Optimality of a class of entanglement witnesses for $3\otimes 3$ systems

TL;DR

This paper characterizes when a specific class of entanglement witnesses for 3x3 quantum systems are optimal, showing they are optimal only under certain parameter conditions involving permutation cycles.

Contribution

It provides a complete characterization of the optimality conditions for a class of entanglement witnesses in 3x3 systems, linking optimality to permutation structure and parameter values.

Findings

Hermitian matrix $W_{\

induced by $\

Abstract

Let $\Phi_{t,\pi}: M_3({\mathbb C}) \rightarrow M_3({\mathbb C})$ be a linear map defined by $\Phi_{t,\pi}(A)=(3-t)\sum_{i=1}^3E_{ii}AE_{ii}+t\sum_{i=1}^3E_{i,\pi(i)}AE_{i,\pi(i)}^\dag-A$, where $0\leq t\leq 3$ and $\pi$ is a permutation of $(1,2,3)$. We show that the Hermitian matrix $W_{\Phi_{t,\pi}}$ induced by $\Phi_{t,\pi}$ is an optimal entanglement witness if and only if $t=1$ and $\pi$ is cyclic.