Universal Scaling of the Minimum Error Probability in Qualification of Quantum States
Abstract
Qualification of quantum states judges which of two sets of quantum states an unknown state lies in, where the two sets are labeled by two distinct parameter regions. We formulate this problem as a composite quantum hypothesis test and uncover universal scaling laws for the minimum error probability for N copies. Taking polarization-direction qualification and purity qualification as examples, we show that the N-copy permutation symmetry and the geometric symmetries of the parameter regions identify the optimal measurements and the "worst pairwise states". The minimum error probability scales as N−3/2exp(−Nξ) for disjoint regions and as (NF)−1/2 for adjacent regions, where ξ and F are the quantum Chernoff divergence and quantum Fisher information associated with the "worst pairwise states", respectively. With the minimum error probability serving as an order parameter, the transition between the scaling behaviors becomes a second-order phase transition as N→∞. Our approach determines whether a quantum state belongs to a given set without full state tomography, thereby enabling qualification of large ensembles using finite samples.
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BibTeX
@misc{fei2026universal,
title = {{Universal Scaling of the Minimum Error Probability in Qualification of Quantum States}},
author = {Zhaoyu Fei and Yaotian Li and Weicheng Huang and Xiaoguang Wang and Y. -M. Du},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.04870},
eprint = {2608.04870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04870},
}
RIS
TY - UNPB
TI - Universal Scaling of the Minimum Error Probability in Qualification of Quantum States
AU - Zhaoyu Fei
AU - Yaotian Li
AU - Weicheng Huang
AU - Xiaoguang Wang
AU - Y. -M. Du
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.04870
UR - https://arxiv.org/abs/2608.04870
AB - Qualification of quantum states judges which of two sets of quantum states an unknown state lies in, where the two sets are labeled by two distinct parameter regions. We formulate this problem as a composite quantum hypothesis test and uncover universal scaling laws for the minimum error probability for N copies. Taking polarization-direction qualification and purity qualification as examples, we show that the N-copy permutation symmetry and the geometric symmetries of the parameter regions identify the optimal measurements and the "worst pairwise states". The minimum error probability scales as N^−3/2exp(−Nξ) for disjoint regions and as (NF)^−1/2 for adjacent regions, where ξ and F are the quantum Chernoff divergence and quantum Fisher information associated with the "worst pairwise states", respectively. With the minimum error probability serving as an order parameter, the transition between the scaling behaviors becomes a second-order phase transition as N→∞. Our approach determines whether a quantum state belongs to a given set without full state tomography, thereby enabling qualification of large ensembles using finite samples.
ER -