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

We discuss the procedure of different partitions in the finite set of $N$ integer numbers and construct generic formulas for a bijective map of real numbers $s_y$, where $y=1,2,\dots ,N$, $N={\prod }_{k=1}^n{X}_k$, and $X_k$ are positive integers, onto the set of numbers $s(y(x_1,x_2,\dots ,x_n))$. We give the functions used to present the bijective map, namely, $y(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ and $x_k(y)$ 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.