The controlled-NOT leaves computational-basis signals unentangled:

∣0 ⟩_{A} ∣0 ⟩_{E} \mapsto ∣00 ⟩, ∣1 ⟩_{A} ∣0 ⟩_{E} \mapsto ∣11 ⟩ .

For the two Hadamard-basis choices it gives

∣ + ⟩_{A} ∣0 ⟩_{E} \mapsto \frac{∣00 ⟩ + ∣11 ⟩}{2} = ∣ ϕ^{+} ⟩,

∣ - ⟩_{A} ∣0 ⟩_{E} \mapsto \frac{∣00 ⟩ - ∣11 ⟩}{2} = ∣ ϕ^{-} ⟩ .

ii

Solution

Conditioned on a sifted-key position, Alice and Bob used the computational basis with probability

1/2

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

\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} .

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