Sharp Continuity Bound for Quantum Conditional Entropy

Precise statement

What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound $h_2(\delta) + \delta \log(d^2 - 1)$ up to $\delta = 1 - d^{-2}$, conjectured by Wilde, is proved.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT-5.6 Sol

The key proof idea, adapting the tight classical argument of Alhejji and Smith to the fully quantum setting, was developed with ChatGPT-5.6 Sol.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Five-author arXiv preprint. Not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-27

   arXiv:2607.24687 - Sharp continuity of quantum conditional entropy

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.