Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 15D / b / Solution

Alice adjoins $|0{⟩}_{A^{\prime }}$ and applies a local unitary whose action on the relevant inputs isMeasuring $A^{\prime }$ and obtaining zero leaves the unnormalized stateso the normalized state is the Bell state $|{\varphi }^{+}⟩$. The success probability is $2s^2>0$.

Applying this entanglement concentration independently to $n$ pairs yields an expected $2s^{2}n$ Bell pairs. Alice and Bob measure successful pairs in the computational basis to obtain identical secret random bits; public communication identifies successful positions without revealing outcomes. These bits form a secret key for a one-time pad of expected length $2s^{2}n$.

Solved by gpt-5.6-sol high.