Polarization Tomography of Quantum Radiation: Theoretical Aspects and Operator Approach

We present theoretical foundations for the quantum tomography of polarization states of light fields as a method for measuring their polarization density operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat R$$ \end{document}, which characterizes only the polarization degrees of freedom of the radiation. We mainly attend to the method in which the tomographic observables (the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat R$$ \end{document} “measurement instruments”) are polarizable in nature. We show that the quantum nature of this method can be adequately expressed using the quasispectral tomographic decompositions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat R$$ \end{document} in special operator bases, which are finite sums of partially orthogonal projection operators determining the probability distributions of tomographic observables as the decomposition coefficients. We obtain the matrix versions of such “tomographic” representations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat R$$ \end{document}, in particular, by projecting them on semiclassical operator bases determining the polarization quasiprobability functions. We briefly discuss the information aspects of the schemes considered in the paper.