# Sequential measurements: Busch-Gleason theorem and its applications

## TL;DR

This paper extends Gleason's theorem to sequential quantum measurements, deriving probability measures and quantum channels using Liouville space, with applications to quantum cryptography protocols like BB84.

## Contribution

It introduces a new axiomatic framework for sequential quantum measurements using super-Liouville space, expanding the applicability of Gleason-type theorems.

## Key findings

- Derived probability measures for sequential measurements
- Constructed super-Liouville space with Bayesian interpretation
- Provided axiomatic derivation of BB84 protocol results

## Abstract

Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important application of the new method is demonstrated, providing an axiomatic derivation of important results of the BB84 protocol in quantum cryptography.