Memory Structure in Measured Stochastic Processes of Quantum States

A time series of qubits, when measured by a classical observer, generically result in highly complex observed classical time series. The measurement can both increase or decrease the inherent randomness of the observed classical stochastic process with respect to the underlying stochastic process of qubits. Even when the underlying stochastic process of qubits can be predicted with a finite memory resources, the act of measuring it will generically result in a classical stochastic process that requires infinite memory resources to predict. We discuss the causes for this divergence in memory requirements and present a method to quantify the rate of this divergence, the statistical complexity dimension of the measured process. This represents a quantifier for the structure of the measured classical process, and smoothly varies with the choice of measurement.