On the structure of a reversible entanglement generating set for three partite states

Transcription

1 On the structure of a reversile entanglement generating set for three partite states A. Acín 1, G. Vidal and J. I. Cirac 3 1 GAP-Optique, University of Geneva, 0, Rue de l École de Médecine, CH-111 Geneva 4, Switzerland Institute for Quantum Information, California Institute of Technology, Pasadena, CA 9115, USA 3 Max Planck Institut für Quantenoptik, Hans Kopfermann Str. 1, D Garching, Germany (Feruary 1, 00) We show that Einstein Podolsky Rosen Bohm (EPR) and Greenerger Horne Zeilinger Mermin (GHZ) states can not generate, through local manipulation and in the asymptotic limit, all forms of three partite pure state entanglement in a reversile way. The techniques that we use suggest that there may e a connection etween this result and the irreversiility that occurs in the asymptotic preparation and distillation of ipartite mixed states. PACS Nos a, Bz To identify the fundamentally inequivalent ways quantum systems can e entangled is a major goal of quantum information theory. In the case of systems shared y two parties, Alice and Bo, there is only one type of entanglement, namely that contained in the Einstein Podolsky Rosen Bohm (EPR) state EPR = 1 ( ), (1) in the sense that, in the limit of large N, Alice and Bo can reversily transform N copies of any other state ψ into EPR states y using only local operations and classical communication (LOCC) [1]. This simple picture ecomes much richer in systems shared y more than two parties, since also genuine multipartite entanglement exists []. In particular, the Greenerger Horne Zeilinger Mermin (GHZ) state GHZ = 1 ( ) () can not e reversily generated from EPR states pairwise distriuted among Alice, Bo and a third partie Claire [3]. In the terminology of Ref. [], this implies that EPR states alone do not form a minimal reversile entanglement generating set (MREGS) for three partite states. The results of [3] left open the question whether, instead, the set G 3 = { GHZ, EPR, EPR AC, EPR BC } (3) constitutes a MREGS. Denoting y an asymptotically (i.e. in the large N limit) reversile transformation using LOCC, this question amounts to assessing the feasiility of a transformation of the form ψ N C GHZ gn EPR xn (4) EPR yn AC EPR zn BC, where g, x, y, z 0, for any three partite state ψ C. If this were the case, then entanglement in three partite systems could e regarded as consisting only of GHZ and EPR correlations. In the meantime it has een proved that not all four partite states can e reversily generated from a distriution of EPR and three and four partite GHZ states [4]. However, no evidence has een found contradicting the following conjecture. Conjecture: G 3 is a MREGS for three partite states. On the contrary, all reversile transformations of three partite states so far reported, involving Schmidt decomposale states [], ut also a whole class of more elaorated states [5], seem to support it. In this Letter we construct a three partite state Ψ C that can not e reversily generated only with states of the set G 3, thus disproving the aove conjecture. We also show that even a reversile transformation of states of G 3 into the state Ψ and states of G 3 is impossile. That is, we show that there are cases where the transformation of Eq. (4) can not e made reversile even if the coefficients g, x, y, z are eventually allowed to e negative [6]. Notice that such a possiility, not previously excluded in four partite systems, would have allowed a slightly different description of multipartite entanglement, also ased exclusively on EPR and GHZ correlations. These results, therefore, indicate the need to extend the set G 3 in order to eventually otain a MREGS, either in its original formulation or in the extended sense descried aove. We would like to note, however, that the notion of a non trivial MREGS implicitly assumes that the manipulation of multipartite pure states can e made reversile. This is, admittedly, an appealing idea, ut has not yet een proved. In this sense, our results can e just interpreted as to indicate that a fundamental irreversiility occurs during the process of comining EPR and GHZ entanglements into the three partite pure state Ψ. It is natural to inquire into the origin of such an irreversiility, which is somewhat analogous to the one that characterizes the cycle of preparing and distilling ipartite mixed states [7]. Actually, the argument that will 1

