Entropic uncertainty relations in quantum phase space

The seminal uncertainty relation introduced by Heisenberg was expressed in terms of variances. Nevertheless, uncertainty can also be measured in terms of entropy. The first entropic uncertainty relation is due to Białynicki-Birula and Mycielski in 1975. In this talk, I present several improvements of this entropic uncertainty relation. Our first novel entropic uncertainty relation takes x-p correlations into account and is consequently saturated by all pure Gaussian states in an arbitrary number of modes, improving on the original formulation by Białynicki-Birula and Mycielski. Our second main result is the derivation of an entropic uncertainty relation that holds for any two vectors of not-necessarily canonically conjugate variables based on the matrix of their commutators. We then define a general form of the entropic uncertainty principle that combines both previous results. It expresses the incompatibility between two vectors of arbitrary variable and is saturated by all pure Gaussian states.