Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the R\'enyi entropies. The proof of these inequalities requires the use of the exact expression for the
(p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in
N-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.
We investigate the computational hardness of estimating the quantum
α-Rényi entropy
SαR(ρ)=1−αlnTr(ρα) and the quantum
q-Tsallis entropy
SqT(ρ)=q−11−Tr(ρq), 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
SqT(ρ), is at least
τ1 or at most
τ2, where
τ1−τ2 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=∞, 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=∞, and for all real orders
0<q≤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.