Bayes’ rule is often introduced in textbooks through examples involving urns filled with colored balls. It is striking, however, that the same formula works just as well in situations totally unrelated to urns or counting. In fact, Bayes’ rule applies also to how we update beliefs, learn from data, and draw inferences about the world. The fact that its success extends far beyond probability puzzles suggests that Bayes’ rule captures something fundamental about rational reasoning itself.
Indeed, there are several ways to justify Bayes’ rule as the only possible rule of consistent updating. De Finetti, Cox, Jeffreys, Jeffrey, and many other argued that, if an agent deviates from Bayes’ rule, they open themselves to “attacks” that would make them “lose” (money, time, resources, etc.) with probability one in the long run. However, all such approaches rely on axioms, and while some axioms may seem natural and acceptable to one researcher, they may not seem as convincing to another.
Interestingly, there exists however another way (similar to Jaynes’ maximum entropy principle, in fact) to justify Bayes’ rule: the principle of minimum change. When new data arrive, we should only revise our prior beliefs as much as needed to remain consistent with the new evidence. This is a conservative stance toward knowledge that avoids bias by preferring the smallest possible adjustment compatible with the facts.
In a work recently published on Physical Review Letters, together with Ge Bai and Valerio Scarani, we asked how this principle might extend to the quantum world. Quantum systems do not possess definite properties before measurement, and probabilities are replaced by density matrices that describe our partial knowledge of outcomes. Updating such knowledge is therefore not straightforward. What does it mean to change a quantum state “as little as possible” while incorporating new information?
To address this, we reformulated the minimum change principle directly at the level of quantum processes rather than their individual states. We used quantum fidelity to measure how similar two processes are and searched for the update that maximizes this similarity. The result turned out to coincide, in many important cases, with a well-known mathematical transformation in quantum information theory called the Petz recovery map.
This finding provides a new interpretation of the Petz map: it is not merely a technical tool, but rather, the natural quantum analogue of Bayes’ rule. Of all the possible updates, this one changes our quantum description in the smallest consistent way. This explains why, despite often being described as merely “pretty good,” the Petz map keeps reappearing in many different contexts. It has been independently rediscovered in quantum error correction, statistical mechanics, and even quantum gravity because it expresses the same logic that makes Bayes’ rule universal. It is not an approximation, but rather, the quantum form of rational inference itself.
The quantum version of Bayes’ rule does more than provide a new mathematical identity. It offers a systematic way to reason about quantum systems, to retrodict past states from observed data, and to quantify how much information about the past is lost over time. This connects directly to our previous studies on observational entropy and retrodictability, where the second law of thermodynamics emerges as a statement about the progressive loss of our ability to reconstruct the past.
Seen from this perspective, learning, inference, and even entropy growth are aspects of a single story about how information evolves. The quantum Bayes’ rule sits at the centre of that story, providing a bridge between classical reasoning and the probabilistic structure of quantum theory. A rule that is often illustrated with urns and balls finds a new form in quantum theory, where neither urns nor balls exist. This perhaps suggests that rational updating is not tied to any particular physical model (classical, quantum, etc.) but expresses a deeper logic of information itself.
Quantum Quia 