VibeMathedMath problems solved with AI
Sign in

Secret key from all entangled states: does every entangled state have positive distillable secret key? (Kruger-Werner Problem 24)

For a bipartite state ρAB\rho_{AB}, the distillable secret key KD(ρ)K_D(\rho) is the best rate at which Alice and Bob, using local operations and authenticated public communication on many copies, can extract bits that agree and are independent of an eavesdropper holding a purification and the public record. Separable states have KD=0K_D=0, and Horodecki et al. showed that some bound entangled (PPT) states nevertheless have KD>0K_D>0, so key and distillable entanglement differ. The converse was posed as Problem 24, 'Secret key from all entangled states', in Kruger and Werner's 2005 collection of open problems in quantum information theory (contact P. Horodecki). Is KD(ρ)>0K_D(\rho)>0 for every entangled state ρ\rho?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Quantum cryptography: distillable secret key and bound entanglement
Posed by
P. Horodecki (contact), Problem 24 'Secret key from all entangled states' in O. Kruger and R. F. Werner (eds.), Some open problems in quantum information theory (2005)
Year posed
2005
Years open
21y
Solved
2026-09-27
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is an explicit entangled density operator ρ\rho on C10⊗C10\mathbb C^{10}\otimes\mathbb C^{10} whose range contains no nonzero product vector, such that for every nn and every admitted completed bit-output protocol on ρ⊗n\rho^{\otimes n} the output is at trace-norm distance at least 1/51/5 from an ideal secret bit; hence KD(ρ)=0K_D(\rho)=0. The state is the normalized Choi matrix of a composition of two PPT maps; a coherence inequality for such states bounds secrecy by disagreement probability. So the answer to Problem 24 is no. Not shown: minimality of dimension ten, or a zero-key example for protocol models beyond the paper's admitted class.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Kruger-Werner Problem 24. The theorem gives an explicit entangled state on C10⊗C10\mathbb C^{10}\otimes\mathbb C^{10} with KD=0K_D=0 for the paper's class of admitted protocols, which includes ordinary finite two-way public-discussion protocols with round counts growing with block length, plus specified countable processes; readers should note the answer is stated relative to that formal protocol model. The proof was not refereed. The release's Lean formalization (lean/docs/272.md) covers only the PPT-composition consequences, and says explicitly that the zero-key statement is outside it, so this entry is unreviewed. Entanglement of the state rests on an exact Galois-group computation, with a supplementary certificate output in the release.

Sources

Changelog1 change

Discussion