Simultaneous Estimation of Nonlinear Functionals of a Quantum State

• URL: http://arxiv.org/abs/2505.16715v1
• Date: Thu, 22 May 2025 14:23:48 GMT
• Title: Simultaneous Estimation of Nonlinear Functionals of a Quantum State
• Authors: Kean Chen, Qisheng Wang, Zhan Yu, Zhicheng Zhang,
• Abstract summary: We consider a fundamental task in quantum information theory, estimating the values of $operatornametr(Orho)$, $operatornametr(Orho2)$,..., $operatornametr(Orhok)$ for an observable $O$ and a quantum state $rho$.<n>We show that $widetildeTheta(k)$ samples of $rho$ are sufficient to simultaneously estimate all the $k$ values.
• Score: 12.191929463091354
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider a fundamental task in quantum information theory, estimating the values of $\operatorname{tr}(O\rho)$, $\operatorname{tr}(O\rho^2)$, ..., $\operatorname{tr}(O\rho^k)$ 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, $\operatorname{tr}(O\rho^k)$. 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.

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