Quadratic Stability of Entropy Minimizers under Block-Separable Convex Constraints

We investigate entropy minimization problems for quantum states subject to convex block-separable constraints. Our principal result is a quantitative stability theorem: under a natural confining (fixed-support) hypothesis, if a state has entropy within ε of the minimum permitted by the constraint, then it must lie within Oε^1/2 in trace norm of the set of entropy minimizers. We show that this rate is optimal and cannot be improved uniformly. The analysis is entirely finite-dimensional and exploits the block-separable structure of the constraint set, which induces a natural decomposition of entropy into a marginal (classical) component and conditional (internal) components. Quadratic stability emerges from the curvature of Shannon entropy on the marginal polytope and of von Neumann entropy on the constrained block states, yielding explicit stability constants determined by the geometry of the constraint. We further demonstrate that this stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed a priori and the entropy minimizer arises intrinsically from the constraint geometry. The framework is abstract and independent of any arithmetic input, and provides a general quadratic stability principle for entropy minimization under structured convex constraints.