Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 10D (Quantum Information and Computation)

In the BB84 protocol, Alice independently chooses $n$ random bits and, for each bit, a random computational or Hadamard basis. She sends the corresponding states $|0⟩,|1⟩,|+⟩,|-⟩$ through the noiseless quantum channel. Bob independently chooses one of the two bases for each qubit and measures. Over the public classical channel they reveal only their basis choices and retain positions where the choices agree. Each position survives with probability $1/2$, so the expected secret-key length is $n/2$. In the ideal no-eavesdropper setting their retained bits agree exactly.

The controlled-NOT leaves computational-basis signals unentangled:For the two Hadamard-basis choices it givesBoth outputs are maximally entangled Bell states.

Conditioned on a sifted-key position, Alice and Bob used the computational basis with probability $1/2$ and the Hadamard basis with probability $1/2$. Eve causes no error in the computational basis. In the Hadamard basis Bob's reduced state is maximally mixed, so his bit is wrong with probability $1/2$. The final bit-error rate is therefore