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

TL;DR

This paper introduces a hierarchy of entanglement cones using rank-constrained $C^*$-convex hulls, revealing new structural insights into quantum entanglement and separability through generalized convexity concepts.

Contribution

It develops a novel hierarchy of intermediate quantum cones via $k$-$C^*$-convexity, connecting to Schmidt number cones and proposing new conjectured structures for PPT cones.

Findings

The $C^*$-convex hull of the separable cone collapses to positive semidefinite matrices.

The hierarchy $ ext{MCL}_k(ullet)$ recovers known Schmidt number cones.

New conjectured intermediate cones for the PPT cone are proposed.

Abstract

This paper systematically investigates the geometry of fundamental quantum cones, the separable cone ($\mathscr{P}_+$) and the Positive Partial Transpose (PPT) cone ($\mathcal{P}_{\mathrm{PPT}}$), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing $C^*$-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, $\mathcal{C}_{\mathrm{PPT}, k}$.