Geometric Rigidity of Entropy Minimizers under Block Constraints

We establish a sharp quantitative stability theorem for the von Neumann entropy under a natural class of convex constraints induced by orthogonal block decompositions of finite-dimensional Hilbert spaces. Given a block–convex set of block-diagonal quantum states, we prove that any state whose entropy is within ε of the minimal value must lie within O(ε)O(\sqrt{\varepsilon})O(ε) in trace norm of the manifold of entropy minimizers. The stability exponent is shown to be optimal, and purely quadratic stability is characterized precisely in the confining (fixed-support) regime. The proof relies on a canonical decomposition of entropy into classical (Shannon) and conditional components, combined with sharp relative entropy estimates and geometric properties of the marginal polytope. As an application, we obtain quantitative non-concentration bounds for spectral measures associated with finite spectral decompositions of self-adjoint operators. The framework is entirely finite-dimensional, independent of arithmetic or dynamical assumptions, and provides a general geometric stability principle for entropy minimization under structured convex constraints.