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The positive partial transpose test is one of the main criteria for detectingentanglement, and the set of states with positive partial transpose isconsidered as an approximation of the set of separable states. However, we donot know to what extent this criterion, as well as the approximation, areefficient. In this paper, we show that the positive partial transpose testgives no bound on the distance of a density matrix from separable states. Moreprecisely, we prove that, as the dimension of the space tends to infinity, themaximum trace distance of a positive partial transpose state from separablestates tends to 1. Using similar techniques, we show that the same result holdsfor other well-known separability criteria such as reduction criterion,majorization criterion and symmetric extension criterion. We also bring anevidence that the sets of positive partial transpose states and separablestates have totally different shapes.

Approximating the Set of Separable States Using the Positive Partial Transpose Test