{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"af5776a05b32679b77cb533b58ab65695d77f2eb893ecaac496728475ddfd707","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["739762f12407ed0e7d7603081db99791c540b8c9c650118d7c95395904b30d49","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"927309035dfdb9ca1ee9d1c85039c77c93335da75e45dcbba8af7816c5fa6329","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"qi.quantum-shannon-theory.npt-bound-entanglement","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"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.","posed_since":"2000","precise":"Does there exist a bipartite state $\\rho$ on $C^{d}\\,\\otimes\\,C^{d}$, $d \\ge 3$, 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, $(P\\,\\otimes\\,Q)(\\rho^{\\Gamma})^{\\otimes\\,n}(P\\,\\otimes\\,Q) \\ge 0$? 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":"739762f12407ed0e7d7603081db99791c540b8c9c650118d7c95395904b30d49","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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