Simultaneous Estimation of Nonlinear Functionals of a Quantum State

Abstract:We consider a fundamental task in quantum information theory, estimating the values of $tr(O\rho)$, $tr(O\rho^2)$, $\ldots$, $tr(O\rho^k)$ for an observable $O$ and a quantum state $\rho$. We show that $\widetilde\Theta(k)$ samples of $\rho$ 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, $tr(O\rho^k)$. 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.

28 pages. v2: minor revision

arXiv:2505.16715 [quant-ph]

(or arXiv:2505.16715v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2505.16715 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.1109/TIT.2026.3699531 Focus to learn more DOI(s) linking to related resources

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