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Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds

Author Yupan Liu

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LIPIcs.STACS.2026.66.pdf
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ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Mathematics of computing → Information theory
  • Theory of computation → Quantum complexity theory
Keywords
  • computational hardness
  • quantum state testing
  • quantum Rényi entropy
  • quantum Tsallis entropy
  • von Neumann entropy

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Abstract

We investigate the computational hardness of estimating the quantum α-Rényi entropy S^𝚁_α(ρ) = (ln Tr(ρ^α))/(1-α) and the quantum q-Tsallis entropy S^𝚃_q(ρ) = (1-Tr(ρ^q))/(q-1), both converging to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEA_α) and Quantum q-Tsallis Entropy Approximation (TsallisQEA_q) ask whether S^𝚁_α(ρ) or S^𝚃_q(ρ), respectively, is at least τ_Y or at most τ_N, where τ_Y - τ_N is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 1) and some cases of the quantum q-Tsallis entropy, while existing approaches do not readily extend to other orders. 
We establish that for all positive real orders, the rank-2 variants Rank2RényiQEA_α and Rank2TsallisQEA_q are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), our results imply:  
- For all real order α > 0 and 0 < q ≤ 1, LowRankRényiQEA_α and LowRankTsallisQEA_q are BQP-complete, where both are restricted versions of RényiQEA_α and TsallisQEA_q with ρ of polynomial rank. 
- For all real order q > 1, TsallisQEA_q is BQP-complete. 
Our hardness results stem from reductions based on new inequalities relating the α-Rényi or q-Tsallis binary entropies of different orders, where the reductions differ substantially from previous approaches, and the inequalities are also of independent interest.

Cite As Get BibTex

Yupan Liu. Computational Hardness of Estimating Quantum Entropies via Binary Entropy Bounds. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 66:1-66:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.66

Author Details

Yupan Liu
  • School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Switzerland
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

This work was supported in part by funding from the Swiss State Secretariat for Education, Research and Innovation (SERI), in part by MEXT Q-LEAP grant No. JPMXS0120319794, and in part by JSPS KAKENHI grant No. JP24H00071.

Acknowledgements

The author is grateful to François Le Gall and Qisheng Wang for helpful discussions, and especially to Qisheng Wang for mentioning references [Li and Wu, 2019; Bun et al., 2020]. The author also appreciates ChatGPT for suggesting the use of generalized binomial coefficients in proving Lemma 27. Additionally, the author thanks the anonymous reviewers for their constructive comments.

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