An Ergodic Theorem for Quantum Processes with Applications to Matrix Product States

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels. Under a natural irreducibility condition, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk.