A general algorithm for manipulating non-linear and linear entanglement witnesses by using exact convex optimization

TL;DR

This paper introduces a universal convex optimization-based algorithm for constructing and manipulating linear and nonlinear entanglement witnesses, applicable to any N-partite quantum system for improved entanglement detection.

Contribution

The paper presents a novel, general framework using convex optimization to derive and manipulate both linear and nonlinear entanglement witnesses for complex quantum systems.

Findings

Successfully applied to detect entanglement in unextendible product bases, W-states, and GHZ states.

Demonstrated the ability to manipulate existing entanglement witnesses using the proposed method.

Provided examples for three-qubit systems showcasing classification and detection capabilities.

Abstract

A generic algorithm is developed to reduce the problem of obtaining linear and nonlinear entanglement witnesses of a given quantum system, to convex optimization problem. This approach is completely general and can be applied for the entanglement detection of any N-partite quantum system. For this purpose, a map from convex space of separable density matrices to a convex region called feasible region is defined, where by using exact convex optimization method, the linear entanglement witnesses can be obtained from polygonal shape feasible regions, while for curved shape feasible regions, envelope of the family of linear entanglement witnesses can be considered as nonlinear entanglement witnesses. This method proposes a new methodological framework within which most of previous EWs can be studied. To conclude and in order to demonstrate the capability of the proposed approach, besides providing some nonlinear witnesses for entanglement detection of density matrices in unextendible product bases, W-states, and GHZ with W-states, some further examples of three qubits systems and their classification and entanglement detection are included. Also it is explained how one can manipulate most of the non-decomposable linear and nonlinear three qubits entanglement witnesses appearing in some of the papers published by us and other authors, by the method proposed in this paper. Keywords: non-linear and linear entanglement witnesses, convex optimization. PACS number(s): 03.67.Mn, 03.65.Ud