Entanglement without distillable secret key

The explanation

Picture Alice and Bob in two labs, sharing many copies of a quantum state. They want a secret key: random bits that agree at both ends and that an eavesdropper, Eve, cannot predict. The ideal one-bit key is 00 with chance one half and 11 with chance one half, with Eve learning nothing. They may measure locally, talk over a public line as long as they like, and process all copies together. Eve hears every public message and holds the rest of the quantum system.

The state is the resource. If it is separable, meaning it could be built from ordinary shared randomness, no key is possibleSource 1. A state that is not separable is called entangled. Some weakly entangled states, of the kind called PPT, still give a positive key rateSource 2. So a natural guess was that every entangled state gives some key. Krüger and Werner's 2005 list of open problems posed this as Problem 24Source 3. The obstacle is that a no-key proof must defeat every possible protocol, of any length.

The paper claims an explicit entangled state on two ten-level systems with zero distillable secret keySource 4. Distillable key is the best rate of nearly perfect key bits per copy. The claim is stronger than a zero rate. For any number of copies and any admitted protocol, the output bit stays at trace distance at least 1/10 from a perfect bitSource 5. Trace distance measures how well any test can tell two states apart. There is no limit on the rounds of talk inside the modelSource 6.

The core is a short inequality. Let e be the chance that the two bits differ. Let f measure how alike Eve's views are when the key is 00 and when it is 11. For states built in the paper's way, f is at most √(e(1 − e)) + e. A near-perfect key needs small e and f close to one half. Try a protocol whose output sits at trace-norm distance 0.2. Then e is at most 0.1, so f is at most √0.09 + 0.1 = 0.4. Secrecy needs f to be at least 0.4. Anything closer than 0.2 makes the two demands clashSource 7.

To make the state entangled, the paper checks that its range holds no product vector. The proof uses twenty special directions in four dimensions. A quadratic in four variables has ten coefficients, so nine points can always be fitted. The paper shows no nonzero quadratic vanishes at ten of them. Two quadratics then cover at most nine directions each, and 9 + 9 < 20Source 8.

The same construction gives two PPT maps whose composition is not entanglement breaking. This is a negative answer to the two-map PPT-composition conjecture of Christandl, Müller-Hermes and WolfSource 9. A separate channel on 21-level systems has a square that is not entanglement breakingSource 10. Earlier work settled the composition question for two maps on three-level systemsSource 11. The authors do not claim either dimension is minimalSource 12.

How sure is this? Lean, a computer proof checker, has formal statements for the dimension-ten pair of maps and for the 21-level channel. These cover the missing product vectors and the failure of entanglement breaking. The zero-key theorem itself is outside the Lean statements. It rests on the paper's long protocol analysis, which has no published specialist review.

What it doesn't show. This does not show the dimension ten is the smallest possible, and it does not cover protocols outside the paper's admitted model.