Optimal von Neumann Entropy Estimation

We show that the von Neumann entropy of an unknown $d$-dimensional quantum state $ρ$ can be estimated to within additive error $\varepsilon$ using \[ Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)(1+\varepsilon\log^2(d))} + \frac{\log^2(d)}{\varepsilon^2}\right) \] samples of $ρ$, matching the lower bound