A functorial characterization of von Neumann entropy
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We classify the von Neumann entropy as a certain concave functor from finite-dimensional non-commutative probability spaces and state-preserving *-homomorphisms to real numbers. This is made precise by first showing that the category of non-commutative probability spaces has the structure of a Grothendieck fibration with a fiberwise convex structure. The entropy difference associated to a *-homomorphism between probability spaces is shown to be a functor from this fibration to another one involving the real numbers. Furthermore, the von Neumann entropy difference is classified by a set of axioms similar to those of Baez, Fritz, and Leinster characterizing the Shannon entropy difference. The existence of disintegrations for classical probability spaces plays a crucial role in our classification.
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