Entangleability of cones

We solve a long-standing conjecture by Barker, proving 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. Here, given two proper cones $\C_1$, $\C_2$, their minimal tensor product is the cone generated by products of the form $x_1\otimes x_2$, where $x_1\in \C_1$ and $x_2\in \C_2$, while their maximal tensor product is the set of tensors that are positive under all product functionals $\varphi_1\otimes \varphi_2$, where $\varphi_1|_{\C_1}\geq 0$ and $\varphi_2|_{\C_2}\geq 0$. Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.