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.


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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[1] C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), no. 4, 520–546. · Zbl 1106.46047
[2] M. D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), no. 3, 285–290. · Zbl 0327.15018
[3] U. Haagerup and M. Musat, Factorization and dilation problems for completely positive maps on von Neumann algebras, Comm. Math. Phys. 303 (2011), no. 2, 555–594. · Zbl 1220.46044
[4] U. Haagerup and M. Musat, An asymptotic property of factorizable completely positive maps and the Connes embedding problem, Comm. Math. Phys. 338 (2015), no. 2, 721–752. · Zbl 1335.46059
[5] B. Kümmerer, Markov dilations on the \(2×2\) matrices, In Operator algebras and their connections with topology and ergodic theory (Busteni, 1983), Lect. Notes in Math. 1132, Berlin, Springer, 1985, 312–323.
[6] E. Ricard, A Markov dilation for self-adjoint Schur multipliers, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4365–4372. · Zbl 1166.46037
[7] J. A. Smolin, F. Verstraete, and A. Winter, Entanglement of assistance and multipartite state distillation, Phys. Rev. A (3) 72 (2005), 0523171–10.
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