The expectation monad in quantum foundations

Abstract

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 Świrszcz. These convex compact Hausdorff spaces are dually equivalent to Banach (complete) order unit spaces, via a result of Kadison, 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 Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.