We study the commutation structure within the Pauli groups built on all
decompositions of a given Hilbert space dimension q, containing a square,
into its factors. The simplest illustrative examples are the quartit (q=4)
and two-qubit (q=22) systems. It is shown how the sum of divisor function
σ(q) and the Dedekind psi function ψ(q)=q∏p∣q(1+1/p) enter
into the theory for counting the number of maximal commuting sets of the qudit
system. In the case of a multiple qudit system (with q=pm and p a prime),
the arithmetical functions σ(p2n−1) and ψ(p2n−1) count the
cardinality of the symplectic polar space W2n−1(p) that endows the
commutation structure and its punctured counterpart, respectively. Symmetry
properties of the Pauli graphs attached to these structures are investigated in
detail and several illustrative examples are provided.Comment: Proceedings of Quantum Optics V, Cozumel to appear in Revista
Mexicana de Fisic