Every PPT channel has finite entanglement-breaking index
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Statement
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.
Context
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.
The eventual entanglement-breaking question for PPT channels sits directly beside the PPT-squared conjecture, one of entanglement theory's best-known open problems since 2012. Resolving eventual-EB in full generality is a real advance the field will use; the PPT-squared conjecture itself remains open, which caps it below the 30 band.
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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.
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