Quantum Advantage in Postselected Metrology

Abstract: We show that postselection offers a nonclassical advantage in metrology. In every parameter-estimation experiment, the final measurement or the postprocessing incurs some cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from an experimental trial) to cost. This improvement, we show, stems from the negativity of a quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution can be expressed as real and nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasiprobabilities. The distribution's nonclassically negative values enable postselected experiments to outperform even postselection-free experiments whose input states and final measurements are optimized: Postselected quantum experiments can yield anomalously large information-cost rates. We prove that this advantage is genuinely nonclassical: no classically commuting theory can describe any quantum experiment that delivers an anomalously large Fisher information. Finally, we outline a preparation-and-postselection procedure that can yield an arbitrarily large Fisher information. Our results establish the nonclassicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool.