  Quantum physics constrains the accuracy of joint measurements of incompatible
observables. Here we test tight measurement-uncertainty relations using single
photons. We implement two independent, idealized uncertainty-estimation
methods, the 3-state method and the weak-measurement method, and adapt them to
realistic experimental conditions. Exceptional quantum state fidelities of up
to 0.99998(6) allow us to verge upon the fundamental limits of measurement
uncertainty.
