Bayesian rule, as a method for computing probabilities, has been widely applied in numerous fields since its proposal in 1763. Now, an international research team has successfully demonstrated how Bayesian rule operates in the quantum world. The study was published on August 28, 2025, in Physical Review Letters, with co-authors including Assistant Professor Ge Bai from The Hong Kong University of Science and Technology and Professor Francesco Buscemi from Nagoya University, Japan.
Professor Valerio Scarani, Deputy Director of the Centre for Quantum Technologies, stated: “I believe this is a breakthrough in the field of mathematical physics.” For 250 years, Bayes' rule has helped people make more informed guesses, and this team has now endowed it with “quantum techniques.” Although previous researchers had proposed quantum analogs of Bayes' rule, this team is the first to derive the quantum Bayes' rule from first principles. Bayes' rule, named after Thomas Bayes, interprets probability as the degree of belief in an event and follows the principle of minimal change: new information updates beliefs in the way that changes them the least. In the quantum world, quantum states determine probabilities, and measurement updates beliefs with particle position information. Professors Scarani, Assistant Professor Ge Bai, and Professor Buscemi started from the quantum simulation of the minimal change principle. They used quantum fidelity to quantify change and derived the quantum Bayes' rule by maximizing the fidelity between two objects representing the forward and reverse processes. They found that in certain cases, the equation matches the Petz recovery map.
Professor Scarani said: “This is the first time we have derived it from higher principles, which can serve as a validation for using the Petz map.”
The Petz map holds great potential in quantum computing and can be applied to tasks such as quantum error correction and machine learning. The team plans to continue exploring the application of the minimal change principle to other quantum measurements, hoping to uncover more solutions.
