A dedicated global model structure for K-linear ∞-local systems is constructed via simplicial chain complexes, monoidal for base 1-types under the external tensor product.
Entanglement of Sections: The pushout of entangled and parameterized quantum information
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abstract
A question raised by Freedman & Hastings (2023) 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 and through our work in quantum programming theory but has not otherwise found due attention in quantum theory yet.
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Hermitian forms on Hilbert spaces arise from the monoid structure of complex conjugation in Z/2-equivariant real linear types within LHoTT, requiring only a negative unit term.
Flux quantization of the M5-brane tensor field in twisted Cohomotopy yields Pontrjagin homology observables that reproduce abelian Chern-Simons theory and braid actions on defect anyons.
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A Global Model Structure for $\mathbb{K}$-Linear $\infty$-Local Systems
A dedicated global model structure for K-linear ∞-local systems is constructed via simplicial chain complexes, monoidal for base 1-types under the external tensor product.
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Quantum and Reality
Hermitian forms on Hilbert spaces arise from the monoid structure of complex conjugation in Z/2-equivariant real linear types within LHoTT, requiring only a negative unit term.
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Engineering of Anyons on M5-Probes via Flux Quantization
Flux quantization of the M5-brane tensor field in twisted Cohomotopy yields Pontrjagin homology observables that reproduce abelian Chern-Simons theory and braid actions on defect anyons.