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arXiv:quant-ph/0305106

Information-Theoretic Comparison of Quantum Many-Body Systems

K. Ch. Chatzisavvas, C. P. Panos, S. E. Massen

Published May 19, 2003• 2 citations
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Abstract

An information measure inspired by Onicescu's information energy and Uffink's information measure (recently discussed by Brukner and Zeilinger) are calculated as functions of the number of particles $N$ for fermionic systems (nuclei and atomic clusters) and correlated bosonic systems (atoms in a trap). Our results are compared with previous ones obtained for Shannon's information entropy, where a universal property was derived for atoms, nuclei, atomic clusters and correlated bosons. It is indicated that Onicescu's and Uffink's definitions are finer measures of information entropy than Shannon's.

DOI
10.48550/arxiv.quant-ph/0305106

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An information measure inspired by Onicescu's information energy and Uffink's information measure (recently discussed by Brukner and Zeilinger) are calculated as functions of the number of particles NNN for fermionic systems (nuclei and atomic clusters) and correlated bosonic systems (atoms in a trap). Our results are compared with previous ones obtained for Shannon's information entropy, where a universal property was derived for atoms, nuclei, atomic clusters and correlated bosons. It is indicated that Onicescu's and Uffink's definitions are finer measures of information entropy than Shannon's.
We will study fermionic systems like atomic nuclei and bosonic systems like the correlated atoms in a trap from an information-theoretical point of view. The Shannon and Onicescu information measures are calculated for the above systems by comparing the correlated and uncorrelated cases as functions of the strength of the short range correlations. One-body and two-body density and momentum distributions are employed. Thus, the effect of short-range correlations on the information content is evaluated. The magnitude of distinguishability between the correlated and uncorrelated densities is also discussed employing suitable measures for the distance of states i.e. the well known Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. We will see that the same information-theoretical properties hold for quantum many-body systems obeying Bose–Einstein and Fermi–Dirac (statistics).
A direct connection of information entropy SSS and kinetic energy TTT is obtained for nuclei and atomic clusters, which establishes TTT as a measure of the information in a distribution. It is conjectured that this is a universal property for fermionic many-body systems. We also check rigorous inequalities previously found to hold between S and T for atoms and verify that they hold for nuclei and atomic clusters as well. These inequalities give a relationship of Shannon's information entropy in position-space with an experimental quantity i.e. the rms radius of nuclei and clusters.
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.
A correlated many-body calculation is presented to characterize the Shannon information entropy of trapped interacting bosons. We reformulate the one-body Shannon information entropy in terms of the one-body probability density. The minimum limit of the entropy uncertainty relation is approached by making NNN very small in our numerical work. We examine the effect of correlations in the calculation of information entropy. Comparison with the mean-field result shows that the correlated basis function is indeed required to characterize the important features of the information entropies. We also accurately calculate the point of critical instability of an attractive BEC, which is in close agreement with the experimental value. Next, we calculate two-body entropies in position and momentum spaces and study quantum correlations in the attractive BEC.
In our previous papers, we proposed an analysis of the relations between information theory and observation theory. Referring to the parameters of the observation and using Shannon's formulation, we defined and calculated the information associated with a result of observation and the entropy of a random variable for a process of observation. This paper presents an extension of this theory for other definitions of entropy (Onicescu, hyperbolic). The method allows in particular to define Onicescu and hyperbolic entropies of continuous random variables. We examine the main properties of these new functions and propose a comparative analysis.
Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.