Tensor products and teleportation protocols for abstract state spaces

In a well-known generalization of classical probability theory, arbitrary compact convex sets serve as abstract ``state spaces'' for (hypothetical) physical systems, with classical systems corresponding to simplices and quantum systems, to state spaces of C*-algebras. One can define natural tensor products for abstract state spaces, modeling composite systems subject to a no-signaling condition. Remarkably, many basic quantum-information theoretic phenomena, including the no-cloning and no-broadcasting theorems, already appear at this level of generality. However, the existence of a teleportation protocol is a strong constraint, moving us closer to quantum theory. In this talk, after a brief summary of the framework, I will outline what we currently understand about teleportation in this setting. This represents recent and ongoing joint work with Howard Barnum, Jonathan Barrett and Matthew Leifer