Characterizations of homomorphisms among unital completely positive maps

• URL: http://arxiv.org/abs/2403.07229v2
• Date: Sun, 19 Jan 2025 01:51:59 GMT
• Title: Characterizations of homomorphisms among unital completely positive maps
• Authors: Andre Kornell,
• Abstract summary: We prove that a unital completely positive map between finite-dimensional C*algebras is a homomorphism if only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about maps between finite sets.
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• Abstract: We prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C*-algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C*-algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.

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