Invariance of Stationary and Ergodic Properties of a Quantum Source under Memoryless Transformations

We prove that the stationarity and the ergodicity of a quantum source are preserved by any trace-preserving completely positive linear map of the tensor product form ${\cal E}^{\otimes m}$, where a copy of ${\cal E}$ acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources.