Maximum Bell Violations via Genetic Algorithm Search
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.