Symmetries of finite Heisenberg groups for multipartite systems

TL;DR

This paper extends the analysis of symmetry groups of finite Heisenberg groups from bipartite to multipartite quantum systems, revealing new symmetries and providing a proof related to mutually unbiased bases in prime power dimensions.

Contribution

It generalizes the symmetry analysis of finite Heisenberg groups to multipartite systems and describes new symmetry groups as generalizations of symplectic groups over modular rings.

Findings

Detailed description of normalizers and symmetry groups for multipartite systems.

Identification of a new class of symmetry groups generalizing symplectic groups.

New proof of the existence of mutually unbiased bases in prime power dimensions.

Abstract

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_{i}}, i=1,...,k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces of dimensions n_{1},...,n_{k}. Symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.