Nature Is Stingy: Universality of Scrooge Ensembles in Quantum Many-Body Systems
4 affiliations
- 1California Institute of Technology
- 2Technology Innovation Institute
- 3National University of Singapore
- 4AWS Center for Quantum Computing
Abstract
Recent advances in quantum simulators allow direct experimental access to ensembles of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics governed by maximum-entropy principles, known as deep thermalization. At infinite temperature, this universality is characterized by Haar-random ensembles. More generally, physical constraints such as finite temperature or conservation laws lead to Scrooge ensembles, which are maximally entropic distributions of pure states consistent with these constraints. Here, we introduce Scrooge k -designs, which approximate Scrooge ensembles, and we use this framework to sharpen the conditions under which Scrooge-like behavior emerges. We first show that global Scrooge designs arise from long-time chaotic unitary dynamics alone, without measurements. Second, we show that measuring a complementary subsystem of a scrambled global state drawn from a global Scrooge 2 k -design induces a local Scrooge k -design. Third, we show that a local Scrooge k -design arises from an arbitrary entangled state when the complementary system is measured in a scrambled basis induced by a unitary drawn from a Haar 2 k -design. These results show that the resources required to generate approximate Scrooge ensembles scale only with the desired degree of approximation, enabling efficient implementations. Complementing our analytical results, numerical simulations identify coherence, entanglement, nonstabilizerness, and information scrambling as essential ingredients for the emergence of local Scrooge-like behavior. Together, our findings advance theoretical explanations for maximally entropic, information-stingy randomness in quantum many-body systems.
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BibTeX
@article{mok2026nature,
title = {{Nature Is Stingy: Universality of Scrooge Ensembles in Quantum Many-Body Systems}},
author = {Wai‐Keong Mok and Tobias Haug and Wen Wei Ho and John P. Preskill},
journal = {Physical Review X},
year = {2026},
month = aug,
doi = {10.1103/tb52-jxmx},
url = {https://doi.org/10.1103/tb52-jxmx},
note = {CC BY},
}
RIS
TY - JOUR
TI - Nature Is Stingy: Universality of Scrooge Ensembles in Quantum Many-Body Systems
AU - Wai‐Keong Mok
AU - Tobias Haug
AU - Wen Wei Ho
AU - John P. Preskill
PY - 2026
DA - 2026/08/11
JO - Physical Review X
DO - 10.1103/tb52-jxmx
UR - https://doi.org/10.1103/tb52-jxmx
AB - Recent advances in quantum simulators allow direct experimental access to ensembles of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics governed by maximum-entropy principles, known as deep thermalization. At infinite temperature, this universality is characterized by Haar-random ensembles. More generally, physical constraints such as finite temperature or conservation laws lead to Scrooge ensembles, which are maximally entropic distributions of pure states consistent with these constraints. Here, we introduce Scrooge k -designs, which approximate Scrooge ensembles, and we use this framework to sharpen the conditions under which Scrooge-like behavior emerges. We first show that global Scrooge designs arise from long-time chaotic unitary dynamics alone, without measurements. Second, we show that measuring a complementary subsystem of a scrambled global state drawn from a global Scrooge 2 k -design induces a local Scrooge k -design. Third, we show that a local Scrooge k -design arises from an arbitrary entangled state when the complementary system is measured in a scrambled basis induced by a unitary drawn from a Haar 2 k -design. These results show that the resources required to generate approximate Scrooge ensembles scale only with the desired degree of approximation, enabling efficient implementations. Complementing our analytical results, numerical simulations identify coherence, entanglement, nonstabilizerness, and information scrambling as essential ingredients for the emergence of local Scrooge-like behavior. Together, our findings advance theoretical explanations for maximally entropic, information-stingy randomness in quantum many-body systems.
SN - 2160-3308
ER -