Information theoretical measures from cumulative and survival densities in quantum systems

Abstract

Entropic uncertainty and statistical correlation measures, based on survival and cumulative densities, are explored in some representative quantum systems. We illustrate how the cumulative residual entropy in the quantum well system recovers the correct classical behavior for larger quantum numbers while the Shannon entropy does not. Two interacting and noninteracting oscillators are used to examine two-particle entropies and their related correlation measures. The joint cumulative residual entropy does distinguish between symmetric and antisymmetric wave functions in interacting systems as the interaction is turned on. The joint Shannon entropy does not distinguish between the symmetries even in the presence of interaction. Conversely, the joint Shannon entropy and joint cumulative residual entropy are both unable to distinguish between the symmetries for certain states of the noninteracting oscillators. As measures of statistical correlation, the mutual information and the cross cumulative residual entropy yield similar behaviors as a function of the strength of the interparticle interaction. K E Y W O R D S correlation measures, cumulative residual entropy, mutual information, quantum systems, Shannon entropy with q(x) the positive probability density defined over the entire range of the real-valued one-dimensional x variable. It should be mentionedthat this definition can be extended to higher dimensions. In this case, the normalization constraint implies that q(x)dx 1, but q(x) is not bounded, thus, it could and does give rise to negative values of the entropy. This is not a serious objection to the differential entropy if one takes care to treat it as a relative measure with the interpretation that given at least two possible systems or two possible states of the system, the entropy measures the response of the density as a delocalization/localization effect. Furthermore, there is no meaning associated with the probability at a certain point, x, in a continuous distribution since what is important is the probability in the element between x and Int J Quantum Chem. 2017;117:e25387.

The effect of stochastic barrier fluctuation on semiclassical transmission probability and Shannon entropy of a symmetric double well potential

“…Since it is introduction in 1948, the Shannon entropy has been applied in a wide variety of fields . The Shannon entropy is a measure of the randomness in the distribution.…”

“…We have calculated the WKB like transmission probability by taking the continuous potential energy function and the barrier width varying with the grid point. We have also calculated the Shannon entropy [ , P i = population of i th state] of the system which clearly builds up with time only when the stochasticity is introduced. However, this Shannon entropy has a strong dependence on the frequency of the stochastic fluctuation.…”

Information and quantum theories: an analysis in one-dimensional systems

“…Initially identified as an autonomous area, the mathematical theory of communication or information theory now has its fundamental concepts utilized in distinct fields of expertise, mainly through the informational entropy [9][10][11]. Information theory in the atomic, molecular and chemical physics context has generated a reasonable number of analyses [12][13][14][15][16][17][18][19]. Works about ions in plasma environments [20] and correlation measurements [21,22] show great results in favor of the informational treatment.…”

Correlaciones estadísticas de órdenes altos en sistemas cuánticos de tres partículas

“…Las definiciones entrópicas para la densidad acumulativa cu x (a) también se han considerado [71]. Las interpretaciones obtenidas de a y S x son de naturaleza consistente cuando se aplican a sistemas cuánticos de variables continuas aunque existen diferencias [72].…”

“…E[ (b|x 1 )] = E[ (a|x 2 )] por tratarse de partículas indistinguibles. Los comportamientos de S Γ y JCRE han sido comparados y contrastados en sistemas cuánticos de dos partículas [72]. Una forma alternativa para definir entropías modificadas de dos variables es a través de la entropía residual acumulativa conjunta modificada (mJCRE)…”