Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.28202

Purifications for Convex Cones

Felix Campidell, Tim Netzer

quant-phmath.FASubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations
arXivHTMLPDF

Abstract

Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of $C$ is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, $k$-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.

Related papers

Annihilating Entanglement Between Cones

Guillaume Aubrun, Alexander Müller‐Hermes

Abstract Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform thek-fold maximal tensor product of a cone $${\textsf{C}}_1$$ C1 into thek-fold minimal tensor product of a cone $${\textsf{C}}_2$$ C2 , and the pair $$({\textsf{C}}_1,{\textsf{C}}_2)$$ (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that $$({\textsf{C}}_1,{\textsf{C}}_2)$$ (C1,C2) is resilient if either $${\textsf{C}}_1$$ C1 or $${\textsf{C}}_2$$ C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

arXiv:2110.11825•Communications in Mathematical Physics•Dec 27, 2022•8 citations

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

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

Convex semidefinite tensor optimization and quantum entanglement

Liding Xu, Ye-Chao Liu, Sebastian Pokutta

The cone of positive-semidefinite (PSD) matrices is fundamental in convex optimization, and we extend this notion to tensors, defining PSD tensors, which correspond to separable quantum states. We study the convex optimization problem over the PSD tensor cone. While this convex cone admits a smooth reparameterization through tensor factorizations (analogous to the matrix case), it is not self-dual. Moreover, there are currently no efficient algorithms for projecting onto or testing membership in this cone, and the semidefinite tensor optimization problem, although convex, is NP-hard. To address these challenges, we develop methods for computing lower and upper bounds on the optimal value of the problem. We propose a general-purpose iterative refinement algorithm that combines a lifted alternating direction method of multipliers with a cutting-plane approach. This algorithm exploits PSD tensor factorizations to produce heuristic solutions and refine the solutions using cutting planes. Since the method requires a linear minimization oracle over PSD tensors, we design a spatial branch-and-bound algorithm based on convex relaxations and valid inequalities. Our framework allows us to study the white-noise mixing threshold, which characterizes the entanglement properties of quantum states. Numerical experiments on benchmark instances demonstrate the effectiveness of the proposed methods.

math.OCCross-listedarXiv:2511.05258•Nov 10, 2025•0 citations

Entanglement in the Dicke subspace

Aabhas Gulati, Ion Nechita, Clément Pellegrini

We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.

quant-pharXiv:2602.15800•Quantum•Feb 18, 2026•1 citation

Classical theories with entanglement

Giacomo Mauro D’Ariano, Marco Erba, Paolo Perinotti

We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e., simplicial theories whose pure states are jointly perfectly discriminable. The usual classical theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving noncausal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement and classify their system-composition rules. We conclude by proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of $n$-local discriminability---no mixed state admits a purification. Our results hold also in the general case where the sets of states fail to be convex.

arXiv:1909.07134•Physical Review A•Apr 20, 2020•23 citations
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of CCC is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, kkk-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.
Abstract Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform thek-fold maximal tensor product of a cone C1{\textsf{C}}_1C1​ C1 into thek-fold minimal tensor product of a cone C2{\textsf{C}}_2C2​ C2 , and the pair (C1,C2)({\textsf{C}}_1,{\textsf{C}}_2)(C1​,C2​) (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that (C1,C2)({\textsf{C}}_1,{\textsf{C}}_2)(C1​,C2​) (C1,C2) is resilient if either C1{\textsf{C}}_1C1​ C1 or C2{\textsf{C}}_2C2​ C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.
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.
The cone of positive-semidefinite (PSD) matrices is fundamental in convex optimization, and we extend this notion to tensors, defining PSD tensors, which correspond to separable quantum states. We study the convex optimization problem over the PSD tensor cone. While this convex cone admits a smooth reparameterization through tensor factorizations (analogous to the matrix case), it is not self-dual. Moreover, there are currently no efficient algorithms for projecting onto or testing membership in this cone, and the semidefinite tensor optimization problem, although convex, is NP-hard. To address these challenges, we develop methods for computing lower and upper bounds on the optimal value of the problem. We propose a general-purpose iterative refinement algorithm that combines a lifted alternating direction method of multipliers with a cutting-plane approach. This algorithm exploits PSD tensor factorizations to produce heuristic solutions and refine the solutions using cutting planes. Since the method requires a linear minimization oracle over PSD tensors, we design a spatial branch-and-bound algorithm based on convex relaxations and valid inequalities. Our framework allows us to study the white-noise mixing threshold, which characterizes the entanglement properties of quantum states. Numerical experiments on benchmark instances demonstrate the effectiveness of the proposed methods.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.
We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e., simplicial theories whose pure states are jointly perfectly discriminable. The usual classical theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving noncausal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement and classify their system-composition rules. We conclude by proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of nnn-local discriminability---no mixed state admits a purification. Our results hold also in the general case where the sets of states fail to be convex.