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Strong correspondence principle for joint measurement of conjugate observables

It is demonstrated that the the statistics for a joint measurement of two conjugate variables in Quantum Mechanics are expressed through an equation identical to the classical one, provided that joint classical probabilities are substituted by Wigner functions and that the interaction between system and detectors is accounted for. This constitutes an extension of Ehrenfest correspondence principle, and it is thereby dubbed strong correspondence principle. Furthermore, it is proved that the detectors provide an additive term to all the cumulants, and that if they are prepared in a Gaussian state they only contribute to the first and second cumulant.