Semiclassical Ehrenfest paths in open quantum systems

Referee: [Section 2] Derivation of the Fokker-Planck equation for the mixing measure (Section 2, around the explicit FP form): The closure of the FP equation for μ assumes the quantum state remains exactly representable as a Gaussian mixture at all times. This holds only for linear/quadratic open-system dynamics; for nonlinear H or non-linear jump operators the evolution produces non-Gaussianity, so the FP equation does not capture the full dynamics. This assumption is load-bearing for the central claim of a general phase-space picture.

Authors: The referee is correct that the Fokker-Planck equation for the mixing measure μ closes in explicit form precisely when the open-system dynamics preserve the Gaussian manifold, which occurs for quadratic Hamiltonians and linear Lindblad operators. Our derivation begins from the master equation under the Gaussian mixture ansatz and yields the closed FP equation for μ in that setting. For general nonlinear systems the representation becomes approximate, and the FP equation then supplies an effective semiclassical description rather than an exact evolution. In the revised manuscript we will add an explicit statement of these conditions in Section 2, together with a brief discussion of the approximate validity outside the quadratic regime. This qualification directly addresses the load-bearing assumption without altering the derivation itself. revision: yes

Referee: [Section 3] Embedding of the generalized Ehrenfest theorem (Section 3): The separation of coherent and irreversible contributions to observable expectations is performed with respect to the Gaussian mixture. Without explicit equations demonstrating that the embedding remains exact when the mixture evolves under the derived FP (rather than requiring further approximations), the microscopic separation is not guaranteed to be rigorous beyond the quadratic case.

Authors: The generalized Ehrenfest theorem (as cited) holds exactly for any open quantum system and already separates the coherent and irreversible contributions to d⟨A⟩/dt. We apply this theorem to expectation values taken with respect to the Gaussian mixture whose mixing measure evolves according to the derived FP equation. To make the embedding fully transparent, the revised Section 3 will contain the explicit substitution of the FP drift and diffusion terms into the Ehrenfest relation, confirming that the separation follows directly from the theorem and the FP evolution without further approximations inside the mixture representation. When the state remains Gaussian the separation is exact; otherwise it is consistent with the semiclassical approximation employed throughout the work. We therefore view the embedding as rigorous within the stated framework. revision: partial

Referee: [Introduction] Scope of the claims (Introduction and Conclusion): The abstract and introduction present the method for 'open quantum systems' without qualification, yet the Gaussian mixture representation with closed FP evolution fails to be preserved under general Lindblad dynamics. This overstatement of applicability is central to the claimed 'transparent interpretation' and requires either explicit restriction to quadratic systems or additional justification.

Authors: We acknowledge that the abstract, introduction, and conclusion employ the phrase 'open quantum systems' without sufficient qualification regarding the exactness of the Gaussian-mixture closure. The manuscript's central contribution is a phase-space interpretation that is exact for quadratic open systems and serves as a transparent semiclassical tool more generally. In the revision we will modify the abstract, introduction, and conclusion to state the precise conditions for exact closure (quadratic Hamiltonians and linear dissipators) while noting the utility of the approach as an interpretive framework for broader classes of open dynamics. These changes will align the claims with the mathematical scope and preserve the claimed transparent view within the appropriate regime. revision: yes