Local Fisher information quantifies snapshot distinguishability, whereas monitoredquantum-state discrimination often depends on what a later measurement recordreveals. For finite-dimensional linearized dynamics with additive Gaussian detector noise,we show that the Fisher information of the record is the finite-time, noise-weighted observabilityGramianGT =Z T0Φ(t, 0)†C(t)†N−1C(t)Φ(t, 0) dt.This gives a detector-specific dynamic Fisher metric for initial-state and parameter discrimination.In the stable time-independent case, the infinite-window limit satisfies a Lyapunovequation, motivating the term Lyapunov-Fisher metric only for that limit. A monitoredresonatorexample shows analytically how an initially invisible Liouville coordinate is transducedinto a homodyne-like record: the snapshot Fisher information is zero, but the recordFisher information builds up and directly lowers the optimal binary decision error. Theconstruction is distinct from quantum Fisher information, thermodynamic Lyapunov functionals,and persistent readout-rate operators; it is a finite-time discrimination geometryof specified measurement records.
Local Fisher information quantifies snapshot distinguishability, whereas monitoredquantum-state discrimination often depends on what a later measurement recordreveals. For finite-dimensional linearized dynamics with additive Gaussian detector noise,we show that the Fisher information of the record is the finite-time, noise-weighted observabilityGramianGT =Z T0Φ(t, 0)†C(t)†N−1C(t)Φ(t, 0) dt.This gives a detector-specific dynamic Fisher metric for initial-state and parameter discrimination.In the stable time-independent case, the infinite-window limit satisfies a Lyapunovequation, motivating the term Lyapunov-Fisher metric only for that limit. A monitoredresonatorexample shows analytically how an initially invisible Liouville coordinate is transducedinto a homodyne-like record: the snapshot Fisher information is zero, but the recordFisher information builds up and directly lowers the optimal binary decision error. Theconstruction is distinct from quantum Fisher information, thermodynamic Lyapunov functionals,and persistent readout-rate operators; it is a finite-time discrimination geometryof specified measurement records.
Understanding how thermal fluctuations modify or suppress quantum-enhanced sensing is a central problem in quantum metrology. Closely related questions arise in quantum thermodynamics, particularly regarding the relation between the work induced by a signal and the information acquired by a sensor. Here, we establish general relations among the sensitivity of thermal quantum states, as quantified by different quantum Fisher information metrics, their temperature dependence, and the work induced by a unitary signal. We further show that, at high temperature, all monotone QFI metrics coincide to leading order with a universal quantity that can be expressed as the variance of a simple observable and itself defines a lesser-known QFI metric. This emergent uniqueness is reminiscent of the uniqueness of classical Fisher information in information geometry. In continuous-variable systems, the infinite-temperature limit is finite and generically non-zero for quadratic Gaussian signals, while it diverges for higher-degree signals. Remarkably, for any purely quadratic signal generator, the infinite-temperature QFI is at least twice the zero-temperature SLD QFI or, equivalently, at least eight times the ground-state variance of the generator. Consequently, the Cramér--Rao lower bound on the estimator variance is reduced by at least a factor of two. We illustrate these results using harmonic chains of bosonic modes relevant to trapped-ion platforms and quantum field-theoretic models.
Protected and peripheral Liouvillian modes identify which components of an open quantum system can survive, but they do not determine whether those components are readable by a specified detector, syndrome channel, or feedback record. This preprint introduces a readout-rate operator for finite-dimensional semistable quantum dynamics. The operator isolates the coefficient of the linear large-window growth of the finite-time readout Gramian and provides a detector-relative criterion for persistent readout visibility. For semisimple neutral spectrum, the readout-rate operator is expressed in terms of the Riesz projectors onto the neutral frequency blocks and the experimentally specified readout map. Thus the Liouvillian spectrum identifies persistent coordinates, whereas the readout-rate operator identifies the subset of those coordinates that produces a persistent measurement record. Under additive Gaussian readout noise, the same operator becomes the classical Fisher-information-rate matrix for estimating the persistent coordinate from the specified record. The construction is intended as a diagnostic and design principle for decoherence-free subspaces, quantum memories, syndrome records, feedback architectures, and stable auxiliary transduction paths in open quantum systems. It is detector-relative and does not claim optimization over all possible quantum measurements. The main theorem applies to finite-dimensional semistable generators with semisimple neutral spectrum. Keywords: open quantum systems; semistable Liouvillian dynamics; quantum readout; observability Gramian; decoherence-free subspaces; Fisher information; quantum feedback; syndrome extraction; Riesz projectors.
We introduce a quantum Fisher information based measurement-induced nonlocality (QFI-MIN), which quantifies the maximal statistical distinguishability induced by locally invariant unitary dynamics. The proposed measure inherits desirable properties including positivity, local unitary invariance, monotonicity under local operations, and immunity to the local ancilla problem. Analytical expressions are obtained for pure states, arbitrary two-qubit states, and two-qubit X states, revealing a direct connection with entanglement for pure systems. We further establish clear operational interpretations of QFI-MIN in quantum parameter estimation, local channel discrimination, and correlation-assisted communication. Its behavior under amplitude damping, depolarizing, and generalized amplitude damping channels demonstrates robustness against environmental noise. The proposed framework provides a physically consistent and operationally meaningful quantifier of quantum correlations, linking nonlocality, quantum metrology, and quantum information processing
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
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 2×2 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.
Chentsov studied Riemannian metrics on the set of probability measures from the point of view of decision theory. He proved that up to a constant factor the Fisher information is the only metric which is monotone under stochastic transformations. The present paper deals with monotone metrics on the space of finite density matrices on the basis motivated by quantum mechanics. A characterization of those metrics is given in terms of operator monotone functions. Several concrete metrics are constructed and analyzed, in particular, instead of the uniqueness in the probabilistic case, there is a large class of monotone metrics. Some of those appeared already in the physics literature a long time ago. A limiting procedure to pure states is discussed as well.