Abstract
We take a first step towards establishing a link between the topos approach to quantum theory and the monoidal approach to quantum theory. The topos approach to quantum theory makes extensive use of categories of commutative \(C^*\)-algebras and their corresponding Gelfand spectrum. We generalise these categories of \(C^*\)-algebras and generalise the notion of Gelfand spectrum via defining the abstract spectral presheaf. We then characterise this spectral presheaf for the category of sets and relations, and examine how this relates to the notion of observable in this category as studied in the monoidal approach to quantum theory.
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References
Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Symposium on Logic in Computer Science, pp. 415–425 (2004)
Isham, C., Butterfield, J.: A topos perspective on the Kochen-Specker theorem: I. quantum states as generalised valuations
Doering, A., Isham, C.: What is a thing? In: Coecke, B. (ed.) New Structures in Physics, vol. 813, pp. 753–940. Springer, Heidelberg (2011)
Spekkens, R.W.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A. 75 (2007)
Coecke, B., Edwards, B.: Toy quantum categories (extended abstract). Electr. Notes Theor. Comput. Sci. 270, 29–40 (2011)
Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. 23, 555–567 (2013)
Mitchell, B.: Theory of Categories. Academic Press, New York (1965)
Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980)
Heunen, C.: Semimodule enrichment. Electr. Notes Theor. Comput. Sci. 218, 193–208 (2008)
Pavlovic, D.: Quantum and classical structures in nondeterminstic computation. In: Bruza, P., Sofge, D., Lawless, W., Rijsbergen, K., Klusch, M. (eds.) QI 2009. LNCS (LNAI), vol. 5494, pp. 143–157. Springer, Heidelberg (2009). doi:10.1007/978-3-642-00834-4_13
Evans, J., Duncan, R., Lang, A., Panangaden, P.: Classifying all mutually unbiased bases in Rel (2009)
Coecke, B., Edwards, B., Spekkens, R.W.: Phase groups and the origin of non-locality for qubits. Electr. Notes Theor. Comput. Sci. 270, 15–36 (2011)
Coecke, B., Duncan, R., Kissinger, A., Wang, Q.: Strong complementarity and non-locality in categorical quantum mechanics. In: Logic in Computer Science, pp. 245–254 (2012)
Gogioso, S., Zeng, W.: Mermin non-locality in abstract process theories. In: EPTCS, vol. 195, 228–246 (2015)
Abramsky, S., Barbosa, R.S., Kishida, K., Lal, R., Mansfield, S.: Contextuality, cohomology and paradox. In: Logic in Computer Science, pp. 211–228 (2015)
Staton, S., Uijlen, S.: Effect algebras, presheaves, non-locality and contextuality. In: Automata, Languages, and Programming, pp. 401–413 (2015)
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Dunne, K. (2017). A New Perspective on Observables in the Category of Relations: A Spectral Presheaf for Relations. In: de Barros, J., Coecke, B., Pothos, E. (eds) Quantum Interaction. QI 2016. Lecture Notes in Computer Science(), vol 10106. Springer, Cham. https://doi.org/10.1007/978-3-319-52289-0_20
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DOI: https://doi.org/10.1007/978-3-319-52289-0_20
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