4 research outputs found
Relating Operator Spaces via Adjunctions
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 form tr(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
The Expectation Monad in Quantum Foundations
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.Comment: In Proceedings QPL 2011, arXiv:1210.029
Coreflections in Algebraic Quantum Logic
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