Progress towards a unified approach to entanglement distribution

PHYSICAL REVIEW A 92, 012335 (2015) Progress towards a unified approach to entanglement distribution Alexander Streltsov,1,*Remigiusz Augusiak,1Maciej Demianowicz,1and Maciej Lewenstein1,2 1ICFO–Institut de Ci` encies Fot` oniques, Av. C.F. Gauss, 3, E-08860 Castelldefels, Spain 2ICREA–Instituci´ o Catalana de Recerca i Estudis Avanc¸ats, Lluis Companys 23, E-08010 Barcelona, Spain (Received 19 December 2014; published 30 July 2015) Entanglement distribution is key to the success of secure communication schemes based on quantum mechanics, and there is a strong need for an ultimate architecture able to overcome the limitations of recent proposals such as those based on entanglement percolation or quantum repeaters. In this work we provide a broad theoretical background for the development of such technologies. In particular, we investigate the question of whether entanglement distribution is more efficient if some amount of entanglement—or some amount of correlations in general—is available prior to the transmission stage of the protocol. We show that in the presence of noise the answer to this question strongly depends on the type of noise and on the way the entanglement is quantified. On the one hand, subadditive entanglement measures do not show an advantage of preshared correlations if entanglement is established via combinations of single-qubit Pauli channels. On the other hand, based on the superadditivity conjecture of distillable entanglement, we provide evidence that this phenomenon occurs for this measure. These results strongly suggest that sending one half of some pure entangled state down a noisy channel is the best strategy for any subadditive entanglement quantifier, thus paving the way to a unified approach for entanglement distribution which does not depend on the nature of noise. We also provide general bounds for entanglement distribution involving quantum discord and present a counterintuitive phenomenon of the advantage of arbitrarily little entangled states over maximally entangled ones, which may also occur for quantum channels relevant in experiments. DOI: 10.1103/PhysRevA.92.012335 PACS number(s): 03.67.Mn,03.67.Hk,03.65.Ud I. INTRODUCTION Considered as a curiosity in the early days of quantum theory [1], entanglement has now been recognized as the essential ingredient for a growing number of applications in quantum technologies [2,3]. Among them we find, for example, the celebrated quantum cryptography [4] allowing for a provably secure communication between distant parties, and quantum teleportation [5], which offers the possibility of an intact transmission of a state of a particle over an arbitrarily long distance using preshared entanglement and classical communication. Entanglement is also necessary for quantum nonlocality, which is an even stronger resource for certain information-processing tasks, including the above-mentioned secure key distribution [4,6,7] and certified quantum randomness generation [8–10]. A common assumption behind entanglement-based protocols is that long-distance or at least medium-distance entanglement is available beforehand. Several remedies against this drawback have been recently proposed, with the most promising one being based on quantum repeaters [11] and entanglement distillation [12]. However, the necessity of powerful quantum memories appears as the main limiting factor in this proposal (cf. [13]). Another method is based on entanglement percolation [14], but it also suffers problems when considered in realistic situations in the presence of noise and decoherence [15]. The aim of the present work is to explore different realistic scenarios in which the long-distance entanglement can be distributed. The general framework for such a task we adopt here is the following (see Fig. 1). Two parties, Alice and Bob, *alexander[email protected] initially share a three-particle quantum system. Two of the particles are with Alice, and the remaining one is in Bob’s hands. In the most general situation we allow Alice and Bob to share some correlations established before the beginning of the protocol. The distribution of entanglement is then achieved with the aid of a quantum channel which is used to transmit one of Alice’s particles to Bob. A remarkable result with respect to such general entanglement distribution protocols has been obtained in [16]. There, it was shown that the process is even possible without sending entanglement directly: for successful entanglement distribution the exchanged particle does not need to be entangled with the rest of the system. This phenomenon has been termed “entanglement distribution with separable states,” and its experimental verification has also been reported recently [17–20]. These results suggest that such a distribution procedure may be advantageous in the presence of noise: it could be possible to surpass the fragileness of entanglement by sending a separable particle. Despite considerable attempts to understand this phenomenon [21–26], one of the most important questions remains unresolved: Can noisy entanglement distribution with separable states provide an advantage when compared to sending one-half of the maximally entangled state through the same noisy channel? Note that the answer for this question also has a direct importance for the theory and practice of quantum repeaters and quantum percolation where the intermediatedistance entanglement between the involved nodes must be established in some way. In this work we attack this problem by focusing on the following closely related questions: (1) Given a noisy quantum channel, what is the maximal amount of entanglement that can be distributed with and without preshared correlations? 