Monogamy of entanglement between cones

Abstract

A separable quantum state shared between parties A and B can be symmetrically extended to a quantum state shared between party A and parties \(B_1,\ldots ,B_k\) for every \(k\in \textbf{N}\). Quantum states that are not separable, i.e., entangled, do not have this property. This phenomenon is known as “monogamy of entanglement”. We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones \(\textsf{C}_A\) and \(\textsf{C}_B\): The elements of the minimal tensor product \(\textsf{C}_A\otimes _{\min } \textsf{C}_B\) are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product \(\textsf{C}_A\otimes _{\max } \textsf{C}^{\otimes _{\max } k}_B\) for every \(k\in \textbf{N}\). Equivalently, the minimal tensor product of two cones is the intersection of the nested sets of k-extendible tensors. It is a natural question when the minimal tensor product \(\textsf{C}_A\otimes _{\min } \textsf{C}_B\) coincides with the set of k-extendible tensors for some finite k. We show that this is universally the case for every cone \(\textsf{C}_A\) if and only if \(\textsf{C}_B\) is a polyhedral cone with a base given by a product of simplices. Our proof makes use of a new characterization of products of simplices up to affine equivalence that we believe is of independent interest.