# Factorizable maps and traces on the universal free product of matrix algebras

## TL;DR

This paper explores the relationship between factorizable quantum channels on matrix algebras and traces on their free product, revealing new parametrizations and geometric structures of trace spaces.

## Contribution

It establishes a connection between factorizable maps and traces on the free product of matrix algebras, including finite dimensional and hyperlinear traces, and describes the geometric structure of the trace space.

## Key findings

- Factorizable maps are parametrized by traces on the free product of matrix algebras.
- Finite dimensional factorizable maps correspond to finite dimensional traces.
- The set of hyperlinear traces equals the closure of finite dimensional traces.

## Abstract

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