On calculating the mean values of quantum observables in the optical tomography representation

Given a density operator $\hat ρ$ the optical tomography map defines a one-parameter set of probability distributions $w_{\hat ρ}(X,ϕ),\ ϕ\in [0,2π),$ on the real line allowing to reconstruct $\hat ρ$. We introduce a dual map from the special class $\mathcal A$ of quantum observables $\hat a$ to a special class of generalized functions $a(X,ϕ)$ such that the mean value $<\hat a>_{\hat ρ} =Tr(\hat ρ\hat a)$ is given by the formula $<\hat a>_{\hat ρ}= \int \limits_{0}^{2π}\int \limits_{-\infty}^{+\infty}w_{\hat ρ}(X,ϕ)a(X,ϕ)dXdϕ$. The class $\mathcal A$ includes all the symmetrized polynomials of canonical variables $\hat q$ and $\hat p$.

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