Measuring quantum relative entropy with finite-size effect

We study the estimation of relative entropy $D(\rho \mathrm{\parallel }\sigma )$ when $\sigma$ is known. We show that the Cramér-Rao type bound equals the relative varentropy. Our estimator attains the Cramér-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 $O(d^6\text{polylog}d)$. Our proposed estimator unifiedly works under both settings.