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Mathematical Methods in the Applied Sciences

Uncertainty Principles for Rényi Entropy Under Continuous and Discrete Quaternion Fourier Transforms

Shenzhou Zheng

Published Oct 26, 2025• 1 citation
Publisher

Abstract

ABSTRACT We first prove the sharp Hausdorff–Young inequality for the right‐sided continuous quaternion Fourier transform (RCQFT), and the Hausdorff–Young inequalities for the discrete quaternion Fourier transform (DQFT) and for the quaternion Fourier series (QFS), respectively. Next, on the basis of the measure of uncertainty for the Rényi entropy in the quantum world, the uncertainty principle (UP) of the Rényi entropy is established for position and momentum. Moreover, similar uncertainty relations are also derived for the ‐level systems and for the angle and angular momentum, respectively. Finally, all uncertainty relations are shown more attractive while they are expressed in terms of the symmetrized Rényi entropy.

DOI
10.1002/mma.70265

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ABSTRACT We first prove the sharp Hausdorff–Young inequality for the right‐sided continuous quaternion Fourier transform (RCQFT), and the Hausdorff–Young inequalities for the discrete quaternion Fourier transform (DQFT) and for the quaternion Fourier series (QFS), respectively. Next, on the basis of the measure of uncertainty for the Rényi entropy in the quantum world, the uncertainty principle (UP) of the Rényi entropy is established for position and momentum. Moreover, similar uncertainty relations are also derived for the ‐level systems and for the angle and angular momentum, respectively. Finally, all uncertainty relations are shown more attractive while they are expressed in terms of the symmetrized Rényi entropy.
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)(p,q)(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 NNN-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.
Based on the definition and properties of discrete fractional Fourier transform (DFRFT), we introduced the discrete Hausdorff-Young inequality. Furthermore, the discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. Also, the condition of equality via Lagrange optimization was developed, as shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations reach their lowest bounds. In addition, the resolution analysis via the uncertainty is discussed as well.
In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).
The Rényi and Shannon entropies are information‐theoretic measures which have enabled to formulate the position‐momentum uncertainty principle in a much more adequate and stringent way than the (variance‐based) Heisenberg‐like relation. Moreover, they are closely related to various energetic density‐functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment 〈r2〉 for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1‐dimensional) quantum particle in terms of variances: 〈‖x‖2〉〈‖u‖2〉⩾1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon’s or Rényi’s entropies). As another alternative, we consider moments‐based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy under statistical moments’ constraints (〈‖x‖a〉 and 〈‖u‖b〉). Our result improves a relation proposed very recently by Dehesa et al.. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a =b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance‐based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d‐dimensional hydrogenic systems.