Emergence of the Scrooge Ensemble in the Sachdev-Ye-Kitaev Model

Abstract

The probabilistic nature of quantum measurement provides a direct window into the structure and complexity of many-body wave functions. When only part of a system is measured, the remaining degrees of freedom form an ensemble of post-measurement states whose statistical structure can reveal a stronger form of thermalization, known as deep thermalization. Recent numerical evidence suggests that this phenomenon is characterized by convergence of the projected ensemble to the Scrooge ensemble, a maximally random ensemble compatible with a given density matrix. In this Letter, we use the solvable Sachdev-Ye-Kitaev (SYK) model to unveil the mechanism by which the Scrooge ensemble emerges in many-body systems. By formulating measurement probabilities and post-measurement states in terms of path integrals, we analytically characterize all moments of the projected ensemble and show that they exactly match those of the Scrooge ensemble, even at short evolution times. We further connect this result to the saddle-point structure of the measurement path integral, which naturally generates the replica permutations underlying Scrooge statistics. Our results establish the solvable SYK model as a tractable setting for exploring universal statistics of quantum measurements in chaotic many-body dynamics.