The Sheaf-Theoretic Structure of Non-Locality and Contextuality

          @misc{ pirsa_PIRSA:11110108,
            doi = {10.48660/11110108},
            url = {https://pirsa.org/11110108},
            author = {Abramsky, Samson},
            keywords = {Quantum Foundations},
            language = {en},
            title = {The Sheaf-Theoretic Structure of Non-Locality and Contextuality},
            publisher = {Perimeter Institute},
            year = {2011},
            month = {nov},
            note = {PIRSA:11110108 see, \url{https://pirsa.org}}
          }
          

Abstract

We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, which generalizes the familiar probability tables used in non-locality theory to cover Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to *obstructions to the existence of global sections*. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. Some ongoing work with Shane Mansfield and Rui Soares Barbosa is described, which identifies *cohomological obstructions* to the existence of global sections, opening the possibility of applying the powerful methods of cohomology to non-locality and contextuality.