Result 272, Mathematical physics

Entanglement without distillable secret key

Constructs an entangled state on C10⊗C10\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 M21(C)M_{21}(\mathbb C) whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Quantum entanglement need not support a positive rate of secret-key generation. The manuscript claims an explicit example, showing why the presence of this quantum correlation alone may not certify a usable secrecy resource.

What changes?

The manuscript reports an entangled state, a joint quantum state not expressible as a mixture of independent local states, for two ten-dimensional systems. It yields no positive asymptotic rate of shared secret bits under the specified local-instrument protocols that finish with probability one. These permit joint local processing of all copies and unlimited authenticated two-way public communication. No other shared private resource is allowed; the eavesdropper holds both a purification of the input and the full public record.

What does that help mathematicians do?

A separate claimed counterexample uses a trace-preserving quantum channel on 21-dimensional systems satisfying the positive-partial-transpose (PPT) condition. Applying it twice still need not destroy entanglement with a reference system. This contradicts Christandl's PPT-square conjecture: researchers cannot infer that two successive uses erase all such entanglement merely from the PPT restriction. The state construction also contradicts the unrestricted two-map PPT-composition conjecture.

Are there practical applications?

For quantum cryptography, this is a limitation result, not a new key-generation method. Its immediate value is foundational: it separates possessing entanglement from obtaining secret bits at a positive rate under the stated rules. Assessments of cryptographic resources must therefore account for the particular state and allowed protocols, rather than treating entanglement as a sufficient certificate.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Entanglement with zero distillable secret key in local dimension ten

September 27, 2026 53 pages

We construct an entangled state on C10⊗C10\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 M21(C)M_{21}(\mathbb C) has a square that is not entanglement breaking.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/272.md.

Entanglement without distillable secret key

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper's entanglement construction yields counterexamples to PPT-composition claims. The linked formalization covers these consequences: two explicitly specified PPT maps on 10×1010\times10 complex matrices have a composition that is not entanglement breaking, and the nonzero Choi matrix of that composition has no nonzero product vector in its range. A separate trace-preserving PPT channel on 21×2121\times21 matrices has a square that is not entanglement breaking.

The paper's zero-distillable-secret-key statement is outside these two selected Comparator statements.

Comparator links

Result Comparator statement
PPT channel on dimension 21 whose square is not entanglement breaking DimensionTenChannel.lean
Dimension-ten PPT pair with non-entanglement-breaking composition DimensionTenPair.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.