Foundations of Physics

, Volume 47, Issue 8, pp 1042–1059 | Cite as

SICs and Algebraic Number Theory

  • Marcus ApplebyEmail author
  • Steven Flammia
  • Gary McConnell
  • Jon Yard


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.


SIC-POVMs Algebraic number theory Hilbert’s 12th problem 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Centre for Engineered Quantum Systems, School of PhysicsUniversity of SydneySydneyAustralia
  2. 2.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Controlled Dynamics Theory GroupImperial CollegeLondonUK
  4. 4.Institute for Quantum Computing, Dept. of Combinatorics and OptimizationUniversity of Waterloo and Perimeter Institute for Theoretical PhysicsWaterlooCanada

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