Precise measurements underpin scientific and technological advancements. Quantum mechanics provides an avenue to enhance precision, but it comes with a restriction: Incompatible observables, such as position and momentum, cannot be simultaneously measured to arbitrary accuracy as decreed by Heisenberg's uncertainty principle. This restriction can be bypassed by instead measuring commuting modular observables, which are counterparts to the naturally incompatible observables. Here, we measure modular observables to estimate small changes in position and momentum with a single-mode multiparameter sensor. We deterministically prepare grid states in the mechanical motion of a trapped ion and demonstrate uncertainties in position and momentum below the standard quantum limit (SQL). Further, we examine another pair of incompatible observables-number and phase. We prepare a different resource-number-phase states-and demonstrate a metrological gain over the SQL. These results introduce previously unidentified measurement capabilities unavailable to classical systems and mark a substantial step in quantum metrology.
Valahu et al. (Fri,) studied this question.