Peer review Referee Report on Is it Gaussian Testing bosonic quantum states David Gross Princeton

This work presents a formal critical assessment of the article “Is it Gaussian? Testing bosonic quantum states” by Girardi et al. The manuscript under review develops operational tests and sample complexity bounds for identifying Gaussian bosonic quantum states based on invariance under orthogonal transformations. While the mathematical analysis is internally consistent within the stated assumptions, the present assessment identifies a fundamental conceptual limitation. Specifically, the original work implicitly treats Gaussianity as a physically privileged property without deriving it from any variational principle, dynamical closure condition, or stability requirement. The proposed criteria therefore characterize representational symmetry rather than physical necessity. It is shown that invariance under transformation groups does not imply physical relevance, and that many dynamically stable and experimentally admissible non-Gaussian states are excluded by construction. Moreover, the exponential intractability of Gaussianity testing for mixed states, explicitly acknowledged in the original manuscript, indicates that Gaussianity is not a robust physical invariant in realistic noisy systems. The absence of demonstrated dynamical closure further undermines the interpretation of Gaussianity as a structurally meaningful category. The conclusion is that the work under review addresses a well-posed methodological problem but does not establish Gaussianity as a physically necessary or fundamental property of bosonic quantum states. The results are therefore mathematically valid but physically unmotivated in their current form.