2012/10/03 by D. M. Appleby, Ingemar Bengtsson, Stephen Brierley +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Algebraic and Geometric Analysis #Clifford algebra #Combinatorics #Dimension (graph theory) #Geometry #Group (periodic table) #Group representation #Linear subspace #Mathematics #Monomial #Pure mathematics #Quantum Mechanics and Applications #Quantum mechanics #Representation (politics) #Square (algebra) #quant-ph
paper · pdf · doi:10.26421/qic14.3-4-9
published as Quantum Information and Computation, Vol. 14, No. 3 & 4, 0339-0360 (2014) · 28 pages, AMS latex
arxiv created 2012/10/03 · openalex publication_date 2014/03/01 · arxiv updated 2022/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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).