What was already known
Christandl’s 2012 PPT-squared question asked whether a PPT channel becomes entanglement breaking after composition with itself. Prior work established special cases, including low-dimensional channels, but not the general claim. The same manuscript’s dimension-ten construction uses two PPT maps and addresses a different, unrestricted two-map statement.
AI involvement
The manuscript credits OpenAI as author and its release credits an internal frontier model with the mathematics. The available account does not establish task-only autonomy or disclose every human input.
OpenAI is the manuscript author and its release attributes the mathematics to an internal frontier model. The available disclosure does not establish task-only autonomy.
- Supporting source (openai.com) — Mathematics release and model attribution
Importance
Supplies a counterexample to the same-channel PPT-squared conjecture in dimension 21.
Limitations
This is Theorem 1.3, distinct from the paper’s dimension-ten two-map counterexample and its zero-distillable-secret-key state. It does not assert a minimal counterexample dimension. The Lean proof covers the channel claim; no Lean check is claimed for the separate secret-key theorem.
Technical verification
The official Lean solution module proves `exists_channel_fin21`: a 21-dimensional map is PPT and trace preserving, and its square is not entanglement breaking. The formalization note identifies this as the paper’s channel theorem. This is source-reported; the development was not rebuilt here.
Formal checking
Official solution module proves exists_channel_fin21 with a completed proof term, using main_channel_fin21 and publishedChoi21; the theorem asserts PPT, trace preservation, and non-entanglement-breaking square.
- Checker
- Lean kernel, as reported by OpenAI
- Scope
- An explicit trace-preserving PPT channel on M21(C) whose square is not entanglement breaking.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No named human account explaining the channel construction’s central reasoning was identified. This is the default for missing documentation, not a finding that people cannot understand it.
Correction and publication history
Distinct Theorem 1.3 claim, prior baseline, attribution and exact Lean proof scope reviewed for non-notifying local inclusion.
Read this published revision