Entanglement of Sections: The pushout of entangled and parameterized quantum information

• URL: http://arxiv.org/abs/2309.07245v2
• Date: Tue, 21 Nov 2023 08:48:16 GMT
• Title: Entanglement of Sections: The pushout of entangled and parameterized quantum information
• Authors: Hisham Sati and Urs Schreiber
• Abstract summary: Recently Freedman & Hastings asked for a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/-structure. We make precise a form of the relevant pushout diagram in monoidal category theory. We show how this model category serves as categorical semantics for the linear-multiplicative fragment of Linear Homotopy Type Theory.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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