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### Covariant mutually unbiased bases

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.Comment: 44 pages, some remarks and references added in v

### Weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The
properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of
these systems, are studied. Weak mutually unbiased bases in these systems are
defined as bases for which the overlap of any two vectors in two different
bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$
(where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the
geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense
that there is a duality between the weak mutually unbiased bases and the
maximal lines through the origin. In the special case of prime $d$, there are
no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are
mutually unbiased bases

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