The finite Heisenberg-Weyl groups in radar and communications
EURASIP Journal on Applied Signal Processing
On Weyl-Heisenberg orbits of equiangular lines
Journal of Algebraic Combinatorics: An International Journal
Equiangular lines, mutually unbiased bases, and spin models
European Journal of Combinatorics
Computing Equiangular Lines in Complex Space
Mathematical Methods in Computer Science
High-resolution radar via compressed sensing
IEEE Transactions on Signal Processing
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
Quantum Information & Computation
On the quantumness of a hilbert space
Quantum Information & Computation
Equiangular spherical codes in quantum cryptography
Quantum Information & Computation
Quantumness, generalized 2-desing and symmetric informationally complete POVM
Quantum Information & Computation
The monomial representations of the Clifford group
Quantum Information & Computation
Decoherence-Insensitive Quantum Communication by Optimal -Encoding
IEEE Transactions on Information Theory
Systems of imprimitivity for the clifford group
Quantum Information & Computation
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Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension ≤ 67. However, a proof that they exist in every finite dimension has yet to be constructed. In this paper we examine the Galois group of SICs covariant with respect to the Weyl-Heisenberg group (or WH SICs as we refer to them). The great majority (though not all) of the known examples are of this type. Scott and Grassl have noted that every known exact WH SIC is expressible in radicals (except for dimension 3 which is exceptional in this and several other respects), which means that the corresponding Galois group is solvable. They have also calculated the Galois group for most known exact examples. The purpose of this paper is to take the analysis of Scott and Grassl further. We first prove a number of theorems regarding the structure of the Galois group and the relation between it and the extended Clifford group. We then examine the Galois group for the known exact fiducials and on the basis of this we propose a list of nine conjectures concerning its structure. These conjectures represent a considerable strengthening of the theorems we have actually been able to prove. Finally we generalize the concept of an anti-unitary to the concept of a g-unitary, and show that every WH SIC fiducial is an eigenvector of a family of g-unitaries (apart from dimension 3).