From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability

When can a quantum optical source be replaced by a coherent state with controlled error in every subsequent experiment? We turn its Husimi phase-space entropy excess into an error bound for the entire quantum state. For every fixed Rényi order $p>0$ and every density operator on any finite number of modes, D(ρ)²≤ Cₚδₚ(ρ), D(ρ)=inf_α\|ρ-|αα|\|₁, where δₚ is the total Rényi–Wehrl deficit and Cₚ is independent of mode count, energy, rank and photon-number support. One coherent replacement then bounds every measurement-probability error after any quantum channel. We identify the source structure that determines the finer error near the coherent manifold: after optimising displacement, D a+b, where a is the population outside the reference vacuum and b its coherence with the remaining state. This distinguishes linear thermal response from square-root squeezing response and yields intermediate rates. The guarantee holds throughout the positive-order range, but its optimal coefficient necessarily diverges at small order: Cₚᵒᵖᵗ~4/[plog(1/p)] as p↓0. A reduction of fractional Husimi moments to one active mode proves this sharp limit. The results specify both when entropy certifies a multimode coherent approximation and how its accuracy depends on the structure of the source. Version 1.1 — 9 October 2026 This revision extends the original mode-uniform Wehrl theorem to every positive Rényi order and develops its use as a coherent-state approximation guarantee. All positive orders and arbitrary density operators. The full trace-norm bound is uniform over every finite mode number, without a finite-energy assumption. Explicit sufficient coefficients include C₁=64/log 2 and C₂=16. Source-dependent accuracy. A local population–coherence law separates thermal, squeezed and intermediate responses. The same coherent reference controls subsequent quantum channels and measurement probabilities. Sharp small-order limit. The optimal global coefficient satisfies plog(1/p)Cₚᵒᵖᵗ→4. The complete proof uses a one-active-mode reduction of fractional Husimi moments. Expanded presentation. Two vector figures explain source-dependent error, the role of total entropy deficit and the order-dependent bounds. The 19-page PDF includes the complete manuscript, proofs and updated references. This is an expanded revision of Uniform Wehrl stability across all bosonic mode numbers: total entropy deficit controls full trace-norm distance to coherent states, first published on 6 October 2026.