Complete set of unitary irreps of Discrete Heisenberg Group $HW_{2^s}$

TL;DR

This paper explicitly constructs all unitary irreducible representations of the discrete finite Heisenberg-Weyl group over a finite phase space, detailing their characters and fusion rules, with implications for quantum mechanics and computation.

Contribution

It provides a complete explicit construction of all unitary irreducible representations of the discrete Heisenberg-Weyl group $HW_{2^s}$, including characters and fusion rules.

Findings

Explicit representations of $HW_{2^s}$ constructed

Characters and fusion rules determined

Potential applications in finite quantum mechanics and quantum computing

Abstract

Following the method of induced group representations of Wigner-Mackay, the explicit construction of all the unitary irreducible representations of the discrete finite Heisenberg-Weyl group $HW_{2^s}$ over the discrete phase space lattice $Z_{2^s}$ $\otimes$ $Z_{2^s}$ is presented. We explicitly determine their characters and their fusion rules. We discuss possible physical applications for finite quantum mechanics and quantum computation.