Simultaneous Estimation of Nonlinear Functionals of a Quantum State

TL;DR

This paper introduces a method to efficiently estimate multiple nonlinear functionals of a quantum state simultaneously, significantly reducing the sample complexity compared to estimating each functional separately, with applications in quantum information processing.

Contribution

It demonstrates that a near-optimal number of samples suffices for simultaneous estimation of multiple functionals, extending to general functionals via polynomial approximation.

Findings

Sample complexity is nearly independent of the number of functionals k.

Method improves entanglement spectroscopy and quantum many-body system analysis.

Extends to general functionals using polynomial approximation.

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.