Solving large-scale optimization problems related to Bell’s Theorem

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.

Efficient Device-Independent Entanglement Detection for Multipartite Systems

“…Indeed, for the scenarios with m ¼ 2, 3, we are able to detect nonlocality for systems of up to N ¼ 29 and N ¼ 15, respectively, thus overcoming the current limits of Ref. [32].…”

“…Nevertheless, finding a convex decomposition in the multipartite scenario is generally an intractable problem because the number of deterministic strategies grows as d mN . Already in the simplest cases in which each party measures only m ¼ 2, 3 dicothomic measurements, the best approach currently known stops at N ¼ 11 and N ¼ 7, respectively [32].…”

Qutrit witness from the Grothendieck constant of order four

“…However, we expect that the presented algorithm in combination with other powerful methods (such as Gilbert's distance algorithm) may find applications beyond XOR nonlocal games as well. Such possible tasks concern Bell nonlocality with more inputs [47,48], more outcomes [49] or genuine nonlocality in the case of multipartite settings [50,51]. Note a recent method [52] based on the Navascues-Pironio-Acin hierarchy [53], which tackles these problems in a different way.…”

On the Solution of Linear Programming Problems in the Age of Big Data

“…Gondzio [3] presents a certain class of large-scale optimization problems arising in quantum information science and related to Bell's theorem. These problems are two-level optimization problems.…”