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arXiv:1509.07955

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

Willi-Hans Steeb, Yorick Hardy

Published Sep 26, 2015• 0 citations
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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.

DOI
10.48550/arxiv.1509.07955

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{\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-12\frac1221​ and the case spin-1. The spin-12\frac1221​, spin-1, spin-32\frac3223​ 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.
{\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 12\frac1221​ and the case spin 1. The spin-12\frac1221​, spin-1, spin-32\frac3223​ and spin-2 cases are studied in detail.
An N-tangle can be defined for the finite dimensional Hilbert space H = C2N , with N = 3 or N even.We give an orthonormal basis which is fully entangled with respect to this measure.We provide a spin Hamilton operator which has this entangled basis as normalized eigenvectors if N is even. From these normalized entangled states a Bell matrix is constructed and the cosine-sine decomposition is calculated. If N is odd the normalized eigenvectors can be entangled or unentangled depending on the parameters.
Abstract We consider a hierarchy of Hamilton operators Ĥ N in finite dimensional Hilbert spaces C2N \mathbb{C}^{2^N } C2N. We show that the eigenstates of Ĥ N are fully entangled for N even. We also calculate the unitary operator U N(t) = exp(—Ĥ N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.