Simultaneous Estimation of Nonlinear Functionals of a Quantum State

Understand

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.