Scientists Derive Quantum Version of Bayes’ Rule in Major Breakthrough

An international team of researchers has successfully derived a quantum version of Bayes’ rule, a fundamental concept in probability theory. Their groundbreaking discovery was published in *Physical Review Letters* on August 28, 2025, and explores how beliefs are updated within the quantum realm, where conventional physical laws do not apply.

The research was spearheaded by Professor Valerio Scarani from the Centre for Quantum Technologies at the National University of Singapore, alongside 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

Developed in the 18th century by mathematician Thomas Bayes, Bayes’ rule provides a method for updating the probability of a hypothesis based on new evidence. It finds applications in various fields, including medical diagnosis, weather forecasting, and machine learning. Essentially, Bayes’ rule allows individuals to adjust their expectations as new information becomes available. For instance, if someone suspects they have the flu and later receives a positive test result, Bayes’ rule quantifies how much more likely it is that they are indeed ill.

While the classical version of Bayes’ rule is well established, applying it in quantum contexts has proven challenging. Quantum systems operate under principles that differ significantly from classical mechanics; they are characterized by probabilities and wavefunctions that detail the likelihood of finding a particle in a specific state.

Quantum Applications of Bayes’ Rule

Previously, researchers had proposed various quantum analogues of Bayes’ rule. However, these were not drawn from fundamental quantum mechanics principles. The current team approached the problem by examining how beliefs should adjust following new quantum measurements, adhering closely to the principle known as minimum change. This principle asserts that when presented with new information, beliefs should be modified as minimally as necessary to accommodate the new facts.

In traditional Bayes’ rule, this is mathematically represented by minimizing the distance between the prior and updated probability distributions. To transition this concept to the quantum landscape, the researchers employed a measure known as quantum fidelity. This metric assesses how closely two quantum states resemble one another. Their objective was to maximize fidelity, thereby determining the smallest adjustment in belief that still incorporates the observed data.

Connecting to the Petz Map

The researchers discovered that their newly formulated equations corresponded with a well-established mathematical tool in quantum information theory: the Petz recovery map. Introduced in the 1980s, this map has been regarded as a promising candidate for a quantum interpretation of Bayes’ rule due to its advantageous properties. Notably, it had never before been derived from fundamental principles.

With this recent study, the team has confirmed the Petz map’s significance in quantum reasoning and paved the way for potential applications in areas such as quantum error correction and quantum machine learning.

Implications for Future Research

The researchers are now investigating whether the principle of minimum change can yield additional quantum analogues by exploring it within different mathematical frameworks. Their findings may help bridge the gap between classical and quantum reasoning, contributing to the foundational knowledge necessary for future advancements in quantum technologies.

This research not only enhances the understanding of probability in quantum systems but also suggests promising avenues for practical applications that could revolutionize fields reliant on quantum mechanics.