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    Furber, Robert and Jacobs, Bart 2015. Towards a Categorical Account of Conditional Probability. Electronic Proceedings in Theoretical Computer Science, Vol. 195, Issue. , p. 179.

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  • Print publication year: 2016
  • Online publication date: June 2016

6 - Relating operator spaces via adjunctions

Summary

Abstract. This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the formtr(A−), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum structures, via a dual adjunction between convex sets and effect modules. The approach systematically uses categories of modules, via their description as Eilenberg-Moore algebras of a monad.

Introduction. There is a recent exciting line of work connecting research in the semantics of programming languages and logic, and research in the foundations of quantum physics, including quantum computation and logic, see [9] for an overview. This paper fits in that line of work. It concentrates on operators (on Hilbert spaces) and organises and relates these operators according to their algebraic structure. This is to a large extent not more than a systematic presentation of known results and connections in the (modern) language of category theory. However, the approach leads to clarifying results, like Theorem 14 that relates density operators and effects via a dual adjunction between convex sets and effect modules (extending earlier work [25]). It is in line with many other dual adjunctions and dualities that are relevant in programming logics [31, 1, 30]. Indeed, via this dual adjunction we can put the work [11] on quantum weakest preconditions in perspective (see especially Remark 15).

The article begins by describing the familiar sets of operators (bounded, selfadjoint, positive) on a (finite-dimensional) Hilbert space in terms of functors to categories of modules. The dual adjunctions involved are made explicit, basically via dual operation VV, see Section 2. Since the algebraic structure of these sets of operators is described in terms of modules over various semirings, namely over complex numbers C (for bounded operators), over real numbers R (for self-adjoint operators), and over non-negative real numbers R≥0 (for positive operators), it is useful to have a uniform description of such modules. It is provided in Section 3, via the notion of algebra of a monad (namely the multiset monad).

