Refined sample complexities from the tomographic rate function
Abstract
Protocols for quantum state tomography can be characterized either by their sample complexity or by the tomographic rate function that governs the large deviation behaviour of error estimates. We establish a framework for the analysis of covariant protocols which allows us to compare these measures. First, we derive the rate functions for several protocols, including random purification-based protocols and the sample-optimal protocol of Haah et al. These rate functions are expressed in terms of a new family of divergences which includes the reverse sandwiched Rényi divergence and Keyl's annealed quantum relative entropy as special cases. We show that these rate functions obey a strict ordering, even though the sample complexity of each protocol is order optimal. We consider a different version of sample complexity based on Wasserstein distance rather than trace distance, and show that the ordering of the rate functions implies analogous strict inequalities for the Wasserstein-type sample complexity. Thus the tomographic rate function is a more fine-grained indicator of the performance of a tomography protocol than the usual sample complexity.
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BibTeX
@misc{grootveld2026refined,
title = {{Refined sample complexities from the tomographic rate function}},
author = {Arick Grootveld and Alexander Maloney and Jason Pollack and Peixue Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40271},
eprint = {2609.40271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40271},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Refined sample complexities from the tomographic rate function
AU - Arick Grootveld
AU - Alexander Maloney
AU - Jason Pollack
AU - Peixue Wu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.40271
UR - https://arxiv.org/abs/2609.40271
AB - Protocols for quantum state tomography can be characterized either by their sample complexity or by the tomographic rate function that governs the large deviation behaviour of error estimates. We establish a framework for the analysis of covariant protocols which allows us to compare these measures. First, we derive the rate functions for several protocols, including random purification-based protocols and the sample-optimal protocol of Haah et al. These rate functions are expressed in terms of a new family of divergences which includes the reverse sandwiched Rényi divergence and Keyl's annealed quantum relative entropy as special cases. We show that these rate functions obey a strict ordering, even though the sample complexity of each protocol is order optimal. We consider a different version of sample complexity based on Wasserstein distance rather than trace distance, and show that the ordering of the rate functions implies analogous strict inequalities for the Wasserstein-type sample complexity. Thus the tomographic rate function is a more fine-grained indicator of the performance of a tomography protocol than the usual sample complexity.
ER -