A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces

TL;DR

This paper investigates the conditions under which decomposable entanglement witnesses are optimal, revealing a specific equivalence in qubit-qunit systems and demonstrating limitations in higher-dimensional spaces.

Contribution

It establishes a new equivalence condition for optimality of decomposable entanglement witnesses in qubit-qunit systems and explores its failure in higher dimensions.

Findings

In qubit-qunit systems, optimality, spanning product vectors, and partial transposition are equivalent.

In C^{3}⊗C^{3} systems, the equivalence breaks down, showing limitations of the characterization.

The paper provides insights into the structure of entanglement witnesses and their optimality conditions.

Abstract

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit-qunit decomposable EW (DEW) W the three statements are equivalent: (i) the set of product vectors obeying \bra{e,f}W\ket{e,f}=0 spans the corresponding Hilbert space, (ii) W is optimal, (iii) W=Q^{\Gamma} with Q denoting a positive operator supported on a completely entangled subspace (CES) and \Gamma standing for the partial transposition. While, implications $(i)\Rightarrow(ii)$ and $(ii)\Rightarrow(iii)$ are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in C^{2}\otimes C^{n} span after partial conjugation the whole space. On the other hand, already in the case of C^{3}\otimes C^{3} Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii), or (ii) does not imply (iii), and the above transparent characterization obeyed by qubit-qunit DEWs, does not hold in general.