On quantum tomography on locally compact groups

TL;DR

This paper develops a framework for quantum tomography on locally compact Abelian groups, introducing new mathematical tools and examples for reconstructing quantum states and observables using group-based characteristic functions.

Contribution

It introduces a novel approach to quantum tomography on locally compact Abelian groups, including the construction of linear maps, dual maps, and explicit examples for different groups.

Findings

Defined quantum tomograms as sets of probability distributions

Provided explicit examples for groups R, Z_n, and T

Calculated quantum tomograms for Weyl channel outputs

Abstract

We introduce quantum tomography on locally compact Abelian groups $G$. A linear map from the set of quantum states on the $C^*$-algebra $A(G)$ generated by the projective unitary representation of $G$ to the space of characteristic functions is constructed. The dual map determining symbols of quantum observables from $A(G)$ is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which $G={\mathbb R}$ (the optical tomography), $G={\mathbb Z}_n$ (corresponding to measurements in mutually unbiased bases) and $G={\mathbb T}$ (the tomography of the phase). As an application we have calculated the quantum tomogram for the output states of quantum Weyl channels.