Schreiber Entanglement of Sections

An article that we are finalizing:

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• Urs Schreiber$\;$ with $\;$ Hisham Sati $\;$ at $\;$ CQTS:

  
  

  Entanglement of Sections:

  The pushout of entangled and parameterized quantum information

  
  

  download:

  • arXiv:2309.07245

  • pdf

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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. Concretely, we present an integral model category of simplicial functors into simplicial $\mathbb{K}$-chain complexes which conveniently presents the $\infty$-category of parameterized $H\mathbb{K}$-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 “Motivic Yoga”-fragment of the recently constructed linear homotopy type theory (LHoTT, [Ri22]), which in [SS23b] we describe as a quantum programming language with classical control, dynamic lifting and topological effects. In particular, this serves to express the homotopical construction of topological anyon braid quantum gates recently described in [MSS23].

Talk presentation:

• at Quantum Information and Quantum Matter

  NYU Abu Dhabi, 22-26 May 2023

Related talks:

• Quantum Data Types via Linear Homotopy Type Theory

  talk at Workshop on Quantum Software @ QTML 2022

  Naples, 12 Nov 2022

  slides: pdf (view on full screen)

• Effective Quantum Certification via Linear Homotopy Types,

  talk at Colloquium of the Topos Institute

  Part II: 24 Aug 2023

  video: YT

  slides: pdf (view full screen)

Related articles:

• Topological Quantum Gates in Homotopy Type Theory

• The Quantum Monadology

• Quantum and Reality

all part of the project:

• Quantum Certification via Linear Homotopy Types

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