Additivity of entanglement of purification
For a density operator $\rho_{AB}$ on $A \otimes B$, with $A$ and $B$ finite-dimensional complex Hilbert spaces, the entanglement of purification is $$E_P(A{:}B)_\rho = \min S\bigl(\operatorname{Tr}_{BB'} |\psi\rangle\langle\psi|\bigr),$$ the minimum over finite-dimensional spaces $A', B'$ and unit vectors $\psi \in A \otimes A' \otimes B \otimes B'$ with $\operatorname{Tr}_{A'B'} |\psi\rangle\langle\psi| = \rho_{AB}$, where $S(\tau) = -\operatorname{Tr}(\tau \log_2 \tau)$ is the von Neumann entropy; the minimum is attained. For a second state $\sigma_{CD}$, the product of optimal purifications of $\rho_{AB}$ and $\sigma_{CD}$ shows $E_P(AC{:}BD)_{\rho \otimes \sigma} \le E_P(A{:}B)_\rho + E_P(C{:}D)_\sigma$, where $A, C$ are held by one party and $B, D$ by the other; equality holds when $\rho_{AB}$ and $\sigma_{CD}$ are both pure, and when both are product states. Terhal, Horodecki, Leung and DiVincenzo proved that the regularization $\lim_{n \to \infty} \frac{1}{n} E_P(\rho^{\otimes n})$ is the entanglement cost of preparing $\rho_{AB}$ by local operations with a vanishing rate of communication. Chen and Winter proved a convexity property of the regularization minus the entropy and, combined with numerical computations for two-qubit Werner states, gave strong evidence that the entanglement of purification differs from its regularization. Negari and Baghali Khanian proved that the Rényi analogue of order $\alpha$ is not additive for every $0 \le \alpha < 1$, already for two copies of a classical two-qubit state.
Problem. For all finite-dimensional complex Hilbert spaces $A, B, C, D$ and all density operators $\rho_{AB}$ and $\sigma_{CD}$, is $$E_P(AC{:}BD)_{\rho \otimes \sigma} = E_P(A{:}B)_\rho + E_P(C{:}D)_\sigma \, ?$$
Source: Terhal, Horodecki, Leung and DiVincenzo (2002), Section VI, for two copies of one state; for two arbitrary states, Chen and Winter (2012).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- IT
- Keywords
- additivity, AIM Open Applied Problems, entanglement of purification, von Neumann entropy
References
@article{terhal2002,
author = {Terhal, Barbara M. and Horodecki, Micha{\l} and Leung, Debbie W. and DiVincenzo, David P.},
title = {The entanglement of purification},
journal = {J. Math. Phys.},
volume = {43},
number = {9},
year = {2002},
pages = {4286--4298},
doi = {10.1063/1.1498001},
eprint = {quant-ph/0202044},
archivePrefix = {arXiv},
}
@misc{chen2012,
author = {Chen, Jianxin and Winter, Andreas},
title = {Non-additivity of the entanglement of purification (beyond reasonable doubt)},
year = {2012},
eprint = {1206.1307},
archivePrefix = {arXiv},
}
@misc{negari2026,
author = {Negari, Amir-Reza and Baghali Khanian, Zahra},
title = {R{\'e}nyi entanglement of purification is non-additive},
year = {2026},
eprint = {2608.28897},
archivePrefix = {arXiv},
}
@misc{aim-entanglement-of-purification-additivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Additivity of entanglement of purification}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/342-entanglement-of-purification-additivity.md},
note = {Entry 342 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that the von Neumann entanglement of purification is nonadditive. For two-qubit Werner states $W(f)$ with singlet weight $1/64\le f\le1/4$, we show that canonical purification is globally optimal: its entanglement, also known as reflected entropy, gives the exact entanglement of purification. In contrast, the regularized entanglement of purification lies strictly below the reflected entropy for $0<f<f_\ast$, where $f_\ast\approx0.017$. Together, …
The entanglement of purification $E_P$, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-15 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.03630v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00402, "Additivity of entanglement of purification", clopen.solutions.