Abstract
We investigate the use of coalgebra to represent quantum systems, thus providing a basis for the use of coalgebraic methods in quantum information and computation. Coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics.
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Notes
For those concerned with set-theoretic foundations, we shall on a couple of occasions refer to ‘superlarge’ categories such as CAT, the category of ‘large categories’ such as Set. If we think of large categories as based on classes, superlarge categories are based on entities ‘one size up’—‘conglomerates’ in the terminology of [19]. This can be formalized in set theory with a couple of Grothendieck universes.
References
Abramsky, S. (2012). Big toy models: representing physical systems as Chu spaces. Synthese, 186(3), 697–718.
Abramsky, S., & Jung, A. (1994). Domain theory. In S. Abramsky, D. Gabbay, T.S.E. Maibaum (Eds.), Handbook of logic in computer science (pp. 1–168). Oxford: Oxford University Press.
Abramsky, S., Gay, S.J., Nagarajan, R. (1996). Interaction categories and the foundations of typed concurrent programming. In M. Broy (Ed.), NATO ASI DPD (pp. 35–113).
Barr, M. (1979). *-Autonomous categories. Lecture notes in mathematics (Vol. 752). Springer.
Barr, M. (1998). The separated extensional Chu category. Theory and Applications of Categories, 4(6), 137–147.
Barwise, J., & Seligman, J. (1997). Information flow: the logic of distributed systems. Cambridge University Press.
Blank, J., Exner, P., Havlicek, M. (2008). Hilbert space operators in quantum mechanics (2nd ed.). New York: Springer.
Calin, G., Myers, R., Pattinson, D., Schröder, L. (2007). Coloss: the coalgebraic logic satisfiability solver. Department of Computing, Imperial College. Technical report. Download and documentation.
Chu, P.-H. (1979). Constructing *-autonomous categories. Lecture notes in mathematics [4] (Vol. 752, pp. 103–137).
Dirac, P.A.M. (1947). The principles of quantum mechanics (3rd ed.). Oxford: Oxford University Press.
Droste, M., & Zhang, G.-Q. Bifinite Chu spaces. In Mossakowski et al. [22] (pp. 179–193).
Faure, C.-A., & Frölicher, A. (2000). Modern projective geometry. Dordrecht: Kluwer Academic Publishers.
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science (TCS), 50, 1–102.
Giuli, E., & Tholen, W. (2007). A topologist’s view of Chu spaces. Applied Categorical Structures, 15(5–6), 573–598.
Grothendieck, A. (1970). Catégories fibrées et descente (exposé VI). In A. Grothendieck (Ed.), Revêtement Etales et Groupe Fondamental (SGA1). Lecture notes in mathematics (Vol. 224, pp. 145–194). New York: Springer.
Gumm, H.P., & Schröder, T. (2005). Types and coalgebraic structure. Algebra Universalis, 53, 229–252.
Gunter, C.A., & Scott, D.S. (1990). Semantic domains. In Handbook of theoretical computer science, volume B: formal models and sematics (B) (pp. 633–674). Elsevier.
Hansen, H.H., Kupke, C., Pacuit, E. (2007). Bisimulation for neighbourhood structures. In Proceedings of the 2nd conference on algebra and coalgebra in computer science (CALCO 2007), Bergen, Norway. Springer LNCS (Vol. 4624, pp. 279–293). Springer.
Herrlich, H., & Strecker, G. (1973). Category theory: an introduction. Boston: Allyn and Bacon.
Ivanov, L. (2008). Modeling non-iterated system behavior with Chu spaces. In H.R. Arabnia (Ed.), CDES (pp. 145–150). Iona: CSREA Press.
Lafont, Y., & Streicher, T. (1991). Games semantics for linear logic. In LICS (pp. 43–50). IEEE Computer Society.
Mossakowski, T., Montanari, U., Haveraaen, M. (Eds.) (2007). Algebra and coalgebra in computer science, second international conference, CALCO 2007, Bergen, Norway, August 20–24, 2007, proceedings. Lecture notes in computer science (Vol. 4624). Springer.
Nguyen, N., Nguyen, H.T., Berlin, W., Kreinovich, V. (2001). Chu spaces: towards new foundations for fuzzy logic and fuzzy control, with applications to information flow on the world wide web. JACIII, 5(3), 149–156.
Palmigiano, A., & Venema, Y. Nabla algebras and Chu spaces. In Mossakowski et al. [22] (pp. 394–408).
Papadopoulos, B.K., & Syropoulos, A. (2000). Fuzzy sets and fuzzy relational structures as Chu spaces. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8(4), 471–479.
Pavlovic, D. (1997). Chu I: cofree equivalences, dualities and *-autonomous categories. Mathematical Structures in Computer Science, 7(1), 49–73.
Pratt, V.R. (1995). The Stone gamut: a coordinatization of mathematics. In LICS (pp. 444–454). IEEE Computer Society.
Pratt, V.R. (2003). Transition and cancellation in concurrency and branching time. Mathematical Structures in Computer Science, 13(4), 485–529.
Rutten, J.J.M.M. (2000). Universal coalgebra: a theory of systems. Theoretical Computer Science, 249(1), 3–80.
Seely, R.A.G. (1989). Linear logic, *-autonomous categories and cofree coalgebras. In Categories in computer science and logic. Contemporary mathematics (Vol. 92, pp. 371–382). Am. Math. Soc.
Tews, H. (2000). Coalgebras for binary methods. Electrical Notes Theory on Computer Science, 33, 83–111.
van Benthem, J. (2000). Information transfer across Chu spaces. Logic Journal of the IGPL, 8(6), 719–731.
Vannucci, S. (2004). On game formats and Chu spaces. Department of Economics University of Siena 417, Department of Economics, University of Siena.
von Neumann, J. (1955). Mathematical foundations of quantum mechanics. Princeton University Press. Translated from Mathematische Grundlagen der Quantenmechanik, Springer, 1932.
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Abramsky, S. Coalgebras, Chu Spaces, and Representations of Physical Systems. J Philos Logic 42, 551–574 (2013). https://doi.org/10.1007/s10992-013-9276-4
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DOI: https://doi.org/10.1007/s10992-013-9276-4