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arXiv:2109.11671 · Indiana University Mathematics Journal

Matricial Archimedean order unit spaces and quantum correlations

Roy Araiza, Travis B. Russell, Mark Tomforde

Published Jan 1, 2023• 5 citations
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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.

DOI
10.1512/iumj.2023.72.9576

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We introduce a matricial analogue of an Archimedean order unit space, which we call a kkk-AOU space. We develop the category of kkk-AOU spaces and kkk-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 kkk-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional kkk-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjecture in terms of operator systems and kkk-AOU spaces.
Abstract This paper is devoted to the classification of quantum systems among the quantum spaces. In the normed case we obtain a complete solution to the problem when an operator space turns out to be an operator system. The min and max quantizations of a local order are described in terms of the min and max envelopes of the related state spaces. Finally, we characterize min-max-completely positive maps between Archimedean order unit spaces and investigate entanglement breaking maps in the general setting of quantum systems. Bibliography: 34 titles.
Abstract We construct a family of operator systems and kkk-AOU spaces generated by a finite number of projections satisfying a set of linear relations. This family is universal in the sense that the map sending the generating projections to any other set of projections which satisfy the same relations is completely positive. These operator systems are constructed as inductive limits of explicitly defined operator systems. By choosing the linear relations to be the nonsignalling relations from quantum correlation theory, we obtain a hierarchy of ordered vector spaces dual to the hierarchy of quantum correlation sets. By considering another set of relations, we also find a new necessary condition for the existence of SIC-POVM’s.
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