Maximum Bell Violations via Genetic Algorithm Search

TL;DR

This paper employs a genetic algorithm to optimize measurement settings in Bell inequality experiments, identifying quantum extremes and bounds for various matrix configurations, advancing understanding of quantum non-locality.

Contribution

It introduces a genetic algorithm approach to find measurement operators that maximize Bell violations, extending analysis to a class of weight matrices with specific structural properties.

Findings

GA finds measurement settings reaching quantum bounds for N=2 to 10.

Maximum Bell violations are between the Bell threshold and quantum bound.

Zero quantum gap matrices are characterized by row and column sum criteria.

Abstract

Bell inequality experiments measure the correlation coefficients of two spatially separated systems. In an EPR setup, at one location Alice has $N_a\geq 2$ observables $A =\{\A_j\}_1^{N_a}$ while at a second remote location Bob has $N_b \geq2 $ observables $B= \{\B_k\}_1^{N_b}$. Within this bipartite environment each real $N_a \times N_b$ weight matrix $W$ constructs a Bell operator $\widehat{S}_W$ defined by the $jk$ sum of $W_{jk}\, \A_j \otimes \B_k$. Operator $\widehat{S}_W$ has the Bell non-locality boundary given by a hidden variable norm of $W$. As the $(A,B)$ composition varies, quantum extremes arise when the $\widehat{S}_W$ operator norm has the greatest possible Bell violation. A genetic algorithm (GA) search over all $(A,B)$ is used to find examples of the Alice and Bob operators that realize quantum extremes. A class $\XX_N$ of weights of special interest is given by the square $N_a=N_b=N$ matrices having two $\pm 1$ entries in each row and column with an odd number of minus signs. The class $\XX_N$ is a natural extension of the $2 \times 2 $ CHSH family. For dimensions $N=2\sim10$ the GA search finds that both the EPR correlation matrices and the Bell operator extremes do saturate their respective quantum bounds. Maximum Bell operator expectations fall between two benchmarks: the Bell inequality threshold and the quantum bound. The difference between these benchmarks is the quantum gap. Weight matrices $W$ that have zero quantum gap are determined by a row, column sum criteria.