Covariant Mappings for the Description of Measurement, Dissipation and Decoherence in Quantum Mechanics

TL;DR

This paper reviews the formalism of quantum measurement and dynamics using covariant mappings, analyzing their symmetry properties and applying them to various quantum systems and decoherence phenomena.

Contribution

It introduces a covariant framework for describing quantum state transformations, including measurement and evolution, with applications to master equations and decoherence models.

Findings

Analysis of covariant master equations for different quantum systems

Connection between decoherence behaviors and classical Lévy processes

General structure of translation-covariant quantum dynamical maps

Abstract

The general formalism of quantum mechanics for the description of statistical experiments is briefly reviewed, introducing in particular position and momentum observables as POVM characterized by their covariance properties with respect to the isochronous Galilei group. Mappings describing state transformations both as a consequence of measurement and of dynamical evolution for a closed or open system are considered with respect to the general constraints they have to obey and their covariance properties with respect to symmetry groups. In particular different master equations are analyzed in view of the related symmetry group, recalling the general structure of mappings covariant under the same group. This is done for damped harmonic oscillator, two-level system and quantum Brownian motion. Special attention is devoted to the general structure of translation-covariant master equations. Within this framework a recently obtained quantum counterpart of the classical linear Boltzmann equation is considered, as well as a general theoretical framework for the description of different decoherence experiments, pointing to a connection between different possible behaviours in the description of decoherence and the characteristic functions of classical L\'evy processes.