An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states

Quantum channels represent the most general physical changes of a quantum system. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. Such maps vastly generalize stochastically independent maps (e.g., random independence) or equality of the channel maps (i.e., translation invariance). The repeated composition of an ergodic sequence of maps could represent the effect of repeated application of a given quantum channel subject to arbitrary correlated noise or decoherence. Under such a hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one -- entanglement breaking-- channel. As an application, we describe the thermodynamic limit of ergodic Matrix Product States and derive a formula for the expectation value of a local observable and prove that the 2-point correlations of local observables in such states decay exponentially in the bulk with their distance.