Inequalities Detecting Quantum Entanglement for $2\otimes d$ Systems

TL;DR

This paper introduces a set of inequalities based on local observables to detect quantum entanglement in $2\otimes d$ systems, providing both necessary and sufficient conditions for certain cases and an experimental detection method.

Contribution

It develops new inequalities for entanglement detection in $2\otimes d$ systems, extending the criteria to mixed states and offering an experimentally feasible approach.

Findings

Inequalities are necessary and sufficient for $2\otimes 2$ and $2\otimes 3$ systems.

The inequalities detect all non-positive partial transpose entangled states.

They provide an experimental method for entanglement detection.

Abstract

We present a set of inequalities for detecting quantum entanglement of $2\otimes d$ quantum states. For $2\otimes 2$ and $2\otimes 3$ systems, the inequalities give rise to sufficient and necessary separability conditions for both pure and mixed states. For the case of $d>3$, these inequalities are necessary conditions for separability, which detect all entangled states that are not positive under partial transposition and even some entangled states with positive partial transposition. These inequalities are given by mean values of local observables and present an experimental way of detecting the quantum entanglement of $2\otimes d$ quantum states and even multi-qubit pure states.