About the Dedekind psi function in Pauli graphs

TL;DR

This paper explores the mathematical structure of Pauli groups in qudit systems, revealing how divisor and Dedekind psi functions describe the counting of maximal commuting sets and the symmetries of associated graphs.

Contribution

It introduces a novel connection between arithmetical functions and the commutation structure of Pauli groups in qudit systems, with detailed analysis of symmetry properties.

Findings

Divisor and Dedekind psi functions quantify maximal commuting sets.

Symmetry properties of Pauli graphs are characterized.

Examples include quartit and two-qubit systems.

Abstract

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 ($q=2^2$) systems. It is shown how the sum of divisor function $\sigma(q)$ and the Dedekind psi function $\psi(q)=q \prod_{p|q} (1+1/p)$ 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 $q=p^m$ and $p$ a prime), the arithmetical functions $\sigma(p^{2n-1})$ and $\psi(p^{2n-1})$ count the cardinality of the symplectic polar space $W_{2n-1}(p)$ 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.