Positive finite rank elementary operators and characterizing entanglement of states

TL;DR

This paper constructs a class of indecomposable positive finite rank elementary operators to provide a simple criterion for separability of pure states and identify new entangled states undetectable by PPT and realignment criteria.

Contribution

It introduces a new class of elementary operators that characterize entanglement and offers a straightforward separability criterion for pure states in bipartite systems.

Findings

Provides a necessary and sufficient separability criterion for pure states.

Identifies new entangled states not detectable by PPT or realignment.

Constructs indecomposable positive finite rank elementary operators of order (n,n).

Abstract

In this paper, a class of indecomposable positive finite rank elementary operators of order $(n,n)$ are constructed. This allows us to give a simple necessary and sufficient criterion for separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order $(2,2)$ and get some new mixed entangled states that can not be detected by the positive partial transpose (PPT) criterion and the realignment criterion.