# On tensors of factorizable quantum channels with the completely depolarizing channel

Document Type: Original Article

Author

Department of Mathematics Hokkaido University, Japan

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.

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### Histroty

• Receive Date: 29 March 2018
• Revise Date: 21 May 2018
• Accept Date: 24 May 2018