## On tensors of factorizable quantum channels with the completely depolarizing channel.(English)Zbl 1457.81022

Summary: 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.

### MSC:

 81P47 Quantum channels, fidelity 46L07 Operator spaces and completely bounded maps 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47C15 Linear operators in $$C^*$$- or von Neumann algebras 47L07 Convex sets and cones of operators
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