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

TL;DR

This paper investigates the Riccati equation in decoherence, showing how environment symmetry influences solutions and connecting these findings to the standard Kraus representation, especially under initial correlations.

Contribution

It demonstrates that environment invariance under antilinear transformation leads to unique Riccati solutions and explores the implications for decoherence dynamics beyond standard methods.

Findings

Environment invariance yields special Riccati solutions.

Standard Kraus representation may not apply with initial correlations.

Explicit solutions are derived for commuting environment operators.

Abstract

In this paper we revisit the problem of decoherence applying the block operator matrices analysis. Riccati algebraic equation associated with the Hamiltonian describing the process of decoherence is studied. We prove that if the environment responsible for decoherence is invariant with respect to the antylinear transformation then the antylinear operator solves Riccati equation in question. We also argue that this solution leads to neither linear nor antilinear operator similarity matrix. This fact deprives us the standard procedure for solving linear differential equation (e.g, Schrodinger equation). Furthermore, the explicit solution of the Riccati equation is found for the case where the environment operators commute with each other. We discuss the connection between our results and the standard description of decoherence (one that uses the Kraus representation). We show that reduced dynamics we obtain does not have the Kraus representation if the initial correlations between the system and its environment are present. However, for any initial state of the system (even when the correlations occur) reduced dynamics can be written in a manageable way.