Bounding the Sample Fluctuation for Pure States Certification with Local Random Measurement

Remarkable breakthroughs in quantum science and technology are demanding for more efficient methods in analyzing quantum many-body states. A significant challenge in this field is to verify whether a quantum state prepared by quantum devices in the lab accurately matches the desired target pure state. Recent advancements in randomized measurement techniques have provided fresh insights in this area. Specifically, protocols such as classical shadow tomography and shadow overlap have been proposed. Building on these developments, we investigate the fundamental properties of schemes that certify pure quantum states through random local Haar measurements. We derive bounds for sample fluctuations that are applicable regardless of the specific estimator construction. These bounds depend on the operator size distribution of either the observable used to estimate fidelity or the valid variation of the reduced density matrix for arbitrary observables. Our results unveil the intrinsic interplay between operator complexity and the efficiency of quantum algorithms, serving as an obstacle to local certification of pure states with long-range entanglement.