2 lead to disprove the aove conjecture would fail if mixed state entanglement could e reversily distilled. This fact suggests a connection etween the two irreversile processes. Our strategy consists in showing that a conservation law oeyed in reversile asymptotic entanglement transformations [3] would e violated if EPR and GHZ states could generate Ψ reversily. Let Ψ C denote an aritrary three partite pure state shared y Alice, Bo and Claire, and let ρ e the mixed state resulting from tracing out Claire s susystem. The relative entropy of entanglement of ρ [8], E Ω (ρ ) min S(ρ σ ), (5) σ Ω where Ω is some convex set of states (typically, that of separale states) invariant under LOCC and S(ρ σ) tr(ρ log ρ ρ log σ) is the quantum relative entropy, was originally introduced to quantify the entanglement of ipartite mixed states. Its regularized version, Ω (ρ ) lim ), (6) N N is a lower ound for the entanglement cost E c [9,10] of ρ, or numer of EPR states per copy of ρ needed to asymptotically prepare copies of ρ. It is also an upper ound for its distillale entanglement E d [9,11], or numer of EPR states per copy of ρ that can e asymptotically distilled from copies of ρ. Indeed, Ω fulfills the postulates required in [1] for an entanglement measure and therefore [1,13] E c (ρ ) Ω (ρ ) E d (ρ ). (7) Particularly relevant in the context of this work will e the fact that, as showed in [3], the relative entropy of entanglement of (say) susystems, E Ω () muste conserved during any reversile pure-state transformation of the system C. Applied to transformation (4) this law reads )=E Ω([EPR] xn ), (8) [EPR] EPR EPR, where we have used that when tracing out part C, only EPR gives a non-separale contriution [14]. Thus, in the large N limit we are left with the condition Ω (ρ ) =x, (9) where x is the numer of EPR states per copy of ρ that should e availale on the rhs of Eq. (4), and we have used that E Ω ([EPR] ) = 1. Similarly, if instead we allow now for states of G 3 to appear simultaneously in oth sides of transformation (4), we otain [EPR] x1n )=E Ω([EPR] xn ), (10) x 1,x 0, which implies the condition lim [EPR] x1n ) = x. (11) N N Now, there are several possile elections of the set Ω. Here we will consider only the set Sep of separalestates, and the set PPT of states with positive partial transposition. Each of these choices leads to a different constraint. In particular, Eq. (9) ecomes two conditions, Sep (ρ ) =x, (1) PPT (ρ ) =x. (13) We will next consider a pure state Ψ C such that its reduced density matrix for systems, ρ, is a PPT ound entangled state [15], and therefore PPT (ρ )=0. First we will prove that Sep (ρ ) > 0, which leads to the contradiction 0 = x>0, indicating that Ψ C can not e reversily generated with states of G 3 [16]. Notice that when applied to the PPT state ρ, Eq. (11) for Ω = PPT implies that x 1 = x [17]. We will also prove that [EPR] x1n ) lim >x 1, (14) N N that y sustitution in Eq. (11) for Ω = Sep implies that x >x 1. Therefore, we must have x 1 = x >x 1, which is again a contradiction, this time meaning that the states of G 3 can not reversily generate the state Ψ and states of G 3. We construct the three partite state Ψ C C 3 C 3 C 4 as a purification of the PPT ound entangled state ρ introduced in [18] and employed in [7] to prove the irreversiility of the preparation distillation cycle of ipartite mixed states. More specifically, let P e a projector onto the orthogonal complement in C 3 C 3 of the suspace spanned y the following vectors: 0 ( ), ( ), ( 1 + ), ( 1 + ) 0, ( ) ( ). The state ρ is proportional to the projector P, ρ P /4. Its three partite purification Ψ C, such that ρ =tr C Ψ Ψ,reads Ψ 1 4 φ i i C, (15) i=1 where { i } 4 i=1 is an orthonormal asis in C4 and the orthonormal set { φ i } 4 i=1 fulfills P = i φ i φ i. In order to proceed, we need the following result.

