Do bound entangled states with negative partial transpose exist
In plain words
Some entangled states cannot be purified into perfect entangled pairs, no matter how many copies are used. All known examples pass a simple test called positive partial transpose, and it is open whether states that fail the test can also be impossible to purify.
Precise statement
Does there exist a bipartite state $\rho$ on $C^{d}\,\operatorname{tensor}\,C^{d}$, $d \ge 3$, with negative partial transpose $\rho^{\Gamma}$ that is not distillable, i.e. for every n and all rank-2 projectors P, Q on the n-copy spaces, $(P\,\operatorname{tensor}\,Q)(\rho^{\Gamma})^{\operatorname{tensor}\,n}(P\,\operatorname{tensor}\,Q) \ge 0$? It suffices to decide this for Werner states $\rho(d, \alpha) = (1 + \alpha V)/(d^{2} + \alpha d)$, $V$ the swap operator, with $-1/2 \le \alpha < -1/d$ (for $\alpha < -1/2$ the state is already 1-copy distillable).
What would settle it
A proof that some NPT Werner state is n-copy undistillable for all n, or a distillation protocol for every NPT state.
Status in the literature
Listed as open with a prize in 2022 (Horodecki, Rudnicki and Zyczkowski, PRX Quantum 3, 010101).