Systems of imprimitivity for the clifford group

Authors:
D. M. Appleby;Ingemar Bengtsson;Stephen Brierley;Åsa Ericsson;Markus Grassl;Jan-Åke Larsson
Affiliations:
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada;Stockholms Universitet, Stockholm, Sweden;Heilbronn Institute for Mathematical Research, Department of Mathematics, Bristol, UK;Matematiska institutionen, Linköpings Universitet, Linköping, Sweden;Centre for Quantum Technologies, National University of Singapore, Singapore;Institutionen för Systemteknik, Linköpings Universitet, Linköping, Sweden
Venue:
Quantum Information & Computation
Year:
2014

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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).