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arXiv:cond-mat/0501322 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Information in statistical physics

Roger Balian

Published Apr 26, 2005• 40 citations
arXivHTMLPDF
DOI
10.1016/j.shpsb.2005.02.001

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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.
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
The locus of probability flow in Quantum Mechanics and information is explored. We explore loops, loop sequences and loop surfaces in the statistical geodesics. Having known about the loop character of the statistical geodesics in probability space, we discuss wider physical aspects of consequences of the geometry of the probabilities in the form of loop sequences and loop surfaces. These are interpreted as the locus of probability flow. A deeper philosophical observation reveals that these loops, loop sequences and loop surfaces are the information grids. These investigations in many ways could be significant not only in the research in Statistics, but also in Mathematics, and Physics in general, and entropy, gravity and information theory in specific.