Quadratic Stability of Entropy Minimizers under Block-Separable Convex Constraints

We study quadratic stability of entropy minimization under convex constraints with block-separable structure. For constraint sets of the formC = ⊕ᵢ pᵢ ρᵢ: p ∈ Π, ρᵢ ∈ Cᵢ, we establish explicit stability estimates showing that, under a confining (fixed-support) hypothesis, the entropy gap controls the squared trace-norm distance to the set of entropy minimizers, with constants determined by the geometry of the constraint. The stability constant admits a natural decomposition into marginal and conditional components. The marginal contribution is governed by the curvature of Shannon entropy on the marginal polytope Π at extreme points, while the conditional contribution is governed by the curvature of von Neumann entropy at entropy-minimizing states within each block. We prove that the quadratic exponent is optimal by constructing explicit examples for which no linear stability bound holds uniformly. Exploiting the block-separable structure, the analysis reduces the global stability problem to independent marginal and conditional subproblems, yielding a geometric characterization of the stability constants in terms of constraint curvature. This stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed and the minimizer emerges intrinsically from the constraint geometry.