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

  • Xinyu Song
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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 κν∈[1,d+1] denote the normalized trace-zero frame spread; κν=1 for exact complex projective $2$-designs. For any fixed rank-one measurement and n≤ c2d2/κν, the minimax mean-squared error is at least c1log2{d2/(κνn)}. For bounded frame spread and every fixed 0<a<2, the minimax risk is of order log2 d throughout n≲ d2−a; hence fixed uniform accuracy requires Ω(d2) observations. Existing tomography bounds give bounded risk from O(d3log3 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.

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Cite

BibTeXRIS
BibTeX
@misc{song2026framespread,
  title = {{A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements}},
  author = {Xinyu Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00270},
  eprint = {2610.00270},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00270},
}
RIS
TY  - UNPB
TI  - A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements
AU  - Xinyu Song
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00270
UR  - https://arxiv.org/abs/2610.00270
AB  - We study estimation of the von Neumann entropy of a d-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let κ_ν∈[1,d+1] denote the normalized trace-zero frame spread; κ_ν=1 for exact complex projective $2$-designs. For any fixed rank-one measurement and n≤ c_2d^2/κ_ν, the minimax mean-squared error is at least c_1log^2{d^2/(κ_νn)}. For bounded frame spread and every fixed 0<a<2, the minimax risk is of order log^2 d throughout n≲ d^2−a; hence fixed uniform accuracy requires Ω(d^2) observations. Existing tomography bounds give bounded risk from O(d^3log^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.
ER  -