Mathematical aspects of the decomposition of diagonal U(N) operators

Abstract: We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are introduced, providing clear visualizations of the structure of these decompositions. We also discuss symmetries of the suggested decomposition. Methods and representations developed in this paper can be applied in different areas, including optimization of quantum computing algorithms, complex biological analysis, crystallography, optimization of AI models, and others.