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 \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where \(x_1\in {\mathcal {C}}_1\) and \(x_2\in {\mathcal {C}}_2\), while their maximal tensor product is the set of tensors that are positive under all product functionals \(\varphi _1\otimes \varphi _2\), where \(\varphi _1|_{{\mathcal {C}}_1}\geqslant 0\) and \(\varphi _2|_{{\mathcal {C}}_2}\geqslant 0\). 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.

我们用Barker解决了一个长期的猜想，证明了当且仅当两个圆锥中的一个由线性独立集合生成时，两个有限维固有圆锥的最小和最大张量积才重合。在此，给定两个适当的圆锥\（{\ mathcal {C}} _​​ 1 \），\（{\ mathcal {C}} _​​ 2 \），它们的最小张量积是形式为\ {x_1 \ otimes的积生成的圆锥x_2 \），其中\（x_1 \ in {\ mathcal {C}} _​​ 1 \）和\（x_2 \ in {\ mathcal {C}} _​​ 2 \）中，而它们的最大张量积是一组正数张量在所有产品功能\（\ varphi _1 \ otimes \ varphi _2 \）下，其中\（\ varphi _1 | _ {{\ mathcal {C}} _​​ 1} \ geqslant 0 \）和\（\ varphi _2 | _ {{\ mathcal {C}} _​​ 2} \ geqslant 0 \）。我们的证明技术涉及凸几何，基本代数拓扑以及受量子信息理论启发的计算的混合。我们的动力来自物理学的基础：作为一种应用，我们证明了由一般概率理论建模的任何两个非经典系统都可以被纠缠。