Symmetries of finite Heisenberg groups for k-partite systems

TL;DR

This paper extends the understanding of symmetries in finite Heisenberg groups to multipartite quantum systems, describing their symmetry groups as generalizations of finite symplectic groups over modular rings.

Contribution

It provides a comprehensive description of the symmetry groups for composite quantum systems with k subsystems, generalizing previous results to multipartite cases.

Findings

Symmetry groups are described as quotients of normalizers in finite Heisenberg groups.

The symmetry groups generalize finite symplectic groups over modular rings.

Results apply to systems with arbitrary dimensions n_i in Z_{n_i}.

Abstract

Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in Z_{n_i}, i=1,...,k. Their Hilbert spaces are given by k-fold tensor products of Hilbert spaces of dimensions n_1,...,n_k. Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.