-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).

DOI: 10.26421/qic14.3-4-9

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Systems of imprimitivity for the Clifford group

Appleby, D.M., Bengtsson, I., Brierley, S. +3 more

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).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).

Mar 1, 201414 citations

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The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, every known SIC-POVM fiducial vector is an eigenstate of an order 3 unitary in the Clifford group. I show that every order 3 unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and construct a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with numerical evidence.The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, every known SIC-POVM fiducial vector is an eigenstate of an order 3 unitary in the Clifford group. I show that every order 3 unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and construct a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with numerical evidence.

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A remarkable representation of the Clifford group

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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.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.

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D. M. Appleby

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arXiv:quant-ph/0412001Journal of Mathematical PhysicsApr 15, 2005304 citations

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