Optimal simultaneous measurements of incompatible observables of a single photon

The ultimate limits of measurement precision are dictated by the laws of quantum mechanics. One of the most fascinating results is that joint or simultaneous measurements of noncommuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for determining what an “optimal” joint measurement is, with corresponding different trade-off relations for the measurements. It is generally a nontrivial task to devise or implement a strategy that minimizes the joint-measurement uncertainty. Here, we implement the simplest possible technique for an optimal four-outcome joint measurement and demonstrate a type of optimal measurement that has not been realized before in a photonic setting. We experimentally investigate a joint-measurement uncertainty relation that is more fundamental in the sense that it refers only to probabilities and is independent of values assigned to measurement outcomes. Using a heralded single-photon source, we demonstrate quantum-limited performance of the scheme on single quanta. Since quantum measurements underpin many concepts in quantum information science, this study is both of fundamental interest and relevant for emerging photonic quantum technologies.