One of the most important question in Quantum Information Theory was to figure out whether the so-called Minimum Output Entropy (MOE) was additive. In this talk I will start by defining the counter-example originally built by Belinschi, Collins and Nechita. Then I will explain how with the help of a novel strategy, we managed with Collins to compute concentration estimate on the probability that the MOE is non-additive and how it yielded some explicit bounds for the dimension of spaces where violation of the MOE occurs. Finally, I will talk more in detail about this novel strategy which consists in interpolating random matrices and free operators with the help of free stochastic calculus.[-]

Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures.
In this talk we first review classical results on variance uncertainty inequalities (in particular Heisenberg, Robertson and Breitenberger inequalities). We then focus on discrete (and in particular finite-dimensional) situations, where variance has to be replaced with more suitable localization measures. We then present recent results on support and entropic inequalities, describing joint localization properties of vector expansions with respect to two frames.

Keywords: uncertainty principle - variance of a function - Heisenberg inequality - support inequalities - entropic inequalities[-]