On Lieb-Robinson Bounds in Open Quantum Systems

Abstract

In 1971, A. Kossakowski \cite{kossakowski} axiomatized the study of the dynamics associated to non-Hamiltonian systems of quantum particles. These have come to be known as Open Systems, and through the work of Lindblad \cite{Lindblad76}, a classification of the generators of the dynamics of such systems in the Heisenberg picture -- for bounded generators -- is known (this is not so for the unbounded case). Following the work of Gorini, Kossakowski, and Sudarshan: \cite{gks}; we prove this classification scheme in finite dimensions, where it is accessible by computational means as in \cite{alickifannes}. With this knowledge, we prove a Lieb-Robinson bound for the irreversible dynamics in the case of time-independent interactions on a countable (possibly infinite) collection of sites. Such a result was proven by \cite{Nachtergaele} for the time-dependent case, in 2011.