1050-2947/2015/92(1)/012335(14) 012335-1 ©2015 American Physical Society STRELTSOV, AUGUSIAK, DEMIANOWICZ, AND LEWENSTEIN PHYSICAL REVIEW A 92, 012335 (2015) boBecilA preshared correlations quantum channel FIG. 1. (Color online) General framework for entanglement distribution. Alice is initially in possession of two particles, while one particle is in Bob’s hands. Alice and Bob further have access to preshared correlations and an additional, possibly noisy quantum channel which is used for entanglement distribution. (2) Are preshared correlations helpful for entanglement distribution via a given quantum channel? Note that a negative answer to the second question also implies that entanglement distribution with separable states is not the best strategy in this situation. As our study reveals, the answers to these questions depend on the way entanglement is quantified. In particular, we show that if the entanglement quantifier is subadditive (that is, its value for a tensor product of any two states is not greater than the sum of the values for the individual states), preshared correlations provide no advantage for single-qubit Pauli channels or any tensor product thereof. In this situation the best distribution strategy is to send one half of the maximally entangled state down the noisy channel. However, not all entanglement quantifiers are subadditive. In particular, it is conjectured that the distillable entanglement is superadditive [27]. Assuming this conjecture holds true, we show that preshared correlations can indeed provide an advantage for the distribution of distillable entanglement. Another surprising result is obtained for the logarithmic negativity: for this entanglement measure states with arbitrarily little entanglement can show better performance for entanglement distribution when compared to maximally entangled states. We further present bounds for noisy entanglement distribution given by quantum discord [28,29], thus significantly extending the results provided in [22,23] to the noisy scenario. Moreover, the results presented in this work strongly suggest that a unified approach to entanglement distribution is indeed possible. In particular, based on our findings it is very reasonable to assume that preshared correlations do not provide an advantage for any subadditive entanglement quantifier, regardless of the type of noisy channel used for the distribution. If this assumption is correct, sending one half of some pure entangled state down a noisy channel will be the best strategy in this very general scenario. However, we also show that maximally entangled states are not necessarily optimal for this process. This paper is organized as follows. In Sec. II we study noiseless entanglement distribution, while the scenario involving noise is considered in Sec. III. In Sec. IV we investigate the optimal entanglement distribution without preshared correlations, i.e., we consider the maximal amount of entanglement that can be distributed via a given noisy channel if Alice and Bob do not share any correlations initially. Finally, the possible advantage of preshared correlations for noisy entanglement distribution is discussed in Sec. V. II. NOISELESS ENTANGLEMENT DISTRIBUTION The starting point of this section is the general scenario for entanglement distribution considered in [22,23]; see also [29] for a detailed discussion. In particular, we assume that two parties, Alice and Bob, have access to a general tripartite quantum state ρ=ρABC. We further assume, without loss of generality, that the entanglement distribution is realized by sending the particle Cfrom Alice to Bob, and that during the entire process particles Aand Bare in possession of Alice and Bob, respectively. If the quantum channel used for the transmission of the particle Cis noiseless, the amount of entanglement distributed in this process is quantified via the difference EA|BC(ρ)−EAC|B(ρ) between the final amount of entanglement EA|BC(ρ) and the initial amount of entanglement EAC|B(ρ). As it was shown in [16], entanglement distribution is also possible by sending a particle which is not entangled with the rest of the system, i.e., there exist states ρ=ρABC such that EC|AB (ρ)=0 and, at the same time, EA|BC(ρ)− EAC|B(ρ)>0. This finding has triggered a debate about the type of correlations which are responsible for entanglement distribution. An important result in this context was provided in Refs. [22] and [23]. The amount of distributed entanglement cannot exceed the amount of quantum discord C|AB between the exchange particle Cand the remaining system AB: C|AB (ρ)⩾EA|BC(ρ)−EAC|B(ρ).(1) At this point, it is also important to notice that in general quantum discord does not vanish on separable states. This inequality was shown to hold for all distance-based quantifiers of entanglement and discord [22]: EX|Y(ρXY )=min σXY ∈S D(ρXY ,σXY ),(2) X|Y(ρXY )=min {X i} DρXY , i X iρXY X i.(3) Here, Sis the set of bipartite separable states, {X i}is a local von Neumann measurement on the subsystem X, and Dcan be any general distance which satisfies the following two properties [22]: (1) Ddoes not increase under quantum operations: D([ρ],[σ]) ⩽D(ρ,σ)(4) for any quantum operation and any pair of quantum states ρand σ, and (2) Dsatisfies the triangle inequality D(ρ,σ)⩽D(ρ,τ)+D(τ,σ)(5) for any three quantum states ρ,σ, and τ. As it was further shown in [22,23], the results presented above also hold for the quantum relative entropy S(ρ||σ)= Tr[ρlog ρ]−Tr[ρlog σ], despite the fact that the relative entropy in general does not satisfy the triangle inequality. The corresponding quantifiers of entanglement and discord in this 012335-2 PROGRESS TOWARDS A UNIFIED APPROACH TO . . . PHYSICAL REVIEW A 92, 012335 (2015) case are known as the relative entropy of entanglement ERand the relative entropy of discord R: EX|Y R(ρXY )=min σXY ∈S S(ρXY ||σXY ),(6) X|Y R(ρXY )=min {X i} SρXY  i X iρXY X i.