Entangled, but no secret?
An AI preprint claims an entangled state with zero distillable key, and a counterexample to the PPT-squared conjecture
Covers result 272 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:08. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Alice and Bob want a secret key that an eavesdropper, Eve, cannot know. Shared entangled pairs, measured the same way on both sides, give matching random bits that no one else can predict, and every secret key of this kind needs entanglement. The question is whether every entangled state can give some key.
Some entanglement is hard to see. The standard test flips one side, a partial transpose: for a perfectly entangled pair it gives a negative eigenvalue of −1/2. But in the late 1990s Paweł Horodecki found entangled states that pass this test, and the Horodeckis called them bound entangled, because no pure entanglement can be distilled from them. Surprisingly, in 2003 three Horodeckis and Oppenheim showed that some of these still give secret key. Whether every entangled state gives key went on a 2005 list of open problems as Problem 24, and a 2018 paper still called it widely open. A second question, asked by Christandl in 2012, is whether two channels in a row, each passing the partial transpose test, must always destroy entanglement.
Result 272 is one OpenAI preprint, 53 pages, dated September 27, 2026. It constructs an explicit entangled state of two ten-level systems that passes the partial transpose test, and from which no secret key can be distilled by a wide class of protocols: local quantum operations on any number of copies, two-way public communication, and an eavesdropper holding a purification and every public message. Any output bit stays at least a fixed distance from an ideal secret bit. The same construction gives counterexamples to the partial-transpose-squared question: two explicit maps on ten levels, each passing the test, whose composition still carries entanglement, and a single channel on 21 levels whose square does. The state is exactly the Choi matrix of that composition, scaled to trace one.
What it doesn’t claim
That bound entanglement never gives key, which is false; protocols outside the admitted class; or the smallest dimension where this happens. In dimension three, Christandl’s conjecture was already proved true.
Where it stands
- Posted as a preprint
- OpenAI, 53 pages, dated September 27, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- In part: the two counterexamples, not the secret-key theorem (docs)
Last checked October 19, 2026. The preprint has not been revised. The two counterexamples, the ten-level pair and the 21-level channel, are formalized in Lean, about 45,000 lines with no unproved steps, which we did not build. The secret-key theorem, the title result, is not in the audited check. The paper ships exact verification programs, which ran and matched on our machine. OSVerify has not reviewed it, and we found no comment from specialists.
How the proof goes
- Twenty special directions. The entanglement proof starts from an integer 6×4 matrix pencil with exactly 20 special directions (all real, by our computation). With Galois theory and arithmetic modulo three primes, the paper shows that no quadric passes through ten of them. A product vector in the range would need two quadrics covering all twenty, but two catch at most eighteen, so the state must be entangled. Standard numerical tests miss it: our realignment value is 0.54, below the threshold of 1.
- The one-fifth wall. The key side uses a four-effect inequality: being secret needs coherence between the both-zero and both-one outcomes, and for these states, being correct forbids it. The two bounds cross exactly at 1/5, so in the paper’s measure no protocol gets closer than 1/5 to a perfect secret bit.
The numbers in the video
- Ours: the partial transpose of a perfectly entangled pair has eigenvalue −1/2.
- Ours: the paper’s 100×100 Choi matrix rebuilt: rank 80, passes the partial transpose test, trace 130,208,350,782; realignment value 0.54.
- Ours: the two bounds of the key argument cross exactly at 1/5, with value 2/5.
Read the source
- Entanglement with Zero Distillable Secret Key in Local Dimension Ten (PDF), 53 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 272.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Alice and Bob want a secret key that an eavesdropper, Eve, cannot know. Quantum mechanics offers a way: shared entangled pairs, measured the same way on both sides, give matching random bits that no one else can predict. Every secret key of this kind needs entanglement. The question is whether every entangled state can give some key. Some entanglement is hard to see. The standard test flips one side, a partial transpose: for a perfectly entangled pair, our computer finds a negative eigenvalue of −1/2. But in the late 1990s Paweł Horodecki found entangled states that pass this test, and the Horodeckis called them bound entangled, because no pure entanglement can be distilled from them. Surprisingly, in 2003, three Horodeckis and Oppenheim showed that some of these still give secret key. So maybe every entangled state gives key. In 2005 this went on a list of open problems in quantum information as Problem 24, and a 2018 paper still called it widely open. A second question, asked by Christandl in 2012, is whether two channels in a row, each passing the partial transpose test, must always destroy entanglement.
1:17The claims
Result 272 is one OpenAI preprint, 53 pages, dated September 27, 2026. It constructs an explicit entangled state of two 10-level systems that passes the partial transpose test, and from which no secret key can be distilled by a wide class of protocols: local quantum operations on any number of copies, two-way public communication, and an eavesdropper holding a purification and every public message. Any output bit stays at least a fixed distance from an ideal secret bit. The same construction answers Christandl’s question with a counterexample: two explicit maps on 10 levels, each passing the test, whose composition still carries entanglement, and a single channel on 21 levels whose square does. The state is exactly the Choi matrix of that composition, scaled to trace one. Not claimed: that bound entanglement never gives key, which is false; protocols outside the admitted class; or the smallest dimension where this happens. In dimension three, Christandl’s conjecture was already proved true.
2:26How the proof goes
The entanglement proof starts from an integer 6×4 matrix pencil with exactly 20 special directions; our computer finds all 20 are real. The paper shows, with Galois theory and arithmetic modulo three primes, that no quadric passes through 10 of them. A product vector in the range would need two quadrics covering all 20, but two catch at most 18, so the state’s range holds no product vector, and the state must be entangled. Standard numerical tests miss it: our realignment value is 0.54, below the threshold of 1. The key side uses a four-effect inequality: being secret needs coherence between the both-zero and both-one outcomes, and for these states, being correct forbids it. The two bounds cross exactly at 1/5, so in the paper’s measure no protocol gets closer than 1/5 to a perfect secret bit.
3:22Where it stands
The preprint is unrefereed and unchanged since release. The two counterexamples, the 10-level pair and the 21-level channel, are formalized in Lean, about 45,000 lines with no unproved steps, which we did not build. The secret-key theorem, the title result, is not in the audited check. The paper ships exact verification programs, which ran and matched on our machine. OSVerify has not reviewed it, and we found no comment from specialists. If it holds, entanglement and secret key are different resources: here is an entangled state from which, by these protocols, not even one nearly secret bit can be squeezed.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.