Braiding transformation, entanglement swapping, and Berry phase in entanglement space

Abstract

We show that braiding transformation is a natural approach to describe quantum entanglement by using the unitary braiding operators to realize entanglement swapping and generate the Greenberger-Horne-Zeilinger states as well as the linear cluster states. A Hamiltonian is constructed from the unitary ${\stackrel{\ifmmode \check{}\else \v{}\fi{}}{R}}_{i,i+1}(\ensuremath{\theta},\ensuremath{\varphi})$ matrix, where $\ensuremath{\varphi}=\ensuremath{\omega}t$ is time-dependent while $\ensuremath{\theta}$ is time-independent. This in turn allows us to investigate the Berry phase in the entanglement space.