On the structure of mirrored operators obtained from optimal entanglement witnesses

TL;DR

This paper explores the structure of mirrored entanglement witnesses, proposing a conjecture about their form when derived from optimal witnesses, which impacts the detection of bound entangled states and offers insights into the separability problem.

Contribution

It introduces a conjecture on the nature of mirrored operators from optimal entanglement witnesses and analyzes their properties, revealing new structural insights into entanglement detection.

Findings

Mirrored operators from optimal EWs are conjectured to be positive or decomposable.

Mirrored EWs from non-optimal witnesses can be non-decomposable.

Mirrored operators from extremal decomposable witnesses are positive semi-definite.

Abstract

Entanglement witnesses (EWs) are a versatile tool in the verification of entangled states. The framework of mirrored EW doubles the power of a given EW by introducing its twin -- a mirrored EW -- whereby two EWs related by mirroring can bound the set of separable states more efficiently. In this work, we investigate the relation between the EWs and its mirrored ones, and present a conjecture which claims that the mirrored operator obtained from an optimal EW is either a positive operator or a decomposable EW, which implies that positive-partial-transpose entangled states, also known as the bound entangled states, cannot be detected. This conjecture is reached by studying numerous known examples of optimal EWs. However, the mirrored EWs obtained from the non-optimal ones can be non-decomposable as well. We also show that mirrored operators obtained from the extremal decomposable witnesses are positive semi-definite. Interestingly, the witnesses that violate the well known conjecture of Structural Physical Approximation, do satisfy our conjecture. The intricate relation between these two conjectures is discussed and it reveals a novel structure of the separability problem.