# Conditional distributions for quantum systems

**Authors:** Arthur J. Parzygnat  
**arXiv:** [2102.01529](https://arxiv.org/abs/2102.01529)  
**Date:** 2021-02-02  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2102.01529v1/

## Key points

- Study of conditional distributions for quantum systems within matrix algebras is a subject of ongoing research
- Extension of results to cases where marginal density matrices are not invertible
- Distinction between Bayesian inverses and the Petz recovery map
- Generalization to encompass general finite-dimensional C*-algebras, including hybrid classical-quantum systems
- Inquiry into necessary and sufficient conditions for positive conditionals, particularly in commutative algebras
- Exploration of joint states admitting positive conditionals and structure of conditionals for multipartite states in quantum systems
- Considerations around constructing multiple conditionals for defining conditional independence or establishing Markov chains using classical random variables

## Summary

The study of conditional distributions for quantum systems, within the framework of matrix algebras, is a subject of ongoing research with promising future directions. This work delves into the intricacies of extending results to cases where marginal density matrices are not invertible, shedding light on the complexities that arise when analyzing disintegrations and Bayesian inversion. The extension also highlights the distinction between Bayesian inverses and the Petz recovery map. Furthermore, there is a significant effort towards generalizing these findings to encompass general finite-dimensional C*-algebras, including hybrid classical-quantum systems represented by direct sums of matrix algebras. <br><br>

Looking ahead, potential inquiries include necessary and sufficient conditions for positive conditionals in such settings, with a particular emphasis on automatically satisfying these conditions for commutative algebras to reproduce the conditional version of Bayes' theorem. The exploration of joint states admitting positive conditionals and the structure of conditionals for multipartite states in quantum systems presents intriguing avenues for further investigation. Additionally, considerations around constructing multiple conditionals for defining conditional independence or establishing Markov chains using classical random variables offer fertile ground for exploration.<br><br>

In summary, this work sets the stage for a deeper understanding of conditional distributions in quantum systems and opens up exciting possibilities for exploring various aspects of quantum information theory and probabilistic modeling within this framework.