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Physical Review A

Physicality of evolution and statistical contractivity as equivalent notions of maps

Matteo Scandi, Paolo Abiuso, Dario De Santis, Jacopo Surace

Published Jul 23, 2025• 0 citations
Publisher

Abstract

Statistical quantifiers are generically required to contract under physical maps, following the intuition that information should be lost under noisy transformations. This principle is so important in statistics that it even allows one to derive uniqueness results based on it: the Chentsov-Petz theorem identifies the Fisher information metrics as the only family, on the space of probability distributions (or density matrices), that contracts under physical maps. This construction could suggest that statistical quantifiers are a derived concept, while the only fundamental objects are the physical evolutions. The aim of this work is to disprove this belief. To this end, we prove a statement dual to the Chentsov-Petz theorem, showing that among all possible linear maps, the only ones that contract the Fisher information (including, in the quantum case, any idle ancilla) are exactly the physical ones. This result proves that, contrary to the common opinion, there is no fundamental hierarchy between physical maps and canonical statistical quantifiers, as either of them can be defined in terms of the other. © 2025 American Physical Society

DOI
10.1103/2wb3-9gsh

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Statistical quantifiers are generically required to contract under physical maps, following the intuition that information should be lost under noisy transformations. This principle is so important in statistics that it even allows one to derive uniqueness results based on it: the Chentsov-Petz theorem identifies the Fisher information metrics as the only family, on the space of probability distributions (or density matrices), that contracts under physical maps. This construction could suggest that statistical quantifiers are a derived concept, while the only fundamental objects are the physical evolutions. The aim of this work is to disprove this belief. To this end, we prove a statement dual to the Chentsov-Petz theorem, showing that among all possible linear maps, the only ones that contract the Fisher information (including, in the quantum case, any idle ancilla) are exactly the physical ones. This result proves that, contrary to the common opinion, there is no fundamental hierarchy between physical maps and canonical statistical quantifiers, as either of them can be defined in terms of the other. © 2025 American Physical Society
Statistical quantifiers are generically required to contract under physical evolutions, following the intuition that information should be lost under noisy transformations. This principle is very relevant in statistics, and it even allows to derive uniqueness results based on it: by imposing their contractivity under any physical maps, the Chentsov-Petz theorem singles out a unique family of metrics on the space of probability distributions (or density matrices) called the Fisher information metrics. This result might suggest that statistical quantifiers are a derived concept, as their very definition is based on physical maps. The aim of this work is to disprove this belief. Indeed, we present a result dual to the Chentsov-Petz theorem, proving that among all possible linear maps, the only ones that contract the Fisher information are exactly the physical ones. This result shows that, contrary to the common opinion, there is no fundamental hierarchy between physical maps and canonical statistical quantifiers, as either of them can be defined in terms of the other.
A non-isolated physical system typically loses information to its environment, and when such loss is irreversible the evolution is said to be Markovian. Non-Markovian effects are studied by monitoring how information quantifiers, such as the distance between physical states, evolve in time. Here we show that the Fisher information metric emerges as a natural object to study in this context; we fully characterize the relation between its contractivity properties and Markovianity, both from the mathematical and operational point of view. We prove, for classical dynamics, that Markovianity is equivalent to the monotonous contraction of the Fisher metric at all points of the set of states. At the same time, operational witnesses of non-Markovianity based on the dilation of the Fisher distance cannot, in general, detect all non-Markovian evolutions, unless specific physical postprocessing is applied to the dynamics. Finally, we show for the first time that non-Markovian dilations of Fisher distance between states at any time correspond to backflow of information about the initial state of the dynamics at time 0, via Bayesian retrodiction. All the presented results can be lifted to the case of quantum dynamics by considering the standard CP-divisibility framework.
The two principal/immediate influences—which we seek to interrelate here—upon the undertaking of this study are papers of Życzkowski and Słomczyński [J. Phys. A 34, 6689 (2001)] and of Petz and Sudár [J. Math. Phys. 37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudár work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudár? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys’ prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions—recently studied by Srednicki [Phys. Rev. A 71, 052107 (2005)]—formed by normalizing the volume elements of the various information metrics, is also discussed in our context.
The importance of the Fisher information metrics and its quantum generalisations is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher informations, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature and provide a cohesive treatment to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending previous works, we show that dynamical properties such as (complete) positivity, Markovianity, detailed balance, retrodictive power of evolution maps can be characterised in terms of their relation with respect to the Fisher information metrics. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.
Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sudár (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024