  We consider the problem of existence of bound entangled states with
non-positive partial transpose (NPT). As one knows, existence of such states
would in particular imply nonadditivity of distillable entanglement. Moreover
it would rule out a simple mathematical description of the set of distillable
states. Distillability is equivalent to so called n-copy distillability for
some n. We consider a particular state, known to be 1-copy nondistillable,
which is supposed to be bound entangled. We study the problem of its two-copy
distillability, which boils down to show that maximal overlap of some projector
Q with Schmidt rank two states does not exceed 1/2. Such property we call the
the half-property. We first show that the maximum overlap can be attained on
vectors that are not of the simple product form with respect to cut between two
copies. We then attack the problem in twofold way: a) prove the half-property
for some classes of Schmidt rank two states b) bound the required overlap from
above for all Schmidt rank two states. We have succeeded to prove the
half-property for wide classes of states, and to bound the overlap from above
by c<3/4. Moreover, we translate the problem into the following matrix analysis
problem: bound the sum of the squares of the two largest singular values of
matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr
A^\dagger A + Tr B^\dagger B = 1/4.
