Quantum Fourier transform, Heisenberg groups and quasiprobability distributions

TL;DR

This paper investigates the deep connections between Heisenberg groups, quantum Fourier transforms, and quasiprobability distributions, providing new insights into their mathematical structure and applications in quantum information processing.

Contribution

It introduces a unified framework linking Heisenberg groups, quantum Fourier transforms, and distribution functions, with explicit reconstruction formulas and applications to quantum tomography.

Findings

Distribution functions relate to group algebra elements.

Wigner distributions' marginals derive from automorphisms of the Heisenberg group.

Explicit inverse Radon transform formulas for Wigner functions.

Abstract

This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of "conjugate" observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of two equivalent representation arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of Wigner function from the marginal distributions via inverse Radon transform giving explicit formulas. We consider applications of our approach to quantum information processing and quantum process tomography.