Measuring quantum relative entropy with finite-size effect

Quantum 9, 1725 (2025).

https://doi.org/10.22331/q-2025-05-05-1725

We study the estimation of relative entropy $D(rho|sigma)$ 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 $O(d^6 polylog~d)$. Our proposed estimator unifiedly works under both settings.