Tight concentration inequalities for quantum adversarial setups exploiting permutation symmetry

Takaya Matsuura; Shinichiro Yamano; Yui Kuramochi; Toshihiko Sasaki; Masato Koashi

doi:10.22331/q-2024-11-27-1540

Tight concentration inequalities for quantum adversarial setups exploiting permutation symmetry

Takaya Matsuura^1,2, Shinichiro Yamano³, Yui Kuramochi⁴, Toshihiko Sasaki^3,5, and Masato Koashi^3,5

¹Centre for Quantum Computation & Communication Technology, School of Science, RMIT University, Melbourne VIC 3000, Australia
²RIKEN Center for Quantum Computing (RQC), Hirosawa 2-1, Wako, Saitama 351-0198, Japan
³Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁴Department of Physics, Faculty of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, Japan
⁵Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Published:	2024-11-27, volume 8, page 1540
Eprint:	arXiv:2403.11719v3
Doi:	https://doi.org/10.22331/q-2024-11-27-1540
Citation:	Quantum 8, 1540 (2024).

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Abstract

We developed new concentration inequalities for a quantum state on an $N$-qudit system or measurement outcomes on it that apply to an adversarial setup, where an adversary prepares the quantum state. Our one-sided concentration inequalities for a quantum state require the $N$-qudit system to be permutation invariant and are thus de-Finetti type, but they are tighter than the one previously obtained. We show that the bound can further be tightened if each qudit system has an additional symmetry. Furthermore, our concentration inequality for the outcomes of independent and identical measurements on an $N$-qudit quantum system has no assumption on the adversarial quantum state and is much tighter than the conventional one obtained through Azuma's inequality. We numerically demonstrate the tightness of our bounds in simple quantum information processing tasks.

Featured image: Geometric relation inside the probability simplex ${\cal P}$ between the set ${\cal A}$ of atypical types, the set ${\cal Q}$ of expected types, the set ${\cal R}$ of types that contains ${\cal Q}$ and all extremal types of ${\cal P}$ except $p_1$, and ${\cal R}'$ that contains every type that is $\sqrt{2}/n$ away from every type in ${\cal R}$ in the Euclidean distance. When independent and identical quantum measurements are performed, the Kullback-Leibler divergence between ${\cal A}$ and ${\cal R}'$ gives the exponent of the probability that the type of measurement outcomes lies in ${\cal A}$.

Popular summary

This research develops new concentration inequalities to estimate the probability of obtaining rare or atypical events in quantum measurements, even in adversarial settings where an adversary may influence the system. The structure we exploit to obtain these results is the permutation symmetry of the multiple quantum systems. These inequalities are indispensable for ensuring the reliability and security of quantum technologies, such as quantum cryptography and quantum state verification.

► BibTeX data

@article{Matsuura2024tightconcentration,
  doi = {10.22331/q-2024-11-27-1540},
  url = {https://doi.org/10.22331/q-2024-11-27-1540},
  title = {Tight concentration inequalities for quantum adversarial setups exploiting permutation symmetry},
  author = {Matsuura, Takaya and Yamano, Shinichiro and Kuramochi, Yui and Sasaki, Toshihiko and Koashi, Masato},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1540},
  month = nov,
  year = {2024}
}

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Cited by

[1] Jia-Xuan Li, Yang-Guang Shan, Rong Wang, Feng-Yu Lu, Zhen-Qiang Yin, Shuang Wang, Wei Chen, De-Yong He, Guang-Can Guo, and Zheng-Fu Han, "Secure quantum key distribution against correlated leakage source", Science Advances 12 15, eaed2420 (2026).

[2] Yang-Guang Shan, Zhen-Qiang Yin, Shuang Wang, Wei Chen, De-Yong He, Guang-Can Guo, and Zheng-Fu Han, "Improved finite-key analysis for side-channel-secure quantum key distribution with de Finetti reduction", Physica Scripta 100 11, 115117 (2025).

[3] Akihiro Mizutani, Shun Kawakami, and Go Kato, "Finite-key security analysis of the decoy-state BB84 QKD with passive measurement", Quantum Science and Technology 11 1, 015010 (2026).

[4] Shlok Nahar, Devashish Tupkary, Yuming Zhao, Norbert Lütkenhaus, and Ernest Y.-Z. Tan, "Postselection Technique for Optical Quantum Key Distribution with Improved de Finetti Reductions", PRX Quantum 5 4, 040315 (2024).

[5] Takaya Matsuura, Shinichiro Yamano, Yui Kuramochi, Toshihiko Sasaki, and Masato Koashi, "Asymptotically tight security analysis of quantum key distribution based on universal source compression", arXiv:2504.07356, (2025).

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