Measuring quantum relative entropy with finite-size effect

We study the estimation of relative entropy D(ρ\|σ) when σ 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(d²) when the dimension d increases. This sample complexity is optimal when σ is the complexity mixed state. Also, it has time complexity O(d⁵ d). Our proposed estimator unifiedly works under both settings.