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arXiv:cond-mat/0409589 · Physical Review B

Fisher information, Wehrl entropy, and Landau diamagnetism

Sergio Curilef, F. Pennini, Angelo Plastino

Published Jan 25, 2005• 19 citations
arXivHTMLPDF

Abstract

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. We reveal a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.

DOI
10.1103/physrevb.71.024420

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Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. We reveal a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.
In the present study we analyze some consequences that come from revised measures as the Wehrl entropy and the Fisher information for the problem of a particle in a magnetic field starting from a complete description of the Husimi function. We discuss in the most complete form (three dimensions) some results related to measures in contrast with the incomplete form (two dimensions) shown in previous contributions. Some limiting cases as high and low temperatures are discussed. From the present reasoning, it is suggested that the formulation in two dimensions is sufficient unto itself to explain the problem whenever the length of the cylindrical geometry of the system is large enough. Otherwise, it is not possible to work in all finite temperatures, a natural lower temperature bound emerges from the analysis when three dimensions are considered.
In this work, the definition of the density operator on quantum states in Hilbert spaces and some of its aspects relevant in thermodynamics and information-theoretical entropy calculations are given. In this framework, a physical model describing an electron in a magnetic field is investigated. The so-called exotic Landau problem in noncommutative plane is also considered. Then, a model related to the fractional quantum Hall effect is revisited. Thanks to the completeness relations verified by the coherent states (CS) in these models, the thermodynamics is discussed by using the diagonal PPP-representation of thedensity operator. Specifically, the Q-Husimi distribution and the Wehrl entropy are determined.
By interpolating between Fisher information and mechanical kinetic energy, we introduce a general notion of kinetic energy with respect to a parameter of Schrödinger wavefunctions from a statistical inference perspective. Kinetic energy is the sum of Fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase. A family of integral inequalities concerning kinetic energy and moments are established, among which the Cramér–Rao inequality and the Weyl–Heisenberg inequality, are special cases. In particular, the integral inequalities involving the negative order moments are relevant to the study of electron systems. Moreover, by specifying the parameter to a scale, we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity.
An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems, such as hydrogen-like atoms and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with m=0, which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.
We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.