Fisher-Information Certification from Bell Statistics: CHSH Bounds, Information Loss, and Finite-Data Guarantees

Abstract

Bell-test data are typically used to certify nonlocal correlations, while quantum metrology is concerned with the sensitivity of measurement statistics to unknown parameters. Here we connect these tasks at the level of experimentally accessible Bell statistics. Using standard moment-based Fisher-information inequalities, we derive lower bounds on the classical Fisher information directly from Bell scores and their parameter dependence, without requiring full state tomography. For CHSH measurements we establish a hierarchy between the information contained in the full outcome record, the set of individual correlators, and the compressed Bell parameter, thereby quantifying the metrological cost of data compression. In polarization-phase sensing, this hierarchy reveals a Bell-metrology tradeoff: the point of maximal CHSH violation need not maximize, and can even suppress, the Fisher information carried by the Bell score itself. We further derive finite-sample lower-confidence bounds from coincidence counts and analyze the effects of biased marginals, accidental coincidences, visibility loss, and multipair emission. The framework provides a practical route for extracting metrological guarantees from Bell-type experiments using only standard correlation measurements.