Given many copies of some unknown quantum state $\rho$, I would like to compute its von Neumann entropy $S(\rho)$. What algorithm could be used for this that minimizes the number of copies required? We require that the estimate of the entropy has to be $\varepsilon-$close and one will need more copies as $\varepsilon\rightarrow 0$.

The naive solution is to do tomography and obtain a classical description of the state. This would require exponentially many copies as we increase the dimension of $\rho$. But the classical description of the state has a lot more information so perhaps there is a smarter way?