Computational hardness of estimating quantum entropies via binary entropy bounds
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Quantum Physics arXiv:2601.03734 (quant-ph) [Submitted on 7 Jan 2026] Title:Computational hardness of estimating quantum entropies via binary entropy bounds Authors:Yupan Liu View a PDF of the paper titled Computational hardness of estimating quantum entropies via binary entropy bounds, by Yupan Liu View PDF HTML (experimental) Abstract:We investigate the computational hardness of estimating the quantum $\alpha$-Rényi entropy ${\rm S}^{\tt R}_{\alpha}(\rho) = \frac{\ln {\rm Tr}(\rho^\alpha)}{1-\alpha}$ and the quantum $q$-Tsallis entropy ${\rm S}^{\tt T}_q(\rho) = \frac{1-{\rm Tr}(\rho^q)}{q-1}$, both converging to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $\alpha$-Rényi Entropy Approximation (RényiQEA$_\alpha$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$_q$) ask whether $ {\rm S}^ {\tt R}_{\alpha}(\rho)$ or ${\rm S}^{\tt T}_q(\rho)$, respectively, is at least $\tau_{\tt Y}$ or at most $\tau_{\tt N}$, where $\tau_{\tt Y} - \tau_{\tt N}$ 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$_\alpha$ and Rank2TsallisQEA$_q$ are ${\sf BQP}$-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 orders $\alpha > 0$ and $0 1$, TsallisQEA$_q$ is ${\sf BQP}$-complete. Our hardness results stem from reductions based on new inequalities relating the $\alpha$-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. Comments: Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC); Information Theory (cs.IT) Cite as: arXiv:2601.03734 [quant-ph] (or arXiv:2601.03734v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.03734 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Yupan Liu [view email] [v1] Wed, 7 Jan 2026 09:25:07 UTC (50 KB) Full-text links: Access Paper: View a PDF of the paper titled Computational hardness of estimating quantum entropies via binary entropy bounds, by Yupan LiuView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 Change to browse by: cs cs.CC cs.IT math math.IT References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)