Abstract: Statistical inference is about retrieving unknown information of a system from imperfect observations. We consider two setups where the system evolves quantum mechanically and where the unknown information is encoded as a mixed initial state.

In the first problem, information is retrieved by continuous monitoring some observable. We relate this problem to quantum generalisations of Kolmogorov-Sinai entropy, and conjecture a Planckian bound on the amount of information gained per unit time.

In the second one, the dynamics is inflationary (having more and more degrees of freedom), and the observations take place at late time, giving rise to a quantum tree reconstruction problem. We generalise the Kesten-Stigum criterion and showcase the existence of two thresholds (between perfect/imperfect/no recovery) in simple models.

Based on
2507.20914 (PRL in press), PRL 136, 090404 (2026), PRA 109, 032226 (2024), PRL 132, 110201 (2024), work in progress.