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

Abstract 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, $$D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,$$ where $\delta_p$ is the total Rényi–Wehrl deficit and $C_p$ 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, $D\asymp a+b$, 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: $C_p^{\rm opt}\sim4/[p\log(1/p)]$ as $p\downarrow0$. 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 $C_1=64/\log 2$ and $C_2=16$. 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 $p\log(1/p)C_p^{\rm opt}\to4$. 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.