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Compositional Quantum Logic

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Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7860))

Abstract

Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles’ heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order is extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation. In fact, our construction is entirely compositional, without any additional assumptions on limits or enrichment. Interpreted in the category of finite-dimensional Hilbert spaces, it yields the projection lattices of arbitrary finite-dimensional C*-algebras. Interestingly, there are models that falsify standardly assumed correspondences, most notably the correspondence between noncommutativity of the algebra and nondistributivity of the order.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Oxford, UK

    Bob Coecke, Chris Heunen & Aleks Kissinger

Authors
  1. Bob Coecke
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  2. Chris Heunen
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  3. Aleks Kissinger
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Parks Road, Wolfson Building, OX1 3QD, Oxford, UK

    Bob Coecke  & Luke Ong  & 

  2. Department of Computer Science, McGill University, 3480 Rue University, H3A 0E9, Montréal, QC, Canada

    Prakash Panangaden

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Coecke, B., Heunen, C., Kissinger, A. (2013). Compositional Quantum Logic. In: Coecke, B., Ong, L., Panangaden, P. (eds) Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science, vol 7860. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38164-5_3

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  • DOI: https://doi.org/10.1007/978-3-642-38164-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38163-8

  • Online ISBN: 978-3-642-38164-5

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