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Monitored quantum dynamics -- unitary evolution interspersed withmeasurements -- has recently emerged as a rich domain for phase structure inquantum many-body systems away from equilibrium. Here we study monitoreddynamics from the point of view of an eavesdropper who has access to theclassical measurement outcomes, but not to the quantum many-body system. Weshow that a measure of information flow from the quantum system to theclassical measurement record -- the informational power -- undergoes a phasetransition in correspondence with the measurement-induced phase transition(MIPT). This transition determines the eavesdropper's (in)ability to learnproperties of an unknown initial quantum state of the system, given a completeclassical description of the monitored dynamics and arbitrary classicalcomputational resources. We make this learnability transition concrete bydefining classical shadows protocols that the eavesdropper may apply to thisproblem, and show that the MIPT manifests as a transition in the samplecomplexity of various shadow estimation tasks, which become harder in thelow-measurement phase. We focus on three applications of interest: Pauliexpectation values (where we find the MIPT appears as a point of optimallearnability for typical Pauli operators), many-body fidelity, and globalcharge in $U(1)$-symmetric dynamics. Our work unifies different manifestationsof the MIPT under the umbrella of learnability and gives this notion a generaloperational meaning via classical shadows.

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