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SICs and Algebraic Number Theory

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Abstract

We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.

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Notes

  1. This is not the sense of the word as it is used in quantum field theory, to refer to a symmetry relating bosons and fermions, or in what is usually meant by supersymmetric quantum mechanics.

  2. There are actually four ray-class fields over \(\mathbb {Q}(\sqrt{D})\) whose conductors have finite part \(d'\). The SIC field \(\mathbb {E}\) generated by a minimal multiplet is the largest of these fields; specifically, the one with ramification allowed at both infinite places. The other three are subfields of \(\mathbb {E}\), and they also play a role in the theory. In particular, the field \(\mathbb {E}_1\) defined earlier is a ray-class field.

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Acknowledgements

We are grateful to John Coates, Brian Conrad, Steve Donnelly, James McKee, Andrew Scott, and Chris Smyth for many useful comments and discussions. This research was supported in part by the Australian Research Council via EQuS project number CE11001013, and in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. SF acknowledges support from an Australian Research Council Future Fellowship FT130101744 and JY acknowledges support from National Science Foundation Grant No. 116143.

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Authors and Affiliations

  1. Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia

    Marcus Appleby & Steven Flammia

  2. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA

    Steven Flammia

  3. Controlled Dynamics Theory Group, Imperial College, London, UK

    Gary McConnell

  4. Institute for Quantum Computing, Dept. of Combinatorics and Optimization, University of Waterloo and Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Jon Yard

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Correspondence to Marcus Appleby.

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Appleby, M., Flammia, S., McConnell, G. et al. SICs and Algebraic Number Theory. Found Phys 47, 1042–1059 (2017). https://doi.org/10.1007/s10701-017-0090-7

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