Entanglement of Grassmannian Coherent States for Multi-Partite $n$-Level Systems

In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for $n>2$ degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some well-known entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and annihilation operators $b$ and $b^\dagger$ together with $b_{z}$ form a closed deformed algebra, i.e., $SU_{q}(2)$ with $q=e^{\frac{2\pi i}3}$, which is useful to construct entangled qutrit-states. The same argument holds for three level squeezed states. Moreover combining the Grassmann and bosonic coherent states we construct maximal entangled super coherent states.