A2 Refereed review article in a scientific journal
Covariant mutually unbiased bases
Authors: Carmeli C, Schultz J, Toigo A
Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD
Publication year: 2016
Journal: Reviews in Mathematical Physics
Journal name in source: REVIEWS IN MATHEMATICAL PHYSICS
Journal acronym: REV MATH PHYS
Article number: ARTN 1650009
Volume: 28
Issue: 4
Number of pages: 43
ISSN: 0129-055X
eISSN: 1793-6659
DOI: https://doi.org/10.1142/S0129055X16500094
Abstract
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension 2(r) covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension 2(r) covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.
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