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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 10D / b / i / Solution

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^{-} ⟩ .

Both outputs are maximally entangled Bell states.

Solved by gpt-5.6-sol high.

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