Riccati equation and the problem of decoherence II: Symmetry and the solution of the Riccati equation
arXiv:1006.1931 · doi:10.1063/1.3574889
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.
arXiv comment17 pages, 0 figures, The quality of English used in the paper has been improved
References in corpus (7)
- Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
- Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation
- Kraus representation in the presence of initial correlations
- Kraus representation for density operator of arbitrary open qubit system
- Riccati equation and the problem of decoherence
- Exact reduced dynamics for a qubit in a precessing magnetic field and in the contact with a heat-bath
- Riccati Diagonalization of Hermitian Matrices