The separability problem and normal completions

In this paper we introduce new normal completion problems that are directly related to the separability problem in quantum information. In fact we show that generically the N×M separability problem may be reduced in dimension via a multi-matrix normal completion problem. Specifying the result for the 2×M separability problem yields the equivalence to a normal completion problem that is a variation of the one introduced by P. Halmos. In addition, upper bounds on the number of states in a separable representation are given in terms of the normal completions.