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Any discrete quantum process is represented by a sequence of quantumchannels. We consider ergodic quantum processes obtained by a map that takesthe points along the trajectory of a discrete ergodic dynamical system to thespace of quantum channels. Under natural irreducibility conditions, we obtain atheorem showing that the state under such a process converges exponentiallyfast to an ergodic sequence depending on the process, but independent of theinitial state. As an application, we describe the thermodynamic limit ofergodic matrix product states and prove that the 2-point correlations of localobservables in such states decay exponentially with their distance in the bulk.

An ergodic theorem for quantum processes with applications to matrix product states