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Reports on Mathematical Physics

The convex structure of the cone of N-fermion positive-semidefinite 1-particle operators

Hubert Grudziński, Jacek Hirsch

Published Jun 1, 2000• 0 citations
Publisher
DOI
10.1016/s0034-4877(00)80002-6

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

Mohsen Kian

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, $\mathcal{C}_{\mathrm{PPT}, k}$.

math-phCross-listedarXiv:2512.05560•Dec 8, 2025•0 citations

On the theory of fermion density operators. I

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Factorization through Lorentz cones

Guillaume Aubrun, Francesca La Piana, Alexander Müller-Hermes

A pair of proper cones $(\mathsf{C}_1,\mathsf{C}_2)$ is said to have the Lorentz factorization property (LFP) if every $(\mathsf{C}_1,\mathsf{C}_2)$-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, $(\mathsf{C}_1,\mathsf{C}_2)$ has the LFP if either $\mathsf{C}_1$ or $\mathsf{C}_2$ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs $(\mathsf{C}_1,\mathsf{C}_2)$ where $\mathsf{C}_1=\mathsf{C}_2$, or in the case where both $\mathsf{C}_1$ and $\mathsf{C}_2$ are polyhedral. We also focus on the case where $\mathsf{C}_1=\mathsf{C}_\square$ is the square-based cone in $\mathbf{R}^3$. Here, we show that $(\mathsf{C}_\square,\mathsf{C})$ has the LFP whenever $\mathsf{C}$ is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of $3\times 3$ positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone $\mathsf{C}$ or for a cone $\mathsf{C}$ with $\text{dim}(\mathsf{C})\leq 5$. Finally, we discuss an application to a problem in quantum information theory.

math.FACross-listedarXiv:2606.27825•Jun 29, 2026

On the Averages of Operators in Finite Fermion Systems

Joseph N. Ginocchio

Springer eBooks•Jan 1, 1980•1 citation

Semidefinite descriptions of separable matrix cones

Roland Hildebrand

Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.

arXiv:math/0701571•Jan 20, 2007•1 citation
A Hierarchy of Entanglement Cones via Rank-Constrained C∗C^*C∗-Convex Hulls
This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (P+\mathscr{P}_+P+​) and the Positive Partial Transpose (PPT) cone (PPPT\mathcal{P}_{\mathrm{PPT}}PPPT​), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C∗C^*C∗-convex hulls of these cones: while P+\mathscr{P}_+P+​ remains stable under local C∗C^*C∗-convex combinations, its global C∗C^*C∗-convex hull collapses entirely to the cone of all positive semidefinite matrices, MCL⁡(P+)=P0\operatorname{MCL}(\mathscr{P}_+) = \mathscr{P}_0MCL(P+​)=P0​. To gain finer control and classify intermediate structures, we introduce the concept of ``kkk-C∗C^*C∗-convexity'', by using the operator Schmidt rank of C∗C^*C∗-coefficients. This constraint defines a new hierarchy of nested intermediate cones, MCL⁡k(⋅)\operatorname{MCL}_k(\cdot)MCLk​(⋅). We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: MCL⁡k(P+)=Tk\operatorname{MCL}_k(\mathscr{P}_+) = \mathcal{T}_kMCLk​(P+​)=Tk​. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, CPPT,k\mathcal{C}_{\mathrm{PPT}, k}CPPT,k​.
A pair of proper cones (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) is said to have the Lorentz factorization property (LFP) if every (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​)-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) has the LFP if either C1\mathsf{C}_1C1​ or C2\mathsf{C}_2C2​ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) where C1=C2\mathsf{C}_1=\mathsf{C}_2C1​=C2​, or in the case where both C1\mathsf{C}_1C1​ and C2\mathsf{C}_2C2​ are polyhedral. We also focus on the case where C1=C□\mathsf{C}_1=\mathsf{C}_\squareC1​=C□​ is the square-based cone in R3\mathbf{R}^3R3. Here, we show that (C□,C)(\mathsf{C}_\square,\mathsf{C})(C□​,C) has the LFP whenever C\mathsf{C}C is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of 3×33\times 33×3 positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone C\mathsf{C}C or for a cone C\mathsf{C}C with dim(C)≤5\text{dim}(\mathsf{C})\leq 5dim(C)≤5. Finally, we discuss an application to a problem in quantum information theory.
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.