Investigating a Class of $2\otimes2\otimes d$ Chessboard Density Matrices via Linear and Non-linear Entanglement Witnesses Constructed by Exact Convex Optimization

TL;DR

This paper develops a method using convex optimization to construct both linear and non-linear entanglement witnesses for a specific class of $2 ext{x}2 ext{x}d$ chessboard density matrices, enabling improved entanglement detection.

Contribution

It introduces a convex optimization framework to precisely construct entanglement witnesses for complex three-partite quantum states, including their optimality and decomposability analysis.

Findings

Successfully constructed various linear and non-linear entanglement witnesses.

Analytical and numerical detection of entanglement in the studied density matrices.

Identified optimal and decomposable entanglement witnesses for the class of states.

Abstract

Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically. {\bf Keywords: chessboard density matrices, optimal non-linear entanglement witnesses, convex optimization} {\bf PACs Index: 03.65.Ud}