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arXiv:quant-ph/0105019 · Physical Review A

Partial recovery of entanglement in bipartite-entanglement transformations

Somshubhro Bandyopadhyay, Vwani Roychowdhury, Farrokh Vatan

Published Apr 10, 2002• 9 citations
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Abstract

Any deterministic bipartite-entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using $2\ifmmode\times\else\texttimes\fi{}2$ auxiliary entangled states, no matter how large the dimensions of the parent states are. For the rest of the special cases of deterministic LOCC transformations, we show that the dimension of the auxiliary entangled state depends on the presence of equalities in the majorization relations of the parent states. We show that genuine recovery is still possible using auxiliary states in dimensions less than that of the parent states for all patterns of majorization relations except only one special case.

DOI
10.1103/physreva.65.040303

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Any deterministic bipartite-entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 auxiliary entangled states, no matter how large the dimensions of the parent states are. For the rest of the special cases of deterministic LOCC transformations, we show that the dimension of the auxiliary entangled state depends on the presence of equalities in the majorization relations of the parent states. We show that genuine recovery is still possible using auxiliary states in dimensions less than that of the parent states for all patterns of majorization relations except only one special case.
Suppose Alice and Bob try to transform an entangled state shared between them into another one by local operations and classical communications. Then in general a certain amount of entanglement contained in the initial state will decrease in the process of transformation. However, an interesting phenomenon called partial entanglement recovery shows that it is possible to recover some amount of entanglement by adding another entangled state and transforming the two entangled states collectively. In this paper, we are mainly concerned with the feasibility of partial entanglement recovery. The basic problem we address is whether a given state is useful in recovering entanglement lost in a specified transformation. In the case where the source and target states of the original transformation satisfy the strict majorization relation, a necessary and sufficient condition for partial entanglement recovery is obtained. For the general case we give two sufficient conditions. We also give an efficient algorithm for the feasibility of partial entanglement recovery in polynomial time. As applications, we establish some interesting connections between partial entanglement recovery and the generation of maximally entangled states, quantum catalysis, mutual catalysis, and multiple-copy entanglement transformation.
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
When an entangled state is transformed into another one with probability one by local operations and classical communication, the quantity of entanglement decreases. This Letter shows that entanglement lost in the manipulation can be partially recovered by an auxiliary entangled pair. As an application, a maximally entangled pair can be obtained from two partially entangled pairs with probability one. Finally, this recovery scheme reveals a fundamental property of entanglement relevant to the existence of incomparable states.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
We study the problem of deterministic transformations of an initial pure entangled quantum state, |ψ〉, into a target pure entangled quantum state, |ϕ〉, by using local operations and classical communication (LOCC). A celebrated result of Nielsen (1999) gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation ψ≺ϕ holds, where ψ and ϕ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an approximate entanglement transformation. Vidal et al. (2000) have proposed a deterministic transformation using LOCC in order to obtain a target state |χopt〉 most approximate to |ϕ〉 in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the majorization lattice. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between ψ and ϕ. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.