Exact Results in Quantum Theory

I will present uncertainty principles for several measurements (on a finite dimensional system) defined by independent random (Haar distributed) unitary matrices. Emphasis will be put on relations expressed in terms of the Shannon entropy (obtained with R. Latała, Z. Puchała, K. Życzkowski) and in terms of the total variation or Hellinger distances. If time permits I will discuss connections with high dimensional probability as well as applications to information locking and quantum data hiding.