Hilbert-Schmidt distance and entanglement witnessing

Palash Pandya, Omer Sakarya, Marcin Wieśniak

Published 2018-11-15, updated 2019-01-23Version 2

Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. In this article we apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance (HSD) between a given state and the set of separable states. While HSD does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide here a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the Closest Separable State (CSS) for the given state. We demonstrate the efficacy of the method with examples.