The commutation relations between the generalized Pauli operators of N-qudits (i. e., Np-level quantum systems),
and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical
pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them
to form the so-called Pauli graph PpN . As per two-qubits (p = 2, N = 2) all basic properties and partitionings
of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The
structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal
with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geometry
of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple
(N > 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of
these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.
Mutually unbiased bases (MUBs), which are such that the inner
product between two vectors in different orthogonal bases is
constant equal to the inverse 1/√d, with d the dimension
of the finite Hilbert space, are becoming more and more studied
for applications such as quantum tomography and cryptography, and
in relation to entangled states and to the Heisenberg-Weil group
of quantum optics. Complete sets of MUBs of cardinality d+1 have
been derived for prime power dimensions d=pm using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less.
The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg and Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters.
Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.