Solving Large-Scale Optimization Problems Related to Bell's Theorem

TL;DR

This paper introduces an improved optimization method using a matrix-free interior point approach to efficiently test the existence of local realistic models for quantum correlations, enabling analysis of larger problems in quantum physics.

Contribution

The paper develops and demonstrates a new implementation of the optimization method using a matrix-free interior point technique, significantly enhancing performance over traditional simplex-based methods.

Findings

Matrix-free interior point method outperforms simplex method in large-scale problems

Enables computation of noise resistance for quantum correlations previously infeasible

Provides extensive data and analysis for quantum non-classicality tests

Abstract

Impossibility of finding local realistic models for quantum correlations due to entanglement is an important fact in foundations of quantum physics, gaining now new applications in quantum information theory. We present an in-depth description of a method of testing the existence of such models, which involves two levels of optimization: a higher-level non-linear task and a lower-level linear programming (LP) task. The article compares the performances of the existing implementation of the method, where the LPs are solved with the simplex method, and our new implementation, where the LPs are solved with a matrix-free interior point method. We describe in detail how the latter can be applied to our problem, discuss the basic scenario and possible improvements and how they impact on overall performance. Significant performance advantage of the matrix-free interior point method over the simplex method is confirmed by extensive computational results. The new method is able to solve problems which are orders of magnitude larger. Consequently, the noise resistance of the non-classicality of correlations of several types of quantum states, which has never been computed before, can now be efficiently determined. An extensive set of data in the form of tables and graphics is presented and discussed. The article is intended for all audiences, no quantum-mechanical background is necessary.