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.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.

arXiv:quant-ph/0608116Physical Review ANov 1, 2006273 citations

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

Shenzhou Zheng

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.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.

Mathematical Methods in the Applied SciencesOct 26, 20251 citation

Uncertainty relations and the Wehrl entropy

Kedar S. Ranade, Wolfgang P. Schleich

Verhandlungen der Deutschen Physikalischen GesellschaftJan 1, 20120 citations

Entropy and the uncertainty principle

Roy B. Leipnik

Information and ControlApr 1, 195949 citations

(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.