22 research outputs found
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
Involutive Categories and Monoids, with a GNS-correspondence
This paper develops the basics of the theory of involutive categories and
shows that such categories provide the natural setting in which to describe
involutive monoids. It is shown how categories of Eilenberg-Moore algebras of
involutive monads are involutive, with conjugation for modules and vector
spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS)
construction is identified as a bijective correspondence between states on
involutive monoids and inner products. This correspondence exists in arbritrary
involutive categories
A Categorical Model for the Geometry of Interaction
We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework