Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity
Abstract
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of "Hilbert-space ergodicity" and "deep thermalization." In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools.
Copyright and License
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.
Acknowledgement
We thank David Huse, Wen Wei Ho, Daniel Ranard, and Saúl Pilatowsky-Cameo for insightful discussions on this work. We acknowledge support by the NSF QLCI Grant No. OMA-2016245, the DOE (DE-SC0021951), the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY-1733907), the Center for Ultracold Atoms, an NSF Physics Frontiers Center (NSF Grant No. PHY-1734011), the DARPA ONISQ program (W911NF2010021), the AFOSR YIP (FA9550-19-1-0044), Army Research Office MURI program (W911NF2010136), NSF CAREER Grant No. 2237244, and NSF CAREER Grant No. 1753386. Support is also acknowledged from the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. A. E. acknowledges funding by the German National Academy of Sciences Leopoldina under Grant No. LPDS 2021-02 and by the Walter Burke Institute for Theoretical Physics at Caltech. F. S. acknowledges support provided by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research (DE-SC0020290), by Amazon Web Services, AWS Quantum Program, and by the DOE QuantISED program through the theory consortium “Intersections of QIS and Theoretical Particle Physics” at Fermilab. J. C. acknowledges support from the Terman Faculty Fellowship at Stanford University.
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Additional details
- National Science Foundation
- OMA-2016245
- United States Department of Energy
- DE-SC0021951
- National Science Foundation
- PHY-1733907
- MIT-Harvard Center for Ultracold Atoms
- National Science Foundation
- PHY-1734011
- Defense Advanced Research Projects Agency
- W911NF2010021
- United States Air Force Office of Scientific Research
- FA9550-19-1-0044
- United States Army Research Office
- W911NF2010136
- National Science Foundation
- DMR-2237244
- National Science Foundation
- PHY-1753386
- National Quantum Information Science Research Centers
- German National Academy of Sciences Leopoldina
- LPDS 2021-02
- Walter Burke Institute for Theoretical Physics
- United States Department of Energy
- DE-SC0020290
- Amazon (United States)
- United States Department of Energy
- QuantISED
- Stanford University
- Terman Faculty Fellowship
- Accepted
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2024-08-28Accepted
- Caltech groups
- AWS Center for Quantum Computing, Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
- Publication Status
- Published