  Uncertainty relations are among the unique fingerprints of quantum physics,
being direct expression of non-commutativity and complementarity. Entropic
uncertainty relations arise in quantum information theory as the most natural
expression of the uncertainty principle and have proven themselves important
tools for cryptographic security proofs. An open question has been how strong
entropic uncertainty relations can be. Here we answer this question by showing
that there exist observables that obey maximally strong entropic uncertainty
relations in high dimensional Hilbert spaces. Actually, we show that maximally
strong uncertainty relations are typically verified if the observables are
distributed somehow uniformly. In particular, we show that phase randomness in
a mutually unbiased basis is sufficient, providing a link between uncertainty
relations and local gauge invariance. For quantum systems of finite dimensions
our approach provides an explicit relation between the strength of the
uncertainty and its probability.
