Experimental investigation of the uncertainty relation in pre- and postselected systems

Uncertainty principle is one of the fundamental principles of quantum mechanics. Exploring such uncertainty relations in pre- and postselected (PPS) systems, where weak measurements on post-selected states have been used as a powerful tool for exploring the foundation of quantum mechanics, has so far eluded experimental effort. In this work, we experimentally investigate the Robertson-Heisenberg-type uncertainty relation for two incompatible observables in a PPS system. Through conducting a von Neumann-type measurement, uncertainty relations between two non-commuting Pauli observables are experimentally unveiled. Such uncertainty relations in the PPS system impose limitations on the joint sharp preparation of pre- and postselected states for two incompatible observables. Our experiments would be useful for the development of amplification techniques for precision measurements.