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On tensors of factorizable quantum channels with the completely depolarizing channel

Yuki Ueda

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.

On tensors of factorizable quantum channels with the completely depolarizing channel

Abstract

In this paper, we obtain results for factorizability of quantum channels. Firstly, we prove that if a tensor of a quantum channel on with the completely depolarizing channel is written as a convex combination of automorphisms on the matrix algebra with rational coefficients, then the quantum channel 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.

Paper Structure

This paper contains 5 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Basic properties of factorizable quantum channels
  4. Proof of Theorem 1.2
  5. Proof of Theorem 1.4

Key Result

Theorem 1.2

Let $T$ be a quantum channel on $M_n(\mathbb{C})$. If there exists a positive integer $k$ such that $T\otimes S_k=\sum_{i=1}^{d(k)} c_i \text{ad}(u_i)\in$conv(Aut($M_n(\mathbb{C})\otimes M_k(\mathbb{C})$)) for some positive integer $d(k)$, unitary matrices $u_1,\cdots ,u_{d(k)}\in\mathcal{U}(M_n(\ma

Theorems & Definitions (12)

  • Theorem 1.2
  • Theorem 1.4
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • Theorem 4.1
  • proof
  • ...and 2 more