Existence of NPT bound entanglement
Let $\rho$ be a density matrix on $\mathbb{C}^{d_A} \otimes \mathbb{C}^{d_B}$, and let $\rho^{T_B}$ be its partial transpose, which transposes matrix entries on the second factor in a fixed product basis; $\rho$ is NPT if $\rho^{T_B}$ has a negative eigenvalue. A vector in the space of $k$ copies, split as $A^k : B^k$ between the two parties, has Schmidt rank at most two if its coefficient matrix has rank at most two. By the criterion of Horodecki, Horodecki and Horodecki, in the form used by DiVincenzo et al. and by Dür et al., $\rho$ can be distilled into maximally entangled pairs by local operations and classical communication exactly when $\langle v, (\rho^{T_B})^{\otimes k} v \rangle < 0$ for some $k \ge 1$ and some such $v$; in particular, entangled states with positive partial transpose cannot be distilled. The Werner states $\varrho(d, \alpha) = (I + \alpha F)/(d^2 + \alpha d)$ on $\mathbb{C}^d \otimes \mathbb{C}^d$, with $F$ the flip and $-1 \le \alpha \le 1$, are NPT exactly when $\alpha < -1/d$. Horodecki and Horodecki showed that an undistillable NPT state exists if and only if some NPT Werner state, for some $d \ge 3$, is undistillable. Fu, Gao and Park, and independently Fraser, Huber, Pozsgay and Vona, proved that $\varrho(d, \alpha)$ is two-copy distillable exactly when it is one-copy distillable, that is, when $\alpha < -1/2$ (2026).
Problem. Does there exist a density matrix $\rho$ on some $\mathbb{C}^{d_A} \otimes \mathbb{C}^{d_B}$ such that $\rho^{T_B}$ has a negative eigenvalue, yet $$\langle v, (\rho^{T_B})^{\otimes k} v \rangle \ge 0$$ for every integer $k \ge 1$ and every vector $v$ of Schmidt rank at most two across $A^k : B^k$?
Source: DiVincenzo, Shor, Smolin, Terhal and Thapliyal (2000) and Dür, Cirac, Lewenstein and Bruß (2000), after Horodecki, Horodecki and Horodecki (1998); Problem 4 of Horodecki, Rudnicki and Życzkowski (2022).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- IT, OA
- Keywords
- AIM Open Applied Problems, bound entanglement, entanglement distillation, Werner states
References
@article{divincenzo2000,
author = {DiVincenzo, David P. and Shor, Peter W. and Smolin, John A. and Terhal, Barbara M. and Thapliyal, Ashish V.},
title = {Evidence for bound entangled states with negative partial transpose},
journal = {Phys. Rev. A},
volume = {61},
number = {6},
year = {2000},
pages = {062312},
doi = {10.1103/PhysRevA.61.062312},
}
@article{dur2000,
author = {D{\"u}r, W. and Cirac, J. I. and Lewenstein, M. and Bru{\ss}, D.},
title = {Distillability and partial transposition in bipartite systems},
journal = {Phys. Rev. A},
volume = {61},
number = {6},
year = {2000},
pages = {062313},
doi = {10.1103/PhysRevA.61.062313},
eprint = {quant-ph/9910022},
archivePrefix = {arXiv},
}
@article{horodecki1998,
author = {Horodecki, Micha{\l} and Horodecki, Pawe{\l} and Horodecki, Ryszard},
title = {Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?},
journal = {Phys. Rev. Lett.},
volume = {80},
number = {24},
year = {1998},
pages = {5239--5242},
doi = {10.1103/PhysRevLett.80.5239},
}
@article{horodecki2022,
author = {Horodecki, Pawe{\l} and Rudnicki, {\L}ukasz and {\.Z}yczkowski, Karol},
title = {Five open problems in quantum information theory},
journal = {PRX Quantum},
volume = {3},
number = {1},
year = {2022},
pages = {010101},
doi = {10.1103/PRXQuantum.3.010101},
eprint = {2002.03233},
archivePrefix = {arXiv},
}
@article{horodecki1999,
author = {Horodecki, Micha{\l} and Horodecki, Pawe{\l}},
title = {Reduction criterion of separability and limits for a class of distillation protocols},
journal = {Phys. Rev. A},
volume = {59},
number = {6},
year = {1999},
pages = {4206--4216},
doi = {10.1103/PhysRevA.59.4206},
}
@article{werner1989,
author = {Werner, Reinhard F.},
title = {Quantum states with {E}instein--{P}odolsky--{R}osen correlations admitting a hidden-variable model},
journal = {Phys. Rev. A},
volume = {40},
number = {8},
year = {1989},
pages = {4277--4281},
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}
@misc{fu2026,
author = {Fu, Jinshi and Gao, Li and Park, Sang-Jun},
title = {A solution to 2-copy distillability of {W}erner states},
year = {2026},
eprint = {2607.21367},
archivePrefix = {arXiv},
}
@misc{fraser2026,
author = {Fraser, Thomas C. and Huber, Felix and Pozsgay, Bal{\'a}zs and Vona, Istv{\'a}n},
title = {On the two-copy distillability of {W}erner states and a new partial trace inequality},
year = {2026},
eprint = {2607.24309},
archivePrefix = {arXiv},
}
@misc{aim-npt-bound-entanglement,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of NPT bound entanglement}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/032-npt-bound-entanglement.md},
note = {Entry 032 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00365, "Existence of NPT bound entanglement", clopen.solutions.