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The expectation monad in quantum foundations. (English) Zbl 1350.68178

Summary: The expectation monad is introduced and related to known monads: it sits between on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. The Eilenberg-Moore algebras of the expectation monad are characterized as convex compact Hausdorff spaces, using a theorem of T. Świrszcz [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 22, 39–42 (1974; Zbl 0276.46036)]. These convex compact Hausdorff spaces are dually equivalent to Banach (complete) order unit spaces, via a result of R. V. Kadison [Mem. Am. Math. Soc. 7, 39 p. (1951; Zbl 0042.34801)], which in turn are equivalent to Banach effect modules. In this way we obtain a close ‘triangle’ relationship between predicates and states for the expectation monad. Moreover, the approach leads to a new reformulation of A. M. Gleason’s theorem [J. Math. Mech. 6, 885–893 (1957; Zbl 0078.28803)], expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.

MSC:

68Q55 Semantics in the theory of computing
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18C20 Eilenberg-Moore and Kleisli constructions for monads
18C50 Categorical semantics of formal languages
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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References:

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