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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2512.05560 |
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| _version_ | 1866912756691632128 |
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| author | Kian, Mohsen |
| author_facet | Kian, Mohsen |
| contents | 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, $\mathcal{C}_{\mathrm{PPT}, k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls Kian, Mohsen Mathematical Physics Functional Analysis Quantum Physics 15A69, 46L07, 81P45 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, $\mathcal{C}_{\mathrm{PPT}, k}$. |
| title | A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls |
| topic | Mathematical Physics Functional Analysis Quantum Physics 15A69, 46L07, 81P45 |
| url | https://arxiv.org/abs/2512.05560 |