Problem
Research questions and friction points this paper is trying to address.
quantum state tomography
sample complexity
tomographic rate function
covariant protocols
Refined sample complexities from the tomographic rate function
🤖 AI Summary
This study addresses the insufficient granularity of sample complexity metrics in quantum state tomography protocols, which hinders the fine-grained differentiation of performance. To overcome this limitation, a covariant protocol analysis framework is established. Methodologically, Wasserstein distance-based sample complexity is introduced, and large deviation theory is employed to derive tomographic rate functions. Furthermore, novel divergence families, including inverse sandwich Rényi divergence and annealed quantum relative entropy, are proposed to compare the large deviation behaviors of protocols. The results establish the strict ordering of these rate functions, demonstrating their capacity to reveal fine-grained rankings among protocols. This work validates that the proposed metric outperforms conventional measures, thereby effectively refining the evaluation system for quantum tomography efficiency.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum state tomography
Tomographic rate function
Sample complexity
Reverse sandwiched Rényi divergence
Wasserstein distance
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