  Recently Freedman & Hastings asked for 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. As a proposed answer to this question, we first make precise a form of
the relevant pushout diagram in monoidal category theory. Then we prove that
the pushout produces 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 bulk of our
result is a further homotopy-theoretic enhancement of the situation to the
"derived category" (infinity-category) of flat infinity-vector bundles
("infinity-local systems") equipped with the "derived functor" of the external
tensor product. Concretely, we present an integral model category of simplicial
functors into simplicial K-chain complexes which conveniently presents the
infinity-category of parameterized HK-module spectra over varying base spaces
and is equipped with homotopically well-behaved external tensor product
structure. In concluding we indicate how this model category serves as
categorical semantics for the linear-multiplicative fragment of Linear Homotopy
Type Theory (LHoTT), which is thus exhibited as a universal quantum programming
language. This is the context in which we recently showed that topological
anyonic braid quantum gates are native objects in LHoTT.
