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Physical Review A

Information entropy and uncertainty inD-dimensional many-body systems

Juan Carlos Angulo

Published Jul 1, 1994• 51 citations
Publisher

Abstract

Several entropic and uncertaintylike results for three-dimensional many-particle systems are extended to the case of arbitrary dimensionality D. This work deals with (i) upper bounds to information entropies and (ii) radial and logarithmic uncertaintylike relationships. The resulting expressions are given in an analytical compact form for any dimensionality. The first few low-dimensional cases (D=1,2,3,4) are emphasized.

DOI
10.1103/physreva.50.311

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Several entropic and uncertaintylike results for three-dimensional many-particle systems are extended to the case of arbitrary dimensionality D. This work deals with (i) upper bounds to information entropies and (ii) radial and logarithmic uncertaintylike relationships. The resulting expressions are given in an analytical compact form for any dimensionality. The first few low-dimensional cases (D=1,2,3,4) are emphasized.
General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.
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Abstract We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. We also discuss relationships between information and quantum theories. In this context we determine the information entropies on the position, S x , and momentum, S p , spaces, in addition to the entropy sum, S t . Here, we use the modified entropic expressions that are dimensionally consistent. We provide original explanations for the behaviors of the S x , S p and S t values analyzing the probability densities and by means of the normalization constants and properties of Fourier transform. We validate the entropic uncertainty relation and inspect the standard deviation and information entropies as measures of quantum uncertainty. The systems of interest are unidimensional one-particle quantum systems in the ground and excited states.
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.
We consider the question of entropic uncertainty relations for prime power dimensions. In order to improve upon such uncertainty relations for higher dimensional quantum systems, we derive a tight lower bound amount of entropy for multiple probability distributions under the constraint that the sum of the collision probabilities for all distributions is fixed. This is purely a classical information theoretical result, however using an interesting result by Larsen \cite{Larsen90} allows us to connect this to an entropic uncertainty relation.