Entangleability of cones

• Published in: Geometric and functional analysis Vol. 31; no. 2; pp. 181 - 205
• Main Authors: Aubrun, Guillaume, Lami, Ludovico, Palazuelos, Carlos, Plávala, Martin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2021; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Cones; Information theory; Mathematical analysis; Mathematics; Mathematics and Statistics; Quantum computing; Quantum phenomena; Tensors; Topology; Tensor product of cones; Secondary: 81P16; General probabilistic theories; Entangleability; Primary: 52A20; 47L07
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-021-00565-5

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 ⊗ x 2 , where x 1 ∈ C 1 and x 2 ∈ C 2 , while their maximal tensor product is the set of tensors that are positive under all product functionals φ 1 ⊗ φ 2 , where φ 1 | C 1 ⩾ 0 and φ 2 | 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.