Measurement-Induced Randomness and Structure in Controlled Qubit Processes

When an observer measures a time series of qubits, the outcomes generate a classical stochastic process. We present a model family of classically controlled qubit time series and show that measurement induces high complexity in these processes in two specific senses: they are inherently unpredictable (positive Shannon entropy rate) and they require an infinite number of features for optimal prediction (divergent statistical complexity). We argue that nonunifilarity is the mechanism underlying the resulting complexities and identify the different contributions to the randomness of the observed process. We examine the influence that the choice of measurement has on the randomness and structure of the measured qubit process and discuss measurement choices of potential interest in obtaining information about the underlying time series of qubits.