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It is well-known that the precision of a phase measurement with aMach-Zehnder interferometer employing strong (macroscopic) classic light can begreatly enhanced with the addition of a weak (microscopic) light field in anon-classical state. The resulting precision is much greater than that possiblewith either the macroscopic classical or microscopic quantum states alone. Inthe context of quantifying non-classicality, the amount by which anon-classical state can enhance precision in this way has been termed its"metrological power". Given the technological difficulty of producinghigh-amplitude non-classical states of light, this use of non-classical lightis likely to provide a technological advantage much sooner than the Heisenbergscaling employing much stronger non-classical states. To date, the enhancementprovided by weak nonclassical states has been calculated only for specificmeasurement configurations. Here we are able to optimize over all measurementconfigurations to obtain the maximum enhancement that can be achieved by anysingle or multi-mode nonclassical state together with strong classical states,for local and distributed quantum metrology employing any linear or nonlinearsingle-mode unitary transformation. Our analysis reveals that the quantumFisher information for \textit{quadrature displacement sensing} is the soleproperty that determines the maximum achievable enhancement in all of thesedifferent scenarios, providing a unified quantification of the metrologicalpower. It also reveals that the Mach-Zehnder interferometer is an optimalnetwork for phase sensing for an arbitrary single-mode nonclassical inputstate, and how the Mach-Zehnder interferometer can be extended to make optimaluse of any multi-mode nonclassical state for metrology.

The power of microscopic nonclassical states to amplify the precision of macroscopic optical metrology