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

M. A. Jafarizadeh, K. Aghayar+1 more|arXiv (Cornell University)|Jan 28, 2009

Quantum Information and Cryptography2 references4 citations

TL;DR

This paper presents a general convex optimization framework for constructing both linear and nonlinear entanglement witnesses (EWs) in arbitrary N-partite quantum systems. By mapping separable states to a feasible region (FR), the method derives linear EWs from polygonal FRs and nonlinear EWs as envelopes of linear EWs for curved FRs, enabling systematic detection of entanglement in complex states like W-states and GHZ-W states.

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

Motivation & Objective

• To develop a universal algorithm for generating entanglement witnesses (EWs) applicable to any N-partite quantum system.
• To address the challenge of detecting entanglement in quantum states where traditional linear EWs may fail, especially for non-decomposable or highly entangled states.
• To unify and generalize existing EW constructions by framing entanglement detection as a convex optimization problem.
• To demonstrate the method’s capability through explicit constructions of nonlinear EWs for W-states, GHZ-W states, and three-qubit systems.
• To provide a systematic way to manipulate and classify non-decomposable linear and nonlinear EWs using convex optimization principles.

Proposed method

• Define a feasible region (FR) as the image of the set of separable density matrices under the trace map $ \mathrm{Tr}(W\rho_{\text{sep}}) $, ensuring convexity due to the convex structure of separable states.
• For polygonal FRs, apply exact convex optimization to identify vertices corresponding to linear entanglement witnesses.
• For curved FRs, compute the envelope of the family of linear EWs obtained via convex optimization to define nonlinear entanglement witnesses.
• Utilize Lagrangian duality and Karush-Kuhn-Tucker (KKT) conditions to solve the convex optimization problems, ensuring global optimality and zero duality gap under strong duality.
• Apply interior-point methods for numerical solutions when analytical solutions are infeasible, leveraging the efficiency of convex optimization algorithms.
• Reapply convex optimization with additional constraints to refine EWs for specific target states, such as those in unextendible product bases or W-states.

Experimental results

Research questions

• RQ1 How can a general algorithm be designed to generate both linear and nonlinear entanglement witnesses for arbitrary N-partite quantum systems?
• RQ2 What is the role of the feasible region (FR) in transforming the entanglement detection problem into a convex optimization framework?
• RQ3 How can nonlinear entanglement witnesses be systematically derived from families of linear EWs via envelope construction?
• RQ4 Can this method detect non-decomposable entanglement in three-qubit systems, such as W-states and GHZ-W states?
• RQ5 To what extent can existing linear and nonlinear EWs in the literature be re-derived and manipulated using this unified convex optimization approach?

Key findings

• The method successfully constructs nonlinear entanglement witnesses for states in unextendible product bases, W-states, and GHZ-W states by deriving the envelope of linear EWs from curved feasible regions.
• For three-qubit systems, the framework enables classification and detection of entanglement in various states, including those with non-decomposable EWs.
• The feasible region for a given system is convex by construction, as it arises from the convex set of separable density matrices under linear trace maps.
• The method proves that strong duality holds in the convex optimization formulation, guaranteeing that KKT conditions yield globally optimal solutions with zero duality gap.
• The approach generalizes and unifies prior EW constructions, allowing systematic manipulation of non-decomposable linear and nonlinear EWs reported in earlier works.
• Numerical solutions via interior-point methods are viable when analytical solutions are intractable, ensuring broad applicability across complex quantum systems.