Operator Systems Generated by Projections
Roy Araiza, Travis Russell
TL;DR
The paper develops universal operator systems generated by finitely many projections under linear relations, constructed as inductive limits and endowed with universal CP properties. By selecting relations that encode nonsignalling constraints, it builds a dual hierarchy of k-AOU spaces that mirrors the standard quantum correlation sets, offering a potential approach to Connes’ embedding problem. It also derives new necessary conditions for the existence of SIC-POVMs via universal d-minimal operator systems and formulates a parallel route to understand mutually unbiased bases. Overall, the work provides a unifying operator-system framework to study quantum correlations, SIC-POVMs, and related information-theoretic structures.
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.