A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements

Abstract:We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $\kappa_\nu\in[1,d+1]$ denote the normalized trace-zero frame spread; $\kappa_\nu=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/\kappa_\nu$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(\kappa_\nu n)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $\Omega(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

Quantum Physics (quant-ph); Statistics Theory (math.ST)

arXiv:2610.00270 [quant-ph]

(or arXiv:2610.00270v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00270 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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