Recall from Chap. 2, Sect. 2.6 that two or more agents can communicate using LOCC protocols [1], and that any quantum system accessed by multiple observers—an apparatus, or even a shared environment—can serve as a quantum channel. Here we develop these ideas with greater rigor, employing the formalism of topological quantum field theories (TQFTs) [2, 3], a generic, minimal assumption formalism for describing the unitary evolution of isolated systems through an assumed background time, i.e. the \(t_U\) introduced in Chap. 2. Formally, a TQFT is a functor (see Appendix A for definition) from the category \(\textbf{Cob}\) of cobordisms to the category \(\textbf{Vect}\) of vector spaces. A cobordism is a manifold of dimension \(n + 1\) whose boundary is a union of manifolds of dimension n; for example, a cylinder with two disks as its boundary. A TQFT is an assignment of vector spaces to the manifolds composing the boundary, and linear maps, i.e. vector-space homomorphisms, to the manifold enclosed by the boundary. In the TQFTs of interest here, the boundary vector spaces are Hilbert spaces of some quantum system S, and the manifold of maps comprises all components—all possible paths—by which S can evolve through time. The manifold of maps is often called the “bulk”; we will use this terminology here.