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Logic and Algebraic Structures in Quantum Computing
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  • Book DOI: https://doi.org/10.1017/CBO9781139519687
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[1] S., Abramsky, Domain theory in logical form, Annals of Pure and Applied Logic, vol. 51 (1991), pp. 1–77.
[2] S., Abramsky and B., Coecke, A categorical semantics of quantum protocols, Handbook of Quantum Logic and Quantum Structures: Quantum Logic (K., Engesser, D. M., Gabbay, and D., Lehmann, editors), North Holland, Elsevier, Computer Science Press, 2009, pp. 261–323.
[3] M. A., Arbib and E. G., Manes, Algebraic Approaches to Program Semantics, Texts and Monographs in Computer Science, Springer, Berlin, 1986.
[4] S., Awodey, Category Theory, Oxford Logic Guides, Oxford University Press, 2006.
[5] M., Barr and Ch., Wells, Toposes, Triples and Theories, Springer, Berlin, 1985, revised and corrected version available from URL: www.cwru.edu/artsci/math/wells/pub/ttt.html.
[6] E. J., Beggs and S., Majid, Bar categories and star operations, Algebras and Representation Theory, vol. 12 (2009), pp. 103–152.
[7] F., Borceux, Handbook of Categorical Algebra, Encyclopedia of Mathematics, vol. 50, 51 and 52, Cambridge University Press, 1994.
[8] P., Busch, Quantum states and generalized observables: a simple proof of Gleason's Theorem, Physical Review Letters, vol. 91 (2003), no. 12, p. 120403.
[9] B., Coecke (editor), New Structures for Physics, Lecture Notes in Physics, vol. 813, Springer, Berlin, 2011.
[10] D., Coumans and B., Jacobs, Scalars, monads and categories, Compositional Methods in Physics and Linguistics (C., Heunen and M., Sadrzadeh, editors), Oxford University Press, 2012, See arxiv.org/abs/1003.0585.
[11] E., D'Hondt and P., Panangaden, Quantum weakest preconditions, Mathematical Structures in Computer Science, vol. 16 (2006), pp. 429–451.
[12] E. W., Dijkstra and C., Scholten, Predicate Calculus and Program Semantics, Springer, Berlin, 1990.
[13] E.-E., Doberkat, Eilenberg-Moore algebras for stochastic relations, Information and Computation, vol. 204 (2006), pp. 1756–1781, Erratum and addendum in: 206(12):1476–1484, 2008.
[14] A., Dvurečenskij and S., Pulmannová, New Trends in Quantum Structures, Kluwer, Dordrecht, 2000.
[15] A., Edalat, An extension of Gleason's theorem for quantum computation, International Journal of Theoretical Physics, vol. 43 (2004), pp. 1827–1840.
[16] J. M., Egger, On involutive monoidal categories, Theory and Applications of Categories, vol. 25 (2011), pp. 368–393.
[17] D. J., Foulis, Observables, states and symmetries in the context of CB-effect algebras, Reports on Mathematical Physics, vol. 60 (2007), pp. 329–346.
[18] D. J., Foulis and M. K., Bennett, Effect algebras and unsharp quantum logics, Foundations of Physics, vol. 24 (1994), pp. 1331–1352.
[19] A., Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, vol. 6 (1957), pp. 885–893.
[20] J. S., Golan, Semirings and their Applications, Kluwer, 1999.
[21] S., Gudder, Examples, problems and results in effect algebras, International Journal of Theoretical Physics, vol. 35 (1996), pp. 2365–2376.
[22] T., Heinosaari and M., Ziman, The Mathematical Language of Quantum Theory. From Uncertainty to Entanglement, Cambridge University Press, 2012.
[23] B., Jacobs, Semantics of weakening and contraction, Annals of Pure and Applied Logic, vol. 69 (1994), pp. 73–106.
[24] B., Jacobs, Categorical Logic and Type Theory, North Holland, Amsterdam, 1999.
[25] B., Jacobs, Convexity, duality and effects, IFIP Theoretical Computer Science 2010 (Boston) (C. S., Calude and V., Sassone, editors), IFIP Advances in Information and Communication Technology, vol. 82, Springer, 2010, pp. 1–19.
[26] B., Jacobs, Probabilities, distribution monads and convex categories, Theoretical Computer Science, vol. 412 (2011), pp. 3323–3336.
[27] B., Jacobs, Involutive categories and monoids, with a GNS-correspondence, Foundations of Physics, vol. 42 (2012), pp. 874–895.
[28] B., Jacobs and J., Mandemaker, Coreflections in algebraic quantum logic, Foundations of Physics, (10 May 2012), pp. 932–958, http://dx.doi.org/doi:10.1007/s10701-012-9654-8.
[29] B., Jacobs and J., Mandemaker, The expectation monad in quantum foundations, Quantum Physics and Logic (QPL) 2011 (B., Jacobs, P., Selinger, and B., Spitters, editors), 2012, EPTCS, to appear; see arxiv.org/abs/1112.3805.
[30] B., Jacobs and A., Sokolova, Exemplaric expressivity of modal logics, Journal of Logic and Computation, vol. 20 (2010), pp. 1041–1068.
[31] P. T., Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, 1982.
[32] K., Keimel, The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras, Topology and its Applications, vol. 156 (2008), pp. 227–239.
[33] A., Kock, Closed categories generated by commutative monads, Journal of the Australian Mathematical Society, vol. XII (1971), pp. 405–424.
[34] S., Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1971.
[35] E. G., Manes, Algebraic Theories, Springer, Berlin, 1974.
[36] S., Pulmannová and S., Gudder, Representation theorem for convex effect algebras, Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), pp. 645–659, available from http://dml.cz/dmlcz/119041.
[37] P., Selinger, Towards a quantum programming language, Mathematical Structures in Computer Science, vol. 14 (2004), pp. 527–586.
[38] M. H., Stone, Postulates for the barycentric calculus, Annali di Matematica Pura ed Applicata, vol. 29 (1949), pp. 25–30.
[39] T., Swirszcz, Monadic functors and convexity, Bulletin de l'Acad. Polonaise des Sciences. Sér. des sciences math., astr. et phys., vol. 22 (1974), pp. 39–42.