Determinism and randomness in quantum dynamics

Abstract

A class of models is considered where a finite or infinite quantum dynamical system in a stationary state is probed by sequences of observations acting on a specified finite subsystem. The whole set of such experiments is described by a time-ordered (causal) and stationary quantum correlation kernel. It is shown that any such kernel can be decomposed in a unique way into a convex combination of two kernels, here called the regular and singular components. A singular system has a strong deterministic property, the predictability of the future from the knowledge of the past is limited only by the inevitable indeterminism of quantum measurements. Furthermore, in this case the full set of correlation functions of arbitrary time order, and hence the dynamical system itself, is determined by the causal kernel. Finite systems and infinite systems satisfying the KMS condition at finite temperature are of this type. In the regular case the dynamics contains a shift, there is a genuine asymptotic randomness and the dynamical system cannot be reconstructed in a unique way from the causal kernel. Non-trivial quantum Markov processes are shown to belong to this class.