Colored Weingarten Calculus for Block-Unitary Ensemble
Abstract
Haar randomness is the standard null model for quantum typicality, scrambling, and random-matrix formulations of thermalization. In many physical settings, however, randomness is naturally constrained to subspaces of Hilbert space. This structure arises, in particular, both in systems with symmetry-resolved sectors and in the Eigenstate Thermalization Hypothesis (ETH), where mesoscopic energy windows define subspaces in which nearby energy eigenstates are mixed. Motivated by these settings, we consider a Block-Unitary Ensemble consisting of independently Haar-random unitaries acting on each subspace. We develop a constructive Weingarten Calculus for this ensemble, providing a unified framework across quantum information and many-body physics. This construction reveals a symmetry-resolved Schur-Weyl duality, where the relevant commutant is not the ordinary permutation algebra, but the one of color-preserving permutations. We apply this framework to quantum states under symmetry constraints and to energy-resolved ensembles for ETH.
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BibTeX
@misc{iannotti2026colored,
title = {{Colored Weingarten Calculus for Block-Unitary Ensemble}},
author = {Daniele Iannotti and Elisa Vallini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30385},
eprint = {2609.30385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Colored Weingarten Calculus for Block-Unitary Ensemble
AU - Daniele Iannotti
AU - Elisa Vallini
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.30385
UR - https://arxiv.org/abs/2609.30385
AB - Haar randomness is the standard null model for quantum typicality, scrambling, and random-matrix formulations of thermalization. In many physical settings, however, randomness is naturally constrained to subspaces of Hilbert space. This structure arises, in particular, both in systems with symmetry-resolved sectors and in the Eigenstate Thermalization Hypothesis (ETH), where mesoscopic energy windows define subspaces in which nearby energy eigenstates are mixed. Motivated by these settings, we consider a Block-Unitary Ensemble consisting of independently Haar-random unitaries acting on each subspace. We develop a constructive Weingarten Calculus for this ensemble, providing a unified framework across quantum information and many-body physics. This construction reveals a symmetry-resolved Schur-Weyl duality, where the relevant commutant is not the ordinary permutation algebra, but the one of color-preserving permutations. We apply this framework to quantum states under symmetry constraints and to energy-resolved ensembles for ETH.
ER -