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Operator Algebra Generalization of a Theorem of Watrous and Mixed Unitary Quantum Channels

David W Kribs, Jeremy Levick, Rajesh Pereira, Mizanur Rahaman

Abstract

We establish an operator algebra generalization of Watrous' theorem \cite{watrous2009} on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

Operator Algebra Generalization of a Theorem of Watrous and Mixed Unitary Quantum Channels

Abstract

We establish an operator algebra generalization of Watrous' theorem \cite{watrous2009} on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.
Paper Structure (11 sections, 26 theorems, 81 equations)

This paper contains 11 sections, 26 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Formulation of the Conjecture
  5. Proof of Main Result
  6. Application: All Unital Channels are Eventually Mixed Unitary
  7. Asymptotic Result for Primitive Unital Channels
  8. Irreducible Channels and Peripheral Eigenvalues
  9. The Case of the Diagonal Algebra: Correlation Matrices and Schur Product Channels
  10. Group Theory Approach
  11. Acknowledgements

Key Result

Theorem 2

Let $\Phi: M_d \rightarrow M_d$ be a unital quantum channel. Then for $0 \leq p \leq 1/ (d^2-1)$, the convex combination of maps given by is a mixed unitary channel.

Theorems & Definitions (53)

  • Definition 1
  • Theorem 2
  • Example 3
  • Theorem 4
  • Lemma 5
  • proof
  • Corollary 6
  • Corollary 7
  • proof
  • Lemma 8
  • ...and 43 more