This 250-year-old equation just got a quantum makeover

“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.

Although other researchers had previously suggested quantum versions of Bayes’ rule, this team is the first to derive a true quantum Bayes’ rule based on a core physical principle.

Conditional probability

Bayes’ rule takes its name from Thomas Bayes, who described his method for calculating conditional probabilities in “An Essay Towards Solving a Problem in the Doctrine of Chances.”

Imagine someone who tests positive for the flu. They might have suspected illness already, but this new result changes their assessment of the situation. Bayes’ rule provides a systematic way to update that belief, factoring in the likelihood of the test being wrong as well as the person’s prior assumptions.

The rule treats probabilities as measures of belief rather than absolute facts. This interpretation has sparked debate among statisticians, with some arguing that probability should represent objective frequency rather than subjective confidence. Still, when uncertainty and belief play a role, Bayes’ rule is widely recognized as a rational framework for decision-making. It underpins countless applications today, from medical testing and weather forecasting 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 minimizes 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 maximizing the fidelity between two objects that represent the forward and the reverse process, in analogy with a classical joint probability distribution. Maximizing fidelity is equivalent to minimizing 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.