Riccati equation and the problem of decoherence II: Symmetry and the solution of the Riccati equation

摘要

In this paper we revisit the problem of decoherence applying the blockoperator matrices analysis. Riccati algebraic equation associated with theHamiltonian describing the process of decoherence is studied. We prove that ifthe environment responsible for decoherence is invariant with respect to theantylinear transformation then the antylinear operator solves Riccati equationin question. We also argue that this solution leads to neither linear norantilinear operator similarity matrix. This fact deprives us the standardprocedure for solving linear differential equation (e.g, Schrodinger equation).Furthermore, the explicit solution of the Riccati equation is found for thecase where the environment operators commute with each other. We discuss theconnection between our results and the standard description of decoherence (onethat uses the Kraus representation). We show that reduced dynamics we obtaindoes not have the Kraus representation if the initial correlations between thesystem and its environment are present. However, for any initial state of thesystem (even when the correlations occur) reduced dynamics can be written in amanageable way.