On tensors of factorizable quantum channels with the completely depolarizing channel

TL;DR

This paper investigates the conditions under which quantum channels can be exactly factored through matrix algebras, focusing on channels combined with the completely depolarizing channel and those with finite-dimensional von Neumann algebra factorizations.

Contribution

It establishes new criteria for the factorizability of quantum channels involving tensor products with the depolarizing channel and finite-dimensional von Neumann algebras.

Findings

Factorization of tensor products with the depolarizing channel implies exact factorization of the original channel.

Channels with factorizations through finite-dimensional von Neumann algebras also admit factorizations through matrix algebras.

Results are proven for channels with rational convex combinations of automorphisms and states.

Abstract

In this paper, we obtain results for factorizability of quantum channels. Firstly, we prove that if a tensor $T\otimes S_k$ of a quantum channel $T$ on $M_n(\mathbb{C})$ with the completely depolarizing channel $S_k$ is written as a convex combination of automorphisms on the matrix algebra $M_n(\mathbb{C})\otimes M_k(\mathbb{C})$ with rational coefficients, then the quantum channel $T$ has an exact factorization through some matrix algebra with the normalized trace. Next, we prove that if a quantum channel has an exact factorization through a finite dimensional von Neumann algebra with a convex combination of normal faithful tracial states with rational coefficients, then it also has an exact factorization through some matrix algebra with the normalized trace.