Unitary Correlation Sets

TL;DR

This paper explores the properties of unitary correlation sets and their relation to Connes' embedding problem, establishing equivalences with the equality of certain sets and norms, thus connecting operator algebras to fundamental questions in quantum information.

Contribution

It demonstrates the equivalence of Connes' embedding problem with the equality of smaller unitary correlation sets and cross norms on matrix tensor products, providing new perspectives on the problem.

Findings

Connes' embedding problem is equivalent to equality of two smaller unitary correlation sets.

Connes' embedding problem is equivalent to equality of two cross norms on $M_n imes M_n$ for all $n ",

Abstract

The unitary correlation sets defined by the first author in conjunction with tensor products of $\mathcal{U}_{nc}(n)$ are further studied. We show that Connes' embedding problem is equivalent to deciding whether or not two smaller versions of the unitary correlation sets are equal. Moreover, we obtain the result that Connes' embedding problem is equivalent to deciding whether or not two cross norms on $M_n \otimes M_n$ are equal for all $n \geq 2$.