Learnability transitions in monitored quantum dynamics via eavesdropper's classical shadows

The dynamics of quantum many-body systems subject to repeated measurements has recently emerged as a rich subject for non-equilibrium physics. Remarkably, these systems can exhibit "measurement-induced phase transitions" (MIPTs) in the structure of quantum correlations, such as entanglement, as a function of the rate or strength of measurements versus unitary interactions. We present an alternative point of view on these phenomena, based not on the structure of correlations in post-measurement states of the quantum system, but rather on the information content of the measurement outcomes themselves. The MIPT maps onto a phase transition in the ability of an eavesdropper to learn properties of an unknown state of the system by monitoring its dynamics. This learnability phase transition can be quantified within the framework of "classical shadow tomography"--a paradigm for learning many properties of quantum states from randomized measurements--where it arises as an abrupt change in the number of experimental repetitions required to learn various properties. This point of view unifies distinct manifestations of the MIPT under a common denominator, and points to new order parameters that could be used for its experimental detection.