Linking a distance measure of entanglement to its convex roof

TL;DR

This paper establishes a link between geometric and distance measures of entanglement, providing new formulas and bounds for quantifying entanglement in quantum states, including a closed form for two qubits.

Contribution

It introduces a new expression for the geometric measure of entanglement based on maximal fidelity, connecting it to a distance measure, and derives bounds and conditions for optimal decompositions.

Findings

New formula for geometric measure of entanglement in terms of fidelity

Closed expression for Bures measure of entanglement of two qubits

Upper bound on the number of elements in optimal decomposition

Abstract

An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglement of two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.