Some open problems in quantum information theory are famous because they are hard to state. This one is easy to state and has been hard to settle: if you take two copies of a quantum state, does the entanglement of purification simply double? A new single-author preprint, “The entanglement of purification is not additive” by Artus Krohn-Grimberghe (Percivio Ltd.), claims a rigorous no, turning a 2012 numerical conjecture into a computer-certified theorem.
This article was drafted with Claude and fact-checked against the cited primary sources. The paper is a v1 preprint and has not yet been peer reviewed; what follows describes what the paper states and proves, with my interpretation labelled separately.
Why this paper is worth your attention
The entanglement of purification, EP, was introduced by Terhal, Horodecki, Leung and DiVincenzo in 2002 as a measure of the total correlations in a bipartite state: the smallest amount of entanglement any purification of the state must carry across the cut. Its regularized version, EP∞, has a clean operational meaning shown in that original work: it equals the asymptotic entanglement cost of preparing the state with local operations and a vanishing rate of classical communication.
Whether EP is additive on tensor powers, that is whether EP(ρ⊗n) = n·EP(ρ), has been open since 2002. In 2012 Jianxin Chen and Andreas Winter gave strong numerical evidence that it is not, titling their preprint “beyond reasonable doubt”, and explicitly left a completely rigorous proof to future work. The new paper states that it supplies that proof, for exactly the state Chen and Winter proposed.
What the paper claims
The counterexample is the two-qubit Werner state W(f) at singlet fraction f = 1/200. The main theorem states
EP∞(W) ≤ T+ = 0.96632… < 97/100 < EP(W),
where T+ is given as an explicit 60-digit rational. Because EP∞ is the infimum of EP(ρ⊗n)/n, a strict gap means that some finite number of copies has a strictly lower per-copy cost than a single copy. That is strict non-additivity.
The author is careful about scope. The proof does not identify which n exhibits the violation (not even whether n = 2 works), gives no upper bound on n, and does not compute the exact value of EP or EP∞ at f = 1/200. The author also credits the phenomenon, the state family, the convexity mechanism and the parameter choice to Chen and Winter; the stated contribution is the rigorous certification.
The key idea
Proving the upper bound on EP∞ only requires exhibiting good purifications. The hard part is the lower bound on the single-copy EP: you must show that every purification has entanglement entropy above 0.97. That is a minimization over a continuous, high-dimensional space.
The paper’s strategy is to avoid searching that space directly. Using a known result of Ibinson, Linden and Winter, it restricts to ancillas of dimension 4, so every purification corresponds to a 16×4 isometry. It then proves that the ordered Schmidt spectrum (p1, p2, p3, …) of any such purification must satisfy four necessary inequalities. That collapses the problem to a three-dimensional box of possible spectra, where an exact computer check becomes feasible.
How it works
An orientation inequality. The paper shows that a qubit channel whose Bloch matrix reverses orientation has |det L| ≤ 1/27. Channels built from the two leading Schmidt vectors must reproduce the Werner state’s correlations, and this forces a polynomial inequality on p1, p2, p3.
A symmetric-extension inequality. The triplet part of any purification has a symmetric extension. Using the two-qubit symmetric-extension criterion of Chen, Ji, Kribs, Lütkenhaus and Zeng plus Mirsky’s singular-value perturbation bound, the paper derives √p1 ≤ √p2 + √p3 + √Q + 1/√50, where Q is the remaining tail weight.
Two elementary facts. The tail satisfies Q ≤ 5p3, and merging the tail into one entry can only lower the entropy.
A certified subdivision. The box of possible (p1, p2, p3) is split by exact-rational bisection into 25,383 leaves. Each leaf is certified either to contain no physical spectrum or to have entropy above 97/100. Every accepting check reduces to an integer or rational comparison.
The upper bound. At f = 0 the paper proves EP∞ = 1 exactly. At f = 1/100 an explicit “Bell-to-Bell” purification gives EP ≤ 0.92261…. Since W(1/200) is the equal mixture of these two endpoints, Chen and Winter’s convexity inequality for EP∞ − S gives EP∞(W(1/200)) ≤ 0.96632…. Because the Chen–Winter preprint only outlined that convexity proof, the new paper includes a full proof from published ingredients in an appendix.
A supplementary artifact on Zenodo contains two Python programs, using only the standard library, that rebuild the whole subdivision from the split rule and check every certificate. The paper reports run times of about 6 seconds and 1 second, and says the verifiers reject deliberately corrupted inputs. I did not run the verifier for this article.
Why it matters
If the proof holds, it settles a question that has been open for more than two decades. It confirms that, for this correlation measure, the single-copy value does not give the full operational story: sharing many copies can be strictly cheaper per copy. The paper places this alongside two contrasts. For the entanglement of formation, non-additivity was shown by Hastings through random channel constructions, but the author notes that this machinery has no known transfer to EP. For a Rényi-2 variant of EP, recent work has proved additivity on relevant classes, so the new result points to a genuinely von Neumann phenomenon on this family.
Technical perspective (interpretation)
This is my reading rather than a claim in the paper. The most transferable part may be the method: replace an intractable optimization over a manifold with a handful of provable necessary conditions on a low-dimensional summary of the optimizer, then close the gap with an exact, integer-checkable certificate. That pattern could be useful for other entropic optimization problems where numerics have long suggested an answer. It is also a reminder that a numerical conjecture with a clear, concrete target, as Chen and Winter left, can make a later rigorous proof much more tractable.
Limitations and open questions
Not yet refereed. This is a single-author v1 preprint. The analytic reductions in the appendices are where expert scrutiny matters most, since the computer check only verifies the finite calculations those reductions produce.
AI assistance. The paper’s contribution statement says AI systems were used for code, calculations, mechanical derivation of proof steps and drafting text under the author’s direction. The author argues that the verification chain is independent of how the derivations were produced.
Non-constructive in n. The number of copies needed to see the violation is unknown, as are the exact values of EP and EP∞ at f = 1/200.
One state family. The result is a single counterexample. How widespread non-additivity is, and how large the gap can be, remain open.
Holography. The paper notes that the conjectured link between EP and the entanglement wedge cross-section concerns a different regime, and says nothing here bears on it beyond ruling out additivity as a general theorem.
Paper information
Title: The entanglement of purification is not additive
Author: Artus Krohn-Grimberghe (Percivio Ltd.)
arXiv: 2609.29539v1 [quant-ph], submitted 25 August 2026, listed 25 September 2026
Verification artifact: 10.5281/zenodo.22097511 (CC BY 4.0)
Length: 24 pages
Primary sources
A. Krohn-Grimberghe, The entanglement of purification is not additive, arXiv:2609.29539 (HTML full text)
B. M. Terhal, M. Horodecki, D. W. Leung, D. P. DiVincenzo, The entanglement of purification, J. Math. Phys. 43, 4286 (2002)
J. Chen, A. Winter, Non-Additivity of the Entanglement of Purification (Beyond Reasonable Doubt), arXiv:1206.1307 (2012)