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.Comment: 67 pages, 41 figures; minor modifications to text and updated
references; abstract shortened to meet arxiv requirements, see pdf for full
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