Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q = 4) and two-qubit (q = 2) systems, the octit (q = 8), qubit/quartit (q = 2× 4) and three-qubit (q = 2) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12, . . ., one defines a bijection between the σ(q) maximal commuting sets [with σ[q) the sum of divisors of q] of Pauli observables and the maximal submodules of the modular ring Zq , that arrange into the projective line P1(Zq) and a independent set of size σ(q)−ψ(q) [with ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q = 2, 2, 3, . . ., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2, 2) (if q = 2) and GQ(3, 3) (if q = 3). More precisely, in dimension p (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field Fp. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p) of cardinality σ(p ), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n−1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information. PACS numbers: 03.67.Lx, 02.10.Ox, 02.20.-a, 02.10.De, 02.40.Dr