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Journal of Russian Laser Research

The Partition Formalism and New Entropic-Information Inequalities for Real Numbers on an Example of Clebsch–Gordan Coefficients

Vladimir I. Man’ko, Z. Seilov

Published Jan 1, 2017• 7 citations
Publisher
DOI
10.1007/s10946-017-9619-7

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The Partition Formalism and New Entropic-Information Inequalities for Real Numbers on an Example of Clebsch-Gordan Coefficients

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We discuss the procedure of different partitions in the finite set of NNN integer numbers and construct generic formulas for a bijective map of real numbers sys_ysy​, where y=1,2,…,Ny=1,2,\ldots,Ny=1,2,…,N, N=∏k=1nXkN=\prod \limits_{k=1}^{n} X_kN=k=1∏n​Xk​, and XkX_kXk​ are positive integers, onto the set of numbers s(y(x1,x2,…,xn))s(y(x_1,x_2,\ldots,x_n))s(y(x1​,x2​,…,xn​)). We give the functions used to present the bijective map, namely, y(x1,x2,...,xn)y(x_1,x_2,...,x_n)y(x1​,x2​,...,xn​) and xk(y)x_k(y)xk​(y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of "hidden gates" for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch--Gordan coefficients as an example of the found relations for real numbers.
Semiclassical integral representations, analogous to initial value expressions for the propagator, are presented for the Clebsch-Gordan angular momentum coupling coefficients. Two forms (L and R types) of the approximation are presented. For each form, new non-Gaussian expressions, which are specifically adapted to the nature of angular momentum variables, are proposed in place of the familiar Gaussian coherent state functions. With these non-Gaussian kernels, it is found that the present treatments are capable of accuracy similar to that obtained from a uniform Airy approximation. Although the present semiclassical approximations involve only real-valued angle variables, associated with sets of angular momenta that are related by ordinary, real, classical transformations, the treatments produce accurate results not only for classically allowed choices of quantum numbers but also for very strongly classically forbidden values.