The structure of partial isometries

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It is well known that the "quantum logic" apporach to the foundations of quantum mechanics is based on the subspace ordering of projectors on a Hilbert space. In this paper, we show that this is a special case of an ordering on partial isometries, introduced by Halmos and McLaughlin. Partial isometries have a natural physical interpretaton, however, they are notoriously not closed under composition. In order to take a categorical approach, we demonstrate that the Halmos-McLaughlin partial ordering, together with tools from both categorical logic and inverse categories, allows us to form a category of partial isometries.

This category can reasonably be considered a "categorification" of quantum logic - we therefore compare this category with Abramsky and Coecke's "compact closed categories" approach to foundations and with the "monoidal closed categories" view of categorical logic. This comparison illustrates a fundamental incompatibility between these two distinct approaches to the foundations of quantum mechanics.
Original languageEnglish
Title of host publicationSemantic Techniques in Quantum Computation
EditorsS. Gay, I. Mackie
Place of PublicationCambridge
PublisherCambridge University Press
Number of pages28
ISBN (Print)9780521513746
Publication statusPublished - Nov 2009

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