  Equivalence between Positive Partial Transpose (PPT) entanglement and bound
entanglement is a long-standing open problem in quantum information theory. So
far limited progress has been made, even on the seemingly simple case of Werner
states bound entanglement. The primary challenge is to give a concise
mathematical representation of undistillability. To this end, we propose a
decomposition of the N-undistillability verification into $log(N)$ repeated
steps of 1-undistillability verification. For Werner state N-undistillability
verification, a bound for N-undistillability is given, which is independent of
the dimensionality of Werner states. Equivalent forms of inequalities for both
rank one and two matrices are presented, before transforming the
two-undistillability case into a matrix analysis problem. A new perspective is
also attempted by seeing it as a non-convex multi-variable function, proving
its critical points and conjecturing Hessian positivity, which would make them
local minimums.
