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AIP conference proceedings

Reconstruction of a density from its entropic moments

E. Romera

Published Jan 1, 2002• 12 citations
Publisher

Abstract

We describe a general reconstruction method of the density of a physical system from a finite number of entropic moments. These statistical quantities, which are integrals of the density to a power α∈R, may also represent some fundamental and/or experimentally accessible quantities of quantum-mechanical systems for specific values of α. We take advantage of the strategy recently used by us to solve the Hausdorff and Stieltjes entropic moment problems, where the main role is played by the inverse function of the density. In our method we first calculate such inverse function by use of an algorithm of minimization of the Fisher information measure of the density, and then we invert it. Two particular cases are discussed to illustrate the applicability of the method.

DOI
10.1063/1.1477065

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 19, 1999•0 citations
We describe a general reconstruction method of the density of a physical system from a finite number of entropic moments. These statistical quantities, which are integrals of the density to a power α∈R, may also represent some fundamental and/or experimentally accessible quantities of quantum-mechanical systems for specific values of α. We take advantage of the strategy recently used by us to solve the Hausdorff and Stieltjes entropic moment problems, where the main role is played by the inverse function of the density. In our method we first calculate such inverse function by use of an algorithm of minimization of the Fisher information measure of the density, and then we invert it. Two particular cases are discussed to illustrate the applicability of the method.
We propose a method for reconstruction of the density matrix from measurable time-dependent (probability) distributions of physical quantities. The applicability of the method based on least-squares inversion is---compared with other methods---very universal. It can be used to reconstruct quantum states of various systems, such as harmonic and anharmonic oscillators including molecular vibrations in vibronic transitions and damped motion. It also enables one to take into account various specific features of experiments, such as limited sets of data and data smearing owing to limited resolution. To illustrate the method, we consider a Morse oscillator and give a comparison with other state-reconstruction methods suggested recently.
Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi-function in analogy with quantum mechanics. The psi-function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi-function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population. The method proposed may be applied to its full extent to solve the statistical inverse problem of quantum mechanics (root estimator of quantum states). In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). The maximum likelihood technique and likelihood equation are generalized in order to analyze quantum mechanical experiments. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition making it possible to choose systems described by quantum mechanics from all statistical models consistent, on average, with the laws of classical mechanics.
A new method of solving classical and quantum problems of statistical data analysis based on the symbiosis of notions of quantum theory and mathematical statistics is considered. Particular attention is given to the specificity of quantum problems, determined by mutually complementary measurements (according to the Bohr complementarity principle), when, for example, a spatial-temporal picture is complemented by a momentum-energy one. The possibility of construction of multiparametric statistical models admitting a stable reconstruction of the parameters from observations (the inverse statistical problem) is studied. In this case, the only universal model of such a kind is the root model, based on the representation of the probability density as the square of the modulus of some function (called the psi function by analogy with quantum mechanics). The psi function is represented as an expansion in terms of an orthonormal basis, with the expansion coefficients being estimated by the maximum likelihood technique. The root approach makes it possible to represent the Fisher information matrix, covariance matrix, and statistical properties of the estimates of the reconstructed states in the simplest and a universal form. Being asymptotically efficient, the method allows one to reconstruct the states with an accuracy close to the theoretically attainable accuracy. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition, which makes it possible to choose, from among all the statistical models, which, on the average, are consistent with the laws of classical mechanics, those systems that are described by quantum mechanics.
Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi function in analogy with quantum mechanics. The psi function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms.
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in 1/D in similar systems with a non-standard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-D limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The D-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the D-dimensional hydrogenic atom at the limit of large D. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.