Optimal von Neumann Entropy Estimation

Abstract:We show that the von Neumann entropy of an unknown $d$-dimensional quantum state $\rho$ can be estimated to within additive error $\varepsilon$ using \[ \Theta\!\left(\frac{d^2}{\varepsilon\log^2(d)(1+\varepsilon\log^2(d))} + \frac{\log^2(d)}{\varepsilon^2}\right) \] samples of $\rho$, matching the lower bound due to Wang (2026). Our estimator is based on a high-degree approximation polynomial and the main technical challenge is bounding its variance despite the large polynomial degree. We overcome this using a quantum Hoeffding decomposition, which preserves cancellations between different polynomial terms and reduces the analysis to derivative bounds for the approximation polynomial.

22 pages, 1 table, 1 algorithm. Improved the sample complexity to optimal

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2608.11151 [quant-ph]

(or arXiv:2608.11151v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.11151 Focus to learn more arXiv-issued DOI via DataCite

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