3 and Zh-X Wang 1 College of Mathematcs and Computatonal Scence, Chna Unversty of Petroleum, Dongyng, Chna Department of Mathematcs, Captal Normal Unversty, Bejng, Chna 3 Max-Planck-Insttute for Mathematcs n the Scences, Lepzg, Germany Abstract We study the entanglement of a multpartte quantum state. An nequalty between the bpartte concurrence and the multpartte concurrence s obtaned. More effectve lower and upper bounds of the multpartte concurrence are obtaned. By usng the lower bound, the entanglement of more multpartte states are detected. PACS numbers: a, 0.0.Hj, w As a potental resource for communcaton and nformaton processng, quantum entanglement has rghtly been the subject of much study n recent years [1]. However the boundary between the entangled states and the separable states, states that can be prepared by means of local operatons and classcal communcatons [], s stll not well characterzed. Entanglement detecton turns out to be a rather tantalzng problem. A more general queston s to calculate the well defned quanttatve measures of quantum entanglement such as entanglement of formaton (EOF) [3] and concurrence [4, 5]. A seres of excellent results have been obtaned recently. There have been some (necessary) crtera for separablty, the Bell nequaltes [6], PPT (postve partal transposton) [7] (whch s also suffcent for the cases and 3 bpartte systems [8]), realgnment [9 11] and generalzed realgnment [1], as well as some necessary and suffcent operatonal crtera for low rank densty matrces[13 15]. Further more, separablty crtera based on local uncertanty relaton [16 19] and the correlaton matrx [0, 1] of the Bloch representaton for a quantum state have been derved, whch are strctly stronger than or ndependent of the PPT and realgnment crtera. The calculaton of entanglement of formaton or concurrence s complcated except for systems [] 1

4 or for states wth specal forms [3]. For general quantum states wth hgher dmensons or multpartte case, t seems to be a very dffcult problem to obtan analytcal formulas. However, one can try to fnd the lower and the upper bounds to estmate the exact values of the concurrence [4 7]. In ths paper, we focus on the concurrence. We derve new lower and upper bounds of concurrence for arbtrary quantum states. From the bounds we can detect more entangled states. Detaled examples are gven to show that the new bounds of concurrence are better than that have been obtaned before. For a pure N-partte quantum state ψ H 1 H H N, dmh = d, = 1,..., N, the concurrence of bpartte decomposton between subsystems 1 M and M +1 N s defned by C ( ψ ψ ) = (1 Tr{ρ 1 M }) (1) where ρ 1 M = Tr M+1 N{ ψ ψ } s the reduced densty matrx of ρ = ψ ψ by tracng over subsystems M + 1 N. On the other hand, the concurrence of ψ s defned by [5] C N ( ψ ψ ) = 1 N ( N ) α Tr{ρ α}, () where α labels all dfferent reduced densty matrces. For a mxed multpartte quantum state, ρ = p ψ ψ H 1 H H N, the correspondng concurrence of (1) and () are then gven by the convex roof: C (ρ) = mn {p, ψ } p C ( ψ ψ ), (3) C N (ρ) = mn {p, ψ } p C N ( ψ ψ ). (4) We now nvestgate the relaton between the two knds of concurrences. Lemma 1: For a bpartte densty matrx ρ H A H B, one has 1 Tr{ρ } 1 Tr{ρ A} + 1 Tr{ρ B}, (5) where ρ A/B = Tr B/A {ρ} be the reduced densty matrces. Proof: Let ρ = j λ j j j be the spectral decomposton, where λ j 0, j λ j = 1.

5 Then ρ 1 = j λ j,ρ = j λ j j j. Therefore 1 Tr{ρ A} + 1 Tr{ρ B} 1 + Tr{ρ } = 1 Tr{ρ A} Tr{ρ B} + Tr{ρ } = ( j λ j ),j,j λ j λ j,,j λ j λ j + j λ j = ( λ j + λ j λ j + λ j λ j + λ j λ j ) ( λ j + λ j λ j ) =,j=j =,j j,j=j,j j,j=j,j j ( λ j + λ j λ j ) + =,j,j,j = λ j λ j 0.,j j λ j The same result n ths lemma has also been derved n [7, 8] to prove the subaddtvty of the lnear entropy. Here we just gve a smpler proof. In the followng we compare the band mult-partte concurrence n (3)(4) by usng the lemma. Theorem 1: For a multpartte quantum state ρ H 1 H H N wth N 3, the followng nequalty holds, C N (ρ) max 3 N C (ρ), (6) where the maxmum s taken over all knds of bpartte concurrence. Proof: Wthout lose of generalty, we suppose that the maxmal bpartte concurrence s attaned between subsystems 1 M and (M + 1) N. For a pure multpartte state ψ H 1 H H N, Tr{ρ 1 M } = Tr{ρ (M+1) N }. From (5) we have CN( ψ ψ ) = N (( N ) Tr{ρ α}) 3 N (N α 3 N (1 Tr{ρ 1 M} + 1 Tr{ρ (M+1) N}).e. C N ( ψ ψ ) 3 N C ( ψ ψ ). Let ρ = One has = 3 N (1 Tr{ρ 1 M}) = 3 N C ( ψ ψ ), N Tr{ρ k}) p ψ ψ attan the mnmal decomposton of the multpartte concurrence. C N (ρ) = p C N ( ψ ψ ) 3 N 3 N mn {p, ψ } k=1 p C ( ψ ψ ) p C ( ψ ψ ) = 3 N C (ρ). 3

