Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 53 / 4 / A / 2 / Solution

The normalized ray produces a secure key; the literal printed vector is not normalized. The squared norm of the coefficients in the PDF is $8/9+1/81=73/81$. Thus the physical pure state on that ray has normalized coefficients $6\sqrt{2}/\sqrt{73}$ and $1/\sqrt{73}$. This is a genuine entangled state with two nonzero Schmidt coefficients, and Eve is independent of it by purity decouples a subsystem from its purification.

For an explicit way to obtain uniform key bits, Alice applies a two-outcome local measurement in quantum mechanics with Kraus operatorsTheir squared products sum to the identity. The successful unnormalized state is $(|00⟩+|11⟩)/\sqrt{73}$, soAlice announces only success or failure. Success leaves an exact Bell pair, yielding one uniform secret bit; failure leaves $|00⟩$ and is discarded. With many copies the successful fraction is positive. Alternatively, direct computational basis measurements give a shared biased bit with probabilities $72/73$ and $1/73$, from which privacy amplification can extract uniform secret bits. The coefficient $1/9$ has not been silently replaced by $1/3$.