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

TL;DR

This paper explores the entanglement properties of Grassmannian coherent states for multi-partite systems with n>2 levels, demonstrating how to construct various well-known entangled states and analyzing their algebraic structures.

Contribution

It introduces a method to generate entangled states from Grassmannian coherent states and reveals the underlying algebraic structure for three-level systems, including qutrits.

Findings

Constructed GHZ, W, Bell, cluster, and bi-separable states from GCSs.

Identified an $SU_q(2)$ algebra structure in three-level systems.

Developed maximal entangled super coherent states combining Grassmann and bosonic states.

Abstract

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 {\bf GHZ}, {\bf 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 $b^\dag$ 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.