Abstract
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.
Highlights
A quantum system whose unitary dynamics is interspersed with repeated measurements follows a random trajectory through Hilbert space [1,2,3,4,5], determined both by the unitary part of the dynamics and by the sequence of measurement outcomes
Many variants of the measurement transition can be imagined; how do we sort them into universality classes? Are there simplifying limits where exact results are possible? Are there useful continuum field theories for the measurement-induced phase transitions (MPTs) and related problems, that allow us to apply the tools of the renormalization group?. This statistical mechanics problem is closely connected to an entanglement transition that takes place in random tensor networks [14,35,36] and the same questions apply in that setting
We suggest alternative ways of thinking about these effective models, making connections with ideas from disordered magnetism: in particular, we suggest a construction of order parameters for the MPT and for entanglement transitions in random tensor networks, based on overlap of Feynman trajectories
Summary
A quantum system whose unitary dynamics is interspersed with repeated measurements follows a random trajectory through Hilbert space [1,2,3,4,5], determined both by the unitary part of the dynamics and by the sequence of measurement outcomes. We apply all the approaches mentioned above (tree approximations, simulations, replica field theories) in the setting of generic quantum circuits for spin 1/2, as well as related random tensor networks, giving results for scaling properties in the entangled phase and close to the critical point. In this setting we are able to obtain the exact location of the entanglement transition (and exact critical properties) for a tree tensor network that is relevant to dynamics with Haar-random gates This result may be useful for further investigations: studies of the MPT in systems with generic unitaries are often hampered by the restriction of numerics to small sizes, which make it difficult to accurately pinpoint entanglement transitions. We contrast these free systems (which have a continuous, rather than discrete, replica symmetry) with generic models, and we discuss some other variants of the MPT
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