Nonpositive-transpose entanglement and bound entanglement: From distillability sets to inequalities and multivariable insights

Equivalence between positive partial transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. Limited progress has been made so far, even on the seemingly simple case of the bound entanglement of Werner states. 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 $\text{log}\left(N\right)$ repeated steps of 1-undistillability verification. For Werner states' $N$-undistillability verification, a parameter interval for $N$-undistillability is given, which is independent of the dimensionality of Werner states. Equivalent forms of inequalities for both rank 1 and 2 matrices are presented, before transforming the 2-undistillability case into a matrix analysis problem. A new perspective is also attempted by seeing it as a nonconvex multivariable function, proving its critical points and conjecturing Hessian positivity, which would make them local minimums.