  The positive partial transpose test is one of the main criteria for detecting
entanglement, and the set of states with positive partial transpose is
considered as an approximation of the set of separable states. However, we do
not know to what extent this criterion, as well as the approximation, are
efficient. In this paper, we show that the positive partial transpose test
gives no bound on the distance of a density matrix from separable states. More
precisely, we prove that, as the dimension of the space tends to infinity, the
maximum trace distance of a positive partial transpose state from separable
states tends to 1. Using similar techniques, we show that the same result holds
for other well-known separability criteria such as reduction criterion,
majorization criterion and symmetric extension criterion. We also bring an
evidence that the sets of positive partial transpose states and separable
states have totally different shapes.
