Matricial Archimedean order unit spaces and quantum correlations

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.