Cones with a mapping cone symmetry in the finite-dimensional case

Mark Girard, Seung-Hyeok Kye, Erling Størmer

Łukasz Skowronek

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.

Nathaniel Johnston, Łukasz Skowronek, Erling Størmer

Mohsen Kian, Fuad Kıttaneh

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.

Francesca La Piana, Alexander Müller‐Hermes

Linear maps between finite-dimensional ordered vector spaces with orders induced by proper cones C A and C B are called entanglement breaking if their partial application sends the maximal tensor product K ⊗ max C A into the minimal tensor product K ⊗ min C B for any proper cone K . We study the larger class of Lorentz-entanglement breaking maps where K is restricted to be a Lorentz cone of any dimension, i.e., any cone over a Euclidean ball. This class of maps appeared recently in the study of asymptotic entanglement annihilation and it is dual to the linear maps factoring through Lorentz cones. Our main results establish connections between these classes of maps and operator ideals studied in the theory of Banach spaces. For operators u : X → Y between finite-dimensional normed spaces X and Y we consider so-called central maps which are positive with respect to the cones C A = C X and C B = C Y . We show how to characterize when such a map factors through a Lorentz cone and when it is Lorentz-entanglement breaking by using the Hilbert-space factorization norm γ 2 and its dual ⁎ γ 2 ⁎ . We also study the class of Lorentz-entanglement annihilating maps whose local application sends the Lorentzian tensor product C A ⊗ L C A into the minimal tensor product C B ⊗ min C B . When C A is a cone over a finite-dimensional normed space and C B is a Lorentz cone itself, the central maps of this kind can be characterized by the 2-summing norm π 2 . Finally, we prove interesting connections between these classes of maps for general cones, and we identify examples with particular properties, e.g., cones with an analogue of the 2-summing property.

Guillaume Aubrun, Ludovico Lami, Carlos Palazuelos +1 more