On the shape of correlation matrices for unitaries

TL;DR

This paper investigates the set of correlation matrices derived from unitaries in finite-dimensional von Neumann algebras, proving it is not closed for all dimensions n ≥ 8, refining previous results for n ≥ 11.

Contribution

It improves the known dimension threshold for the non-closure of the set of correlation matrices from n ≥ 11 to n ≥ 8.

Findings

The set of correlation matrices from unitaries is not closed for n ≥ 8.

This extends previous results that established non-closure for n ≥ 11.

The result has implications for the structure of correlation matrices in finite-dimensional operator algebras.

Abstract

For a positive integer $n$, we study the collection $\mathcal{F}_{\mathrm{fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\leq i,j\leq n$, can be written as $a_{ij}=\tau(U_j^*U_i)$ for some $n$-tuple $U_1, U_2, \ldots, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal{M}$ with tracial state $\tau$. We show that $\mathcal{F}_{\mathrm{fin}}(n)$ is not closed for every $n\geq 8$. This improves a result by Musat and R{\o}rdam which states the same for $n\geq 11$.