  Following recent work of Beigi and Shor, we investigate PPT states that are
"heavily entangled." We first exploit volumetric methods to show that in a
randomly chosen direction, there are PPT states whose distance in trace norm
from separable states is (asymptotically) at least 1/4. We then provide
explicit examples of PPT states which are nearly as far from separable ones as
possible. To obtain a distance of 2-{\epsilon} from the separable states, we
need a dimension of 2^{poly(\log(1/\epsilon))}, as opposed to
2^{poly(1/\epsilon)} given by the construction of Beigi and Shor. We do so by
exploiting the so called {\it private states}, introduced earlier in the
context of quantum cryptography. We also provide a lower bound for the distance
between private states and PPT states and investigate the distance between pure
states and the set of PPT states.
