Dual-Use Bell Experiments for Quantum Polarimetry: Fisher-Information Certification from CHSH Data

Bell-CHSH experiments with polarization-entangled photons are routinely performed to certify nonlocal correlations. We show that the \emph{same} dichotomic coincidence data can also serve as a quantitative metrological certificate for estimating a physical polarization parameter. Focusing on a local birefringent retardance $\phi$ applied to one photon, we (i) express the classical Fisher information (FI) for $\phi$ entirely in terms of CHSH correlators $E_{xy}(\phi)$, and (ii) derive an explicit lower bound on FI using only the observed CHSH data and the differential response $S'(\phi)$ of the Bell parameter. In the ideal two-qubit, unbiased-marginals regime, we further show a \emph{dual-use optimality}: the standard CHSH measurement settings simultaneously saturate the quantum Fisher information (QFI) for retardance estimation. We analyze robustness under depolarizing noise (visibility $v$) and outline a practical SPDC-based implementation requiring only standard polarization analysis. Our results provide an operational bridge between Bell nonlocality certification and precision metrology, while clarifying that large QFI and Bell nonlocality are not universally equivalent resources.