Verifying single-mode nonclassicality beyond negativity in phase space

While negativity in phase space is a well-known signature of nonclassicality, wide variety nonclassical states require their characterization beyond negativity. We establish framework nonclassicality that addresses comprehensively with direct experimental evidence. This includes the phase-space distribution as special case and further analyzes quantum positive distributions effectively. prove it detects all Gaussian non-Gaussian arbitrary dimension remarkably by examining three points only. Our formalism also provides an experimentally accessible lower bound for measure based on trace distance. Importantly, this foundational approach can be adapted to constitute practical tests two directions looking into particle wave nature bosonic systems, via array nonideal on-off detectors coarse-grained homodyne measurement, respectively. All these are practically powerful characterizing nonclasssical reliably against noise, making versatile tool broad range systems technologies.