A Hierarchy of Entanglement Cones via Rank-Constrained $C^{\ast }$-Convex Hulls

Abstract

This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (${\mathcal{P}}_{+}$) and the Positive Partial Transpose (PPT) cone (${\mathcal{P}}_{\mathrm{PPT}}$), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing $C^{\ast }$-convex hulls of these cones: while ${\mathcal{P}}_{+}$ remains stable under local $C^{\ast }$-convex combinations, its global $C^{\ast }$-convex hull collapses entirely to the cone of all positive semidefinite matrices, $\mathrm{MCL}({\mathcal{P}}_{+})={\mathcal{P}}_0$. To gain finer control and classify intermediate structures, we introduce the concept of ``$k$-$C^{\ast }$-convexity'', by using the operator Schmidt rank of $C^{\ast }$-coefficients. This constraint defines a new hierarchy of nested intermediate cones, ${\mathrm{MCL}}_k(\cdot )$. We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: ${\mathrm{MCL}}_k({\mathcal{P}}_{+})={\mathcal{T}}_k$. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, ${\mathcal{C}}_{\mathrm{PPT},k}$.