Estimating Quantum Entropy.

The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of estimating the von Neumann entropy, $S(\rho)$ , and Rényi entropy, $S_{\alpha }(\rho)$ of an unknown mixed quantum state $\rho $ in $d$ dimensions, given access to independent copies of $\rho $ . We provide algorithms with copy complexity $O(d^{2/\alpha })$ for estimating $S_{\alpha }(\rho)$ for $\alpha < 1$ , and copy complexity $O(d^{2})$ for estimating $S(\rho)$ , and $S_{\alpha }(\rho)$ for non-integral $\alpha >1$ . These bounds are at least quadratic in $d$ , which is the order dependence on the number of copies required for estimating the entire state $\rho $ . For integral $\alpha >1$ , on the other hand, we provide an algorithm for estimating $S_{\alpha }(\rho)$ with a sub-quadratic copy complexity of $O(d^{2-2/\alpha })$ , and we show the optimality of the algorithms by providing a matching lower bound.