# Ordering states and channels based on positive Bayesian evidence

## TL;DR

This paper introduces a new partial order on quantum states based on positive Bayesian evidence, which helps identify optimal approximations and minimal state changes during Bayesian updates, and extends to quantum channels.

## Contribution

It defines a novel order on quantum states and channels that captures Bayesian updating and minimal distinguishability, with proven structural properties and extensions to channels.

## Key findings

- The order preserves convex mixtures and tensor products.
- It characterizes optimal approximations to purer states.
- Extends to non-trivial orderings on quantum channels.

## Abstract

In this paper we introduce a new partial order on quantum states that considers which states can be achieved from others by updating on `agreeing' Bayesian evidence. We prove that this order can also be interpreted in terms of minimising worst case distuinguishability between states using the concept of quantum max-divergence. This order solves the problem of which states are optimal approximations to their more pure counterparts and it shows in an explicit way that a proposed quantum analogue of Bayes' rule leads to a Bayesian update that changes the state as little as possible when updating on positive evidence. We prove some structural properties of the order, specifically that the order preserves convex mixtures and tensor products of states and that it is a domain. The uniqueness of the order given these properties is discussed. Finally we extend this order on states to one on quantum channels using the Jamiolkowski isomorphism. This order turns the spaces of unital/non-unital trace-preserving quantum channels into domains that, unlike the regular order on channels, is not trivial for unital trace-preserving channels.