We develop a finite-corner framework for convex cones of normal maps between \(\mathcal B(\mathcal H)\) and \(\mathcal B(\mathcal K)\). The basic structural assumption is stability under finite cut--pad operations. We prove that every point-ultraweakly closed cut--pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut--pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.
We develop a finite-corner framework for convex cones of normal maps between (\mathcal B(\mathcal H)) and (\mathcal B(\mathcal K)). The basic structural assumption is stability under finite cut--pad operations. We prove that every point-ultraweakly closed cut--pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut--pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.
A Hierarchy of Entanglement Cones via Rank-Constrained C∗-Convex Hulls
This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (P+) and the Positive Partial Transpose (PPT) cone (PPPT), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C∗-convex hulls of these cones: while P+ remains stable under local C∗-convex combinations, its global C∗-convex hull collapses entirely to the cone of all positive semidefinite matrices, MCL(P+)=P0. 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, MCLk(⋅). We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: MCLk(P+)=Tk. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, CPPT,k.
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.
In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Stormer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamiolkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Stormer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.
We study mapping cones of positive maps of B(H) into itself, i.e., cones which are closed under composition with completely positive maps. As applications we obtain characterizations of linear functionals with strong positivity properties with respect to so-called symmetric mapping cones, with special emphasis on separable and PPT states.
A state over time (stote) pairs a state ρ with a channel E as the Hermitian, generally non-positive operator (ρ⊗1)⋆JE. Characterizing the convex cone Q these operators generate was posed as an open problem in Hoogsteder-Riera, Calsamiglia and Winter (2026). We exhibit a linear copy of the bipartite separable cone as an exposed face of Q: for output systems of the form B=A′R with A′ a copy of the input, there is an explicit witness W0∈Q∗ and an explicit injective linear map L with σ∈SEP+(A:R)⟺L(σ)∈Q(A:A′R). Composing with the NP-hardness of separability shows that exact membership, weak membership, and weak optimization are NP-hard for Q and its dual. A quantitative stability bound makes the weak-membership reduction valid at inverse-polynomial accuracy. Thus polynomial-time membership tests do not exist unless P=NP. Independently of this complexity assumption, Fawzi's theorem implies that Q(A:A′R) admits no finite semidefinite representation when d=dimA≥2, r=dimR≥2, and d+r>5.