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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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