Matrix N-dilations of quantum channels

TL;DR

This paper proves that certain unital quantum channels derived from automorphisms of matrix algebras can be extended to finite automorphisms on larger systems, allowing repeated channel applications to be viewed as automorphic evolutions.

Contribution

It introduces the concept of finite N-dilations for unital quantum channels obtained from automorphisms, enabling a new perspective on their repeated applications.

Findings

Existence of finite matrix N-dilations for these channels.

Repeated applications correspond to automorphic evolution on larger systems.

Provides a framework for understanding quantum channel dynamics as automorphisms.

Abstract

We study unital quantum channels which are obtained via partial trace of a $*$-automorphism of a finite unital matrix $*$-algebra. We prove that any such channel, $q$, on a unital matrix $*$-algebra, $\mathcal{A}$, admits a finite matrix $N-$dilation, $\alpha _N$, for any natural number N. Namely, $\alpha _N$ is a $*$-automorphism of a larger bi-partite matrix algebra $\mathcal{A} \otimes \mathcal{B}$ so that partial trace of $M$-fold self-compositions of $\alpha _N$ yield the $M$-fold self-compositions of the original quantum channel, for any $1\leq M \leq N$. This demonstrates that repeated applications of the channel can be viewed as $*$-automorphic time evolution of a larger finite quantum system.