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

TL;DR

This paper introduces a dual map for quantum observables in optical tomography, enabling the calculation of mean values directly from probability distributions, including all symmetrized polynomials of canonical variables.

Contribution

It presents a novel dual map from quantum observables to generalized functions, facilitating mean value calculations in optical tomography representation.

Findings

Derived a formula for mean values using optical tomograms.

Included all symmetrized polynomials of canonical variables in the class of observables.

Established a dual map linking observables to probability distributions.

Abstract

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