A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.
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October 2020
Research Article|
October 09 2020
Quantum sets
Andre Kornell
Andre Kornell
a)
Department of Mathematics, University of California
, Davis, California 95616, USA
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E-mail: [email protected]
J. Math. Phys. 61, 102202 (2020)
Article history
Received:
August 29 2018
Accepted:
October 28 2019
Citation
Andre Kornell; Quantum sets. J. Math. Phys. 1 October 2020; 61 (10): 102202. https://doi.org/10.1063/1.5054128
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