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Bell-certified retardance polarimetry from CHSH correlators

Juan Sumaya-Martinez

Published May 7, 2026• 0 citations
Publisher

Abstract

Bell--CHSH measurements with polarization-entangled photons are usually reported only as nonlocality witnesses. Here we show that, for retardance sensing, the same coincidence data can also provide a quantitative metrological certificate. For a local birefringent phase $\phi$ encoded on one photon, we derive the classical Fisher information directly from experimentally accessible CHSH correlators $E_{xy}(\phi)$ and obtain an explicit lower bound in terms of the measured correlators and the phase slope $S'(\phi)$ of the Bell parameter. In the unbiased-marginal two-qubit regime, this leads to a dual-use result: the canonical CHSH analyzer settings are simultaneously Bell-optimal and Fisher-optimal, saturating the quantum Fisher information for retardance estimation. We further show that the certificate remains informative under visibility loss, finite counts, and moderate departure from the Bell-violation threshold, so useful retardance sensitivity can persist even when nonlocality certification is weak or absent. Relative to tomography-based entangled-photon polarimetry, the method converts a standard Bell-calibration run into a sensitivity statement with rigorous lower confidence bounds using only coincidence correlators. The scheme is implementable with standard SPDC sources, wave plates, and polarizing beam splitters.

DOI
10.1364/opticaopen.32177004

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Bell--CHSH measurements with polarization-entangled photons are usually reported only as nonlocality witnesses. Here we show that, for retardance sensing, the same coincidence data can also provide a quantitative metrological certificate. For a local birefringent phase ϕ\phiϕ encoded on one photon, we derive the classical Fisher information directly from experimentally accessible CHSH correlators Exy(ϕ)E_{xy}(\phi)Exy​(ϕ) and obtain an explicit lower bound in terms of the measured correlators and the phase slope S′(ϕ)S'(\phi)S′(ϕ) of the Bell parameter. In the unbiased-marginal two-qubit regime, this leads to a dual-use result: the canonical CHSH analyzer settings are simultaneously Bell-optimal and Fisher-optimal, saturating the quantum Fisher information for retardance estimation. We further show that the certificate remains informative under visibility loss, finite counts, and moderate departure from the Bell-violation threshold, so useful retardance sensitivity can persist even when nonlocality certification is weak or absent. Relative to tomography-based entangled-photon polarimetry, the method converts a standard Bell-calibration run into a sensitivity statement with rigorous lower confidence bounds using only coincidence correlators. The scheme is implementable with standard SPDC sources, wave plates, and polarizing beam splitters.
Bell-CHSH measurements with polarization-entangled photons are commonly used as nonlocality witnesses. Here we show that the same coincidence data can also provide a quantitative metrological certificate for optical retardance sensing. For a local birefringent phase encoded on one photon, we derive the classical Fisher information directly from experimentally accessible CHSH correlators and obtain a lower bound expressed in terms of the measured correlations and the phase derivative of the CHSH parameter. In the unbiased-marginal two-qubit regime, the canonical CHSH analyzer settings are shown to be simultaneously Bell-optimal and Fisher-optimal, saturating the quantum Fisher information for retardance estimation in the ideal case. We further analyze reduced visibility, phase noise, analyzer misalignment, and finite coincidence counts, and construct conservative lower confidence bounds that can be evaluated without quantum-state tomography. The results also clarify that Bell violation and metrological usefulness are distinct resources: nonzero Fisher information can persist below the CHSH-violation threshold. The protocol therefore enables a standard Bell-test data set to provide both a nonlocality benchmark and a statistically controlled sensitivity certificate for quantum polarimetry.
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 ϕ\phiϕ applied to one photon, we (i) express the classical Fisher information (FI) for ϕ\phiϕ entirely in terms of CHSH correlators Exy(ϕ)E_{xy}(\phi)Exy​(ϕ), and (ii) derive an explicit lower bound on FI using only the observed CHSH data and the differential response S′(ϕ)S'(\phi)S′(ϕ) 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 vvv) 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.
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.
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 ϕ\phiϕ applied to one photon, we (i) express the classical Fisher information (FI) for ϕ\phiϕ entirely in terms of CHSH correlators Exy(ϕ)E_{xy}(\phi)Exy​(ϕ), and (ii) derive an explicit lower bound on FI using only the observed CHSH data and the differential response S′(ϕ)S'(\phi)S′(ϕ) 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 vvv) 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.
Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified from continuous time-series data. Here we develop a general theoretical framework showing that Bell-CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak.
Abstract Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified from continuous time-series data. Here we develop a general theoretical framework showing that Bell–CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak.