# Quantum metrics from the trace on full matrix algebras

## TL;DR

This paper demonstrates that specific quantum metrics on matrix algebras of size n are positively separated in the Gromov-Hausdorff propinquity sense when n is composite, highlighting a distinction based on the algebra's size.

## Contribution

It establishes a new separation result for quantum metrics on matrix algebras, depending on whether n is prime or not.

## Key findings

- Quantum metrics on n×n matrix algebras are separated when n is not prime.
- Positive distance in Gromov-Hausdorff propinquity distinguishes composite from prime matrix sizes.
- The result advances understanding of metric geometry in noncommutative spaces.

## Abstract

We prove that, in the sense of the Gromov-Hausdorff propinquity, certain natural quantum metrics on the algebras of $n\times n$-matrices are separated by a positive distance when n is not prime.