On non-commutative operator graphs generated by covariant resolutions of identity

TL;DR

This paper investigates non-commutative operator graphs generated by covariant resolutions of identity, focusing on identifying error-correcting codes (anticliques) and their structure under group symmetries, especially the circle group.

Contribution

It introduces a method to find anticliques in covariant operator graphs and reveals their tensor product and entangled vector structure.

Findings

Identified orthogonal projections as anticliques in covariant operator graphs.

Established a tensor product structure for the representation space.

Demonstrated that anticliques can be generated by entangled vectors.

Abstract

We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.