# Conditioning of Quantum Open Systems

## TL;DR

This paper explores the quantum probabilistic framework for open systems, emphasizing the importance of observable order and conditions for filtering within von Neumann algebra formulations.

## Contribution

It provides a detailed formulation of quantum conditioning using von Neumann algebras and identifies conditions for non-demolition filtering in quantum systems.

## Key findings

- Quantum conditioning depends on observable compatibility.
- Filtering is possible under specific non-demolition conditions.
- The framework clarifies the role of non-commuting observables in quantum probability.

## Abstract

The underlying probabilistic theory for quantum mechanics is non-Kolmogorovian. The order in which physical observables will be important if they are incompatible (non-commuting). In particular, the notion of conditioning needs to be handled with care and may not even exist in some cases. Here we layout the quantum probabilistic formulation in terms of von Neumann algebras, and outline conditions (non-demolition properties) under which filtering may occur.