Hierarchy of Spin Operators, Quantum Gates, Entanglement, Tensor Product and Eigenvalues

TL;DR

This paper explores the spectral properties of two hierarchies of spin Hamilton operators, their relation to quantum gates, entanglement, and eigenvector bases across various spin cases, revealing spectral equivalences and structural insights.

Contribution

It demonstrates that two different hierarchies of spin Hamilton operators share the same spectrum and analyzes their implications for quantum gates and entanglement across multiple spin cases.

Findings

Both Hamilton operator hierarchies have identical spectra.

Detailed analysis of spin-1/2, spin-1, spin-3/2, and spin-2 cases.

Investigation of entanglement and mutually unbiased bases.

Abstract

{\bf Abstract.} We show that two hierarchies of spin Hamilton operators admit the same spectrum. Both Hamilton operators play a central role for quantum gates in particular for the case spin-$\frac12$ and the case spin-1. The spin-$\frac12$, spin-1, spin-$\frac32$ and spin-2 cases are studied in detail. Entanglement and mutually unbiased bases of the eigenvectors is discussed. Two triple Hamilton operators are also investigated. Both are also admitting the same spectrum.