# On non-commutative operator graphs generated by reducible unitary representation of the Heisenberg-Weyl group

## TL;DR

This paper introduces a non-commutative operator graph derived from a reducible unitary representation of the Heisenberg-Weyl group, revealing spectral projections as quantum error-correcting codes based on entangled vectors.

## Contribution

It constructs a new class of operator graphs from the Heisenberg-Weyl group and identifies spectral projections as quantum error-correcting codes.

## Key findings

- Spectral projections serve as anticliques (quantum error-correcting codes).
- Codes are linear envelopes of entangled vectors.
- The approach links group representations with quantum error correction.

## Abstract

We consider a reducible unitary representation of Heisenberg-Weyl group in a tensor product of two Hilbert spaces. A non-commutative operator graph generated by this representation is introduced. It is shown that spectral projections of unitaries in the representation are anticliques (quantum error-correcting codes) for this graph. The obtained codes are appeared to be linear envelopes of entangled vectors.