On quantum tomography on locally compact groups

We introduce quantum tomography on locally compact Abelian groups $G$. A linear map from the set of quantum states on the $C^{\ast }$-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.