aa r X i v : . [ qu a n t - ph ] A ug Lower bounds on the squashed entanglement for multi-party system
Wei Song
Institute for Condensed Matter Physics, School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China
Squashed entanglement is a promising entanglement measure that can be generalized to multipar-tite case, and it has all of the desirable properties for a good entanglement measure. In this paperwe present computable lower bounds to evaluate the multipartite squashed entanglement. We alsoderive some inequalities relating the squashed entanglement to the other entanglement measure.
PACS numbers: 03.67.-a, 03.67.Mn, 03.65.UdKeywords: entanglement measure; squashed entanglement; lower bound
Entanglement has been recognized as a key resourceand ingredient in the field of quantum information andcomputation science. As a result, a remarkable researcheffort has been devoted to characterizing and quantifyingit (see, e.g., Ref.[1, 2] and references therein). Despite alarge number of profound results obtained in this field,e.g., [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], there isstill no general solution to the simplest case, namely thetwo partite case. It is usually accepted that the followingtwo axioms[19]are satisfied for an appropriate entangle-ment measure. One natural axiom is that an entangle-ment measure should not increase under local operationsand classical communication [8]. The other is that everyentanglement measure should vanish on the set of separa-ble quantum states. Some other useful but not necessar-ily properties require the entanglement measures shouldbe convex, additive, and a continuous function in thestate. The issue of entanglement measure for multipar-tite states poses an even greater challenge[31], and mostof existing entanglement measures are constructed for bi-partite state except that the quantum relative-entropyof entanglement[5] and squashed entanglement[21] canbe generalized to multipartite case. Among the existingtwo-partite entanglement measures, additivity only holdsfor squashed entanglement and logarithmic negativity[20]and is conjectured to hold for entanglement of forma-tion, but the quantum relative-entropy of entanglement isnonadditive [32]. Squashed entanglement was introducedby [33]and then independently by Christandl and Winter[21], who showed that it is monotone, and proved its ad-ditivity. It has all of the desirable properties for a goodentanglement measure: it is convex, asymptotically con-tinuous, additive on tensor products and superadditive ingeneral. It is upper bounded by entanglement cost, lowerbounded by distillable entanglement. Very recently, thesquashed entanglement was extended to multipartite caseby Yang et al [34]and similar ideas have also been devel-oped independently in Ref.[35]. Furthermore, in a recentpaper[36], the squashed entanglement is given the oper-ational meaning with the aid of conditional mutual in-formation. Thus the squashed entanglement is a promis-ing candidate among the different kinds of entanglementmeasures. However, it is still very difficult to compute the squashed entanglement and no analytic formula ex-ists even for bipartite states. In fact, it is usually noteasy to evaluate entanglement measures. Entanglementof formation is efficiently computable only for two-qubits[6]. Other measures are usually computable for stateswith high symmetries, such as Werner states, isotropicstate, or the family of ”iso-Werner” states, and squashedentanglement can only be evaluated for so called specialflower states[37].In this paper our aim is to explore a computable lowerbound to evaluate the multipartite squashed entangle-ment. Firstly we briefly review the definition of multi-partite q -squashed entanglement introduced in Ref.[34].Before describing the details of multipartite squashed en-tanglement, it is necessary to recall the definition of mul-tipartite mutual information. In this paper we will adoptthe function I ( A : A : . . . : A n ) = S ( A ) + S ( A ) + . . . + S ( A n ) − S ( A A . . . A n ) as a multipartite mu-tual information, where S ( X ) is the von Neumann en-tropy of system X . This version of multipartite mu-tual information has an interesting feature: it can berepresented as a sum of bipartite mutual informations: I ( A : A : . . . : A n ) = I ( A : A ) + I ( A : A A ) + I ( A : A A A ) + . . . + I ( A n : A A . . . A n − ). Analo-gous to the definition of bipartite conditional mutual in-formation I ( A : B | E ) = S ( AE ) + S ( BE ) − S ( ABE ) − S ( E ), we can also define the multipartite conditional mu-tual information I ( A : A : . . . A N | E ). For the N-partystate ρ A ...A N , the multipartite q -squashed entanglementis defined as E qsq ( ρ A ...A N ) = inf I ( A : A : . . . : A N | E ) , (1)where the infimum is taken over states σ A ...A N ,E , thatare extensions of ρ A ...A N , i.e. T r E σ = ρ . If the extensionstates σ A ,...,A N ,E takes the form P i p i ρ i A ,...,AN ⊗ | i i E h i | ,we call it c-squashed entanglement. Here, we denoteq-squashed entanglement and c-squashed entanglementboth as E sq ( ρ A ...A N ) due to our derivation is irrele-vant to the form of the extension states. We begin byconsidering tri-partite state and later generalize the re-sults to the case of multi-party subsystem. Notice that I ( A : A : . . . : A N | E ) can be represented as the sum ofthe following terms: I ( A : A : . . . : A N | E ) = I ( A : A | E ) + I ( A : A A | E )+ I ( A : A A A | E ) + · · · + I ( A N : A A A . . . A N − | E ) . (2)Now we can prove the following: Lemma 1.
