Experimental Test of Heisenberg's Measurement Uncertainty Relation Based on Statistical Distances

Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [Phys. Rev. Lett. 111, 160405 (2013); Phys. Rev. A 89, 012129 (2014)]. Here we reform their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible but typically non-commutative observables on one qubit are measured simultaneously.