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, ${\xi }_P(x,p)<0$, 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, ${\xi }_P(x,p)\ge 0$, everywhere in phase space. We generalize $\xi$, giving it an appealing form, $\mathcal{S}$, which allows for slight improvements in certification, but $\mathcal{S}$ also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of $\mathcal{S}$ over $\xi$ is that we showed how very sensitive $\xi$ and $\mathcal{S}$ are, and more important still is our simple derivation of $\mathcal{S}$ allowing us to generalize $\xi$ and $\mathcal{S}$ to multiple modes.

arXiv comment10 pages, 3 Figures