摘要
If the phase space-based Glauber-Sudarshan distribution,Pᵨ , has negative values the quantum state, ρ , it describes is nonclassical. Due toP '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, ξ[P] , for the detection of non-classical behaviour of quantum statesPᵨ . 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 functions is non-negative,ξ(x,p) ≥ 0 , everywhere in phase space. We generalizeξgiving 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.