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 they exhibit universal, maximally entropic statistics — deep thermalization — described by Haar ensembles at infinite temperature and by constrained Scrooge ensembles more generally.

 

Here we introduce Scrooge k-designs, which approximate Scrooge ensembles, and sharpen the conditions under which Scrooge-like behavior emerges. We show that global Scrooge designs arise from long-time chaotic unitary dynamics, and that measuring complementary subsystems of scrambled states induces local Scrooge designs. Our results demonstrate that the resources required scale only with the desired approximation, supported by extensive numerics identifying coherence, entanglement, non-stabilizerness and information scrambling as key ingredients.

 

References:

[1] Mok, Haug, Shaw, Endres, and Preskill. Phys. Rev. Lett. 134, 180403 (2025)

[2] Mok, Haug, Ho, and Preskill. arXiv:2601.00266 (2026)