Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements
Abstract
We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least (d/n)log2{n/(4d)} when d≥ C and n≥ Cd, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius r around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by ε<1 from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most d3r2/n+d4/n2+d6r6. Lower bounds under the same frame control yield the local minimax rate d3r2/n+d4/n2 when n≥ Cd2 and r lies in an explicit matching range. For every fixed upper bound on ε, approximation changes only the constants, not the powers of d,n,r. At the critical radius r=n−1/2, the local risk is asymptotically negligible relative to the global risk when nlog2(n/d)≫ d3. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.
Related
- A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements
- Adaptive Local Measurements For Quantum Learning Can Exponentially Outperform Nonadaptive Global Measurements
- Analytic Qubit Separation between POVMs and Projective Measurements
- Cuspidal Cubic Projective Foliations: Global Rigidity, Deformations, and Pull-Backs
- Optimal von Neumann Entropy Estimation
BibTeX
@misc{song2026local,
title = {{Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements}},
author = {Xinyu Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29485},
eprint = {2609.29485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29485},
}
RIS
TY - UNPB
TI - Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements
AU - Xinyu Song
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29485
UR - https://arxiv.org/abs/2609.29485
AB - We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least (d/n)log^2{n/(4d)} when d≥ C and n≥ Cd, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius r around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by ε<1 from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most d^3r^2/n+d^4/n^2+d^6r^6. Lower bounds under the same frame control yield the local minimax rate d^3r^2/n+d^4/n^2 when n≥ Cd^2 and r lies in an explicit matching range. For every fixed upper bound on ε, approximation changes only the constants, not the powers of d,n,r. At the critical radius r=n^−1/2, the local risk is asymptotically negligible relative to the global risk when nlog^2(n/d)≫ d^3. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.
ER -