Induced random mixed states are symmetric Dirichlet mixtures
Abstract
We establish an exact equivalence in distribution between D-dimensional random mixed states induced by partial traces over K-dimensional environments and the Mai-Alquier distribution, a mixture of K independent Haar-random pure states weighted by a symmetric Dirichlet distribution. This identification recasts a class of expectation values for induced ensembles into calculations involving Dirichlet moments and low-order Haar averages on a single-system state space. As applications, we recover exact purity moments up to fourth order, derive the mean Hilbert--Schmidt distance between independent induced ensembles with possibly different environment dimensions, and obtain exact average determinants. These results provide a unified and constructive perspective on induced random mixed states and on the evaluation of quantities such as purity moments, overlaps, and determinants.
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BibTeX
@misc{kirby2026induced,
title = {{Induced random mixed states are symmetric Dirichlet mixtures}},
author = {Brian T. Kirby and Alexander C. B. Greenwood and Sanjaya Lohani and Joseph M. Lukens},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09421},
eprint = {2609.09421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09421},
}
RIS
TY - UNPB
TI - Induced random mixed states are symmetric Dirichlet mixtures
AU - Brian T. Kirby
AU - Alexander C. B. Greenwood
AU - Sanjaya Lohani
AU - Joseph M. Lukens
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09421
UR - https://arxiv.org/abs/2609.09421
AB - We establish an exact equivalence in distribution between D-dimensional random mixed states induced by partial traces over K-dimensional environments and the Mai-Alquier distribution, a mixture of K independent Haar-random pure states weighted by a symmetric Dirichlet distribution. This identification recasts a class of expectation values for induced ensembles into calculations involving Dirichlet moments and low-order Haar averages on a single-system state space. As applications, we recover exact purity moments up to fourth order, derive the mean Hilbert--Schmidt distance between independent induced ensembles with possibly different environment dimensions, and obtain exact average determinants. These results provide a unified and constructive perspective on induced random mixed states and on the evaluation of quantities such as purity moments, overlaps, and determinants.
ER -