We refine the multiplicative ergodic theorem for compositions of random bistochastic completely positive maps and use this to conclude various results about such compositions. In particular, we show that occasionally PPT compositions of bcp maps are asymptotically entanglement breaking almost surely. 

In this paper, we prove ergodic theorems for the natural random environment Markov chain associated to a disordered quantum trajectory. In particular, we establish a law of large numbers for measurement outcomes for disordered quantum trajectories, extending a theorem of Kümmerer and Maassen.

In this paper, we study global dynamical properties of the open quantum dynamical systems defined by a measure-preserving dynamical system and a random quantum channel. We define a notion of irreducibility for such dynamical systems, which we characterize in various ways. We prove a number of ergodic theorems for these dynamical systems. 

We introduce a framework to study quantum trajectories in a dynamical, disordered environment. We show that asymptotic purification is a property intrinsic to the underlying random Kraus operators defining disordered quantum trajectories, in the sense that the disorder set where asymptotic purification occurs has a 0-1 law. We then characterize this property of random Kraus operators in terms of a random subset of the Grassmannian: the so-called dark subspaces. This rigorously extends the notion of dark subspace to the disordered setting and serves as a general framework for further study of the effect of disorder on quantum trajectories.