Title: 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 k parties B1,..., Bk for every k. 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 CA and CB : The elements of the minimal tensor product of CA and CB are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product of CA and k copies of the cone CB for every k. 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 of CA and CB coincides with the set of k-extendible tensors for some finite k. We show that this is universally the case for every cone CA if and only if CB 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.
(joint work with Guillaume Aubrun and Martin Plávala)