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arXiv:2603.12573 · Physical review. E

Pointwise mutual information bounded by stochastic Fisher information

Pedro B. Melo

quant-phcond-mat.stat-mechSubmitted Mar 13, 2026• Announced Mar 16, 2026• 0 citations• Revised Aug 22, 2026 (v3)
v1 · Mar 13, 2026v2 · May 20, 2026v3 · Aug 22, 2026
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Abstract

How much information about an unknown parameter can be assigned to one realized outcome, and how is that information constrained by the sensitivity of the same outcome? We derive upper bounds on the pointwise mutual information in terms of the stochastic Fisher information, both for parameters in a bounded interval and for sufficiently regular differentiable priors, and state the corresponding saturation conditions. Averaging the pointwise inequality and applying Jensen's inequality twice yields the known mutual-information--Fisher-information bound, while making explicit the outcome dependence discarded by the ensemble description. We illustrate the classical result using the instantaneous position of an overdamped Langevin particle as the observed outcome. For a fixed, parameter-independent quantum measurement, the stochastic Fisher information is further bounded by the conditional quantum Fisher information (CQFI), whose probability-weighted mean is the quantum Fisher information. A thermal-qubit field-sensing example evaluates this finite-prior bound deterministically from exact $2\times2$ matrices and, separately, decomposes the projected SLD second moment into population, coherent, and interference contributions. The interference contribution can be negative for an individual probe direction and has zero unweighted trace, but it is not itself an operational measure of readout quality. Neither the CQFI nor its mean selects a measurement basis: the latter is measurement-independent, whereas the relevant basis-dependent local objective is the classical Fisher information of the chosen measurement.

Journal reference
Phys. Rev. E 114, 044141 (2026)
DOI
10.1103/r71y-m4ml
Comments
16 pages, 2 figure

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How much information about an unknown parameter can be assigned to one realized outcome, and how is that information constrained by the sensitivity of the same outcome? We derive upper bounds on the pointwise mutual information in terms of the stochastic Fisher information, both for parameters in a bounded interval and for sufficiently regular differentiable priors, and state the corresponding saturation conditions. Averaging the pointwise inequality and applying Jensen's inequality twice yields the known mutual-information--Fisher-information bound, while making explicit the outcome dependence discarded by the ensemble description. We illustrate the classical result using the instantaneous position of an overdamped Langevin particle as the observed outcome. For a fixed, parameter-independent quantum measurement, the stochastic Fisher information is further bounded by the conditional quantum Fisher information (CQFI), whose probability-weighted mean is the quantum Fisher information. A thermal-qubit field-sensing example evaluates this finite-prior bound deterministically from exact 2×22\times22×2 matrices and, separately, decomposes the projected SLD second moment into population, coherent, and interference contributions. The interference contribution can be negative for an individual probe direction and has zero unweighted trace, but it is not itself an operational measure of readout quality. Neither the CQFI nor its mean selects a measurement basis: the latter is measurement-independent, whereas the relevant basis-dependent local objective is the classical Fisher information of the chosen measurement.
In quantum metrology, precision is typically characterized by an ensemble-averaged quantity, the quantum Fisher information (QFI), which averages over the fluctuations of individual measurement records. Here we introduce the conditional quantum Fisher information (CQFI), a trajectory-level version of the QFI that generalizes the classical stochastic Fisher information to the quantum domain. Defined through the symmetric logarithmic derivative and conditioned on a measurement outcome, the CQFI is a random variable whose average recovers the QFI. Using it, we derive a trajectory-level quantum speed limit, illustrated by the quantum-jump unraveling of a driven thermal qubit. Moreover, the CQFI decomposes into incoherent (population) and coherent (basis-rotation) contributions, together with an interference cross-term. This cross-term vanishes on average but can take negative values along single trajectories, providing a local witness of destructive interference between classical and quantum information channels.
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We derive a general upper bound to mutual information in terms of the Fisher information. The bound may be further used to derive a lower bound for the Bayesian quadratic cost. These two provide alternatives to other inequalities in the literature (e.g., the van Trees inequality) that are useful also for cases where the latter ones give trivial bounds. We then generalize them to the quantum case, where they bound the Holevo information in terms of the quantum Fisher information. We illustrate the usefulness of our bounds with a case study in quantum phase estimation. Here, they allow us to adapt to mutual information (useful for global strategies where the prior plays an important role), the known and highly nontrivial bounds for the Fisher information in the presence of noise. The results are also useful in the context of quantum communication, both for continuous and discrete alphabets.
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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.
We determine the exact local precision limits for single-copy estimation of smooth multiparameter quantum statistical models contained in spin factors, a class of matrix Jordan algebras whose state spaces generalize the qubit Bloch ball. At any parameter point where the symmetric logarithmic derivative (SLD) Fisher information is positive definite, we characterize the entire attainable classical Fisher-information region over all finite-outcome positive-operator-valued measurements. After SLD normalization, this region consists exactly of the real symmetric positive semidefinite matrices with trace at most one, independently of the ambient Hilbert-space dimension. This yields a sharp weighted covariance bound for every positive definite weight, attained by an explicit locally unbiased estimator based on randomized spectral measurements of SLD directions. The proof combines a statistics-preserving positive projection onto the spin factor with the two-eigenvalue structure of its effects, revealing the Jordan-algebraic origin of the tradeoff. The result extends the qubit information tradeoff to models with more than three parameters. Since the optimal measurement depends on the unknown parameter, we simulate an adaptive scheme for a five-parameter model on a four-dimensional Hilbert space and observe performance close to the optimal local benchmark.