An international team of researchers have successfully derived a quantum version of Bayes’ rule, a cornerstone of probability theory
Their discovery was published on August 28, 2025, in Physical Review Letters and examines how beliefs are updated in the quantum world, where normal physics rules no longer apply
The research was conducted by Professor Valerio Scarani from the Centre for Quantum Technologies and the National University of Singapore, Assistant Professor Ge Bai from the Hong Kong University of Science and Technology, and Professor Francesco Buscemi from Nagoya University in Japan.
Understanding Bayes’ rule
Bayes’ rule was developed in the 18th century by mathematician Thomas Bayes and is a method used to update the probability of a hypothesis based on new evidence. It’s used across a range of fields from medical diagnosis and weather forecasting to data science and machine learning.
Put simply, Bayes’ rule enables individuals to update their expectations in response to new information becoming available. For example, if a person believes they might have the flu and then receives a positive test result, Bayes’ rule helps quantify how much more likely it is that they are actually sick.
This rule is grounded in the idea of conditional probability. It works by updating an individual’s prior belief to a new belief (called the posterior) that takes new information into account.
Bayes’ role in the quantum world
Although the classical Bayes’ rule is well understood, its application in the quantum realm has remained elusive. Quantum systems behave differently from classical ones; they are governed by probabilities and wavefunctions that describe the likelihood of finding a particle in a particular state.
Previously, researchers had proposed several quantum analogues of Bayes’ rule, but none had been derived from a fundamental principle of quantum mechanics. The team adopted a new approach by focusing on how beliefs should adapt in response to new quantum measurements, while maintaining as close a connection as possible to the principle of minimum change.
The principle of minimum change
The principle of minimum change states that when new information is received, beliefs should be adjusted as little as necessary to fit the latest facts. In classical Bayes’ rule, this is reflected mathematically by minimising the distance between the original and updated probability distributions.
To translate this into the quantum domain, the team employed a concept known as quantum fidelity, which measures the closeness of two quantum states to each other. Their goal was to maximise fidelity, or in other words, find the slightest change in belief that still accounts for the observed data.
This led them to derive a quantum Bayes’ rule by maximising fidelity between two mathematical objects representing the forward and reverse processes of measurement and belief update.
Connecting to the Petz map
The team found that in specific scenarios, their newly derived equations matched a well-known mathematical tool in quantum information theory, known as the Petz recovery map. This map, introduced in the 1980s, was considered a promising candidate for a quantum version of Bayes’ rule due to its valuable properties. Still, it had never been derived from first principles before.
Now, with this new research, researchers have not only validated the Petz map’s role in quantum reasoning but also opened the door to new applications, such as quantum error correction and quantum machine learning.
Quantum potential: What does this mean for the future?
The researchers are now exploring whether the principle of minimum change can lead to other quantum analogues by applying it to different mathematical measures. Their findings could further close the gap between classical and quantum reasoning, contributing to the foundation of future quantum technologies.