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

SERGIOLI, GIUSEPPE; BELLOMO, GUIDO; BOSYK, GUSTAVO MARTÍN; HOLIK, FEDERICO; FREYTES, HECTOR

PHYSICA A - STATISTICAL AND THEORETICAL PHYSICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2017 vol. 473 p. 404 - 411

0378-4371

We study the problem of deterministic transformations of an extit{initial} pure entangled quantum state, |ψ⟩, into a extit{target} pure entangled quantum state, |ϕ⟩, by using extit{local operations and classical communication} (LOCC). A celebrated result of Nielsen [Phys. Rev. Lett. extbf{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 extit{approximate} entanglement transformation. Vidal extit{et. al} [Phys. Rev. A extbf{62}, 012304 (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 extit{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.