Distance between bound entangled states from unextendible product basis and separable states

Marcin Wieśniak, Palash Pandya, Omer Sakarya, Bianka Woloncewicz

Published 2019-12-13Version 1

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product bases' bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay-Ghosh-Roychowdhury (BGR) construction. This comparison allows to learn about the amount of entanglement and we find a relation between it and a feature of the construction of UPB BE states, namely the size of their central tile. We show that, in vast majority of studied cases, our witnesses are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury.