Exact Haar Statistics of Planar k-Purity in Multipartite Quantum Systems
Abstract
Planar k-uniform states have maximally mixed reductions on every k-site interval of a ring. We introduce planar k-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of n parties with local dimension p, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every 1≤ k≤⌊ n/2⌋. In the balanced qubit case, k=⌊ n/2⌋ and N=2n, the variance is asymptotic to 20/(3nN2) for even n and 6/(nN2) for odd n. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through n=10 validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at n=10. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.
Related
- Extremal States of Multipartite Quantum Spin Systems
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BibTeX
@misc{mirsohi2026exact,
title = {{Exact Haar Statistics of Planar $k$-Purity in Multipartite Quantum Systems}},
author = {Kourosh Mirsohi},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28914},
eprint = {2608.28914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28914},
}
RIS
TY - UNPB
TI - Exact Haar Statistics of Planar k-Purity in Multipartite Quantum Systems
AU - Kourosh Mirsohi
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28914
UR - https://arxiv.org/abs/2608.28914
AB - Planar k-uniform states have maximally mixed reductions on every k-site interval of a ring. We introduce planar k-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of n parties with local dimension p, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every 1≤ k≤⌊ n/2⌋. In the balanced qubit case, k=⌊ n/2⌋ and N=2^n, the variance is asymptotic to 20/(3nN^2) for even n and 6/(nN^2) for odd n. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through n=10 validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at n=10. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.
ER -