Affordable Access

About the Dedekind psi function in Pauli graphs

Authors
  • Planat, Michel R. P.
Type
Published Article
Publication Date
Dec 07, 2010
Submission Date
Dec 07, 2010
Identifiers
arXiv ID: 1012.1461
Source
arXiv
License
Yellow
External links

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.

Report this publication

Statistics

Seen <100 times