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, \(\xi[P]\), 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, \(\xi(x,p) \ge 0\), 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.
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A Sensitive Nonclassicality Certification Functional for Continuous-Variable Systems
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Received: 16 Apr 2026 | Revised: 12 Aug 2026 | Accepted: 01 Sep 2026 | Published: 18 Sep 2026
Abstract
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