Systems of Imprimitivity for the Clifford Group

TL;DR

This paper generalizes the structure of the Clifford group's representation to arbitrary dimensions, revealing systems of imprimitivity of size related to the square-free part of the dimension, and applies this to construct SIC-POVMs in specific dimensions.

Contribution

It extends the known monomial representation of the Clifford group from perfect square dimensions to all dimensions by identifying systems of imprimitivity based on the square-free part.

Findings

Established systems of imprimitivity of size k in arbitrary dimensions

Constructed exact SIC-POVM solutions in dimensions 8, 12, and 28

Provided a method to analyze Clifford group representations in general dimensions

Abstract

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).