Francesco Buscemi - Nagoya University

Quantum Bayes' rule and retrodictive uncertainty relations

This work develops a quantum counterpart of Bayesian updating based on the principle of minimum change. In classical probability, this principle uniquely leads to Bayes’ rule: beliefs are updated so as to incorporate new data while departing minimally from the prior. In quantum theory, however, an analogous rule is nontrivial, because quantum states are noncommutative and lack canonical joint and conditional probability distributions.

We formulate the update at the level of input-output processes rather than marginal states. By representing quantum processes as bipartite states and comparing them through information-theoretic divergences, we obtain an optimization problem whose solution defines a unique retrodictive map. When change is quantified by fidelity, this solution has a closed form and, for a broad class of physically relevant cases, coincides with the Petz transpose map. This provides a principled derivation of a quantum analogue of Bayes’ rule.

Specializing the framework to quantum measurements, viewed as quantum-to-classical channels, we show that the minimum change principle extends beyond fidelity to a broad class of statistical divergences, all yielding the same retrodictive update. This universality identifies the map as a genuine quantum Bayesian inverse. The resulting retrodicted states define symmetric joint probabilities for pairs of measurements and lead to a notion of mutual retrodictability, measuring how well outcomes can be inferred from one another in a backward-in-time sense. The same framework also yields new entropic uncertainty relations, expressed directly in terms of the prior state and measurement operators, which preserve the usual operational interpretation while providing tighter bounds than existing formulations for broad classes of states and measurements.