# Extending Kolmogorov's axioms for a generalized probability theory on collections of contexts

## TL;DR

This paper generalizes Kolmogorov's probability axioms to include conditional probabilities across different contexts, extending quantum probability models and allowing for broader applications beyond traditional Born rule-based probabilities.

## Contribution

It introduces a formal framework using row stochastic matrices to model conditional probabilities between diverse contexts, broadening the scope of probability theory beyond classical and quantum models.

## Key findings

- Framework generalizes quantum probability models
- Allows for non-Born rule type probabilities
- Provides a new mathematical structure for contextual probabilities

## Abstract

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Auff\'eves and Grangier, which in turn are inspired by Gleason's theorem.