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Title: Do bound entangled states with negative partial transpose exist
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.
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).
Settled by: A proof that some NPT Werner state is n-copy undistillable for all n, or a distillation protocol for every NPT state.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: Listed as open with a prize in 2022 (Horodecki, Rudnicki and Zyczkowski, PRX Quantum 3, 010101).
Posed since: 2000