Composing two PPT channels destroys all entanglement
Let $d \ge 2$, and let $\Phi, \Psi \colon M_d(\mathbb{C}) \to M_d(\mathbb{C})$ be completely positive trace-preserving maps such that $T \circ \Phi$ and $T \circ \Psi$ are also completely positive, where $T$ is transposition in a fixed basis. Christandl, Müller-Hermes and Wolf proved that the composition of any two completely positive and completely copositive maps on $M_3(\mathbb{C})$ is entanglement breaking, and proved the corresponding statement for Gaussian quantum channels; they also showed that the assertion for the square of a single such map in every dimension is equivalent to the assertion for compositions of two such maps between matrix algebras of any sizes.
Problem. Must $\Psi \circ \Phi$ be entanglement breaking, that is, must $(\operatorname{id}_d \otimes (\Psi \circ \Phi))(\rho)$ be separable for every density operator $\rho$ on $\mathbb{C}^d \otimes \mathbb{C}^d$?
Source: Christandl (2012), the PPT-squared conjecture, as stated by Christandl, Müller-Hermes and Wolf (2019), Conjecture I.1, for the square of one completely positive and completely copositive map. Stated by AIM — Open Applied Problems for the composition of two trace-preserving such maps.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- OA
- Keywords
- AIM Open Applied Problems, completely positive maps, entanglement breaking channels, PPT squared conjecture
References
@article{christandl2019,
author = {Christandl, Matthias and M{\"u}ller-Hermes, Alexander and Wolf, Michael M.},
title = {When do composed maps become entanglement breaking?},
journal = {Ann. Henri Poincar{\'e}},
volume = {20},
number = {7},
year = {2019},
pages = {2295--2322},
doi = {10.1007/s00023-019-00774-7},
eprint = {1807.01266},
archivePrefix = {arXiv},
}
@misc{aim-ppt-squared,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Composing two PPT channels destroys all entanglement}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/206-ppt-squared.md},
note = {Entry 206 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We construct an entangled state on $\C^{10}\otimes\C^{10}$ with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, source/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00706, "Composing two PPT channels destroys all entanglement", clopen.solutions.