CP^∞ and beyond: 2-categorical dilation theory

Robert Allen and Dominic Verdon

The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the CP^∞-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the CP^∞-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.

Keywords: 2-categories, dilations, quantum channels, von Neumann algebras

2020 MSC: 18N10, 47A20, 81P47

Theory and Applications of Categories, Vol. 41, 2024, No. 50, pp 1783-1811.

Published 2024-11-08.

http://www.tac.mta.ca/tac/volumes/41/50/41-50.pdf

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