Entanglement Criterion And The Bell Inequality

Einstein and his partners published a paper in 1935 (called EPR paradox in history). In this paper they acutely criticized the classical interpretation of quantum mechanics (Copenhagen interpretation). This was the first time to give the conception what the inseparable states were. After that, Bohm gave a simple expatiation for EPR paradox using a system made of two spin-half particles. In 1965, Bell showed an inequality (named Bell inequality) coming from the local hidden variable (LHV) models, which were turned out from EPR paradox. The inequality held for all LHV models, but QM could show the violation of the Bell inequality. Now one could judge which theory was correct through laboratory because the question could be discussed in quantity. Some factors of misplay or error were considered in laboratory environment for such correlated measure when the CHSH inequality was given. But they just considered the pure states and the spin operator of the 2(?)2 system. However, one usually deals with mixed states due to the uncontrolled interaction with the environment and all kinds of possible operators. Werner was the first person who gave the problem what the inseparability of mixed states was and showed an inseparable state satisfying the Bell inequality. So one needed to solve such a question: what was the relation between "violation of the Bell inequality" and "the state is inseparable"? In this paper, it is shown that one is able to construct an inseparable criterion , which is a constraint for all Bell inequalities atthe same time, for mixed states and pure states in general m(?)n(m,n > 1) system,after checking the arbitrary 2(?)2, 3(?)3 and 4(?)4 systems (not only the spin system). To violate the inseparable criterion has two different physical meanings: first, the state is inseparable in QM; second, a local realism interpretation of QM is impossible for the general m (?) n case in LHV models.