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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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