Spectrum Estimation is Almost as Hard as Tomography

2026-07-31

AI Summary

This work investigates the sample complexity lower bounds for estimating unitarily invariant properties of high-dimensional quantum states, including spectral estimation, von Neumann entropy estimation, and rank testing. By constructing hard instances of mixed states based on normalized Haar-random projectors and innovatively employing Jucys–Murphy elements from the symmetric group algebra—along with their higher-order symmetric functions—to analyze quantum state moments and indistinguishability, the authors combine moment matching with f-divergence bounds to establish, for the first time, nearly tight lower bounds of Ω(d²⁻ᵞ) under constant error. These results demonstrate that the sample complexity of these tasks is nearly as demanding as that of full quantum state tomography.

Abstract

We study the sample complexity of estimating and testing fundamental unitarily invariant properties of unknown quantum states; namely, the tasks of spectrum estimation, von Neumann entropy estimation, and rank-testing. For $d$-dimensional states, and for every $γ>0$, we prove a sample complexity lower bound of $Ω(d^{2-γ})$ for spectrum estimation to constant sorted total-variation error, entropy estimation to constant additive error, and rank-testing to constant trace distance. Our hard instances are constructed from sandwiched products of Haar-random projectors, suitably normalized using a novel technique that lets us derive explicit expressions for high-order tensor moments of the resultant states. These moments can be expressed as symmetric functions of Jucys--Murphy elements of the symmetric group algebra. To show that two such mixtures are indistinguishable, we analyze the log-likelihood ratio and perform moment-matching, i.e., we set its low-order Jucys--Murphy components to zero. Indistinguishability is then obtained by bounding an $f$-divergence through the high-order components; the non-zero high-order terms and concentration of functions of Haar-random unitaries also imply separations in typical spectra, entropies, and ranks, proving all our lower bounds.

Problem

• spectrum estimation
• sample complexity
• quantum state
• von Neumann entropy
• rank testing

Innovation

• sample complexity lower bounds
• Haar-random projectors
• Jucys–Murphy elements
• moment-matching
• unitarily invariant properties