Closest separable state when measured by a quasi-relative entropy

TL;DR

This paper investigates the closest separable state to a given quantum state when measured by a quasi-relative entropy, extending known results for relative entropy to a broader class of functions and states.

Contribution

It generalizes the characterization of the closest separable state from relative entropy to quasi-relative entropy for pure states and certain classes of functions.

Findings

Closest separable state matches the relative entropy case for maximally entangled states.

For certain functions and pure states, the same closest separable state applies.

In two-qubit systems, the closest separable state may differ from the relative entropy case.

Abstract

It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.