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States of Convex Sets

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9034)


State spaces in probabilistic and quantum computation are convex sets, that is, Eilenberg–Moore algebras of the distribution monad. This article studies some computationally relevant properties of convex sets. We introduce the term effectus for a category with suitable coproducts (so that predicates, as arrows of the shape X → 1 + 1, form effect modules, and states, arrows of the shape 1 → X, form convex sets). One main result is that the category of cancellative convex sets is such an effectus. A second result says that the state functor is a “map of effecti”. We also define ‘normalisation of states’ and show how this property is closed related to conditional probability. This is elaborated in an example of probabilistic Bayesian inference.


  • Convex Subset
  • Convex Combination
  • Convex Space
  • Real Vector Space
  • Conditional State

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  1. Asimow, L., Ellis, A.: Convexity Theory and its Applications in Functional Analysis. Academic Press, New York (1980)

    MATH  Google Scholar 

  2. Barr, M., Wells, C.: Toposes, triples and theories, vol. 278. Springer, New York (1985)

    MATH  Google Scholar 

  3. Barnum, H., Wilce, A.: Information processing in convex operational theories. In: Coecke, B., Mackie, I., Panangaden, P., Selinger, P. (eds.) Proceedings of QPL/DCM 2008. Elect. Notes in Theor. Comp. Sci, vol. 270(2), pp. 3–15. Elsevier, Amsterdam (2008)

    Google Scholar 

  4. Flood, J.: Semiconvex geometry. Journal of the Australiam Mathematical Society 30, 496–510 (1981)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Fritz, T.: Convex spaces I: Definition and examples. arXiv preprint arXiv:0903.5522 (2009)

    Google Scholar 

  6. Gudder, S.: Convex structures and operational quantum mechanics. Communic. Math. Physics 29(3), 249–264 (1973)

    CrossRef  MathSciNet  Google Scholar 

  7. Gudder, S.: Convexity and mixtures. Siam Review 19(2), 221–240 (1977)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Heinosaari, T., Ziman, M.: The mathematical language of quantum theory: from uncertainty to entanglement. AMC 10, 12 (2012)

    Google Scholar 

  9. Jacobs, B.: Convexity, duality and effects. In: Calude, C.S., Sassone, V. (eds.) TCS 2010. IFIP AICT, vol. 323, pp. 1–19. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  10. Jacobs, B.: New directions in categorical logic, for classical, probabilistic and quantum logic. arXiv preprint arXiv:1205.3940v3 (2014)

    Google Scholar 

  11. Kock, A.: Bilinearity and cartesian closed monads. Mathematica Scandinavica 29, 161–174 (1971)

    MathSciNet  Google Scholar 

  12. Kock, A.: Closed categories generated by commutative monads. Journal of the Australian Mathematical Society 12(04), 405–424 (1971)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Neumann, W.D.: On the quasivariety of convex subsets of affine spaces. Archiv der Mathematik 21(1), 11–16 (1970)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Świrszcz, T.: Monadic functors and categories of convex sets. PhD thesis, Polish Academy of Sciences (1974)

    Google Scholar 

  15. Stone, M.: Postulates for the barycentric calculus. Annali di Matematica Pura ed Applicata 29(1), 25–30 (1949)

    CrossRef  MATH  Google Scholar 

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Correspondence to Bart Jacobs .

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Jacobs, B., Westerbaan, B., Westerbaan, B. (2015). States of Convex Sets. In: Pitts, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2015. Lecture Notes in Computer Science(), vol 9034. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46677-3

  • Online ISBN: 978-3-662-46678-0

  • eBook Packages: Computer ScienceComputer Science (R0)