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) circle times x(2), where x(1) is an element of C-1 and x(2) is an element of C-2, while their maximal tensor product is the set of tensors that are positive under all product functionals phi(1) circle times phi(2), where phi(1)vertical bar C-1 >= 0 and phi(2)vertical bar C-2 >= 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.

Automated summary

This research solves a long-standing conjecture by Barker, showing that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the cones is generated by a linearly independent set. The proof techniques involve a combination of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. The motivation behind the study comes from foundational physics, demonstrating that any two non-classical systems modeled by general probabilistic theories can be entangled.