Optimal Purity Estimation with Incoherent Measurements
Abstract
In this work, we consider the fundamental task of estimating the purity of an unknown state to within multiplicative error ε. When one can perform general collective measurements, Θ(√d/ε2 + d/ε) copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements. In this setting, the only non-trivial result is a non-adaptive algorithm that uses O( d/ε2 + d2/ε ) copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using O( d/ε2 + d3/2/ε ) copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal. We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for ε = o(1/d). We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is Θ(min{ d/ε2 + d3/2/ε, d2/ε + √d/ε2} ).
Related
- Optimal Quantum Purity Amplification: A Top-Down Approach to Error Suppression via Symmetry and Reversibility
- Near-optimal incoherent tomography of low-rank quantum channels
- Adaptive negativity estimation via collective measurements
- Sublinear Copies Suffice for Fidelity Estimation with Pauli Measurements
- Beyond Shor's Conjecture: Optimal Quantum Measurements at Fixed Error
BibTeX
@misc{lee2026optimal,
title = {{Optimal Purity Estimation with Incoherent Measurements}},
author = {Junseo Lee and Chirag Wadhwa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40248},
eprint = {2609.40248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40248},
}
RIS
TY - UNPB
TI - Optimal Purity Estimation with Incoherent Measurements
AU - Junseo Lee
AU - Chirag Wadhwa
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40248
UR - https://arxiv.org/abs/2609.40248
AB - In this work, we consider the fundamental task of estimating the purity of an unknown state to within multiplicative error ε. When one can perform general collective measurements, Θ(√d/ε^2 + d/ε) copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements. In this setting, the only non-trivial result is a non-adaptive algorithm that uses O( d/ε^2 + d^2/ε ) copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using O( d/ε^2 + d^3/2/ε ) copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal. We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for ε = o(1/d). We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is Θ(min{ d/ε^2 + d^3/2/ε, d^2/ε + √d/ε^2} ).
ER -