(7) The relative entropy of entanglement ERwas originally introduced in [30,31]. By its relation to the relative entropy [32,33], it plays a fundamental role in quantum information theory. ERis known to be an upper bound on the distillable entanglement Ed[34,35] and a lower bound on the entanglement of formation Ef[31]: Ed⩽ER⩽Ef.(8) The distillable entanglement Edquantifies the maximal number of singlets that can be asymptotically obtained per copy of the given state via local operations and classical communication (LOCC) [12]. The entanglement of formation Efis defined as [36] Ef(ρXY )=min  i piE(|ψiXY ),(9) where the minimum is taken over all pure-state decompositions {pi,|ψiXY }of the state ρXY , i.e., ρXY =ipi|ψψ|XY , and E(|ψXY )=S(ρX) is the von Neumann entropy of the reduced state. The relative entropy of discord Rwas originally introduced in [37], where it was called “one-way information deficit” [38]. It quantifies the amount of information which cannot be localized by one-way classical communication between two parties. A. Relation to distillable entanglement and entanglement cost Equipped with these tools, we are now in a position to present the first results of this paper. In particular, we will provide a close connection between the relative entropy of discord R, the distillable entanglement Ed, and the entanglement cost Ec. The latter is defined as the minimal number of singlets per copy required for the asymptotic creation of a bipartite quantum state via LOCC [3], and can also be written as the regularized entanglement of formation [39]: Ec(ρ)=lim n→∞ 1 nEf(ρ⊗n).(10) The aforementioned relation between R,Ed, and Ecis provided in the following theorem. Theorem 1. Given a tripartite state ρ=ρABC, the following inequality holds: C|AB R(ρ)⩾EA|BC d(ρ)−EAC|B c(ρ).(11) Proof. This inequality can be proven by noticing that the inequality (1) also holds for the regularized relative entropy of entanglement and discord: lim n→∞ C|AB R(ρ⊗n) n⩾lim n→∞ EA|BC R(ρ⊗n) n−lim n→∞ EAC|B R(ρ⊗n) n. (12) By applying Eq. (8) and using the fact that the distillable entanglement Eddoes not change under regularization we arrive at the inequality lim n→∞ C|AB R(ρ⊗n) n⩾EA|BC d(ρ)−lim n→∞ EAC|B f(ρ⊗n) n.(13) In the next step we recall that the entanglement cost is equal to the regularized entanglement of formation, see Eq. (10), and thus lim n→∞ C|AB R(ρ⊗n) n⩾EA|BC d(ρ)−EAC|B c(ρ).(14) Finally, the desired inequality (11) follows by observing that the relative entropy of discord does not increase under regularization: R(ρ)⩾lim n→∞R(ρ⊗n)/n. Notice that Eq. (11) has a clear operational interpretation: the relative entropy of discord is an upper bound on the number of singlets gained in the process of entanglement distribution in the asymptotic limit. This is because EA|BC d(ρ) quantifies the number of singlets Alice and Bob can distill after performing the entanglement distribution, while EAC|B c(ρ) quantifies the amount of singlets that Alice and Bob need to create the state ρ=ρABC before the entanglement distribution has been performed, both in the asymptotic limit. Moreover, as already mentioned in the proof of Theorem 1, this statement is also true for the regularized relative entropy of discord lim n→∞R(ρ⊗n)/n. B. Relation to measures of NPT entanglement and distillability The results presented above demonstrate that the relation between entanglement and discord in Eq. (1) is more general than anticipated by the original approach [22,23]. In the following we will go one step further by extending these results to general measures of NPT (nonpositive partial transpose) entanglement. In particular, we consider entanglement quantifiers of the form [3] EX|Y PPT (ρXY )=min σXY ∈PPT D(ρXY ,σXY ),(15) where PPT is the set of states having positive partial transpose, and the distance Dsatisfies Eqs. (4) and (5). The amount of quantum discord is defined in the same way as in Eq. (3): X|Y(ρXY )=min {X i} DρXY , i X iρXY X i.(16) The following theorem shows that inequality (1) also applies to these measures of NPT entanglement. Theorem 2. Given a tripartite state ρ=ρABC, the following inequality holds: C|AB (ρ)⩾EA|BC PPT (ρ)−EAC|B PPT (ρ).(17) Proof. The proof goes along the lines of the one of Eq. (1), first presented in [22]. We start by introducing the state 012335-3 STRELTSOV, AUGUSIAK, DEMIANOWICZ, AND LEWENSTEIN PHYSICAL REVIEW A 92, 012335 (2015) σ=σABC, which is PPT with respect to the bipartition AC|B, and, moreover, we assume that it is the closest PPT state to ρ: EAC|B PPT (ρ)=D(ρ,σ). We then define the states ρ= i C iρC i(18) and σ= i C iσC i(19) to arise from ρand σvia the local von Neumann measurement on C, minimizing the distance between ρand ρ, i.e., C|AB (ρ)=D(ρ,ρ). Furthermore, we use the fact that the distance Dsatisfies the triangle inequality, and thus D(ρ,σ)⩽D(ρ,ρ)+D(ρ,σ).(20) Recalling that Ddoes not increase under quantum operations, it follows that D(ρ,σ)⩽D(ρ,σ),(21) and Eq. (20) becomes D(ρ,σ)⩽C|AB (ρ)+EAC|B PPT (ρ).(22) In the final step we note that the state (19) is PPT with respect to all bipartitions, i.e., EAB|C PPT (σ)=EAC|B PPT (σ)= EA|BC PPT (σ)=0. The fact that EAB|C PPT (σ)=0 is obvious, since σarises by performing a local von Neumann measurement {C i}on the state σ, and thus has the form of a quantum-classical state: σ=iC iσC i=ipiσAB i⊗ |ii|C. Moreover, by the very construction, the state σis PPT with respect to the bipartition AC|B, and so is σ, meaning that EAC|B PPT (σ)=0. This, together with the fact that σis classical on the subsystem C, implies that it is also PPT with respect to the remaining bipartition A|BC:EA|BC PPT (σ)=0. This means that the distance between ρand σis an upper bound on EA|BC PPT (ρ), which, when applied in Eq. (22), completes the proof.  