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 (MPTs) 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 treelike structure in the circuit geometry. For this reason, we make a detour to give general universal results for entanglement phase transitions in a class of random tree tensor networks with bond dimension 2, making a connection with the classical theory of 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 that are sensitive to the amount of information that is propagated between the 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 measurement phase transition, the forced-measurement phase transition, and for entanglement transitions in random tensor networks. This analysis shows a surprising difference between the measurement phase transition and the other cases. We discuss variants of the measurement problem with additional structure (for example free-fermion structure), and questions for the future.
- Received 10 October 2020
- Accepted 19 January 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.010352
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
When quantum particles interact, they generate “entanglement”: a set of quantum correlations that has been likened to an invisible web connecting the particles. These quantum correlations are one of the key differences with the prequantum view of the world. Another key feature of quantum mechanics is that measurement inevitably affects the state of the particles. Whenever the properties of a particular particle are measured, there is a “state collapse,” that disconnects that particle from the web of entanglement. There is therefore a competition in quantum dynamics between interactions, which spin the entanglement web, and measurements, which can break its filaments.
Recent research by the authors and others showed that there is a threshold rate for measurements in a large system. Measuring more frequently than this rate gives a sharp transition at which the web of entanglement falls apart. However, an exact theory of these measurement phase transitions has remained elusive. This paper provides some exact results for measurement transitions, principally by studying the case where particles interact haphazardly with all others in a kind of “qubit soup.” In the approach used here, a special regime, in which the problem really does simplify to the connectivity of a classical “web” in space time, is considered. This special case then suggests an approach to the generic problem, involving treelike quantum networks (with no loops). They give exact results for transitions on such trees, and conjecture that the results carry over to the original qubit soup. They develop an approach to studying this soup numerically, using correlations between initial and final time. Finally, field-theoretic approaches to measurement transitions are introduced, making contact with ideas from the theory of impure magnetic systems.
The results provide a deeper understanding of how quantum information propagates through time. They are relevant to fundamental questions of how hard it is to simulate quantum processes on a classical computer, and may shed light on protocols for quantum computation.
