The Inverse Eigenvalue Problem for Entanglement Witnesses

TL;DR

This paper solves the inverse eigenvalue problem for entanglement witnesses in two-qubit systems, providing a full characterization and new spectral conditions, and explores duality with absolutely separable states.

Contribution

It offers a complete solution for the two-qubit case and introduces new spectral conditions for entanglement witnesses in higher dimensions.

Findings

Complete characterization of spectra for two-qubit entanglement witnesses

New necessary spectral conditions in arbitrary dimensions

Duality established between entanglement witnesses and absolutely separable states

Abstract

We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely solve this problem in the two-qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions. We also establish a natural duality relationship with the set of absolutely separable states, and we completely characterize witnesses (i.e., separating hyperplanes) of that set when one of the local dimensions is 2.