Complexity-Aware Theory Testing from Bell Witnesses

Abstract

Bell statistical-strength analyses and complexity-based model selection are usually treated separately. Here we relate them by showing that a witness obtained from a coarse-graining of full Bell trials yields, through data processing, a lower bound on the Kullback-Leibler (KL) distance to a competitor class in terms of the induced witness distribution. For binary Bell-game witnesses this reduces to a Bernoulli bound, and in the CHSH scenario the local image collapses to a single threshold, giving the closed-form expression D_KL(Bern(omega) || Bern(3/4)) under uniform inputs, with a corresponding extension to known nonuniform designs. A finite-sample Hoeffding argument gives a lower confidence bound under independent trials. We also include a non-CHSH example based on the three-party Mermin-GHZ game. Because the bound is measured in bits per trial, it can be compared directly with an MDL/BIC-type complexity penalty and thereby yields a conservative crossover criterion for when a more expressive competitor becomes worthwhile. For the reproducible four-photon data of Wang et al., the witness certifies a positive information gap against locality, while a full-table comparison across local, no-signaling, saturated, and two compact nonlocal families favors low-dimensional nonlocal descriptions once complexity is charged. A four-parameter unbiased-correlator control shows that the data support compact nonlocality over locality, while only weakly distinguishing the specific cosine structure of the two-parameter model; an AIC comparison instead favors broader nonlocal controls. We also report witness-based benchmarks from additional published CHSH experiments and discuss the interpretational scope of BIC for constrained or non-regular model classes.