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A quantum many-body system whose dynamics includes local measurements at anonzero rate can be in distinct dynamical phases, with differing entanglementproperties. We introduce theoretical approaches to measurement-induced phasetransitions (MPT) and also to entanglement transitions in random tensornetworks. Many of our results are for "all-to-all" quantum circuits withunitaries and measurements, in which any qubit can couple to any other, andrelated settings where some of the complications of low-dimensional models arereduced. We also propose field theory descriptions for spatially local systemsof 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 showthat certain all-to-all measurement circuits allow exact results by exploitinglocal tree-like structure in the circuit geometry. For this reason, we make adetour to give general universal results for entanglement phase transitionsrandom tree tensor networks, making a connection with classical directedpolymers on a tree. We then compare these results with numerics in all-to-allcircuits, both for the MPT and for the simpler "Forced Measurement PhaseTransition" (FMPT). We characterize the two different phases in all-to-allcircuits using observables sensitive to the amount of information propagatedbetween initial and final time. We demonstrate signatures of the two phasesthat can be understood from simple models. Finally we proposeLandau-Ginsburg-Wilson-like field theories for the MPT, the FMPT, andentanglement transitions in random tensor networks. This analysis shows asurprising difference between the MPT and the other cases. We discussmeasurement dynamics with additional structure (e.g. free-fermion structure),and questions for the future.

Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory