How likely you think something is to happen depends on what you already believe about the circumstances. That is the simple concept behind Bayes' rule, an approach to calculating probabilities, first proposed in 1763. Now, an international team of researchers has shown how Bayes' rule operates in the quantum world.
"I would say it is a breakthrough in mathematical physics," said Professor Valerio Scarani, Deputy Director and Principal Investigator at the Centre for Quantum Technologies, and member of the team. His co-authors on the work published on 28 August 2025 in Physical Review Letters are Assistant Professor Ge Bai at the Hong Kong University of Science and Technology in China, and Professor Francesco Buscemi at Nagoya University in Japan.
"Bayes' rule has been helping us make smarter guesses for 250 years. Now we have taught it some quantum tricks," said Prof Buscemi.
While researchers before them had proposed quantum analogues for Bayes' rule, they are the first to derive a quantum Bayes' rule from a fundamental principle.
Conditional probability
Bayes' rule is named for Thomas Bayes, who first defined his rules for conditional probabilities in 'An Essay Towards Solving a Problem in the Doctrine of Chances'.
Consider a case in which a person tests positive for flu. They may have suspected they were sick, but this new information would change how they think about their health. Bayes' rule provides a method to calculate the probability of flu conditioned not only on the test result and the chances of the test giving a wrong answer, but also on the individual's initial beliefs.
Bayes' rule interprets probabilities as expressing degrees of belief in an event. This has been long debated, since some statisticians think that probabilities should be "objective" and not based on beliefs. However, in situations where beliefs are involved, Bayes' rule is accepted as a guide for reasoning. This is why it has found widespread use from medical diagnosis and weather prediction to data science and machine learning.
Principle of minimum change
When calculating probabilities with Bayes' rule, the principle of minimum change is obeyed. Mathematically, the principle of minimum change minimises the distance between the joint probability distributions of the initial and updated belief. Intuitively, this is the idea that for any new piece of information, beliefs are updated in the smallest possible way that is compatible with the new facts. In the case of the flu test, for example, a negative test would not imply that the person is healthy, but rather that they are less likely to have the flu.
In their work, Prof Scarani, who is also from NUS Department of Physics, Asst Prof Bai, and Prof Buscemi began with a quantum analogue to the minimum change principle. They quantified change in terms of quantum fidelity, which is a measure of the closeness between quantum states.
Researchers always thought a quantum Bayes' rule should exist because quantum states define probabilities. For example, the quantum state of a particle provides the probability of it being found at different locations. The goal is to determine the whole quantum state, but the particle is only found at one location when a measurement is performed. This new information will then update the belief, boosting the probability around that location.
The team derived their quantum Bayes' rule by maximising the fidelity between two objects that represent the forward and the reverse process, in analogy with a classical joint probability distribution. Maximising fidelity is equivalent to minimising change. They found in some cases their equations matched the Petz recovery map, which was proposed by Dénes Petz in the 1980s and was later identified as one of the most likely candidates for the quantum Bayes' rule based just on its properties.
"This is the first time we have derived it from a higher principle, which could be a validation for using the Petz map," said Prof Scarani. The Petz map has potential applications in quantum computing for tasks such as quantum error correction and machine learning. The team plans to explore whether applying the minimum change principle to other quantum measures might reveal other solutions.
