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Linear dependencies in Weyl–Heisenberg orbits

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Abstract

Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl–Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

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Acknowledgments

IB and KB thank Markus Grassl for taking an interest in this problem when we visited Singapore, and especially for drawing our attention to the linear dependencies in the “odd” SIC in eight dimensions. We also thank a referee for constructive comments. HBD was supported by the Natural Sciences and Engineering Research Council of Canada and by the U.S. Office of Naval Research (Grant No. N00014-09-1-0247). IB was supported by the Swedish Research Council under contract VR 621-2010-4060. DMA was supported in part by the U.S. Office of Naval Research (Grant No. N00014-09-1-0247) and by the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.

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Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Hoan Bui Dang & D. M. Appleby

  2. Physics Department, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Hoan Bui Dang

  3. Stockholms Universitet, AlbaNova, Fysikum, 106 91, Stockholm, Sweden

    Kate Blanchfield & Ingemar Bengtsson

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  1. Hoan Bui Dang
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  2. Kate Blanchfield
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  3. Ingemar Bengtsson
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  4. D. M. Appleby
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Correspondence to Kate Blanchfield.

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Dang, H.B., Blanchfield, K., Bengtsson, I. et al. Linear dependencies in Weyl–Heisenberg orbits. Quantum Inf Process 12, 3449–3475 (2013). https://doi.org/10.1007/s11128-013-0609-6

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  • DOI: https://doi.org/10.1007/s11128-013-0609-6

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