On the stability of pointer states using Lyapunov theory

Pointer states are states of an open quantum system that are able to survive the constant monitoring of the system by an environment. It has been shown that open systems that are prepared in superpositions of such pointer states quickly decohere and evolve into classical statistical mixtures of (pure) pointer states. In this paper we demonstrate, using appropriate modeling assumptions for the system environment interaction, the following result: An individual trajectory of the system state involves towards a specific pointer state (and not just a statistical mixture of the same) if one monitors the environment state by measuring environmental observables even if only a fraction of these measurement outcomes are known to the observer. The central tool used to demonstrate this is the identification of conserved quantities that correspond to the eigenprojections of the system-environment Hamiltonian. We construct Lyapunov functions using this Hamiltonian to demonstrate the stability of the pointer states.