Breaking the Quadratic Barrier for von Neumann Entropy Estimation

We study the sample complexity of estimating the von Neumann entropy of an unknown $d$-dimensional quantum state. All previously known estimators require $\Omega(d^2)$ samples, and plug-in estimators are known to face a quadratic barrier. We give the first subquadratic-sample estimator: for additive error $\varepsilon$, our estimator uses $$O\!\left(\frac{d^2 \log^2(\log(d)) \log(1/\varepsilon)}{\varepsilon^2 \log^2(d)} + \frac{\log^2(d/\varepsilon)}{\varepsilon^2}\right)$$ samples. In particular, for constant $\varepsilon$, the complexity is $O_\varepsilon(d^2\log^2(\log(d))/\log^2(d))=o(d^2)$. Our analysis introduces a new pinching inequality that bounds the entropy loss under a space direct-sum decomposition, together with a bias-corrected estimator for large eigenvalues and a new bounded-coefficient polynomial estimator for small eigenvalues.