Consider the state |ψ〉 = √2 |0〉 + √2 |1〉 from the previous question. To estimate the phase θ,
we measure |ψ〉 in the sign basis. What is the probability that the outcome of the measurement
is +? Recall that eiθ = cos θ + i sin θ
To estimate the phase θ, we need to measure the state |ψ⟩ in the sign basis. In the sign basis, the two basis states are |+⟩ and |-⟩, which are given by:
|+⟩ = 1/√2 (|0⟩ + |1⟩)
|-⟩ = 1/√2 (|0⟩ - |1⟩)
To find the probability that the outcome of the measurement is +, we need to take the inner product of the state |ψ⟩ with the |+⟩ state and then calculate the absolute value squared of that inner product. Mathematically, it can be expressed as:
P(+) = |⟨+|ψ⟩|²
Substituting the values of |+⟩ and |ψ⟩, we get:
P(+) = |⟨+|ψ⟩|²
= |⟨+|(√2|0⟩ + √2|1⟩)|²
= |1/√2 ⟨+|0⟩ + 1/√2 ⟨+|1⟩|²
= |1/√2 * 1/√2 + 1/√2 * 1/√2|² (since ⟨+|0⟩ = 1/√2 and ⟨+|1⟩ = 1/√2)
= |1/2 + 1/2|²
= 1
Therefore, the probability that the outcome of the measurement in the sign basis is + is 1.