We derive general upper bounds to pointwise mutual information in terms of stochastic Fisher information and show these bounds average to known results in the literature for bounds to mutual information in terms of Fisher information. These results deepen the connection between information-theoretical quantities and are shown to hold in general cases. We test the bounds in classical systems and provide a quantum generalization. Our results are useful for stochastic dynamics and quantum sensing, establishing fundamental theoretical limits for information extraction in single experimental realizations.
We derive general upper bounds to pointwise mutual information in terms of stochastic Fisher information and show these bounds average to known results in the literature for bounds to mutual information in terms of Fisher information. These results deepen the connection between information-theoretical quantities and are shown to hold in general cases. We test the bounds in classical systems and provide a quantum generalization. Our results are useful for stochastic dynamics and quantum sensing, establishing fundamental theoretical limits for information extraction in single experimental realizations.
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.
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.
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.
Open questions from Sarovar and Milburn (2006 J.Phys. A: Math. Gen. 39 8487) are answered. Sarovar and Milburn derived a convenient upper bound for the Fisher information of a one-parameter quantum channel. They showed that for quasi-classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator-valued measures (POVMs) attaining this bound. They asked (i) whether their bound is attainable more generally, (ii) whether explicit expressions for optimal POVMs can be derived from the attainability condition. We show that the symmetric logarithmic derivative (SLD) quantum information is less than or equal to the SM bound, i.e.\ H(θ)≤CΥ(θ) and we find conditions for equality. As the Fisher information is less than or equal to the SLD quantum information, i.e. FM(θ)≤H(θ), we can deduce when equality holds in FM(θ)≤CΥ(θ). Equality does not hold for all channels. As a consequence, the attainability condition cannot be used to test for optimal POVMs for all channels. These results are extended to multi-parameter channels.
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.
Recently various ways to quantify coherence of a quantum state have been proposed. In this paper, several reliable quantum coherence measures based on quantum Fisher information (QFI) are presented. For pure states, this coherence measure is equivalent to that based on skew information, while for mixed states, we first define two natural coherence measures via QFI and find that the two definitions are exactly identical. By virtual of the special structure of qubit, we show that the mutual unbiased bases' (MUBs) qubit coherence is constrained only by the length of the Bloch vector. Then we provide upper bounds of the coherence measure with respect to any given basis and lower bounds in the case of a set of MUBs for both pure states and mixed states of arbitrary dimension. Finally, we investigate the connection between our coherence measures and quantum metrology, thereby providing an operational meaning for these coherence measures.