Characterizations of homomorphisms among unital completely positive maps

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.