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

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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 = 22) systems, the octit (q = 8), qubit/quartit (q = 2 × 4) and three-qubit (q = 23) 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) being the sum of divisors of q) of Pauli observables and the maximal submodules of the modular ring , that arrange into the projective line and an independent set of size σ(q) − ψ(q) (with ψ(q) being the Dedekind psi function). In the multiple qudit case, e.g. q = 22, 23, 32, … , the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2, 2) (if q = 22) and GQ(3, 3) (if q = 32). More precisely, in the dimension pn (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 . In this space, one makes use of the commutator to define a symplectic polar space W2n − 1(p) of cardinality σ(p2n − 1) 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(p2n − 1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, punctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.