On two-distillable Werner states

We consider bipartite mixed states in a \(d\otimes d\) quantum system. We say that \(\rho\) is PPT if its partial transpose \(1 \otimes T (\rho)\) is positive semidefinite, and otherwise \(\rho\) is NPT. The well-known Werner states are divided into three types: (a) the separable states (the same as the PPT states); (b) the one-distillable states (necessarily NPT); and (c) the NPT states which are not one-distillable. We give several different formulations and provide further evidence for validity of the conjecture that the Werner states of type (c) are not two-distillable.