The above theorem extends the range of applications of Eq. (1) to distance-based quantifiers of NPT entanglement. The same arguments can also be applied to measures of distillability defined as [3] EX|Y ND (ρXY )=min σXY ∈ND D(ρXY ,σXY ).(23) Here, ND is the set of nondistillable states, and, as before, the distance Dsatisfies Eqs. (4) and (5). Using the same arguments as in the proof of Theorem 2, we see that Eq. (17) generalizes to these distillability measures: C|AB (ρ)⩾EA|BC ND (ρ)−EAC|B ND (ρ),(24) where the quantum discord C|AB is defined in the same way as in Eq. (16). Finally, the above results also hold for measures of NPT entanglement and distillability based on the relative entropy, although, as mentioned earlier, the latter does not satisfy the triangle inequality in general. The fact that Eqs. (17) and (24) still apply to these measures can be seen using the same arguments as in the proof of Theorem 2by observing that for the relative entropy the inequality (20) becomes equality [22]. C. Relation to Schatten norms The results presented so far hold for a very general class of quantifiers for entanglement and discord. In particular, we have seen that Eq. (1) applies for any entanglement measure E, which is defined via the minimal distance to the set of separable, nondistillable, or PPT states, if the amount of discord is quantified as in Eq. (3). The corresponding distance only needs to satisfy two minimal requirements given in Eqs. (4) and (5): it should not increase under quantum operations and it should satisfy the triangle inequality. On the other hand, we have also seen that Eq. (1) can be still valid even if the distance violates one of these properties. This was demonstrated for the relative entropy which can violate the triangle inequality. In the following we will show that Eq. (1) may also hold for distances violating Eq. (4), i.e., those that are not contractive under quantum operations. To this end we will consider the following distance: Dp(ρ,σ)=||ρ−σ||p,(25) with ·pbeing the Schatten pnorm of an operator Mdefined through ||M||p=(Tr[(M†M)p/2])1/p,(26) with p⩾1. Clearly, D1coincides with the trace distance and thus does not increase under quantum operations [2]. However, contrary to what had been claimed in [40], already D2(socalled Hilbert-Schmidt distance) can increase under quantum operations as shown in [41]. The arguments from Ref. [41] can be further generalized to show this fact for any p>1(see also [42,43] for similar considerations). Now let Epbe defined as EX|Y p(ρXY )=min σXY ∈T Dp(ρXY ,σXY ),(27) with the minimization going over the set T, which here might denote either of the sets: separable, nondistillable, or PPT states. Let further pbe defined by Eq. (3) with the distance taken to be Dp. The following theorem shows that Eq. (1)also holds in this situation. Theorem 3. Given a tripartite state ρ=ρABC, the following inequality holds: C|AB p(ρ)⩾EA|BC p(ρ)−EAC|B p(ρ).(28) Proof. The proof follows exactly the lines of the proof of Theorem 2. The only thing which needs to be proved is the fact that although Dpmay increase under general quantum operations, it does not for those operations that map the states ρand σto ρand σin Eqs. (18) and (19), respectively. For this purpose, we notice that such mapping is unital, i.e., iC i1C i=1,where 1=1ABC is the identity operator, and it was shown in Ref. [42] that no unital map can increase the pnorm for any p⩾1. This implies that for p⩾1, Dpdoes satisfy Eq. (21) for the states of interest, which completes the proof.  While quantifiers of discord based on Schatten norms have been considered only recently [43–54], entanglement quantifiers of this type were studied already more than a decade ago [40,41,55–57]. Despite this fact, it has been an open question if Epis a proper entanglement measure, i.e., 012335-4 PROGRESS TOWARDS A UNIFIED APPROACH TO . . . PHYSICAL REVIEW A 92, 012335 (2015) if it is nonincreasing under LOCC for p>1[41]. In what follows we will put this question to rest by showing that Ep can increase by simply discarding a part of the system. For this purpose, let ρAB be a quantum state such that EA|B p(ρAB )>0. Then, consider its extension to a three-partite state defined as ρABC =ρAB ⊗1C/2, where the particle Cis a qubit. We will now show that the entanglement of ρAB is larger than the entanglement of ρABC: EA|B p(ρAB )>E A|BC p(ρABC) (29) for all p>1. To this end, observe that the amount of entanglement EA|BC p(ρABC) is bounded from above by the distance Dp(ρABC,σ ABC)forσABC =σAB ⊗1C/2, where σAB is the closest separable state to ρAB . Moreover, notice that the distance between ρABC and σABC can also be expressed as [54] Dp(ρABC,σ ABC)= 1C 2p Dp(ρAB ,σAB ).(30) Recalling that the state σAB was defined to be the closest separable state to ρAB and using the fact that ||1C/2||p= 21/p−1, one obtains EA|BC p(ρABC)⩽21/p−1EA|B p(ρAB ).(31) The inequality (29) follows by noting that for p>1the exponent 1/p −1 is negative, and thus 21/p−1<1 in this case. Similar results with respect to quantum discord were also obtained recently [43,54]. In particular, it was pointed out in [43] that the geometric discord X|Y G=(X|Y 2)2can increase under local operations on any of the parties Xor Y, while most quantifiers of discord known in the literature do not increase under quantum operations on the subsystem Y. This result was later extended to all measures of discord pfor p>1 [54]. On the one hand, this observation together with Eq. (29) provides strong constraints for the possible applications of entanglement and discord quantifiers based on Schatten norms. On the other hand, the close relation of E2to the problem of finding optimal entanglement witnesses [56,57] and the connection between Epand pestablished in Theorem 3 demonstrate the use of these quantities for understanding the structure of entanglement from a geometric perspective. III. NOISY ENTANGLEMENT DISTRIBUTION In the scenario considered so far, Alice and Bob aimed at distributing entanglement by having access to a noiseless quantum channel. Since noise is unavoidable in any realistic experiment; we will now consider the more general situation in which the channel used for entanglement distribution is noisy. Similarly to the foregoing discussion, we assume that Alice and Bob have access to a tripartite initial state ρi= ρABC, where Alice is initially in possession of the particles Aand C, and Bob is in possession of the remaining particle B. If Alice uses a noisy channel Cto send her particle Cto Bob, they end up in the final state ρf=C[ρi]. The amount of entanglement distributed in this process is then given by EA|BC(ρf)−EAC|B(ρi). Having introduced the concept of noisy entanglement distribution, we are now in a position to extend Eq. (1)to A B C ΛC 1ΛC 2 FIG. 2. (Color online) Decomposition of a noisy channel Cin two channels C 1and C 2. For an initial state ρi=ρABC the final state after the application of the channel is given by ρf=C[ρi]= C 2(C 1[ρi]). The figure illustrates the intermediate state ˜ρ=C 1[ρi] after the application of C 1only. See main text for details. this general scenario. In the following theorem we will show that noisy entanglement distribution is in general limited by the amount of discord in each of the states ρiand ρf. Theorem 4. Given a quantum channel Cand two states ρi=ρABC and ρf=C[ρi], the following inequality holds: min{C|AB (ρi),C|AB (ρf)}⩾EA|BC(ρf)−EAC|B(ρi). (32) Here, Eand are any quantifiers of entanglement and discord which satisfy Eq. (1). Proof. We first apply Eq. (1) to the state ρi, thus arriving at C|AB (ρi)⩾EA|BC(ρi)−EAC|B(ρi). Then the inequality C|AB (ρi)⩾EA|BC(ρf)−EAC|B(ρi) follows by recalling that entanglement does not increase under local noise, i.e., EA|BC(ρi)⩾EA|BC(ρf). Using analogous reasoning one can also prove the inequality C|AB (ρf)⩾EA|BC(ρf)− EAC|B(ρi). Application of Eq. (1) to the state ρfgivesusthe inequality C|AB (ρf)⩾EA|BC(ρf)−EAC|B(ρf). One then completes the proof by using EAC|B(ρf)⩽EAC|B(ρi), which again follows from the fact that entanglement does not increase under local noise. As quantum discord can increase or decrease under local noise [44,58–60], the claim follows.  A. Divisible channels Let us consider a decomposition of the channel Cinto two channels C 1and C 2such that the successive application of these channels is equivalent to the application of C: ρf=C[ρi]=C 2C 1[ρi].(33) (See also Fig. 2for an illustration.) If such a decomposition is possible with nonunitary C 1and C 2, the channel Cis called divisible [61]. By introducing an intermediate state ˜ρ=C 1[ρi],(34) we will now show that the amount of distributed entanglement is in general bounded above by the amount of discord in the state ˜ρ: C|AB (˜ρ)⩾EA|BC(ρf)−EAC|B(ρi).(35) As in the foregoing discussion, we assume that Eand are quantifiers of entanglement and discord satisfying Eq. (1). Under this assumption, Eq. (35) can be proven using similar arguments as in the proof of Eq. (32). In particular, we can apply Eq. (1) to the intermediate state ˜ρ, thus obtaining the inequality C|AB (˜ρ)⩾EA|BC(˜ρ)−EAC|B(˜ρ). The proof 012335-5 STRELTSOV, AUGUSIAK, DEMIANOWICZ, AND LEWENSTEIN PHYSICAL REVIEW A 92, 012335 (2015) of Eq. (35) is complete by making use of the fact that entanglement does not increase under local noise, leading to the inequalities EA|BC(˜ρ)⩾EA|BC(ρf) and EAC|B(˜ρ)⩽ EAC|B(ρi). B. Markovian time evolution Here we will see that the results presented in the previous section have a nice application in the scenario in which the particle Cused for entanglement distribution is subject to a Markovian time evolution C (t2,t1). If we assume that the process starts with the initial state ρi=ρABC at the time t=0, then for any time t⩾0 the time-evolved state is given by ρt=C (t,0)[ρi].(36) Denoting then by Tthe total time required for the process, the corresponding final state ρfcan be written as ρf=ρT=C (T,0)[ρi].(37) We are now in position to prove that the amount of entanglement distributed via a Markovian time evolution is bounded from above by the amount of discord in the time-evolved state ρtfor any T⩾t⩾0: C|AB (ρt)⩾EA|BC(ρf)−EAC|B(ρi).(38) Here, Eand are quantifiers of entanglement and discord satisfying Eq. (1). To prove the above statement, we use the fact that any Markovian time evolution C (t2,t1)obeys the composition law [62], that is, C (t2,t1)[ρ]=C (t2,t)C (t,t1)[ρ](39) for any state ρand all t2⩾t⩾t1⩾0. This, together with Eqs. (36) and (37), leads us to the following expression for the final state: ρf=C (T,0)[ρi]=C (T,t)[ρt],(40) for all T⩾t⩾0. One then obtains Eq. (38) by applying Eq. (35) with ˜ρ=ρt. Let us notice that the inequality (38) also implies that the distribution of entanglement via a Markovian time evolution is bounded above by the minimal discord mintC|AB (ρt), minimized over all times tranging between 0 and the duration of the total procedure T. On the other hand, any violation of Eq. (38) can also be regarded as a witness for the non-Markovianity of the underlying time evolution. These results support recent attempts to detect and quantify non- Markovianity via quantum entanglement [63] and quantum discord [47,64,65]. Noting that the inequality (38)isvalid for a very general class of quantifiers for entanglement and discord, further investigation in this direction can lead to a better understanding of entanglement and discord in the context of detecting non-Markovianity. IV. OPTIMAL ENTANGLEMENT DISTRIBUTION WITHOUT PRESHARED CORRELATIONS In the foregoing discussion we considered noiseless and noisy entanglement distribution, and presented several tools for bounding the amount of entanglement distributed in this process. In this section we will apply them to the following problem: How much entanglement can be distributed via a given quantum channel? Let us begin with the scenario in which Alice and Bob are not correlated initially, i.e., the initial and the final state are given by ρi=ρAC ⊗ρB(41) and ρf=C[ρAC]⊗ρB,(42) respectively. We assume again that Alice is initially in possession of the particles Aand C, while Bob holds the particle B. In the distribution process, the particle Cis sent from Alice to Bob via the quantum channel C. Thus the initial entanglement between Alice and Bob is zero, and the amount of distributed entanglement is given by EA|C(C[ρAC]). In the following, we are interested in optimal entanglement distribution, i.e., we ask which initial states ρAC lead to the maximal final entanglement after the application of a quantum channel C. Clearly, if the quantum channel Cis noiseless, the optimal distribution strategy is achieved if Alice prepares her particles Aand Cin the maximally entangled state, |φ+AC =1 √dC dC−1  i=0|iiAC ,(43) and sends the particle Cto Bob. Interestingly, as we will see below, this strategy is not always optimal if the quantum channel Cis noisy. In passing, it is crucial to notice that all maximally entangled states show the same performance for entanglement distribution, i.e., E(C[|φmeφme|AC ]) =E(C[|φ+φ+|AC]) (44) is true for any maximally entangled state |φmeAC,any entanglement measure E, and any noisy channel C.This can be seen by first noting that any maximally entangled state |φmeAC can be written as |φmeAC =UA|φ+AC , where UAis a unitary acting on the subsystem A. Then, to get Eq. (44) one uses the facts that UAcommutes with Cand that any entanglement quantifier Eis invariant under local unitaries [3]. A. Relation to entanglement of formation In this section we will show that maximally entangled states are optimal for entanglement distribution for all noisy channels if the exchanged particle Cis a qubit and the amount of entanglement is quantified via the entanglement of formation Ef. We then have the following theorem. Theorem 5. For any mixed state ρAC with dA⩾dC=2 and any channel C, the following inequality holds: Ef(C[|φ+φ+|AC]) ⩾Ef(C[ρAC ]).(45) Proof. We first recall that the entanglement of formation is a convex function of the state. This implies that for any mixed state ρAC there exists a pure state |ψAC which shows at least the same performance for entanglement distribution: Ef(C[|ψψ|AC]) ⩾Ef(C[ρAC ]).(46) 012335-6 PROGRESS TOWARDS A UNIFIED APPROACH TO . . . PHYSICAL REVIEW A 92, 012335 (2015) To complete the proof we will show that the maximally entangled state has the best performance among all pure states, i.e., Ef(C[|φ+φ+|AC]) ⩾Ef(C[|ψψ|AC ]) (47) for any pure state |ψAC with dA⩾2, dC=2, and any single-qubit channel C. At this point, it is important to note that the state |ψAC is effectively a two-qubit state, even if the dimension of the subsystem Ais larger than 2. This follows from the Schmidt decomposition of |ψAC, which due to the fact that the subsystem Cis two-dimensional, is of the form |ψAC =λ0|00+λ1|11. The state C[|ψψ|AC] can thus be regarded as a mixed state of two qubits. With this in mind, we can now use the fact that for all two-qubit states the entanglement of formation admits a simple formula: Ef=g(C), where gis a nondecreasing function and Cis the concurrence [66]. The final ingredient of our proof is the factorization law for concurrence (see Eq. (5)in[67]). Adapted to our notation it reads C(C[|ψψ|AC]) =C(C[|φ+φ+|AC ]) ·C(|ψψ|AC). (48) Since the concurrence is never larger than 1, we arrive at the following inequality: C(C[|φ+φ+|AC]) ⩾C(C[|ψψ|AC ]).(49) Note that this inequality also holds if the concurrence Cis replaced by the entanglement of formation Ef, since the latter is a nondecreasing function of the concurrence. This observation completes the proof [68].  It is worth mentioning that the above result can be generalized to a larger class of entanglement measures, namely, to all those measures which for two qubits can be written as a nondecreasing function of concurrence, that is, E=g(C).(50) This can be seen by exploiting the same argumentation as before. Apart from the entanglement of formation, examples of such measures are the geometric measure of entanglement [69,70], the Bures measure of entanglement [30,31], and the Groverian measure of entanglement [71,72]. For two qubits all those measures reduce to a nondecreasing function of concurrence (see Fig. 4in Ref. [73]). B. Relation to Pauli channel We now show that for an important type of noise, the Pauli channel, the statement made in the previous section can be generalized to all entanglement measures. The action of the Pauli channel reads C p[ρAC]= 3  i=0 piσC iρACσC i,(51) where the exchanged particle Cis a qubit and σiare Pauli matrices with σ0=1. We have the following: Theorem 6. For any mixed state ρAC with dA⩾dC= 2 and any Pauli channel C pthe following inequality A R ΛC p[|φ+φ+|˜ RC ] ρAR ˜ RC A C τAC =Λ C p[ρAC ] FIG. 3. (Color online) The state C p[|φ+φ+|˜ RC] can be used to teleport the particle Rof the state ρAR by performing a joint Bell measurement on Rand ˜ R, and a conditional rotation on C(upper figure). This procedure leaves the subsystem AC in the final state τAC =C p[ρAC ] (lower figure). holds: EC p[|φ+φ+|AC]⩾EC p[ρAC](52) for any entanglement measure E. Proof. Let us start by introducing two additional particles Rand ˜ Rwith dR=d˜ R=2. We will now show that the state C p[|φ+φ+|˜ RC] can be used for teleportation in the following way: if two parties share the state C p[|φ+φ+|˜ RC] and apply the standard teleportation protocol [5] for teleporting the two-dimensional subsystem Rof a total state ρAR, they will end up sharing the state C p[ρAC]. (See also [74] for similar considerations.) This can be seen explicitly by considering the essential steps of the standard teleportation protocol (see Fig. 3). In the first step, a joint Bell measurement is performed on the subsystems Rand ˜ R. Depending on the outcome iof the measurement, the subsystem AC is found in one of the four states C p[σC iρACσC i] with 0 ⩽i⩽3. In the final step, a conditioned unitary rotation σC iis applied on the subsystem C, leading to the final state τAC =σC iC pσC iρACσC iσC i.(53) At this point, it is crucial to note that the Pauli channel commutes with the Pauli matrices σC i, i.e., C pσC iρACσC i=σC iC p[ρAC]σC i,(54) which can be seen by inspection using the anticommutation relation σaσb=−σbσafor 1 ⩽a,b ⩽3. Using Eq. (54)we see that the final state τAC becomes independent from the outcome of the measurement i: τAC =C p[ρAC].(55) Finally, note that all steps mentioned above can be performed by using local operations and classical communication (see Fig. 3). This implies that the final state τAC cannot have more entanglement than the state C p[|φ+φ+|˜ RC], regardless of the 012335-7 STRELTSOV, AUGUSIAK, DEMIANOWICZ, AND LEWENSTEIN PHYSICAL REVIEW A 92, 012335 (2015) entanglement measure Eused to quantify it. This completes the proof.  It should be stressed that the result presented in Theorem 6 can also be extended to the scenario in which the channel used for entanglement distribution is a tensor product of different single-qubit Pauli channels. As an example, consider a fourdimensional particle Cconsisting of two qubits C1and C2. The channel C pis now of the form C p=C1 p⊗ C2 p, where C1 pand  C2 pare two (possibly different) Pauli channels. The action of this channel onto an arbitrary state ρAC =ρAC1C2is given by C p[ρAC]=C1 p⊗ C2 p[ρAC1C2].(56) Using similar lines of reasoning as in the proof of Theorem 6, we see that the best performance in this case is also achieved for the maximally entangled state, i.e., the inequality EC p[|φ+φ+|AC]⩾EC p[ρAC](57) holds for any state ρAC with dA⩾dC=4 and the maximally entangled state |φ+AC =(1/2) 3 i=0|iiAC. This statement is also true if the exchanged particle Cconsists of nqubits, and the channel C pis a combination of n(possibly different) single-qubit Pauli channels. In this case, the best performance is achieved for the maximally entangled state (43) with dA⩾ dC=2n. Finally, we note that similar arguments can also be applied to a more general family of channels defined as follows: C[ρAC]= i piUC iρACUC i†,(58) where the particles Aand Ccan have arbitrary dimensions and UC iare unitary operators that act only on the particle Cand have the following two properties: (1) The unitaries (UC i)†commute with the channel C, i.e., CUC i†ρACUC i=UC i†C[ρAC]UC i.(59) (2) For the maximally entangled state |φ+AC, the states |ψiAC =UC i|φ+AC (60) form a complete orthonormal basis, i.e., ψi|ψj=δij and i|ψiψi|=1. As we will show in the following, the maximally entangled state |φ+AC is optimal for entanglement distribution via a noisy channel given in Eq. (58). We will prove this statement by following the same reasoning as for Pauli channels (see also Fig. 3). In particular, we will show that the state C[|φ+φ+|˜ RC] can be used to teleport the particle Rof dimension not larger than dC, such that for any state ρAR the final state has the form τAC =C[ρAC]. This can be proven by considering the state ρAR ⊗C[|φ+φ+|˜ RC], and applying a joint measurement on the particles Rand ˜ Rin the basis |ψiR˜ R=UR i|φ+R˜ R. Conditioned on the measurement outcome i, the resulting postmeasurement state of the particles Aand Cis then given by C[(UC i)†ρACUC i]. In the final step of the proof, we use Eq. (59) and apply conditional unitary rotations UC i, arriving at the desired final state τAC = C[ρAC]. Using the same reasoning as for the Pauli channels, this proves the optimality of the maximally entangled state |φ+AC for the channels given in Eq. (58). Examples of such channels more general than the Pauli channels are the Weyl-covariant channels. C. Relation to negativity and amplitude damping channel All the results presented so far support the intuition that sending one-half of a maximally entangled state down a noisy quantum channel represents the optimal strategy if two parties wish to distribute entanglement between them. In particular, we have seen that this statement is true for all single-qubit channels if the figure of merit is the entanglement of formation, or any other entanglement measure which for two qubits reduces to a nondecreasing function of concurrence. Moreover, for Pauli channels we saw that this statement becomes completely general: in this case maximally entangled states are the optimal resource, regardless of the entanglement measure used. Quite surprisingly, this intuition is generally not correct [75,76]. In particular, it was shown in Ref. [76] (see Sec. III therein) that maximally entangled states are not optimal for entanglement distribution if the amount of entanglement is quantified by the negativity [77,78], which is defined as N(ρAC)=||ρTA||1−1, where TAdenotes the partial transposition over the system Aand ||M||1=Tr√M†Mis the trace norm of M. In what follows we will recall this result, using, however, a slightly different entanglement monotone which is the logarithmic negativity given by En(ρAC)=log2||ρTA||1=log2[N(ρAC )+1].