Breaking the Quadratic Barrier for von Neumann Entropy Estimation

2026-08-11

AI Summary

This work addresses the long-standing limitation that estimating the von Neumann entropy of high-dimensional quantum states requires at least Ω(d²) samples. We propose the first estimator achieving sub-quadratic sample complexity by combining bias-corrected estimation for large eigenvalues with bounded-coefficient polynomial approximation for small eigenvalues. A novel compression inequality, derived via orthogonal direct-sum decomposition of the state space, is introduced to tightly control entropy estimation error. Under constant additive error, our algorithm achieves a sample complexity of O(d² log²(log d)/log² d + log²(d/ε)/ε²), which is o(d²). This result breaks through the established theoretical barrier and represents the first demonstration of sub-quadratic sample efficiency in quantum entropy estimation.

Abstract

We study the sample complexity of estimating the von Neumann entropy of an unknown $d$-dimensional quantum state. All previously known estimators require $Ω(d^2)$ 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.

Problem

• von Neumann entropy
• sample complexity
• quantum state
• subquadratic estimation
• entropy estimation

Innovation

• von Neumann entropy
• sample complexity
• subquadratic estimation
• pinching inequality
• bias-corrected estimator