The entanglement of purification is not additive
Abstract
The entanglement of purification EP, 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 singlet fraction f = 1/200 is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state W, EP∞(W) ≤ 0.9663… < 97/100 < EP(W), hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.
Related
- Rényi Entanglement of Purification Is Non-additive
- Entanglement of purification for Werner states: canonical purification and nonadditivity
- Rényi-2 entanglement of purification of the symmetric Werner state
- Arbitrary high-dimensional entanglement purification with the superposition of paths
- Entanglement Cost of Optimal Distributed Quantum State Purification
BibTeX
@misc{krohngrimberghe2026entanglement,
title = {{The entanglement of purification is not additive}},
author = {Artus Krohn-Grimberghe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29539},
eprint = {2609.29539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29539},
}
RIS
TY - UNPB
TI - The entanglement of purification is not additive
AU - Artus Krohn-Grimberghe
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29539
UR - https://arxiv.org/abs/2609.29539
AB - 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 singlet fraction f = 1/200 is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state W, E_P^∞(W) ≤ 0.9663… < 97/100 < E_P(W), hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.
ER -