Maximally strong entropic uncertainty relations and gauge invariance

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.