A quantum many-body system whose dynamics includes local measurements at a nonzero rate can be in distinct dynamical phases, with differing entanglement properties. We introduce theoretical approaches to measurement-induced phase transitions (MPT) and also to entanglement transitions in random tensor networks. Many of our results are for all-to-all quantum circuits with unitaries and measurements, in which any qubit can couple to any other, and related settings where some of the complications of low-dimensional models are reduced. We also propose field theory descriptions for spatially local systems of any finite dimensionality. To build intuition, we first solve the simplest minimal cut toy model for entanglement dynamics in all-to-all circuits, finding scaling forms and exponents within this approximation. We then show that certain all-to-all measurement circuits allow exact results by exploiting local tree-like structure in the circuit geometry. For this reason, we make a detour to give general universal results for entanglement phase transitions random tree tensor networks, making a connection with classical directed polymers on a tree. We then compare these results with numerics in all-to-all circuits, both for the MPT and for the simpler Forced Measurement Phase Transition (FMPT). We characterize the two different phases in all-to-all circuits using observables sensitive to the amount of information propagated between initial and final time. We demonstrate signatures of the two phases that can be understood from simple models. Finally we propose Landau-Ginsburg-Wilson-like field theories for the MPT, the FMPT, and entanglement transitions in random tensor networks. This analysis shows a surprising difference between the MPT and the other cases. We discuss measurement dynamics with additional structure (e.g. free-fermion structure), and questions for the future.
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