Characterization and properties of weakly optimal entanglement witnesses

TL;DR

This paper analyzes the properties of weakly optimal entanglement witnesses, showing their mathematical form, relation to eigenvalues of separable states, and applications in constructing witnesses in larger Hilbert spaces.

Contribution

It introduces a specific form for weakly optimal entanglement witnesses and explores their geometric properties and applications in higher-dimensional spaces.

Findings

Weakly optimal witnesses can be expressed as $W^{wopt}=\sigma - c_{\sigma}^{max} I$.

Established the relation between weakly optimal witnesses and eigenvalues of separable states.

Applied weakly optimal witnesses to construct entanglement witnesses in larger Hilbert spaces.

Abstract

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.