Constructs an entangled state on \mathbb C^{10}\otimes\mathbb C^{10} with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on M_{21}(\mathbb C) whose square is not entanglement breaking disproves Christandl's PPT-square conjecture. (Lean)
Manuscript
Entanglement with zero distillable secret key in local dimension ten
September 27, 2026 · source · cite as \cite{OAI:Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026}
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.
Result 272 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md