Galois automorphisms of a symmetric measurement
(pp0672-0720)
D.M.
Appleby, Hulya Yadsan-Appleby, and Gerhard Zauner
doi:
https://doi.org/10.26421/QIC13.7-8-8
Abstracts:
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).
Key words:
Quantum Measurement, Symmetric Informationally Complete
Positive Operator Valued Measure, Clifford Group, Galois Theory |