Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes

Among the predictive hidden Markov models that describe a given stochastic process, the $ϵ\text{-machine }$ is strongly minimal in that it minimizes every Rényi-based memory measure. Quantum models can be smaller still. In contrast with the $ϵ\text{-machine }$ 's unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model. Quantum memory optimization then depends on which memory measure best matches a given problem's circumstance.