Spectral characterizations of entanglement witnesses

TL;DR

This paper systematically analyzes the spectral properties of entanglement witnesses, revealing fundamental differences between decomposable and nondecomposable types and demonstrating their effectiveness in detecting complex quantum states.

Contribution

It provides a detailed spectral characterization of entanglement witnesses, distinguishing decomposable from nondecomposable types and establishing their detection capabilities.

Findings

Infimum of smallest eigenvalue attainable by DEWs but not by NDEWs.

Spectral properties help identify conditions for mirrored entanglement witnesses.

NDEWs can detect non-positive-transpose states beyond two-qubit and qubit-qutrit systems.

Abstract

We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-transpose (NPT) state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.