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

TL;DR

This paper introduces a formalism for partitioning finite sets of real numbers, develops bijective mappings to reveal hidden correlations, and derives new entropic inequalities with applications to Clebsch-Gordan coefficients.

Contribution

It presents a novel partition formalism and explicit bijective maps to analyze correlations and derive entropic inequalities, exemplified by Clebsch-Gordan coefficients.

Findings

Derived new entropic-information inequalities for finite real number sets.

Established explicit bijective maps revealing hidden correlations.

Applied the formalism to Clebsch-Gordan coefficients as an example.

Abstract

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 $y=1,2,\ldots,N$, $N=\prod \limits_{k=1}^{n} X_k$, and $X_k$ are positive integers, onto the set of numbers $s(y(x_1,x_2,\ldots,x_n))$. We give the functions used to present the bijective map, namely, $y(x_1,x_2,...,x_n)$ 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.