Every PPT channel has finite entanglement-breaking index
We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly containing the class of 2-superpositive maps, has entanglement-breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide strong evidence that the PPT-cubed conjecture may hold in full generality.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Entanglement theory
- Posed by
- Matthias Christandl
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-08-13
- Model
- ChatGPT-5.6 Sol
- Vendor
- OpenAI
- Collaborators
- Sang-Jun Park
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version.
What the AI did
The author states that AI tools including ChatGPT were used for exploratory mathematical discussions during development of the manuscript, in addition to language and LaTeX assistance. In particular, some ideas used in the proof-development process in Section 3 arose during interactions with GPT-5.6 Sol. These suggestions were subsequently examined, reformulated, incorporated into the manuscript, and independently verified by the author.
Verification
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.13551, Park, 14 pages). The AI statement is verbatim as this entry describes: ChatGPT for language, LaTeX and exploratory mathematical discussions, with some ideas in the splitting section arising during interactions with GPT-5.6 Sol. That is a vague disclosure, and the site's rule takes the lower tier, so ai-assisted stands. The proof was not checked here. One-day-old preprint, no independent review.
Source
- PaperarXiv
Submitted by VibeGene on