Entanglement with zero distillable secret key in local dimension ten
We construct an entangled state on \(\mathbb C^{10}\otimes\mathbb C^{10}\) with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on \(M_{21}(\mathbb C)\) has a square that is not entanglement breaking.
Cite
@misc{OAI:Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026,
author = {{OpenAI}},
title = {{Entanglement with zero distillable secret key in local dimension ten}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf}{OAI:Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026}},
year = {2026}
}