Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 6 / d

Encode $0$ by phase zero and $1$ by phase $\pi$. Alice and Bob choose their bits independently and keep their choices private. Bob announces the trial numbers with a D0 click over an authenticated public channel. For each retained trial, c guarantees that their bits agree. In the ideal symmetric setup, the two equal-phase cases $00$ and $11$ give the same click probability, so announcing success does not itself reveal which shared bit was used.

The matching-phase cases occur with probability $1/2$, and a matching case succeeds with probability $1/16$. ThereforeFor independent ideal trials, the raw-key length has a binomial distribution with standard deviation $\sqrt{31N}/32$. These are bits before sacrificing test positions or performing privacy amplification.

A missing click does not certify unequal phases: the same phases also fail to click with probability $15/16$. With uniform choices,Thus a no-click trial supplies neither a known equal bit nor a known complemented bit. This is why it cannot be retained by the direct error-free sifting rule. The source's “cannot use” should be understood in this sense: the events are not statistically independent, so an absolute claim that they contain no information would be false.

To obtain a secret final key, reveal and discard a random subset of retained bits to test the observed error rate, then apply reconciliation and privacy amplification as required by a security analysis of the actual implementation. Resolving which time path a photon took removes the accepted histories' quantum coherence and can turn destructive interference into false D0 clicks, providing a simple disturbance test. Equal-phase click correlations alone are not a proof of security against every attack, particularly in the presence of losses and device imperfections. The supplied ideal model fixes the sifted rate $N/32$; it does not fix a positive secure final length without an adversary model and observed parameters.