Uniform Wehrl stability across all bosonic mode numbers: total entropy deficit controls full trace-norm distance to coherent states

Quantitative Wehrl stability is known at each fixed number of bosonic modes; the multimode results reviewed here allow dimension dependence. We prove a full-state stability theorem with one constant that works simultaneously for every finite mode count. For every $m$-mode bosonic density operator $\rho$ with finite mean total photon number, $$\inf_{\alpha\in\mathbb{C}^m}\|\rho-|\alpha\rangle\langle\alpha|\|_1^2\le120\,[W(\rho)-m].$$ The same numerical constant works simultaneously for every finite $m$, independently of the total energy, rank, and occupation support. Thus a vanishing total Wehrl entropy deficit forces convergence of the entire quantum state to a coherent state even when the number of modes varies; by contrast, a vanishing deficit per mode is not sufficient, as shown by an explicit fixed-energy thermal family. The resulting coherent reference is a genuine full-state approximation: the same trace-norm error propagates through every subsequent quantum channel and controls every measurement probability. The proof first converts Wehrl deficit into quantum impurity through a dimension-free directional entropy estimate for a field of rank-one projections, $$1-\operatorname{Tr}\rho^2\le\frac{4}{\log2}\,[W(\rho)-m].$$ A beam-splitter Wehrl-entropy chain rule then links this purity estimate to the known two-copy coherent-state test, yielding the full mixed-state result. As a further consequence, the same bound controls the Rényi entropy of order two generated when a pure noncoherent input is split with vacuum. The theorem therefore turns total Wehrl entropy into a mode-independent error budget for coherent-state replacement in multimode bosonic quantum systems.