Necessary and sufficient conditions for bipartite entanglement

TL;DR

This paper derives comprehensive conditions for bipartite entanglement applicable to all Hilbert spaces, introducing optimized inequalities and methods for experimental detection and identification of bound entangled states.

Contribution

It presents new necessary and sufficient entanglement criteria, including analytical solutions for certain operators and a numerical approach for general cases.

Findings

Derived analytical solutions for a class of projection operators

Proposed a numerical method for entanglement testing

Showed how to identify bound entangled states with positive partial transposition

Abstract

Necessary and sufficient conditions for bipartite entanglement are derived, which apply to arbitrary Hilbert spaces. Motivated by the concept of witnesses, optimized entanglement inequalities are formulated solely in terms of arbitrary Hermitian operators, which makes them useful for applications in experiments. The needed optimization procedure is based on a separability eigenvalue problem, whose analytical solutions are derived for a special class of projection operators. For general Hermitian operators, a numerical implementation of entanglement tests is proposed. It is also shown how to identify bound entangled states with positive partial transposition.