6 hold: Corollary For a trpartte quantum state ρ H 1 H H 3, the followng nequalty C 3 (ρ) maxc (ρ) (7) where the maxmum s taken over all knds of bpartte concurrence. In [4] a lower bound for a bpartte state ρ H A H B, d A d B, has been obtaned, C (ρ) d A (d A 1) [max( T A(ρ), R(ρ) ) 1]. (8) where T A, R and stand for the partal transpose, realgnment, and the trace norm (.e., the sum of the sngular values), respectvely. In [6, 9], from the separablty crtera related to local uncertanty relaton, covarance matrx and correlaton matrx, the followng lower bounds for bpartte concurrence are obtaned: C (ρ) C(ρ) (1 Tr{ρ A }) (1 Tr{ρ B }) da (d A 1) (9) and 8 C (ρ) d 3 A d B (d A 1) ( T(ρ) da d B (d A 1)(d B 1) ), (10) where the entres of the matrx C, C j = λ A λ B j λ A I db I da λ B j, T j = d Ad B λ A λ B j, λ A/B k stands for the normalzed generator of SU(d A /d B ),.e. Tr{λ A/B k λ A/B l } = δ kl and X = Tr{ρX}. It s shown that the lower bounds (9) and (10) are ndependent of (8). Now we consder a multpartte quantum state ρ H 1 H H N as a bpartte state belongng to H A H B wth the dmensons of the subsystems A and B beng d A = d s1 d s d sm and d B = d sm+1 d sm+ d sn respectvely. By usng the corollary, (8), (9) and (10) we have the followng lower bound: Theorem : For any N-partte quantum state ρ, we have: C N (ρ) 3 N max{b1, B, B3}, (11) where B1 = max {} B = max {} B3 = max {} M (M 1) [ max( TA (ρ ), R(ρ ) ) 1 ], C(ρ ) (1 Tr{(ρ A ) }) (1 Tr{(ρ B ) }), M (M 1) 8 M 3 N (M 1) ( T(ρ ) M N (M 1)(N 1) ), 4

7 ρ s are all possble bpartte decompostons of ρ, and M = mn {d s1 d s d sm,d sm+1 d sm+ d sn }, N = max {d s1 d s d sm,d sm+1 d sm+ d sn }. In [7, 30, 31], t s shown that the upper and lower bound of multpartte concurrence satsfy (4 3 N )Tr{ρ } N α Tr{ρ α} C N (ρ) N [( N ) α Tr{ρ α}]. (1) In fact we can obtan a more effectve upper bound for mult-partte concurrence. Let ρ = λ ψ ψ H 1 H H N, where ψ s are the orthogonal pure states and λ = 1. We have C N (ρ) = mn p C N ( ϕ ϕ ) {p, ϕ } λ C N ( ψ ψ ). (13) The rght sde of (13) gves a new upper bound of C N (ρ). Snce λ C N ( ψ ψ ) = 1 N λ ( N ) Tr{(ρ α) } α 1 N ( N ) α Tr{ λ (ρ α) } 1 N ( N ) α Tr{(ρ α ) }, the upper bound obtaned n (13) s better than that n (1). The lower and upper bounds can be used to estmate the value of the concurrence. Meanwhle, the lower bound of concurrence can be used to detect entanglement of quantum states. We now show that our upper and lower bounds can be better than that n (1) by several detaled examples. Example 1: Consder the Dü r-crac-tarrach states defned by [3]: ρ = σ=± λ σ 0 Ψ σ 0 Ψ σ λ j ( Ψ + j Ψ+ j + Ψ j Ψ j ), (14) j=1 where the orthonormal Greenberger-Horne-Zelnger (GHZ)-bass Ψ ± j 1 ( j ± (3 j) ), j 1 j 1 1 j wth j = j 1 j n bnary notaton. From theorem we have that the lower bound of ρ s 1. If we mx the state wth whte nose, 3 ρ(x) = (1 x) I 8 + xρ, (15) 8 by drect computaton we have, as shown n FIG. 1, the lower bound obtaned n (1) s always zero, whle the lower bound n (11) s larger than zero for 0.45 x 1, whch 5

8 C 3 Ρ x FIG. 1: Our lower and upper bounds of C 3 (ρ) from (11)(13)(sold lne) and the upper bound obtaned n (1)(dot lne) whle the lower bound n (1) s always zero. shows that ρ(x) s detected to be entangled at ths stuaton. And the upper bound (dot lne) n (1) s much larger than the upper bound we have obtaned n (13) (sold lne). Example : We consder the depolarzed state [3]: ρ = (1 x) I 8 + x ψ + ψ +, (16) 8 where 0 x 1 representng the degree of depolarzaton, ψ + = 1 ( ). From FIG. one can obvously seen that our upper bound s tghter. For 0 x our lower bound s hgher than that n (1),.e. our lower bound s closer to the true concurrence. Moreover for 0. x , our lower bound can detect the entanglement of ρ, whle the lower bound n (1) not. We have studed the concurrence for arbtrary multpartte quantum states. We derved new better lower and upper bounds. The lower bound can also be used to detect more multpartte entangled quantum states. Acknowledgments Ths work s supported by NSFC under grant , NKBRSFC under grant 004CB [1] Nelsen M A, Chuang I L. Quantum Computaton and Quantum Informaton. Cambrdge: Cambrdge Unversty Press, (000). [] R. F. Werner, Phys. Rev. A 40, 477 (1989). 6

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