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Phys-19-A2 · revision r1 · on Phys-19

OpenAI claim summary for Phys-19

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The explicit 10-by-10 entangled state has zero distillable secret key: every completed finite-copy local-operations/public-communication bit-output protocol stays a uniform trace-distance amount from a private shared bit, even with the purification and transcript included. This refutes the universal positive-key assertion by a single state. Within these assumptions, this is a claimed negative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Secret key from all entangled states
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Secret key from all entangled states

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A bipartite state with local dimension ten is claimed entangled yet to have zero distillable private key under the specified unrestricted-copy, two-way local-instrument protocols.

Mathematical statement of the claimed result

Result 1.
An explicitly specified entangled state $\rho$ on $\mathbb C^{10}\otimes\mathbb C^{10}$ is claimed to have no nonzero product vector in its range and distillable key $K_D(\rho)=0$. For every $n\ge1$ and admitted completed local/public-classical bit-output protocol on $\rho^{\otimes n}$, let $\tau_{K_AK_BE'}$ be its normalized output, retaining the purification and full public transcript in $E'$. The quantitative claim is \[ \left\|\tau-\tfrac12\sum_{i=0}^1|i,i\rangle\langle i,i|\otimes\sigma_{E'}\right\|_1\ge1/5 \quad\text{for every state }\sigma_{E'}. \] Protocols use no initially shared private resource beyond the input; local auxiliaries and private randomness are allowed, and countable completed processes obey the source's reference-compatibility conditions. The conventional halved trace distance is therefore at least $1/10$.

Outline of the argument

A represented PPT-map class obeys a four-effect coherence bound. Tracking complete public transcripts preserves common-effect constraints under adaptive instruments, which obstruct secret-key generation; related compositions yield the advertised PPT channel counterexamples.

Formal verification

For family 272, the selected consequences concern PPT compositions in dimensions ten and twenty-one that remain non-entanglement-breaking. The secret-key obstruction itself is outside those entries. The listed Comparator files are DimensionTenChannel.lean, DimensionTenPair.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources