Approximate transformations of bipartite pure-state entanglement from the majorization lattice

We study the problem of deterministic transformations of an initial pure entangled quantum state, vertical bar psi >, into a target pure entangled quantum state, vertical bar phi >, 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 psi < phi holds, where psi and phi 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 vertical bar chi(opt)> most approximate to vertical bar phi > 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 psi and phi. 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. (C) 2017 Elsevier B.V. All rights reserved.