Minimax Quantum State Tomography in a Known Trace-Distance Ball
Abstract
We study quantum state tomography when the unknown state lies in a trace-distance ball of radius R around a known reference state ρ. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write Γ=d(Tr√ρ)2+d2R. For d≥2, 0<R≤1, and n≥1 copies, the expected risk is Θ(min{R,√(Γ/n)}). At failure probability 0<δ≤1/3, the minimax confidence radius is Θ (min{R,√((Γ+log(1/δ))/n)}). For ε≤ c0R with a universal c0>0, the copy complexity is Θ((Γ+log(1/δ))/ε2). For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.
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BibTeX
@misc{li2026minimax,
title = {{Minimax Quantum State Tomography in a Known Trace-Distance Ball}},
author = {Yonglong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05374},
eprint = {2610.05374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05374},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Minimax Quantum State Tomography in a Known Trace-Distance Ball
AU - Yonglong Li
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05374
UR - https://arxiv.org/abs/2610.05374
AB - We study quantum state tomography when the unknown state lies in a trace-distance ball of radius R around a known reference state ρ. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write Γ=d(Tr√ρ)^2+d^2R. For d≥2, 0<R≤1, and n≥1 copies, the expected risk is Θ(min{R,√(Γ/n)}). At failure probability 0<δ≤1/3, the minimax confidence radius is Θ (min{R,√((Γ+log(1/δ))/n)}). For ε≤ c_0R with a universal c_0>0, the copy complexity is Θ((Γ+log(1/δ))/ε^2). For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.
ER -