A non-zero Kolmogrov–Sinai entropy for a classical dynamical system is a signature of dynamical instability. This chapter presents an approach to quantifying randomizing dynamical behaviour in deterministic quantum systems based on a spin chain model. The starting point is an operational partition that is refined in the course of time. To each partition corresponds a correlation matrix and the dynamics lead eventually to a shift-invariant state on a quantum spin chain with its associated entropy. General properties and bounds are proved, which allow for the computation of the entropy in a number of simple model systems such as finite systems, shift dynamics on a quantum spin chain, free shifts, and Powers–Price shifts.
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