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

Abstract

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 [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 entanglement transformation. Vidal et. al [Phys. Rev. A 62, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state |χ opt 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.

The Optimal Uncertainty Relation

“…The optimization of entropic uncertainty relation is then turned to finding the quantum state whose catalytically majorizes others, which is hard to be solved analytically . It is worth mentioning that majorization lattice has, and may have more, profound applications in the entanglement transformation …”

Detecting coherence via spectrum estimation

“…Thus, our methods can be used to obtain the common resource states that can generate a whole subclass of entangled states. This concept may also be generalized to the case of approximate state transformations [62][63][64].…”

Discrete And Differentiable Entanglement Transformations

“…As shown in Ref. [54], the supremum state optimizes the distance on the majorization lattice in terms of the Shannon entropies with the target states, i.e. d(ξ sup , Φ) = min ξ ∈Ma(Ψ) d(ξ , Φ) (where d(Ψ, Φ) was defined in Section III A), but it does not optimize the fidelity with the target state.…”