Classical-Nonclassical Polarity of Gaussian States*

Gaussian states with nonclassical properties such as squeezing and entanglement serve as crucial resources for quantum information processing. Accurately quantifying these properties within multi-mode Gaussian states has posed some challenges. To address this, we introduce a unified quantification: the 'classical-nonclassical polarity', represented by $mathcal{P}$.

For a single mode, a positive value of $mathcal{P}$ captures the reduced minimum quadrature uncertainty below the vacuum noise, while a negative value represents an enlarged uncertainty due to classical mixtures. For multi-mode systems, a positive $mathcal{P}$ 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 multi-mode 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 such as the consersion between varied kinds of nonclassical properties. Moreover, the conservation relation could give rise to a solution of high-mode entanglement quantification in terms of Gaussian states.