Abstract
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.
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 language | English |
|---|---|
| Title of host publication | Semantic Techniques in Quantum Computation |
| Editors | S. Gay, I. Mackie |
| Place of Publication | Cambridge |
| Publisher | Cambridge University Press |
| Pages | 361-388 |
| Number of pages | 28 |
| ISBN (Print) | 9780521513746 |
| Publication status | Published - Nov 2009 |