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.
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.
We give two upper bounds to the mutual information in arbitrary quantum estimation strategies. The first is based on some simple Fourier properties of the estimation apparatus. The second is derived using the first, but, interestingly, depends only on the Fisher information of the parameter, so it is valid even beyond quantum estimation. We illustrate the usefulness of these bounds by characterizing the quantum phase estimation algorithm in the presence of noise. In addition, for the noiseless case, we extend the analysis beyond applying the bound and we discuss the optimal entangled and adaptive strategies, clarifying inaccuracies appearing on this topic in the literature.
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.
Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.
We derive upper bounds on the quantum Fisher information in interferometry with N subsystems, e.g. two-level atoms or Gaussian modes, in the presence of arbitrarily correlated Gaussian dephasing including independent and collective dephasing. The derived upper bound enables us to analyse the Fisher information asymptotic behaviour when N→∞. Dephasing introduces random phases to subsystems dynamics, which lowers the precision of estimating the phase difference ϕ in an interferometer. The method presented uses Bayesian estimation of the random phases and eliminates dephasing noise by calculating their weighted arithmetic mean, which correponds to the phase ϕ estimated in interferometry.
Abstract The quantum Fisher information (QFI) plays a crucial role in quantum information theory and in many practical applications such as quantum metrology. However, computing the QFI is generally a computationally demanding task. In this work we analyze a lower bound on the QFI which we call the sub-quantum Fisher information (sub-QFI). The bound can be efficiently estimated on a quantum computer for an n -qubit state using 2 n qubits. The sub-QFI is based on the super-fidelity, an upper bound on Uhlmann’s fidelity. We analyze the sub-QFI in the context of unitary families, where we derive several crucial properties including its geometrical interpretation. In particular, we prove that the QFI and the sub-QFI are maximized for the same optimal state, which implies that the sub-QFI is faithful to the QFI in the sense that both quantities share the same global extrema. Based on this faithfulness, the sub-QFI acts as an efficiently computable surrogate for the QFI for quantum sensing and quantum metrology applications. Finally, we provide additional meaning to the sub-QFI as a measure of coherence, asymmetry, and purity loss.
On some interrelations of generalized q-entropies and a generalized Fisher information, including a Cramér-Rao inequality
In this communication, we describe some interrelations between generalized q-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of entropy and Fisher information. More precisely, a generalized Fisher information naturally pops up in the expression of the derivative of the Tsallis entropy. This generalized Fisher information also appears as a special case of a generalized Fisher information for estimation problems. Indeed, we derive here a new Cramér-Rao inequality for the estimation of a parameter, which involves a generalized form of Fisher information. This generalized Fisher information reduces to the standard Fisher information as a particular case. In the case of a translation parameter, the general Cramér-Rao inequality leads to an inequality for distributions which is saturated by generalized q-Gaussian distributions. These generalized q-Gaussians are important in several areas of physics and mathematics. They are known to maximize the q-entropies subject to a moment constraint. The Cramér-Rao inequality shows that the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a fixed moment. Similarly, the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a given q-entropy.