E91 protocol description
The E91 protocol was proposed by Artur Ekert in 1991. His idea: the security of the key can be checked through the violation of a Bell inequality — that is, by making sure the photons are truly entangled with each other and therefore cannot be “linked” to Eve.
The source creates photon pairs in the state |Ψ⁻⟩ = (|HV⟩ − |VH⟩)/√2 and sends one photon to Alice and the other to Bob. If both measure polarization at the same angle, the results are always opposite.
For each pair, Alice randomly picks a polarizer angle from 0°, 22.5°, 45°, and Bob from 22.5°, 45°, 67.5°. Result: 0 — the photon passed the polarizer, 1 — it did not.
If Alice's and Bob's angles are equal, their bits are always opposite. The bigger the angle difference, the rarer that is: at 22.5° — 85% of the time, at 45° — a coin flip, at 67.5° — only 15%. This coordination between distant photons is stronger than anything classical.
After all measurements, Alice and Bob announce over the public channel which angles they used. The results stay secret for now.
Pairs with equal angles (22.5°/22.5° and 45°/45°) go into the key — Bob flips his bits. Pairs with Alice at 0°/45° and Bob at 22.5°/67.5° go into the Bell test. The rest are discarded.
Alice and Bob reveal the test pairs completely and turn them into a game: at 0°/67.5° the bits should match, in the other three combinations they should differ. Ordinary photons with any pre-arranged “cheat sheet” win at most 75% of rounds; entangled ones win about 85%. They also reveal some of the key pairs to count errors.
If the win rate is clearly above 75% and errors are few, entanglement is intact and Eve knows little about the key: the key is corrected and compressed. If the win rate is 75% or lower, the photons behaved like ordinary ones — someone measured them — and the key is discarded.
Why it works: entanglement is “monogamous”. If Alice’s and Bob’s photons are maximally entangled with each other (|S| ≈ 2√2), no third party can be linked to them. Any measurement by Eve turns the pair into ordinary photons — and |S| drops to classical values.
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Stage 1: Pair source
The source in the middle sends out pairs of entangled photons. Start with the warm-up: see how often Alice's and Bob's bits are opposite for different angle differences. Then start the source — Alice and Bob will choose angles at random, as in the real protocol.
What is simplified: Ekert gives the angles 0°, 45°, 90° for spins — for photons they are halved (0°, 22.5°, 45°). The key decision uses the bound of Acín et al. (2007), but without finite-sample corrections. Channel noise is a fixed ~2%; Eve uses the simplest attack — measuring the photon at a random angle. Error correction and compression are shown schematically, and real systems use millions of pairs.