Simultaneous Estimation of Nonlinear Functionals of a Quantum State

Abstract

We consider a fundamental task in quantum information theory, estimating the values of $\mathrm{tr}(O\ddot{I})$, $\mathrm{tr}(O\ddot{I}{}^2)$, ..., $\mathrm{tr}(O\ddot{I}{}^k)$ for an observable $O$ and a quantum state $\ddot{I}$. We show that $\tilde{\hat{I}}˜(k)$ samples of $\ddot{I}$ 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, $\mathrm{tr}(O\ddot{I}{}^k)$. 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.

24 pages, 1 table, 3 figures