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Quantum tomography, phase-space observables and generalized Markov kernels

We construct a generalized Markov kernel which transforms the observable associated with the homodyne tomography into a covariant phase-space observable with a regular kernel state. Illustrative examples are given in the cases of a ‘Schrödinger cat’ kernel state and the Cahill–Glauber s-parametrized distributions. Also we consider an example of a kernel state when the generalized Markov kernel cannot be constructed.