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 behavior, this simple criterion is impractical to use. Recent work has presented a general, sensitive, and noise-tolerant certification functional for the detection of nonclassical behavior of quantum states $\varrho$. There, it was shown that when this functional takes on negative values somewhere in phase space, $ξ_{P}(x,p) < 0$, this is 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 nonnegative, $ξ_{P}(x,p) \geq 0$, everywhere in phase space. We generalize $ξ$, 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 $ξ$ is that we showed how very sensitive $ξ$ and $\cal S$ are, and more important still is our simple derivation of $\cal S$ allowing us to generalize $ξ$ and $\cal S$ to multiple modes.