Operator Systems Generated by Projections

Abstract

We construct a family of operator systems and $k$-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 a SIC-POVM.

To appear in International Mathematics Research Notices

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• The SIC Question: History and State of Play
• Connes' embedding problem and Tsirelson's problem
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Cited by in corpus (1)

• Operator system characterizations of SIC-POVMs and mutually unbiased bases