Factorizable Maps and Traces on the Universal Free Product of Matrix Algebras

Abstract

Abstract We relate factorizable quantum channels on $M_n({\mathbb{C}})$, for $n \ge 2$, via their Choi matrix, to certain matrices of correlations, which, in turn, are shown to be parametrized by traces on the unital free product $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$. Factorizable maps with a finite dimensional ancilla are parametrized by finite dimensional traces on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$, and factorizable maps that approximately factor through finite dimensional $C^\ast $-algebras are parametrized by traces in the closure of the finite dimensional ones. The latter set of traces is shown to be equal to the set of hyperlinear traces on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$. We finally show that each metrizable Choquet simplex is a face of the simplex of tracial states on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$.