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.

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• Quantum foundations
• Quantum information theory