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.

Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph); Probability (math.PR)

arXiv:2609.40271 [quant-ph]

(or arXiv:2609.40271v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.40271 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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