An Easily Computable Measure of Gaussian Quantum Imaginarity

The resource-theoretic frameworks for quantum imaginarity have been developed in recent years. Within these frameworks, many imaginarity measures for finite-dimensional systems have been proposed. However, for imaginarity of Gaussian states in continuous-variable (CV) systems, there are only two known Gaussian imaginarity measures, which exhibit prohibitive computational complexity when applied to multi-mode Gaussian states. In this paper, a computable Gaussian imaginarity measure $I^{G_n}$ is proposed for $n$-mode Gaussian systems. The value of $I^{G_n}$ is simply formulated by the displacement vectors and covariance matrices of Gaussian states. A comparative analysis of $I^{G_n}$ with existing two Gaussian imaginarity measures indicates that $I^{G_n}$ can be used to detect imaginarity in any $n$-mode Gaussian states more efficiently. As an application, the dynamics behavior of $(1+1)$-mode Gaussian states is studied in Gaussian Markovian noise environments for two-mode CV system by utilizing $I^{G_2}$. Moreover, it is proved that, $I^{G_n}$ can induce a quantification of any $m$-multipartite multi-mode CV systems which satisfies all requirements for measures of multipartite multi-mode Gaussian correlations, which unveils that, $n$-mode Gaussian imaginarity can also be regarded as a kind of multipatite multi-mode Gaussian correlation and is a multipartite Gaussian quantum resource.