A remarkable representation of the Clifford group

TL;DR

This paper presents a special representation of the Clifford group that simplifies calculations for SIC POVMs and hints at a preferred tensor product structure, with implications for quantum information theory.

Contribution

It introduces a monomial phase-permutation matrix representation of the Clifford group for square-dimensional Hilbert spaces, simplifying SIC POVM computations.

Findings

Representation exists for square dimensions

Simplifies SIC POVM calculations

Suggests a preferred tensor product structure

Abstract

The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group --- the automorphism group of the Heisenberg group --- is represented by monomial phase-permutation matrices. This has a beneficial influence on the amount of calculation that must be done to find Symmetric Informationally Complete POVMs. I make some comments on the equations obeyed by the absolute values of the components of the SIC vectors, and on the fact that the representation partly suggests a preferred tensor product structure.