nLab quantum relation



Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication



The notion of quantum relations of Weaver 2010, Weaver & Kuperberg 2011 is a “quantum” or noncommutative geometry/noncommutative measure theory-version of the basic notion of relations on sets.

Concretely, they define a quantum relation on a von Neumann algebra 𝒜\mathcal{A} equipped with an embedding 𝒜\mathcal{A} \subset \mathcal{B}\mathscr{H} (into the bounded operators on some Hilbert space) as an operator bimodule over the commutant of 𝒜\mathcal{A} which is *\ast-closed in the weak operator topology, and explain why this (non-obvious) definition is sensible; for instance it has good application to quantum error correction.

According to Kornell 2020, and in mild paraphrase (following the discussion at dependent linear type and quantum circuits via dependent linear types):

With composition the evident matrix multiplication (Kornell 2020 (5)), quantum relations between quantum sets form a category qRelqRel, which is a dagger-compact category.

As such, this serves as categorical semantics for quantum programming languages like Quipper equipped with term recursion, via quantum CPOs (Kornell, Lindenhovius & Mislove 2021).


Last revised on January 20, 2024 at 13:25:30. See the history of this page for a list of all contributions to it.