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We study the entanglement in tripartite quantum systems by using theprincipal basis matrix representations of density matrices. Using the Schmidtdecomposition and local unitary transformation, we first convert the generalstates to simpler forms and then construct some special matrices from thecorrelation tensors of the simplified density matrices. Based on the differentlinear combinations of these matrices, necessary conditions are presented todetect entanglement of tripartite states. Detailed examples show that ourmethod can detect more entangled states than previous ones.

Detection of tripartite entanglement based on principal basis matrix representations