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 B₁,… ,B_k for every k∈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 C_A and C_B: The elements of the minimal tensor product C_A⊗_min C_B are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product C_A⊗_max C^⊗_max k_B for every k∈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 C_A⊗_min 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 C_A if and only if 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.