Closed formula for the relative entropy of entanglement in all dimensions

TL;DR

This paper derives a universal closed-form formula for the relative entropy of entanglement applicable to multipartite states across all dimensions, revealing conditions for the uniqueness of the closest separable state.

Contribution

It provides the first comprehensive closed-form expression for the relative entropy of entanglement for all multipartite states in any dimension, extending previous bipartite results.

Findings

Closed formula for all entangled states on the boundary of separable states

Uniqueness of CSS for full-rank entangled states

Reduction to known bipartite formula in two-qubit case

Abstract

The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find a closed formula for all the entangled state for which this state is a CSS. Quite amazing, our formula holds for multipartite states in all dimensions. In addition we show that if an entangled state is full rank, then its CSS is unique. For the bipartite case of two qubits our formula reduce to the one given in Phys. Rev. A 78, 032310 (2008).