Computational hardness of estimating quantum entropies via binary entropy bounds

Abstract

We investigate the computational hardness of estimating the quantum $α$-Rényi entropy ${\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}$ and the quantum $q$-Tsallis entropy ${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}$, both of which converge 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 $ {\rm S}^ {\tt R}_α(ρ)$ or ${\rm S}^{\tt T}_q(ρ)$, is at least $τ_1$ or at most $τ_2$, where $τ_1 - τ_2$ 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 $α$ and $q$, and also for $α=\infty$, 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), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders $α> 0$ or $α=\infty$, and for all real orders $0 < q \leq 1$, 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. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.