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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.","posed_since":"2000","precise":"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).","problem_ref":null,"references":"","settled_by":"A proof that some NPT Werner state is n-copy undistillable for all n, or a distillation protocol for every NPT state.","status_note":"Listed as open with a prize in 2022 (Horodecki, Rudnicki and Zyczkowski, PRX Quantum 3, 010101).","title":"Do bound entangled states with negative partial transpose 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