The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. Random channels appear in a wide variety of applications, from quantum chaos to holographic dualities in theories of quantum gravity to operator dynamics, to random local circuits for their potential to demonstrate quantum supremacy. We consider general ergodic sequences of stochastic channels with arbitrary correlations and non-negligible decoherence. Ergodicity includes and vastly generalizes random independence. We obtain a theorem which shows that the composition of such a sequence of channels converges exponentially fast to a rank-one (entanglement breaking) channel. Using this, we derive the limiting behavior of translation invariant channels, and stochastically independent random channels. We then use our formalism to describe the thermodynamic limit of ergodic Matrix Product States. We derive formulas for the expectation value of a local observable and prove that the 2-point correlations of local observables decay exponentially. We then analytically compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy (i.e., R{e}nyi or von Neumann) across an arbitrary cut can be computed exactly. Other physical implications of our results are that most Floquet phases of matter are meta-stable, and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned. To obtain these results we bridge quantum information theory to dynamical systems and random matrix theory.
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