Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 15D (Quantum Information and Computation)

Alice and Bob need a uniformly random secret key $K\in \{0,1{\}}^m$ known only to them. Alice publicly sends the one-time pad ciphertext $C=M\oplus K$, and Bob recovers $M=C\oplus K$. For every observed $c$ and possible message $m$ there is exactly one equally likely key $k=c\oplus m$, so $C$ is independent of $M$ and Eve learns nothing.

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 $2s^{2}n$.