Towards a Faithful Quantumness Certification Functional for One-Dimensional Continuous-Variable Systems

If the phase space-based Glauber-Sudarshan distribution, $P_{\rho}$, has negative values the quantum state,~$\rho$, 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 $P_{\rho}$. 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 functions is non-negative, $\xi(x,p) \geq 0$, everywhere in phase space. We generalize $\xi$ giving it an appealing form which allows for improved certification. This way we generate a more sensitive family of certification functions. Yet, also these fail for very weakly nonclassical states, the question how to faithfully certify quantumness remains an open question.