Unitarily inequivalent local and global Fourier transforms in multipartite quantum systems

TL;DR

This paper explores the differences between local and global Fourier transforms in multipartite quantum systems, highlighting conditions under which they are unitarily equivalent or inequivalent, with implications for phase space methods and system evolution.

Contribution

It introduces a formalism comparing local and global Fourier transforms in multipartite quantum systems, analyzing their unitary equivalence and implications for quantum phase space representations.

Findings

Local and global Fourier transforms can be unitarily inequivalent depending on system parameters.

The formalism relates local and global phase space methods in quantum systems.

Discussion of system evolution in terms of both local and global variables.

Abstract

A multipartite system comprised of $n$ subsystems, each of which is described with `local variables' in ${\mathbb Z}(d)$ and with a $d$-dimensional Hilbert space $H(d)$, is considered. Local Fourier transforms in each subsystem are defined and related phase space methods are discussed (displacement operators, Wigner and Weyl functions, etc). A holistic view of the same system might be more appropriate in the case of strong interactions, which uses `global variables' in ${\mathbb Z}(d^n)$ and a $d^n$-dimensional Hilbert space $H(d^n)$. A global Fourier transform is then defined and related phase space methods are discussed. The local formalism is compared and contrasted with the global formalism. Depending on the values of $d,n$ the local Fourier transform is unitarily inequivalent or unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both local and global variables, is discussed. The formalism can be useful in the general area of Fast Fourier transforms.