A class of entanglement witnesses and a realignment-like criterion

TL;DR

This paper introduces a new family of entanglement witnesses based on positive maps, which are shown to be equivalent to a stronger realignment-like criterion, effectively detecting PPT entangled states in various dimensions.

Contribution

It develops a multi-parameter framework for constructing entanglement witnesses that generalizes and strengthens the realignment criterion for bipartite quantum systems.

Findings

All constructed EWs are equivalent to a realignment-like criterion.

The criterion outperforms the original realignment criterion for $d_1 eq d_2$.

Effective detection of PPT entangled states in $3 imes 3$ and $3 imes 4$ systems.

Abstract

We present a multi-parameter family of positive maps between spaces of differing dimensions. This framework facilitates the construction of Entanglement Witnesses (EWs) specifically designed for systems living in $d_1 \times d_2$ dimensions. A sufficient condition for positivity is presented. Interestingly, it is shown that all EWs constructed this way are equivalent to a single realignment-like criterion which for $d_1 \neq d_2$ is in general stronger than the original realignment criterion. We illustrate effectiveness of this criterion considering examples of Positive Partial Transpose (PPT) entangled states in $3 \times 3$ and $3 \times 4$ dimensions.