For any tri-partite state ρ A A A , we have E sq ( ρ A : A : A ) ≥ max { C − S ( A A ) ,C − S ( A A ) , C − S ( A A ) } , (3)where C = P i =1 S ( A i ) − S ( A A A ). Proof : Suppose that E is an optimum exten-sion for system A A A satisfying E sq ( ρ A : A : A ) = I ( A : A : A | E ). Then E sq ( ρ A : A : A ) − E sq ( ρ A : A ) − E sq ( ρ A A : A ) ≥ I ( A : A : A | E ) − I ( A : A | E ) − I ( A A : A | E ) = 0 . (4)Thus we have E sq ( ρ A : A : A ) ≥ E sq ( ρ A : A ) +2 E sq ( ρ A A : A ) . Employing a lower bound of the two-partite squashed entanglement presented in Ref.[21], thuswe obtain: E sq ( ρ A : A : A ) ≥ P i =1 S ( A i ) − S ( A A ) − S ( A A A ). If we permute the indices cyclically we getthree inequalities and obtain the sharpest bound. Thisends the proof.It should be noted that the constant 2 in Eq.(4) isdue to the difference of the definition between bipartitesquashed entanglement and multipartite squashed entan-glement.The measures we propose in the case of two par-ties reduces to twice the original squashed entanglement. Corollary 1:
For any tri-partite state ρ A A A , we have E sq ( ρ A : A : A ) ≥ E sq ( ρ A : A )+2 E sq ( ρ A : A )+2 E sq ( ρ A : A ) . (5) Proof . Notice that the monogamy inequality of two-partite squashed entanglement[38], i.e., E sq ( ρ A : BC ) ≥ E sq ( ρ A : B ) + E sq ( ρ A : C ), the proof is obtained immedi-ately.By taking the average over all combinations of twoparties in Eq. (3) we get the following corollary: Corollary 2:
For any tri-partite states ρ A A A , wehave E sq ( ρ A : A : A ) ≥ S ( A ) + S ( A ) + S ( A ) −
13 [ S ( A A ) + S ( A A ) + S ( A A )] − S ( A A A ) . (6)Eq. (3) and Eq. (6) provide computable lower boundsto evaluate the tri-partite squashed entanglement. Us-ing an inequality presented in Ref.[39], we can also relate the relative-entropy of entanglement to the squashed en-tanglement measure. For tri-partite pure state we have E sq ( ρ A : A : A ) = S ( A ) + S ( A ) + S ( A ). Employingthe inequality (12) in Ref.[39] an immediate corollary isas follows: Corollary 3: E RE ( ρ A : A : A ) ≤ E sq ( ρ A : A : A ) ≤ E RE ( ρ A : A : A ) − E RE ( ρ A : A ) − E RE ( ρ A : A ) − E RE ( ρ A : A ) . (7)for any pure tri-partite state ρ A A A .Furthermore, we can derive an inequality relating theconditional entanglement of mutual information with thesquashed entanglement. Conditional entanglement ofmutual information is a new entanglement measure intro-duced in Ref.[29]. Remarkably, it is additive and has anoperational meaning and can straightforwardly be gen-eralized to multipartite cases. Conditional entanglementof mutual information is defined as follows: Definition.
Let ρ AB be a mixed state on a bipartiteHilbert space H A ⊗ H B . The conditional entanglementof mutual information for ρ AB is defined as C I ( ρ AB ) = inf 12 { I ( AA ′ : BB ′ ) − I ( A ′ : B ′ ) } , (8)where the infimum is taken over all extensions of ρ AB , i.e., over all states satisfying the equation T r A ′ B ′ ρ AA ′ BB ′ = ρ AB , and the factor 1/2 is to makeit equal to the entanglement of formation for the purestate case. Yang et al [29] have proved that C I satisfiedall the desired property of a good entanglement mea-sure and it is easy generalized to the multipartite case.For multipartite mixed state ρ A A ... A n , C I ( ρ A ...A n ) =inf { I n ( A A ′ : . . . : A n A ′ n ) − I n ( A ′ : . . . : A ′ n ) } , where I n = P i S ( A i ) − S ( A · · · A n ). Now we present our resultwhich is the following lemma. Lemma 2 . For any tri-partite state ρ A A A , we have C I ( ρ A : A : A ) ≥ max { C I ( ρ A : A ) + 2 E sq ( ρ A A : A ) , C I ( ρ A : A ) + 2 E sq ( ρ A A : A ) , C I ( ρ A : A ) + 2 E sq ( ρ A A : A ) } . (9) Proof . Suppose that A ′ A ′ A ′ is a minimum ex-tension for system A A A satisfying C I ( ρ A : A : A ) = I ( A A ′ : A A ′ : A A ′ ) − I ( A ′ : A ′ : A ′ ). Then - - FIG. 1: Plot of the lower bound of the squashed entanglementfor the mixed state ρ ( p ) . C I ( ρ A : A : A ) − C I ( ρ A : A ) − E sq ( ρ A A : A ) ≥ C I ( ρ A : A : A ) − C I ( ρ A : A ) − I ( A A : A | A ′ ) ≥ C I ( ρ A : A : A ) − C I ( ρ A : A ) − I ( A A ′ A A ′ : A | A ′ ) ≥ S ( A A ′ ) + S ( A A ′ ) + S ( A A ′ ) − S ( A A ′ A A ′ A A ′ ) − S ( A ′ ) − S ( A ′ ) − S ( A ′ ) + S ( A ′ A ′ A ′ ) − S ( A A ′ ) − S ( A A ′ ) + S ( A A ′ A A ′ )+ S ( A ′ ) + S ( A ′ ) − S ( A ′ A ′ ) − I ( A A ′ A A ′ : A | A ′ )= S ( A A ′ A A ′ ) + S ( A ′ A ′ A ′ ) − S ( A ′ A ′ ) − S ( A A ′ A A ′ A ′ ) ≥ . (10)The last inequality is due to strong subadditivity ofthe von Neumann entropy. Analogously we can provethe other two inequalities.Next we generalize our lower bounds on the squashedentanglement to the N -partite case. Using the similarprocedure as proving Lemma 1, we obtain the followinggeneral result: Lemma 3.
