Universal Uncertainty Principle, Simultaneous Measurability, and Weak Values

In the conventional formulation, it is broadly accepted that simultaneous measurability and commutativity of observables are equivalent. However, several objections have been claimed that there are cases in which even nowhere commuting observables can be measured simultaneously. Here, we outline a new theory of simultaneous measurements based on a state-dependent formulation, in which nowhere commuting observables are shown to have simultaneous measurements in some states, so that the known objections to the conventional theory are theoretically justified. We also discuss new results on the relation between weak values and output probability distributions of simultaneous measurements.