Measuring quantum relative entropy with finite-size effect

TL;DR

This paper develops an estimator for quantum relative entropy that achieves optimal sample complexity and bound attainment, effectively handling finite-size effects in quantum state estimation.

Contribution

It introduces a new estimator for quantum relative entropy that attains the Cramér-Rao bound and has optimal sample complexity, applicable in various dimensions.

Findings

Estimator attains the Cramér-Rao type bound.

Sample complexity is $O(d^2)$, optimal for the maximally mixed state.

Time complexity is $O(d^6 ext{polylog} d)$.

Abstract

We study the estimation of relative entropy $D(\rho\|\sigma)$ when $\sigma$ is known. We show that the Cram\'{e}r-Rao type bound equals the relative varentropy. Our estimator attains the Cram\'{e}r-Rao type bound when the dimension $d$ is fixed. It also achieves the sample complexity $O(d^2)$ when the dimension $d$ increases. This sample complexity is optimal when $\sigma$ is the completely mixed state. Also, it has time complexity $O(d^6 \polylog d)$. Our proposed estimator unifiedly works under both settings.