Universal uncertainty relations

Uncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of uncertainty quantifiers. Our relation is universally valid and captures the essence of uncertainty in quantum mechanics.