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The coefficient matrix is an efficient tool in entanglement classificationunder stochastic local operation and classical communication. In this work, wetake all the ranks of the coefficient matrices into account in the method ofentanglement classification, and give the entanglement classification procedurefor arbitrary-dimensional multipartite pure states under stochastic localoperation and classical communication. As a main application, we study theentanglement classification for quantum systems in the Hilbert space ${\cal H}= \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^d$in terms of the ranks of the coefficient matrices.

Entanglement classification of 2 X 2 X 2 X d quantum systems via the ranks of the multiple coefficient matrices