Measuring quantum relative entropy with finite-size effect

Masahito Hayashi

doi:10.22331/q-2025-05-05-1725

Measuring quantum relative entropy with finite-size effect

School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China
International Quantum Academy, Futian District, Shenzhen 518048, China
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

Published:	2025-05-05, volume 9, page 1725
Editor:	Christos Gagatsos
Eprint:	arXiv:2406.17299v4
Doi:	https://doi.org/10.22331/q-2025-05-05-1725
Citation:	Quantum 9, 1725 (2025).

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Abstract

We study the estimation of relative entropy $D(\rho\|\sigma)$ when $\sigma$ is known. We show that the Cramér-Rao type bound equals the relative varentropy. Our estimator attains the Cramér-Rao type bound when the dimension $d$ is fixed. It also achieves the sample complexity $O(d^2)$ when the dimension $d$ increases. This sample complexity is optimal when $\sigma$ is the completely mixed state. Also, it has time complexity $O(d^6 polylog~d)$. Our proposed estimator unifiedly works under both settings.

Popular summary

Distinguishing between quantum states is a fundamental task in quantum information theory, and quantum relative entropy, denoted as $D(\rho||\sigma)$, is a key tool for quantifying this distinguishability. This paper addresses the problem of estimating $D(\rho||\sigma)$ when a state $\rho$ is generated as an approximation of the target state $\sigma$. The proposed method is useful for identifying how close the generated unknown state $\rho$ is to the target state $\sigma$. The author derives a fundamental limit on the precision of this estimation, known as the Cramér-Rao type bound, and proposes a novel estimation method based on the Schur transform. The paper demonstrates that this method achieves a sample complexity of $O(d^2)$, where $d$ is the dimension of the quantum system. This implies that the number of samples required to estimate the relative entropy grows quadratically with the size of the system, offering a more efficient approach to quantifying the distinguishability of quantum states.

► BibTeX data

@article{Hayashi2025measuringquantum,
  doi = {10.22331/q-2025-05-05-1725},
  url = {https://doi.org/10.22331/q-2025-05-05-1725},
  title = {Measuring quantum relative entropy with finite-size effect},
  author = {Hayashi, Masahito},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1725},
  month = may,
  year = {2025}
}

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Cited by

[1] Masahito Hayashi and Yingkai Ouyang, "The Cramér-Rao approach and global quantum estimation of bosonic states", Quantum 9, 1806 (2025).

[2] Yupan Liu and Qisheng Wang, "On Estimating the Trace of Quantum State Powers", IEEE Transactions on Information Theory 72 8, 5830 (2026).

[3] Qisheng Wang and Zhicheng Zhang, "Time-Efficient Quantum Entropy Estimator via Samplizer", IEEE Transactions on Information Theory 71 12, 9569 (2025).

[4] Kean Chen, Qisheng Wang, Zhan Yu, and Zhicheng Zhang, "Simultaneous Estimation of Nonlinear Functionals of a Quantum State", arXiv:2505.16715, (2025).

[5] Jinge Bao, Minbo Gao, and Qisheng Wang, "On Estimating the Quantum Tsallis Relative Entropy", arXiv:2510.00752, (2025).

[6] Kean Chen, Yupan Liu, and Qisheng Wang, "Trace Estimation of Quantum State Powers: Sample Complexity and Computational Hardness", arXiv:2505.09563, (2025).

[7] Yupan Liu and Qisheng Wang, "On estimating the trace of quantum state powers", arXiv:2410.13559, (2024).

[8] Qisheng Wang and Zhicheng Zhang, "Time-Efficient Quantum Entropy Estimator via Samplizer", arXiv:2401.09947, (2024).

[9] Wang Fang and Qisheng Wang, "Query-Optimal and Sample-Optimal Quantum Algorithms for Estimating Fidelity to a Pure State", arXiv:2506.23650, (2025).

[10] Qisheng Wang, "Towards Minimax Estimation of High-Order Functionals by Quantum Arguments", arXiv:2607.07540, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 14:24:05) and SAO/NASA ADS (last updated successfully 2026-08-10 14:24:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

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