A Sensitive Nonclassicality Certification Functional for Continuous-Variable Systems

Abstract

If the phase space-based Glauber-Sudarshan distribution, $P_{\varrho}$, has negative values the quantum state, $\varrho$, it describes is nonclassical. Due to $P$'s singular behaviour this simple criterion is impractical to use. Recent work [Bohmann and Agudelo, Phys. Rev. Lett. 124, 133601 (2020)] presented a general, sensitive, and noise-tolerant certification functional, $\xi[P]$, for the detection of non\-classical behaviour of quantum states $P_{\varrho}$. There, it was shown that when this functional takes on negative values somewhere in phase space, $\xi[P](x,p) < 0$, this is \emph{sufficient} to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification function is non-negative, $\xi(x,p) \geq 0$, everywhere in phase space. We generalize $\xi$, giving it an appealing form, ${\cal S}$, which allows for slight improvements in certification, but ${\cal S}$ also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of ${\cal S}$ over $\xi$ is that we showed how very sensitive $\xi$ and ${\cal S}$ are, and more important still is our simple derivation of ${\cal S}$ allowing us to generalize $\xi$ and ${\cal S}$ to multiple modes.