Distances between States and between Predicates

TL;DR

This paper systematically explores metrics on probability distributions and predicates, unifying their descriptions through the validity relation and applying the framework to classical and quantum probability contexts.

Contribution

It introduces a uniform approach to metrics on states and predicates using the validity relation, extending the adjunction to metric spaces and effect modules, and applies this to classical and quantum probability.

Findings

Unified description of metrics via validity relation

Extension of adjunction to metric spaces and effect modules

Application to classical and quantum probability models

Abstract

This paper gives a systematic account of various metrics on probability distributions (states) and on predicates. These metrics are described in a uniform manner using the validity relation between states and predicates. The standard adjunction between convex sets (of states) and effect modules (of predicates) is restricted to convex complete metric spaces and directed complete effect modules. This adjunction is used in two state-and-effect triangles, for classical (discrete) probability and for quantum probability.