A class of Bell diagonal entanglement witnesses in $\mathbb{C}^4 \otimes \mathbb{C}^4$: optimization and the spanning property

TL;DR

This paper investigates Bell diagonal entanglement witnesses in four-dimensional bipartite systems, generalizing known witnesses, analyzing their optimality, and revealing complex structural properties.

Contribution

It introduces generalized Choi witnesses in $ ext{C}^4 imes ext{C}^4$, analyzes their optimality and spanning properties, and clarifies the structure of optimal entanglement witnesses.

Findings

Generalized Choi witnesses are not optimal in $ ext{C}^4 imes ext{C}^4$

Operators from the second class are optimal but lack spanning property

Optimization procedures reveal intricate structures of entanglement witnesses

Abstract

Two classes of Bell diagonal indecomposable entanglement witnesses in $\mathbb{C}^4 \otimes \mathbb{C}^4$ are considered. Within the first class, we find a generalization of the well-known Choi witness from $\mathbb{C}^3 \otimes \mathbb{C}^3$, while the second one contains the reduction map. Interestingly, contrary to $\mathbb{C}^3 \otimes \mathbb{C}^3$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors, that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. This analysis sheds a new light into the intricate structure of optimal entanglement witnesses.