QLunch: Tristan Benoist
Title: Purification along quantum trajectories
Abstract: Quantum trajectories model quantum systems repeatedly (indirectly) measured. The resulting evolution is a Markov process. In this talk I will discuss their purification property. Proved in 2005 by Kummerer and Maassen, it shows that, in absence of so called dark subspaces, the system state has a tendency to get closer and closer to a pure state along a quantum trajectory. I will revisit this result. I will explain how it is related to the stability of filters that led me and some collaborators to a proof of uniqueness of invariant measure. I will briefly mention the relationship to error correcting codes. I will then list a few new equivalent criteria for the absence of dark subspaces related to minima of polynomials, singular values and Lyapunov exponents. I will conclude with some interesting conjectures dealing with the likeliness of purification and some outlook on infinite dimensional systems and measurement induced phase transitions.