Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
Abstract
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on ℂd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(dr/ε2mathopmax{1,r/√t}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor √t. Further, measuring order r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
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BibTeX
@misc{nayak2026optimal,
title = {{Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements}},
author = {Ashwin Nayak and Xingyu Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10514},
eprint = {2609.10514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10514},
}
RIS
TY - UNPB
TI - Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
AU - Ashwin Nayak
AU - Xingyu Zhou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.10514
UR - https://arxiv.org/abs/2609.10514
AB - We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on ℂ^d of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(dr/ε^2mathopmax{1,r/√t}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor √t. Further, measuring order r^2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
ER -