Checking the optimality of entanglement witnesses: an application to structural physical approximations

TL;DR

This paper investigates the optimality of entanglement witnesses related to structural physical approximations, demonstrating that a previously claimed counterexample is not actually optimal, thereby supporting the conjecture that SPAs to optimal witnesses are separable.

Contribution

It applies a known method to verify the optimality of entanglement witnesses, clarifying the status of a proposed counterexample and reinforcing the conjecture about SPA separability.

Findings

The entanglement witness in question is not optimal.

The method confirms the non-optimality both analytically and numerically.

Supports the conjecture that SPAs to optimal witnesses are separable.

Abstract

In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking optimality of entanglement witnesses developed already in [M. Lewenstein et al., Phys. Rev. A 62, 052310 (2000)], however, hardly exploited so far in the literature.