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arXiv:quant-ph/0105077

A Relation between Coherent States and Generalized Bell States

Kazuyuki Fujii

Published May 17, 2001• 8 citations
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Abstract

In the first half we show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel. In the latter half we treat generalized coherent states and try to generalize this relation to get several generalized Bell states. Our method is based on a geometry and our task may give a hint to open a deep relation between a coherence and an entanglement.

DOI
10.48550/arxiv.quant-ph/0105077

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In the first half we show an interesting relation between coherent states and the Bell states in the case of spin 1/2, which was suggested by Fivel. In the latter half we treat generalized coherent states and try to generalize this relation to get several generalized Bell states. Our method is based on a geometry and our task may give a hint to open a deep relation between a coherence and an entanglement.
The purpose of this paper is to introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1), and to give some applications of them to quantum information theory for graduate students or non--experts who are interested in both Geometry and Quantum Information Theory. In the first half we make a general review of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1) from the geometric point of view and, in particular, prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space. In the latter half we apply a method of generalized coherent states to some important topics in Quantum Information Theory, in particular, swap of coherent states and cloning of coherent ones. We construct the swap operator of coherent states by making use of a generalized coherent operator based on su(2) and show an "imperfect cloning" of coherent states, and moreover present some related problems. In conclusion we state our dream, namely, a construction of {\bf Geometric Quantum Information Theory}.
We outline the generalization of our earlier result (Phys. Lett. A 372 3792–5) on the connection between coherent states and pure state entanglement given there for the symmetric part of . We show that the set of coherent states in the symmetric subspace of is identical to the set of unentangled states for any K. We point out that the deeper reason for this theorem lies in the fact that coherent states result from a set of local unitary transforms of the so-called lowest weight state that is obviously unentangled.
Abstract Majorana stars, the antipodal directions associated with the coherent states that are orthogonal to a spin state, provide a visualization and a geometric understanding of the structures of general quantum states. For example, the Berry phase of a spin-1/2 is given by half the solid angle enclosed by the close path of its Majorana star. It is conceivable that the Berry phase of higher spins may also be related to the geometry of the Majorana constellation. We find that for a spin-1 state, besides the expected contributions from the solid angles enclosed by the close paths of the two Majorana stars, the Berry phase includes a term related to the twist of the relative position vector around the barycenter vector of the two Majorana stars, i.e., the self-rotation of the constellation. Interestingly, if the spin-1 state is taken as a symmetrized two-qubit state, the extra contribution to the Berry phase is given by the self-rotation of the Majorana constellation weighted by the quantum entanglement of the two qubits. This discovery alludes to the relevance of the Majorana stellar geometry in representing the deep structures of quantum states and of quantum entanglement.