Computational hardness of estimating quantum entropies via binary entropy bounds
Yupan Liu
TL;DR
This work analyzes the computational hardness of estimating quantum entropies of generalized Rényi and Tsallis forms, ${\rm S}^{\tt R}_{\alpha}(\rho)$ and ${\rm S}^{\tt T}_q(\rho)$, framing the task as promise problems with a constant gap. It introduces rank-2 reductions that connect quantum entropies of rank-2 mixtures to binary entropies, and proves that Rank2RényiQEA$_{\alpha}$ and Rank2TsallisQEA$_q$ are ${\sf BQP}$-hard for all positive orders, with ${\sf BQP}$-completeness for LowRank variants when the rank is polynomial. The paper also establishes order-zero results in the ${\sf NQP}$ class and develops a suite of new inequalities relating binary entropies across orders, which are of independent interest. By combining these hardness results with existing rank-dependent quantum-query algorithms, the authors obtain comprehensive hardness landscapes: ${\sf BQP}$-hardness for all $\alpha>0$ and $q>0$, with ${\sf BQP}$-completeness for relevant low-rank regimes and for Tsallis entropy with $q>1$. These findings illuminate fundamental limits on efficiently estimating quantum entropies and provide a new framework for reductions via entropy-order inequalities, with potential implications for quantum cryptography and information-processing tasks that rely on entropy estimation.
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 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 $ {\rm S}^ {\tt R}_α(ρ)$ or ${\rm S}^{\tt T}_q(ρ)$, respectively, is at least $τ_{\tt Y}$ or at most $τ_{\tt N}$, where $τ_{\tt Y} - τ_{\tt N}$ 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$_α$ 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 $α> 0$ and $0 < q \leq 1$, LowRankRényiQEA$_α$ and LowRankTsallisQEA$_q$ are ${\sf BQP}$-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 ${\sf BQP}$-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.