A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls

Abstract: This paper systematically investigates the geometry of fundamental quantum cones, the separable cone ($\mathscr{P}<em>+$) and the Positive Partial Transpose (PPT) cone ($\mathcal{P}</em>{\mathrm{PPT}}$), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing $C<sup>*$-convex hulls of these cones: while $\mathscr{P}<em>+$ remains stable under local $C<sup>*$-convex combinations, its global $C<sup>*$-convex hull collapses entirely to the cone of all positive semidefinite matrices, $$\operatorname{MCL}(\mathscr{P}</em>+) = \mathscr{P}<em>0$$. To gain finer control and classify intermediate structures, we introduce the concept of ``$k$-$C<sup>*$-convexity'', by using the operator Schmidt rank of $C<sup>*$-coefficients. This constraint defines a new hierarchy of nested intermediate cones, $\operatorname{MCL}_k(\cdot)$. We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: $$\operatorname{MCL}_k(\mathscr{P}</em>+) = \mathcal{T}<em>k$$. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, $\mathcal{C}</em>{\mathrm{PPT}, k}$.