Covariant mutually unbiased bases

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 even-prime power dimension 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.

44 pages, some remarks and references added in v2

References in corpus (4)

• Permutation Symmetry Determines the Discrete Wigner Function
• Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions
• Spectra of phase point operators in odd prime dimensions and the extended Clifford group
• Nonuniqueness of phase retrieval for three fractional Fourier transforms

Cited by in corpus (4)

• Measurement uncertainty relations for discrete observables: Relative entropy formulation
• Determining quantum coherence with minimal resources
• Constructing extremal compatible quantum observables by means of two mutually unbiased bases
• Maximally symmetric stabilizer MUBs in even prime-power dimensions