On quantum tomography on locally compact groups

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 = R (the optical tomography), G = Z n (corresponding to measurements in mutually unbiased bases) and G = T (the tomography of the phase). As an application we have calculated the quantum tomogram for the output states of quantum Weyl channels.