A question raised by Freedman & Hastings [16] still stands: To produce a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/bundle-structure via their amalgamation (a hypothetical pushout) along bare quantum (information) theory—a question motivated by the role that vector bundles of spaces of quantum states play in the K-theoretic classification of topological phases of matter. Here, we produce a possible answer to this question. To that end, first, we make precise a form of the relevant pushout diagram in monoidal category theory. With the question thus formalized, we proceed to compute this pushout and prove that it gives what is known as the external tensor product on vector bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those equipped with monodromy encoding topological Berry phases. The external tensor product was recently highlighted in the context of topological phases of matter in [36] and through our work in quantum programming theory [49] but has not otherwise found due attention in quantum theory yet.
