Closed formula for the relative entropy of entanglement

TL;DR

This paper derives a compact formula for the inverse problem of the relative entropy of entanglement in two-qubit states, providing insights into its properties and applications, but suggests a general solution remains elusive.

Contribution

It presents a closed-form solution to the inverse problem of REE for two qubits and analyzes its implications and limitations.

Findings

A compact formula for the inverse problem of REE is obtained.

The formula suggests that a general analytical solution for the original problem is limited to special cases.

Applications include demonstrating additivity of REE and its relation to other entanglement bounds.

Abstract

The long-standing problem of finding a closed formula for the relative entropy of entanglement (REE) for two qubits is addressed. A compact-form solution to the inverse problem, which characterizes an entangled state for a given closest separable state, is obtained. Analysis of the formula for a large class of entangled states strongly suggests that a compact analytical solution of the original problem, which corresponds to finding the closest separable state for a given entangled state, can be given only in some special cases. A few applications of the compact-form formula are given to show additivity of the REE, to relate the REE with the Rains upper bound for distillable entanglement, and to show that a Bell state does not have a unique closest separable state.