Computational hardness of estimating quantum entropies via binary entropy bounds

Abstract: We investigate the computational hardness of estimating the quantum $α$-Rényi entropy $${\rm S}^{\tt R}<em>α(ρ) = \frac{\ln {\rm Tr}(ρ<sup>α)}{1-α}$$ and the quantum $q$-Tsallis entropy $${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ<sup>q)}{q-1}$$, both converging to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $α$-Rényi Entropy Approximation (RényiQEA$</em>α$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$<em>q$) ask whether ${\rm S}^ {\tt R}</em>α(ρ)$ or ${\rm S}^{\tt T}<em>q(ρ)$, respectively, is at least $τ</em>{\tt Y}$ or at most $τ<em>{\tt N}$, where $τ</em>{\tt Y} - τ<em>{\tt N}$ 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$</em>α$ and Rank2TsallisQEA$<em>q$ 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 $α&gt; 0$ and $0 &lt; q \leq 1$, LowRankRényiQEA$</em>α$ and LowRankTsallisQEA$<em>q$ are ${\sf BQP}$-complete, where both are restricted versions of RényiQEA$</em>α$ and TsallisQEA$_q$ with $ρ$ of polynomial rank. - For all real order $q&gt;1$, TsallisQEA$_q$ is ${\sf BQP}$-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.