(61) We will also supplement the results of Ref. [76] by noting that for some quantum channels even arbitrarily little entangled states can outperform the maximally entangled state. The effect of suboptimality of maximally entangled states was demonstrated for the single-qubit amplitude damping channel C ad[ρAC]=K1ρAC K† 1+K2ρACK† 2,(62) with Kraus operators K1=|00|C+√1−γ|11|Cand K2=√γ|01|C, and the damping parameter 0 ⩽γ⩽1. If the initial state is chosen as |αAC =√1−α|00AC +√α|11AC (63) with the real parameter 0 ⩽α⩽1, it is straightforward to verify that for ˜α=(1 −γ)/2 the state |˜αAC shows the same performance as the maximally entangled state, that is, EnC ad[|˜α˜α|AC ]=EnC ad[|φ+φ+|AC].(64) This is illustrated in Fig. 4, where the parameter space of αand γis shown. The dashed line for α=˜α=(1 −γ)/2 divides the parameter space into two parts. For α⩽(1 −γ)/2 (lower-left triangle in Fig. 4), the state |αAC shows no advantage when compared to the maximally entangled state, i.e., EnC ad[|αα|AC]⩽EnC ad[|φ+φ+|AC].(65) However, for (1 −γ)/2<α<1/2 (upper right triangle in Fig. 4) the corresponding state |αAC always outperforms 012335-8 PROGRESS TOWARDS A UNIFIED APPROACH TO . . . PHYSICAL REVIEW A 92, 012335 (2015) advantage no advantage 0.0 0.2 0.4 0.6 0.8 1.0 Γ 0.0 0.1 0.2 0.3 0.4 0.5 Α FIG. 4. The plot shows the relevant parameter regions of the damping parameter γand the parameter αwhich enters the initial state |αAC [see Eq. (63)]. Dashed line α=(1 −γ)/2 separates the parameter space in two parts. For (1 −γ)/2<α<1/2 all states |αAC outperform the maximally entangled state. The solid line shows the value of αmax which leads to the maximal logarithmic negativity for a given damping parameter γ. The dotted line shows α=1/2. the maximally entangled state for the damping parameter 0<γ <1: EnC ad[|αα|AC]>E nC ad[|φ+φ+|AC].(66) For a given damping parameter γwe can further maximize the logarithmic negativity of the state C ad[|αα|AC] with respect to the parameter α. Direct algebra shows that the maximum is achieved for αmax =1 γ √1−γ+2(67) (see the solid line in Fig. 4). The corresponding quantity En(C ad[|αmaxαmax|AC ]) is shown in Fig. 5as a function 0 0.5 1 0 0.02 0.04 0.0 0.2 0.4 0.6 0.8 1.0 Γ 0.0 0.2 0.4 0.6 0.8 1.0 En FIG. 5. The solid line shows the logarithmic negativity En of the state C ad[|αmaxαmax|AC ] as a function of the damping parameter γ. Then the dashed line is the corresponding logarithmic negativity of C ad[|φ+φ+|AC ]. The inset shows the difference En(C ad[|αmaxαmax|AC ]) −En(C ad[|φ+φ+|AC ]). The states |αmaxAC outperform the maximally entangled state |φ+AC in the whole region 0 <γ <1. of the damping parameter γ(solid line). There we also show the logarithmic negativity En(C ad[|φ+φ+|AC]) for the maximally entangled state (dashed line). More interestingly, however, it turns out that for the logarithmic negativity, maximally entangled states can be outperformed even by states with arbitrarily little entanglement, which we prove in the following theorem. Theorem 7. For any ε>0 there exists a state ρAC εwith logarithmic negativity at most εand a channel C εsuch that EnC ε[ρAC ε]>E nC ε[|φ+φ+|AC].(68) Proof. We show this result for the amplitude damping channel C ad given in Eq. (62) and the pure state |αAC in Eq. (63). From the fact that the state |αAC is separable for α=0 and maximally entangled for α=1/2, it follows that for any ε>0 there exists αε∈(0,1/2) such that the logarithmic negativity of the state |αεAC is nonzero and at most ε, i.e., 0<E n(|αεAC)⩽ε. (69) To complete the proof it is enough to show that for any ε> 0 there exists an amplitude damping channel C ad with the damping parameter γεsuch that EnC ad[|αεαε|AC]>E nC ad[|φ+φ+|AC].(70) The existence of such a channel follows directly from the arguments presented above. Precisely, by virtue of the inequality (66) we immediately see that Eq. (70)istrueforany damping parameter γεchosen such that 1 −2αε<γ ε<1.  We have then shown that, in some scenarios, states with very little entanglement are a better resource for noisy entanglement distribution when compared to maximally entangled states if the logarithmic negativity Enis used to quantify entanglement. It is worth mentioning that this entanglement measure is closely related to the PPT entanglement cost, i.e., the entanglement cost under quantum operations preserving the positivity of the partial transpose. Precisely, Enis always a lower bound on the PPT entanglement cost [79], while for all two-qubit states both quantities coincide [80]. For this reason the logarithmic negativity is equivalent to the PPT entanglement cost within the framework presented in this section, and all statements made for one quantity are also valid for the other. Moreover, we point out that the result presented in Theorem 7can also be extended to the multicopy scenario, where Alice and Bob have access to many copies of a quantum channel C. The aim of the process in this case is to distribute the maximal logarithmic negativity per copy of the channel. The aforementioned results together with additivity of the logarithmic negativity [78] imply that for amplitude damping noise maximally entangled states can be outperformed by states with arbitrary little entanglement also in this scenario. Let us finally mention that in Ref. [76] the authors show that the maximally entangled states are optimal for entanglement distribution if the single-qubit channel used to transmit the particle is unital and negativity is used as the entanglement measure. 012335-9