Symmetries of the finite Heisenberg group for composite systems

TL;DR

This paper explores the symmetries of the finite Heisenberg group in composite quantum systems, extending previous work to systems with subsystems of arbitrary dimensions, and characterizes the symmetry group via normalizers.

Contribution

It provides a detailed description of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices for composite systems.

Findings

Characterization of the normalizer in the group of inner automorphisms of GL(nm,C)

Explicit description of the symmetry group as a quotient group

Extension of symmetry analysis to systems with arbitrary subsystem dimensions

Abstract

Symmetries of the finite Heisenberg group represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. As is well known, these symmetries are properly expressed in terms of certain normalizer. This paper extends previous investigations to composite quantum systems consisting of two subsystems - qudits - with arbitrary dimensions n and m. In this paper we present detailed descriptions - in the group of inner automorphisms of GL(nm,C) - of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices of orders n and m. The symmetry group is then given by the quotient group of the normalizer.