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 N integer numbers and construct generic formulas for a bijective map of real numbers sy, where y=1,2,,N, N=∏ ₖ₌₁ⁿ Xₖ, and Xₖ are positive integers, onto the set of numbers s(y(x₁,x₂,,xₙ)). We give the functions used to present the bijective map, namely, y(x₁,x₂,...,xₙ) and 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.