Matricial Archimedean order unit spaces and quantum correlations

TL;DR

This paper introduces a new class of matricial order unit spaces called $k$-AOU spaces, explores their categorical properties, and connects them to quantum correlations and Tsirelson's conjecture.

Contribution

It develops the theory of $k$-AOU spaces, establishes their categorical framework, and relates finite-dimensional quantum correlations to states on these spaces, offering a new perspective on Tsirelson's conjecture.

Findings

Defined $k$-AOU spaces and their morphisms.

Established functors to operator systems and CP maps.

Characterized quantum correlations via states on $k$-AOU spaces.

Abstract

We introduce a matricial analogue of an Archimedean order unit space, which we call a $k$-AOU space. We develop the category of $k$-AOU spaces and $k$-positive maps and exhibit functors from this category to the category of operator systems and completely positive maps. We also demonstrate the existence of injective envelopes and C*-envelopes in the category of $k$-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional $k$-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjecture in terms of operator systems and $k$-AOU spaces.