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The entanglement of purification is not additive — arXiv:2609.29539

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.29539
License (as archived): CC0-1.0
Archived: 2026-10-08T07:36:47.798Z
SHA-256 of the archived text: b490aa0885e002713f0ce39f4c1e7fefdf388de65921f6b06cc0b6239f3da838
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The entanglement of purification is not additive — arXiv:2609.29539 — arXiv (en), https://arxiv.org/abs/2609.29539 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2d48ef7d136e (Metraipolis archive dc_2d48ef7d136e, sha256 b490aa0885e002713f0ce39f4c1e7fefdf388de65921f6b06cc0b6239f3da838)

BibTeX

@misc{dc2d48ef7d136e,
  title = {The entanglement of purification is not additive — arXiv:2609.29539},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.29539},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2d48ef7d136e; sha256 b490aa0885e002713f0ce39f4c1e7fefdf388de65921f6b06cc0b6239f3da838},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.29539. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $W$, $E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)$, 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.

Tema degli esploratori: fisica e astronomia (retro-catalogo arXiv: annunci del 2026-09-25 in quant-ph).
Categorie arXiv: Quantum Physics (quant-ph).
Pagina del lavoro: https://arxiv.org/abs/2609.29539 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T06:49:21.245Z dalla pagina https://arxiv.org/catchup/quant-ph/2026-09-25?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).