Regularity of Quantum Trajectories and Randomized Measurements

Quantum trajectories are Markov chains that model the evolution of quantum systems subjected to repeated indirect measurements. The classification of their invariant measures relies heavily on a purification assumption that, while standard, is difficult to check for a given instrument. Moreover, little is known about the invariant measures themselves, like their support or continuity properties.
With Sascha Lill and Cornelia Vogel we recently investigated these questions when the measurements are randomized. Namely, at each time step, the observable measured on the probe used for the indirect measurement is chosen randomly and independently. Considering this randomization, we give sufficient conditions for purification to hold, the invariant measure’s support is the full set of pure states and the invariant measure is absolutely continuous with respect to the uniform measure. Along the way we introduce a new notion of irreducibility for quantum channels we dub multiplicative primitiveness. We provide examples where it holds and some where it does not. That especially shows that, for some quantum channels, whatever the measurement used on the probes, some pure states will never be approximated by the quantum trajectory.
I will present these results and related open questions.