# Operator Systems Generated by Projections

## TL;DR

The paper constructs universal operator systems generated by projections satisfying specific relations, linking them to quantum correlation hierarchies and SIC-POVM existence conditions.

## Contribution

It introduces a universal construction of operator systems from projections satisfying linear relations, connecting to quantum correlation sets and SIC-POVM criteria.

## Key findings

- Constructed operator systems as inductive limits.
- Linked operator systems to quantum correlation hierarchies.
- Derived a new necessary condition for SIC-POVM existence.

## 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.