Categories of relations as models of quantum theory

@article{Heunen2015CategoriesOR,
  title={Categories of relations as models of quantum theory},
  author={Chris Heunen and Sean Tull},
  journal={ArXiv},
  year={2015},
  volume={abs/1506.05028}
}
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal’cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quantum… 

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References

SHOWING 1-10 OF 48 REFERENCES

Categories of quantum and classical channels

A construction that turns a category of pure state spaces and operators into a categories of observable algebras and superoperators is introduced, providing elegant abstract notions of preparation and measurement.

Categorical Formulation of Finite-Dimensional Quantum Algebras

We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional ‘quantum algebras’. We develop the concept of an involution monoid, and use it to

Toy quantum categories

We show that Rob Spekken’s toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and

Can quantum theory be characterized in information-theoretic terms?

It is shown that the first two equivalences break, and the third holds, in a framework of generalized, possibly nonlinear, C*-algebras, which uncovers a hierarchy of notions of when (quantum) information is classical.

Compositional Quantum Logic

This work introduces a framework in which order-theoretic structure comes with a primitive composition operation, extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation.

Quantum Quandaries: a Category-Theoretic Perspective

General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose

Interacting Quantum Observables

Using a general categorical formulation, it is shown that pairs of mutually unbiased quantum observables form bialgebra-like structures that enable all observables of finite dimensional Hilbert space quantum mechanics to be described.

Interacting quantum observables: categorical algebra and diagrammatics

The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.

Relative Frobenius algebras are groupoids

Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

Use of a Canonical Hidden-Variable Space in Quantum Mechanics and Reasoning about Strategies in Topical Form.