The Expectation Monad in Quantum Foundations

TL;DR

This paper introduces the expectation monad, a new mathematical structure that bridges known monads and models quantum states and effects, providing new insights and a reformulation of Gleason's theorem.

Contribution

It defines the expectation monad abstractly and concretely, explores its algebraic properties, and applies it to quantum foundations, including a novel reformulation of Gleason's theorem.

Findings

Expectation monad sits between distribution and continuation monads.

Algebras of the expectation monad are convex compact Hausdorff spaces.

Provides a new formulation of Gleason's theorem in quantum foundations.

Abstract

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.