Monogamy of entanglement between cones

A separable quantum state shared between parties $A$ and $B$ can be symmetrically extended to a quantum state shared between party $A$ and parties $B_1,\dots ,B_k$ for every $k\in \mathbf{N}$. 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 ${\mathsf{C}}_A$ and ${\mathsf{C}}_B$: The elements of the minimal tensor product ${\mathsf{C}}_A{\otimes }_{\min }{\mathsf{C}}_B$ are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product ${\mathsf{C}}_A{\otimes }_{\max }{\mathsf{C}}_B^{{\otimes }_{\max }k}$ for every $k\in \mathbf{N}$. 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 ${\mathsf{C}}_A{\otimes }_{\min }{\mathsf{C}}_B$ coincides with the set of $k$-extendible tensors for some finite $k$. We show that this is universally the case for every cone ${\mathsf{C}}_A$ if and only if ${\mathsf{C}}_B$ 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.