Purifications for Convex Cones
arXiv:2607.28202
The paper studies purification of states using the geometry of finite‑dimensional convex cones, proving that interior points of indecomposable homogeneous cones (e.g., Lorentz cones) always admit a purification and analyzing conditions for uniqueness and boundary behavior.
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 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, -positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.