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We study the commutation relations within the Pauli groups built on alldecompositions 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 ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case,e.g. $q=4,8,12,...$, one defines a bijection between the $\sigma (q)$ maximalcommuting sets [with $\sigma[q)$ the sum of divisors of $q$] of Pauliobservables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$,that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent setof size $\sigma (q)-\psi(q)$ [with $\psi(q)$ the Dedekind psi function]. In themultiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely onsymplectic polar spaces such as the generalized quadrangles GQ(2,2) (if$q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ aprime) of the Hilbert space, the observables of the Pauli group (modulo thecenter) are seen as the elements of the $2n$-dimensional vector space over thefield $\mathbb{F}_p$. In this space, one makes use of the commutator to definea symplectic polar space $W_{2n-1}(p)$ of cardinality $\sigma(p^{2n-1})$, thatencodes the maximal commuting sets of the Pauli group by its totally isotropicsubspaces. Building blocks of $W_{2n-1}(p)$ are punctured polar spaces (i.e. aobservable and all maximum cliques passing to it are removed) of size given bythe Dedekind psi function $\psi(p^{2n-1})$. For multiple qudit mixtures (e.g.qubit/quartit, qubit/octit and so on), one finds multiple copies of polarspaces, ponctured polar spaces, hypercube geometries and other intricatestructures. Such structures play a role in the science of quantum information.

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