Distance between Bound Entangled States from Unextendible Product Bases and Separable States

Abstract

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendibleproduct basis bound entangled states (UPB BE states). The method relies on the fact that an optimalentanglement witness is given by a plane perpendicular to a line between the reference state, entanglementof which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds anapproximation of CSS. In this article, we investigate if this approximation can be good enough toyield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those givenby Bandyopadhyay-Ghosh-Roychowdhury (BGR) construction. This comparison allows us to learnabout the amount of entanglement and we find a relationship between it and a feature of the constructionof UPB BE states, namely the size of their central tile. We show that in most studied cases, witnessesfound with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay,Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections ina realization of the state.