## 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 |
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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 |