A discrete time parameter quantum process is obtained by iterated compositions of quantum operations. A quantum operation, a generalization of a quantum channel, is a completely positive map that is not necessarily trace preserving. As such, a discrete parameter quantum process is described by a sequence of quantum operations. We study the cases where the sequence is strictly stationary. By allowing the quantum operations to not preserve the trace one may study cases such as moment generating functions for a measurement processes or even measurement processes that discard the system upon a certain measurement outcome. An ergodic theorem describing convergence to equilibrium for the class of ergodic quantum processes was recently obtained by Movassagh and Schenker in [56, 55]. We derive similar theorems for the strictly stationary case. Furthermore, under certain irreducubility and mixing conditions, we see that the assymptotics of such processes are governed by the top (maximal) Lyapunov exponent for the ergodic sequences and we derive a law of large numbers (LLN) and a central limit theorem (CLT). In the continuous time-parameter a quantum process is described through a doubled indexed family of quantum operations such that the compositions of maps preserve the dynamics. Here we generalize the results in [56, 55] for the continuous time-parameter and strictly stationary case.
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