Entangleability of cones

TL;DR

This paper proves a conjecture about tensor products of cones, showing they coincide only when one cone is generated by a linearly independent set, with implications for entanglement in physical systems.

Contribution

It resolves a long-standing conjecture by characterizing when minimal and maximal tensor products of cones are equal, using convex geometry and algebraic topology.

Findings

Minimal and maximal tensor products of cones coincide only under specific conditions.

Any two non-classical systems in probabilistic theories can be entangled.

The proof combines convex geometry, algebraic topology, and quantum information techniques.

Abstract

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 $f_1 \otimes f_2$, where $f_1$ is positive on $C_1$ and $f_2$ is positive on $C_2$. 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.