Entangleability of Cones
byGuillaume Aubrun
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. (Joint work with Ludovico Lami, Carlos Palazuelos, Martin Plavala)
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