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Abstract. Recently Freedman & Hastings asked [arXiv:2304.01072] 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 — a question motivated by the role that vector bundles of spaces of quantum states play in the K-theory classification of topological phases of matter (there: parameterized over the Brillouin torus).

In reply to this question, first we make precise a form of the relevant pushout-diagram in monoidal category theory. Then we prove that it 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. This external tensor product was recently highlighted in discussion of topological phases of matter by B. Mera (2020) but has not otherwise found due attention in quantum theory yet.

The bulk of our result is a further homotopy theoretic enhancement of the situation to the “derived category” ( ∞ \infty -category) of flat ∞ \infty -vector bundles (“ ∞ \infty -local systems”) equipped with the “derived functor” of the external tensor product. We explain how this serves as categorical semantics for the multiplicative fragment of the homotopically typed quantum programming language LHoTT. This is the generality in which we recently showed [arXiv:2303.02382] that topological anyonic braid quantum gates are native objects in the LHoTT-quantum programming language (in which case the parameterization is over the configuration space of defect anyons in the Brillouin zone).

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