Computational hardness of estimating quantum entropies via binary entropy bounds

TL;DR

This paper proves the computational hardness of estimating quantum $ ext{R}^{ ext{R}}_{ ext{Renyi}}$ and Tsallis entropies for various orders, establishing BQP-completeness for certain rank-restricted problems.

Contribution

It introduces new reductions based on inequalities relating binary entropies of different orders, extending hardness results to all positive orders and the infinity case.

Findings

Rank-2 variants of entropy estimation are BQP-hard for all positive orders and infinity.

Low-rank entropy estimation problems are BQP-complete for all orders in specified ranges.

New inequalities between binary entropies underpin the reductions and hardness proofs.

Abstract

We investigate the computational hardness of estimating the quantum $\alpha$-R\'enyi 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 of which converge to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $\alpha$-R\'enyi Entropy Approximation (R\'enyiQEA$_\alpha$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$_q$) ask whether $ {\rm S}^ {\tt R}_{\alpha}(\rho)$ or ${\rm S}^{\tt T}_q(\rho)$, is at least $\tau_1$ or at most $\tau_2$, where $\tau_1 - \tau_2$ 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 $\alpha$ and $q$, and also for $\alpha=\infty$, the rank-$2$ variants Rank2R\'enyiQEA$_\alpha$ 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 $\alpha > 0$ or $\alpha=\infty$, and for all real orders $0 < q \leq 1$, LowRankR\'enyiQEA$_\alpha$ and LowRankTsallisQEA$_q$ are BQP-complete, where both are restricted versions of R\'enyiQEA$_\alpha$ and TsallisQEA$_q$ with $\rho$ 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 $\alpha$-R\'{e}nyi or $q$-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.