Deterministic transformations of bipartite entangled states and majorization

BELLOMO, GUIDO

Conferencia; VII Conference on Quantum Foundations; 2017

In the first part of this talk, we will introduce the notion of majorization between probability vectors and its properties, with particular emphasis in its lattice structure recently demonstrated by Cicalese and Vaccaro [IEEE Trans. Inf. Theory 48,933 (2002)]. In the second part, we will approach the problem of the deterministic interconversion of 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 [Phys. Rev. Lett. 83, 436 (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 transformation protocol. Vidal et al. [Phys. Rev. A 62, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state most approximate to φ in terms of maximal fidelity between them. Here, we are going to discuss a strategy to deal with approximate entanglement transformations based on the properties of the majorization lattice. More precisely, we will 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, in some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.