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We investigate the norms of the Bloch vectors for any quantum state withsubsystems less than or equal to four. Tight upper bounds of the norms areobtained, which can be used to derive tight upper bounds for entanglementmeasure defined by the norms of Bloch vectors. By using these bounds atrade-off relation of the norms of Bloch vectors is discussed. Theses upperbounds are then applied on separability. Necessary conditions are presented fordifferent kinds of separable states in four-partite quantum systems. We furtherpresent a complete classification of quantum states for four qudits quantumsystems.

The norms of Bloch vectors and classification of four qudits quantum states