The Entanglement of Three-mode Spin Coherent States

In this study, the detection method of entanglement of 3-mode spin coherence states is investigated. For this purpose, by properly defining the computational codes, these states are mapped to a three qubit quantum state, and then, by using the Mermin- Kalyshko inequality, the entanglement of these states is studied and an analytical relationship to determine the entanglement region is presented. The results of numerical analysis of the extracted inequality show that the entanglement region of these states depends on the coherence parameters, the amount of spin as well as the phase of the states. For example, in the case where the values of the coherence parameters are equal but with opposite signs, by adjusting the coherence phase, the degree of entanglement can be controlled such that the maximum entanglement occurs . More important, as the value of spin increases, the allowable range of the coherence parameter for entanglement detection increases. These results are consistent with the data reported in the study of the degree of entanglement of 2-mode superposition of spin coherent states using other measures and criteria of entanglement.  The findings of this study can be used in the study of non-classical and quantum systems and quantum correlations in quantum information science.