Abstract Fisher-information inequalities are increasingly used as operational witnesses of nonclassical metrological behavior, yet their interpretation is often tied to special pictures such as segmented dynamics or discrete trajectories. We develop a more general causal account. Given any classical experiment described by a directed acyclic graph, conditional independences, and modular parameter dependence, the associated Fisher informations must satisfy causal Fisher-information inequalities (CFIIs). The key result is a causal-path series law: for an additive parameter transmitted along a classical path A → C → B, inverse Fisher information acts as an information resistance and adds in series. Thus, any CFII violation rules out the entire specified classical causal model class. Moreover, such a violation directly certifies metrological advantage, since it implies a precision unattainable by any model in that class. We trace this advantage to Fisher-information synergy: off-diagonal score correlations excluded by classical modularity. A coherent single-qubit rotation illustrates deterministic CFII violation, estimator-level achievability, robustness to split-optimized classical benchmarks, and chain-amplified gains. Finally, an AI-assisted adversarial finite-data test confirms certifiability under visibility loss and readout error, with optimized modular classical models reaching but never crossing the CFII boundary.
Abstract Fisher-information inequalities are increasingly used as operational witnesses of nonclassical metrological behavior, yet their interpretation is often tied to special pictures such as segmented dynamics or discrete trajectories. We develop a more general causal account. Given any classical experiment described by a directed acyclic graph, conditional independences, and modular parameter dependence, the associated Fisher informations must satisfy causal Fisher-information inequalities (CFIIs). The key result is a causal-path series law: for an additive parameter transmitted along a classical path A → C → B, inverse Fisher information acts as an information resistance and adds in series. Thus, any CFII violation rules out the entire specified classical causal model class. Moreover, such a violation directly certifies metrological advantage, since it implies a precision unattainable by any model in that class. We trace this advantage to Fisher-information synergy: off-diagonal score correlations excluded by classical modularity. A coherent single-qubit rotation illustrates deterministic CFII violation, estimator-level achievability, robustness to split-optimized classical benchmarks, and chain-amplified gains. Finally, an AI-assisted adversarial finite-data test confirms certifiability under visibility loss and readout error, with optimized modular classical models reaching but never crossing the CFII boundary.
Fisher-information inequalities have recently been used as operational witnesses of nonclassical metrological behavior, but their physical meaning is often tied to a particular narrative, such as, segmented dynamics or discrete trajectories. We show that a broader interpretation is available and, in fact, more natural: once an experiment is assumed to admit a classical causal model specified by a directed acyclic graph, conditional independences, and modular parameter dependence, the corresponding Fisher informations are forced to obey causal Fisher-information inequalities (CFIIs). The backbone result is a causal-path series law: for an additive causal parameter that propagates through a classical path A→C→B, the inverse Fisher information behaves as an information resistance and must add in series. Consequently, any CFII violation is a rigorous falsification of the entire classical causal model class. We then show that the violation is automatically a metrological resource certificate, because it implies a precision that no member of the classical causal class can attain. The gain mechanism is identified as Fisher-information synergy, i.e. off-diagonal score correlations that classical modularity forbids. A single-qubit coherent-rotation example demonstrates the deterministic CFII violation, estimator-level achievability of the resulting gain, robustness against split-optimized classical benchmarks, and a chain-amplified advantage in long causal decompositions. Finally, an AI-assisted adversarial finite-data stress test shows that the witness remains certifiable under the realistic visibility loss and readout error, while optimized modular classical causal models saturate but do not cross the CFII frontier.
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.
Claims of quantum advantage should remain robust even when classical strategies have access to side information correlated with the benchmark under evaluation, just as Bell certification must account for measurement dependence. We formalize such correlations as benchmark dependence, a task-level generalization of measurement dependence. For every bounded-reward task, we show that the optimal benchmark-dependent classical score obeys Sη≤min{1,Scl+η}, and construct a family of tasks that saturates this bound, showing that the linear dependence on η is tight without further assumptions. For repeated product tasks with roundwise dependence, we obtain the stronger multiplicative bound Sη(n)≤(ωc+η)n, and extend the framework to finite-sample data, mutual-information constraints, multipartite tasks, and correlations distributed along a causal path. Applying these results to aggregated IBM hardware data, we obtain positive raw-count cycle-product certificates of 0.0812 for CHSH and 0.2178 for Mermin--GHZ, while the nine-context magic-square construction remains uncertified; readout-mitigated values are reported separately as sensitivity estimates. We also analyze a non-Bell quantum-kernel benchmark, where a label-construction variable has measured conditional dependence ηλ(Y)=0.5, above the threshold ηreq=0.375, required to close the reported score gap, and yields perfect classical classification. The framework therefore converts a quantum--classical score separation into a quantitative lower bound on the benchmark-correlated classical information required to explain the score separation.
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.
Quantum-enhanced sensing is commonly benchmarked using the quantum Fisher information (QFI), often interpreted as a direct indicator of achievable precision. However, such an interpreta- tion is only justified within a fully specified inference framework that consistently incorporates state preparation, measurement design, resource accounting, estimator construction, prior information, and finite data effects. Here we develop an operational, end-to-end framework for quantum sensing under finite resources and identify general principles required for meaningful performance assess- ment. A central conceptual point is that the relevant unit of estimation is not a single detection event, but the complete data set required to construct a consistent estimator. Rather than relying on abstract statistical formalism, we analyze a set of paradigmatic and experimentally relevant ex- amples that expose common pitfalls in widely discussed quantum sensing strategies. These include quantum interferometry, sensing with squeezed states, and criticality-inspired protocols. Our results clarify when large Fisher information or nonclassical resources translate into genuine metrological advantage, and when they do not. They further provide a practical methodology for designing and evaluating quantum sensing protocols under realistic experimental constraints.
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in N identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.