3 Theorem 1 [7]: A positive constant α<1existssuch that, for all N 1, max a N N P N a N N α N, (16) a N N where a N N C 3N C 3N denotes a product state. The following theorem provides us with a ound for the relative entropy of entanglement with respect to the set Sep and together with theorem 1 is the key to the main result. Theorem : Let P e the projector onto the support of a mixed state ρ of a ipartite system C d C d,let a C d C d denote a product vector and let β e β max a P a. (17) a The relative entropy of entanglement with respect to separale states is ounded elow y E Sep (ρ ) log β. (18) Proof: Let σ Sep e the separale state such that E Sep (ρ )=S(ρ σ ). The quantum relative entropy can only decrease under a trace preserving completely positive map E [19]. In particular, let us consider We find E(τ) PτP +(I P )τ(i P ). (19) S(ρ σ ) S(E(ρ ) E(σ )) = tr(ρ log ρ ρ log Pσ P ), (0) where in the last step we have used that ρ is invariant under E and that we can ignore the contriution (I P )σ (I P ) ecause its support I P is orthogonal to P. Indeed, notice that for positive operators N,M 1 and M, log(m 1 M )=logm 1 log M, and therefore tr[(n 0) log(m 1 M )] = tr(n log M 1 ). Define t tr(pσ ), (1) σ 1 t Pσ P. () Then, ecause σ = i p i a i i a i i is a separale state, we have that t β. We finally otain, S(ρ σ ) tr(ρ log ρ tσ )= log t + S(ρ σ ) log t log β, (3) where we have used that for positive operators N,M and a positive constant k tr(n log km) =tr(n log M)+ (trn)logk, and the positivity of the quantum relative entropy [19]. We need only to concatenate theorems 1 and to find that ) log α N (4), and therefore Sep (ρ ) log α>0, (5) which disprove the initial conjecture for G 3. Notice that we can use this result and the inequalities (7) to recover the result of [7] that E c (ρ ) > E d (ρ ). Indeed, we have 0 = PPT (ρ ) < Sep (ρ ), and oth quantities are etween the entanglement cost E c and the distillale entanglement E d. Let us move now to prove Eq. (14). We need the following two lemmas. Lemma 1: Let P e a projector onto a suspace V of C d C d, and let a C d C d e a product state. Then max a P a =max λ 1(ψ), (6) a ψ V where λ 1 (ψ) denotes the largest coefficient λ i in the Schmidt decomposition of ψ, ψ = i λi u i v i, λ 1 λ i+1. Proof: For any product vector a, let us define the normalized vector γ V as P a / P a. Then a P a = a γ λ 1 (γ), (7) where in the last step we have used lemma 1 of [0]. Let ψ e the vector for which the maximum in the rhs of Eq. (6) is attained, and let i λ i u i v i, λ i λ i+1, e its Schmidt decomposition. Then max λ 1(ψ) =λ 1 = ψ V u 1 v 1 P u 1 v 1, (8) which finishes the proof. Lemma : Let P e a projector onto a suspace V of C d C d and let P Φ e a projector onto a ipartite pure state Φ C d C d with Schmidt decomposition d i=1 λi u i v i, λ i λ i+1. Finally, let α p e α p max a P a, (9) a where a C d C d denotes a product state. Then, max c d c d P P Φ c d = α p λ 1, (30) where the maximization is made over product vectors c d C d+d C d+d. Proof: Notice that P P Φ projects onto a suspace spanned y vectors of the form ψ Φ, ψ V,and that the largest coefficient λ 1 in a Schmidt decomposition fulfills λ 1 (ψ Φ) = λ 1 (ψ)λ 1 (Φ). Then Eq. (30) follows from lemma 1. We would like to ound elow the relative entropy of entanglement E Sep of 3

4 ρ N [EPR] M. (31) The projector onto its support is given y P N [EPR] M and we can use lemma and theorem 1 to otain Antonio.Acin@physics.unige.ch vidal@cs.caltech.edu Ignacio.Cirac@mpq.mpg.de N max a P [EPR] M a αn a M, (3) where (1/) M corresponds to λ 1 (EPR M ). Then we can apply theorem to otain [EPR] M ) N log α + M, (33) which implies Eq. (14). This finishes the proof of the fact that it is not possile to reversily transform states of G 3 into the state Ψ and states of G 3. In this work we have showed y means of a counter example that GHZ and EPR states alone cannot e used to reversily generate all three partite pure states. This result leaves several questions open. It would e interesting to understand the mechanisms that lead to this irreversiility. Recall that in the asymptotic limit some non-trivial three partite states can e reversily generated from EPR and GHZ states [5]. We ignore which conditions determine that a three partite pure state transformation can e performed in a reversile way. The following two facts suggest, however, that there may e a connection etween this question and the irreversiility that takes place during the preparation distillation cycle of ipartite mixed states. (i) All known three-partite reversile transformations [,5] involve pure states whose ipartite reduced mixed states can e distilled and prepared in a reversile way [1]. (ii) The proof that G 3 is not a MREGS relies on the irreversiility that occurs in ipartite mixed state manipulation. Indeed, suppose that E c and E d would not disagree for ρ. Then, ecause of Eq. (7), and Sep PPT would also have een equal, and this would jeopardize our argument. Finally, a major open question is whether a finite MREGS exists for three-partite states and, if so, which kind of states must include. These are difficult issues that certainly deserve further investigation. We cautiously conclude the present work y noting that the states of an eventual MREGS must have ipartite reduced density matrices ale to reproduce the discrepancies etween relative entropies displayed y ρ, and must therefore carry themselves the signature of mixed state irreversiility. A. A. thanks J. Preskill and the IQI for hospitality. We thank W. Dür, E. Jané, N. Linden, Ll. Masanes and S. Popescu for discussion. This work was supported y the European project EQUIP (IST ), y the ESF, y the Swiss FNRS and OFES, and y the NSF (of the United States of America), Grant. No. EIA [1] C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53 (1996), 046. [] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin and A. V. Thapliyal, Phys. Rev. A 63 (001), [3] N. Linden, S. Popescu, B. Schumacher and M. Westmoreland, quant-ph/ [4] S. Wu and Y. Zhang, quant-ph/ [5] G. Vidal, W. Dür and J. I. Cirac, Phys. Rev. Lett. 85 (000), 658. [6] A negative value for, say, the coefficient x of Eq. (4) can e used to symolize that the states EPR must appear on the lhs of that transformation. In the present work this possiility is considered as a special case of allowing any state of G 3 to appear simultaneously and in aritrary proportions on oth sides of Eq. (4). [7] G. Vidal and J. I. Cirac, Phys. Rev. Lett. 86 (001), [8] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight, Phys.Rev.Lett.78 (1997), 75; V. Vedral and M. B. Plenio, Phys. Rev. A 57 (1998), [9] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A 54 (1996), 384. [10] P. M. Hayden, M. Horodecki and B. M. Terhal, J. Phys. A 34 (001), [11] E. M. Rains, Phys. Rev. A 60, 173 (1999). [1] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. 84 (000), 014. [13] An heuristic justification for the second of these inequalities can e found in M. Plenio and V. Vedral, Contemp. Phys. 39, 431 (1998). [14] The relative entropy of entanglement fulfills E Ω(ρ ρ s)=e Ω(ρ ) for any separale state ρ s as a result of its monotonicity under LOCC [8]. [15] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. 80 (1998), 539. [16] The relation etween (i) the equivalence of PPT and separale relative entropies of entanglement and (ii) the question whether G 3 is a MREGS for three partite states, has een previously considered y E. F. Galvao, M. B. Plenio and S. Virmani, J. Phys. A 33 (000), [17] For ρ ppt a PPT state we have E PPT(ρ ρ ppt) = E PPT(ρ ), since (i) y means of LOCC we can get rid of ρ ppt and LOCC can only decrease E PPT, so that E PPT(ρ ρ ppt) E PPT(ρ ), and (ii) for any PPT state π ppt we have S(ρ ρ ppt π ppt ρ ppt) = S(ρ π ppt), which guarantees that E PPT(ρ ρ ppt) is not going to e larger than E PPT(ρ ). [18] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin and B. M. Terhal, Phys. Rev. Lett. 8 (1999), [19] See for instance M. A. Nielsen and I. L. Chuang, Quan- 4

5 tum Computation and Quantum Information, Camridge University Press (000). [0] G. Vidal, D. Jonathan and M. A. Nielsen, Phys. Rev. A 6, (000). [1] This can e checked y noticing that the ipartite reduced density matrices of the states discussed in [5] (which contain the Schmidt decomposale states of []) consist of a mixture of locally orthogonal pure states [5] (either product or entangled). Thus, the entanglement of the mixed state can e distilled without losses y means of a projective local measurement that proailistically picks up one of the pure states of the mixture. 5