Optimal decomposable witnesses without the spanning property

TL;DR

This paper investigates the existence of optimal decomposable entanglement witnesses that do not have the spanning property, providing systematic examples and generalizing previous results to higher-dimensional bipartite systems.

Contribution

It demonstrates the existence of such witnesses without the spanning property and extends prior characterizations to larger bipartite Hilbert spaces.

Findings

Existence of decomposable witnesses without spanning property confirmed.

Provided systematic examples of these witnesses.

Generalized characterization results to m,n ≥ 3 dimensions.

Abstract

One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces H_{m,n}=C^m\otimes C^n such that the product vectors obeying <e,f|W|e,f>=0 do not span H_{m,n}. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of 2\otimes n optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H_{m,n} with m,n\geq 3.