A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls
Mohsen Kian
TL;DR
This work investigates the geometry of fundamental quantum cones, notably the separable cone $\mathscr{P}_+$ and the PPT cone $\mathcal{P}_{\mathrm{PPT}}$, under non-commutative $C^*$-convexity. The authors show a sharp dichotomy: the global $C^*$-convex hull of $\mathscr{P}_+$ collapses to the full PSD cone $\mathscr{P}_0$, while the local hull leaves separability intact. They introduce a $k$-$C^*$-convexity hierarchy based on operator Schmidt rank, establishing that $\operatorname{MCL}_k(\mathscr{P}_+) = \mathcal{T}_k$ (Schmidt-number cones) and $\operatorname{CMCL}_k(\mathscr{P}_+) = \mathcal{T}_k$, thereby providing an algebraic characterization of Schmidt-number structure. Applied to the PPT cone, they define $\mathcal{C}_{\mathrm{PPT},k} = \operatorname{MCL}_k(\mathcal{P}_{\mathrm{PPT}})$, proving stability at $k=1$ and collapse at $k=\min(m,n)$, with conjectured nontrivial intermediate cones for $1<k<d$. The framework offers a new lens for entanglement classification and potential tensor-rank witnesses for bound entanglement.
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, $\mathcal{C}_{\mathrm{PPT}, k}$.