"The complexity of learning quantum states (with applications to face recognition)."

"The complete characterization of a quantum system by physical measurements seems to be a conceptually simple task and is routinely carried out experimentally. It is thus all the more surprising that many fundamental questions pertaining to this procedure remain unanswered. (And, what is more, lead to highly non-trivial mathematical problems). A prime example is determining the sample complexity of quantum state estimation: under realistic conditions, how many experimental runs does one need in order to obtain an estimate for an unknown quantum state with acceptable error bars? Simple answers based on asymptotic statistics turn out to be highly inaccurate (in fact, way too pessimistic). I will report recent progress on this and related problems. It is both based on, and has contributed to, new developments in classical statistics and machine learning theory. I will mention proposals for tasks as varied as face recognition and prediction of online behavior which have been influenced by methods from quantum state tomography."