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Zeitschrift für Naturforschung A

Triple Spin Interaction and Entanglement

Willi-Hans Steeb

Published Feb 1, 2013• 2 citations
Publisher

Abstract

We study a Hamilton operator Ĥ for spin-1/2 with triple spin interactions. The eigenvalues and eigenvectors are determined and the unitary matrices exp(-iHt/ħ) are found. Entanglement of the eigenvectors is investigated. A Hamilton operator K̂ for spin-1 and triple spin interaction is also discussed.
DOI
10.5560/zna.2012-0087

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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.
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.
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.