Title: Entangleability of cones
Abstract: We prove that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set, solving a long-standing conjecture by Barker. Here, given two proper cones C1, C2, their minimal tensor product is the cone generated by products of the form x1 ⊗ x2, where x1 ∈ C1 and x2 ∈ C2, while their maximal tensor product is the set of tensors that are positive under all product functionals f1 ⊗ f2, where f1 is positive on C1 and f2 is positive on C2. Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. (Joint with Ludovico Lami, Carlos Palazuelos and Martin Plavala)