Entropy Stability under Block-Separable Constraints: Fixed-Weight Rank-Lattice Envelopes and Convexification

Abstract

For block-diagonal quantum states with prescribed block weights and dimensions, we determine the minimum von Neumann entropy excess at each trace-norm distance from the blockwise-pure states. The one-block extremal is the classical reverse-Fano profile; the new problem is its constrained assembly when the block weights and dimensions cannot be redistributed. The resulting value function $Φ_{q,n}(D)$ is attained on the full admissible interval and is the lower envelope of finitely many rank-lattice edges. Every minimizing distance allocation has at most one block between consecutive integer ranks, and the active edge can switch nonmonotonically with $D$. We also determine the lower convex envelope of $Φ_{q,n}$ explicitly: it is the polygonal curve through synchronized rank vertices and admits a one-parameter dual representation. The exact nonconvex envelope differs from this convex surrogate by at most $q_\diamond\log(5/4)$, with a sharper stage-dependent bound, and the constant is optimal. For binomial charge-sector weights the uniform gap is $O(L^{-1/2})$. The framework yields a symmetry-sector purity certificate; in a three-qubit phase-damping example the fixed-weight certificate remains nontrivial where the universal classical entropy-error envelope is already vacuous.