Optimal Lewenstein-Sanpera decomposition of two-qubit states using Semidefinite Programming

TL;DR

This paper develops a semidefinite programming framework to determine the optimal Lewenstein-Sanpera decomposition of two-qubit states, providing both analytical solutions for special cases and efficient numerical methods for general states.

Contribution

It introduces a unified semidefinite programming approach to derive necessary and sufficient conditions for optimal LSD of 2-qubit states, bridging gaps across different rank cases.

Findings

Derived conditions for full-rank states using Wellens-Kus equations

Established necessary and sufficient conditions for rank-3 states

Provided an analytic expression for a special class of rank-3 states

Abstract

We use the language of semidefinite programming and duality to derive necessary and sufficient conditions for the optimal Lewenstein-Sanpera Decomposition (LSD) of 2-qubit states. We first provide a simple and natural derivation of the Wellens-Kus equations for full-rank states. Then, we obtain a set of necessary and sufficient conditions for the optimal decomposition of rank-3 states. This closes the gap between the full-rank case, where optimality conditions are given by the Wellens-Kus equations, and the rank-2 case, where the optimal decomposition is analytically known. We also give an analytic expression for the optimal LSD of a special class of rank-3 states. Finally, our formulation ensures efficient numerical procedures to return the optimal LSD for any arbitrary 2-qubit state.