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

This paper systematically investigates the geometry of fundamental quantum cones, the separable cone $mathscr{P}_+$ and the Positive Partial Transpose (PPT) cone $mathcal{P}_PPT$, under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C^*-convex hulls of these cones: while mathscr{P}_+ remains stable under local C^*-convex combinations, its global C^*-convex hull collapses entirely to the cone of all positive semidefinite matrices, operatorname{MCL}\(mathscr{P}_+\) = mathscr{P}₀. To gain finer control and classify intermediate structures, we introduce the concept of "k-C^*-convexity", by using the operator Schmidt rank of C^*-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\(mathscr{P}_+\) = mathcal{T}_k. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, mathcal{C}_{PPT, k}.