For any N -partite state ρ A A ...A N , we have E sq ( ρ A : A : ... : A N ) ≥ N X i =1 , ,...,N S ( A i ) − X M =2 ,...,N − (cid:18) NM (cid:19) N X i <...
For any multipartite state ρ A A ...A N E sq (cid:0) ρ A : A : ... :( A N − A N ) (cid:1) ≥ E sq (cid:0) ρ A : A : ... : A N − (cid:1) + E sq (cid:0) ρ ( A A ...A N − ): A N (cid:1) . (12) Proof.
Suppose that E is a minimum extension forstate ρ A A ...A N , then - - - - - FIG. 2: Plot of the lower bound of the squashed entanglementfor the mixed state ρ W ( p ) . E sq (cid:0) ρ A : A : ... :( A N − A N ) (cid:1) = I ( A : A : . . . : ( A N − A N ) | E )= I ( A : A : . . . : A N − | E )+ I (( A A . . . A N − ) : A N | A N − E ) ≥ E sq (cid:0) ρ A : A : ... : A N − (cid:1) + E sq (cid:0) ρ ( A A ...A N − ): A N (cid:1) . (13)Below we give some examples to show the applicationof Eq.(11). Example 1.
Consider a family of mixed 4-qubit state ρ ( p ) = p | GHZ i h
GHZ | + (1 − p ) | W i h W | ,where | GHZ i = √ ( | i + | i ), and | W i = ( | i + | i + | i + | i ). In order to eval-uate the multipartite entanglement of ρ ( p ), we plot thelower bound of the squashed entanglement as a functionof p in Fig.1. We find the lower bound for 0 ≤ p < . . < p ≤ ρ ( p ) is anentangled state in these cases. It should be noted thatthe analytic expression of the 3-tangle for the 3-qubitstate ρ ( p ) have been obtained in Ref.[40] recently, andthe 3-tangle can be used as an entanglement measure forthe genuine 3-party entanglement. However, their resultsonly restricted to the 3-qubit state and it is not obvi-ously to generalize the 3-tangle to the multipartite case.In contrast, our lower bound can be used to evaluate thesquashed entanglement for arbitrary party systems. Example 2 . Consider a class of generalizedWerner states[41, 42] for 2 ⊗ ⊗ ρ W ( p ) = p I ⊗ I ⊗ I + (1 − p ) | ψ i h ψ | , where | ψ i = √ (2 | i − | i − | i )[43]. The tripar-tite mixed state ρ W ( p ) are invariant under ρ W → R dU U ⊗ U ⊗ U ρ W U † ⊗ U † ⊗ U † and can be regardedas generalized tripartite Werner states. Now we employthe lower bound to evaluate the squashed entanglementof ρ W ( p ). The lower bound is plotted in Fig. 2. We canstill get a positive lower bound for 0 ≤ p < . [1] M.B. Plenio and S. Virmani, Quant. Inf. Comp. , 1(2007).[2] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, e-print quant-ph/0702225.[3] A. Shimony, Ann. NY. Acad. Sci. , 675 (1995).[4] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K.Wootters, Phys. Rev. A , 3824 (1996).[5] V. Vedral, M.B. Plenio, M.A. Rippin, and P.L. Knight,Phys. Rev. Lett. , 2275 (1997).[6] W. K. Wootters, Phys. Rev. Lett 80, 2245 (1998).[7] E. M. Rains, eprint quant-ph/9809078.[8] G. Vidal, J. Mod. Opt. , 355 (2000).[9] B. M. Terhal, Phys. Lett. A , 319 (2000).[10] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev.A , 052306 (2000).[11] A. Uhlmann, Open Sys. Inf. Dyn. , 209. (1998); A.Uhlmann, Phys. Rev. A , 032307 (1998).[12] A. Acin, R. Tarrach, G. Vidal, Phys. Rev. A
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