Sporadic SICs and the Normed Division Algebras

@article{Stacey2016SporadicSA,
  title={Sporadic SICs and the Normed Division Algebras},
  author={Blake C. Stacey},
  journal={Foundations of Physics},
  year={2016},
  volume={47},
  pages={1060-1064}
}
  • Blake C. Stacey
  • Published 4 May 2016
  • Mathematics, Physics
  • Foundations of Physics
Symmetric informationally complete quantum measurements, or SICs, are mathematically intriguing structures, which in practice have turned out to exhibit even more symmetry than their definition requires. Recently, Zhu classified all the SICs whose symmetry groups act doubly transitively. I show that lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups. 
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References

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Symmetric informationally complete measurements (SICs in short) are highly symmetric structures in the Hilbert space. They possess many nice properties which render them an ideal candidate forExpand
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TLDR
It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim. Expand
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We report on a new computer study into the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and areExpand
Symmetric informationally complete positive-operator-valued measures: A new computer study
We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are theExpand
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Complex projective t-designs, particularly SICs and full sets of MUBs, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They includeExpand
Constructing exact symmetric informationally complete measurements from numerical solutions
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjecturesExpand
SICs: Extending the list of solutions
Zauner's conjecture asserts that $d^2$ equiangular lines exist in all $d$ complex dimensions. In quantum theory, the $d^2$ lines are dubbed a SIC, as they define a favoured standard informationallyExpand
Introducing the Qplex: a novel arena for quantum theory
Abstract We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of quantum information theory, and in particularExpand
Informational power of the Hoggar SIC-POVM
Among positive operator valued measures (POVMs) representing general quantum measurements, symmetric informationally complete (SIC) POVMs, called by Christopher Fuchs ‘mysterious entities’, play aExpand
Book Review: On quaternions and octonions: Their geometry, arithmetic, and symmetry
Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H) and the octonions (O). The first two areExpand
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