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Gaussian states with nonclassical properties such as squeezing and entanglement serve as crucial resources for quantum information processing. Accurately quantifying these properties within multimode Gaussian states has posed some challenges. To address this, we introduce a unified quantification: the “classical-nonclassical polarity,” represented by P. For a single mode, a positive value of P captures the reduced minimum quadrature uncertainty below the vacuum noise, while a negative value represents an enlarged uncertainty due to classical mixtures. For multimode systems, a positive P indicates bipartite quantum entanglement. We show that the sum of the total classical-nonclassical polarity is conserved under arbitrary linear optical transformations for any two-mode and three-mode Gaussian states. For any pure multimode Gaussian state, the total classical-nonclassical polarity equals the sum of the mean photon number from single-mode squeezing and two-mode squeezing. Our results provide a new perspective on the quantitative relation between single-mode nonclassicality and entanglement, which may find applications in a unified resource theory of nonclassical features. 

Abstract It is well-known that the precision of a phase measurement with a Mach-Zehnder interferometer employing strong classic light can be greatly enhanced with the addition of weak nonclassical light. In the context of quantifying nonclassicality, the amount by which a nonclassical state can enhance precision in this way has been termed its ’metrological power’. To-date, the enhancement provided by weak nonclassical states has been calculated only for specific measurement configurations. Here we are able to optimize over all measurement configurations to obtain the maximum enhancement that can be achieved by any single or multi-mode nonclassical state together with strong classical states, for local and distributed quantum metrology employing any linear or nonlinear single-mode unitary transformation. Our analysis reveals that the quantum Fisher information for quadrature-displacement sensing is the sole property that determines the maximum achievable enhancement in all of these different scenarios, providing a unified quantification of the metrological power.