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arXiv · 2505.16715

Simultaneous Estimation of Nonlinear Functionals of a Quantum State

Kean Chen·Qisheng Wang·Zhan Yu·Zhicheng Zhang
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Abstract

We consider a fundamental task in quantum information theory, estimating the values of $tr(Oρ)$, $tr(Oρ^2)$, $\ldots$, $tr(Oρ^k)$ for an observable $O$ and a quantum state $ρ$. We show that $\widetildeΘ(k)$ samples of $ρ$ are sufficient and necessary to simultaneously estimate all the $k$ values. This means that estimating all the $k$ values is almost as easy as estimating only one of them, $tr(Oρ^k)$. As an application, our approach advances the sample complexity of entanglement spectroscopy and the virtual cooling for quantum many-body systems. Moreover, we extend our approach to estimating general functionals by polynomial approximation.

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Kean Chen, Qisheng Wang, Zhan Yu, Zhicheng Zhang. 2026-09-15. Simultaneous Estimation of Nonlinear Functionals of a Quantum State. https://doi.org/10.1109/tit.2026.3699531

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