  Operator noncommutation, a hallmark of quantum theory, limits measurement
precision, according to uncertainty principles. Wielded correctly, though,
noncommutation can boost precision. A recent foundational result relates a
metrological advantage with negative quasiprobabilities -- quantum extensions
of probabilities -- engendered by noncommuting operators. We crystallize the
relationship in an equation that we prove theoretically and observe
experimentally. Our proof-of-principle optical experiment features a filtering
technique that we term partially postselected amplification (PPA). Using PPA,
we measure a waveplate's birefringent phase. PPA amplifies, by over two orders
of magnitude, the information obtained about the phase per detected photon. In
principle, PPA can boost the information obtained from the average filtered
photon by an arbitrarily large factor. The filter's amplification of systematic
errors, we find, bounds the theoretically unlimited advantage in practice. PPA
can facilitate any phase measurement and mitigates challenges that scale with
trial number, such as proportional noise and detector saturation. By
quantifying PPA's metrological advantage with quasiprobabilities, we reveal
deep connections between quantum foundations and precision measurement.
