s-order nondiagonal and diagonal measurements of noncommuting observables

Two models for the measurement of position and momentum of systems with one degree of freedom are developed: (i) a two-detector model and (ii) a four-detector model. The two-detector model describes measurements with statistics given by s-order diagonal quasiprobabilities ρ’(α,s) with an upper bound s₀≤-1 for s which is system-state dependent. In order to have s-order statistics, both detectors must be prepared in single-mode squeezed vacuum states which are equally squeezed by a factor depending on the value of s. On the contrary, the four-detector model, with statistics given by s-order positive nondiagonal quasiprobabilities ρ_t(α,β,s), replaces the restriction in s by a restriction t₀≤-1 for t (for each order s there is a family of nondiagonal quasiprobabilities parametrized by the parameter t). In order to have s-order statistics, the four detectors must be prepared in multimode squeezed vacuum states, with the squeeze parameters depending on s and t.