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Characterizations of homomorphisms among unital completely positive maps

Andre Kornell

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.

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.
Paper Structure (5 sections, 17 theorems, 37 equations)

This paper contains 5 sections, 17 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Adjusted Choi matrix
  3. Heisenberg picture
  4. Schrödinger picture
  5. Adjusted entropy

Key Result

Theorem 1.1

Let $A$ and $B$ be finite-dimensional $C^*$-algebras, and let $\varphi\colon A \to B$ be a unital completely positive map. The following are equivalent:

Theorems & Definitions (50)

  • Theorem 1.1: also Corollary \ref{['Q']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 40 more