What is the inner product of 13√|0⟩+2√3√|1⟩ and |+⟩?
Rewrite 13√|0⟩+2√3√|1⟩ in the sign basis.
THE #1ST
(1+sqrt(2))/sqrt(6)
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To find the inner product between two vectors, we need to rewrite them in the same basis. In this case, we need to rewrite the vector 13√|0⟩+2√3√|1⟩ in the sign basis.
In the sign basis, the two states |0⟩ and |1⟩ are represented by the vectors |+⟩ and |-⟩, respectively.
To rewrite the vector 13√|0⟩+2√3√|1⟩ in the sign basis, we can use the following relations:
|0⟩ = 1/√2 (|+⟩ + |-⟩)
|1⟩ = 1/√2 (|+⟩ - |-⟩)
Substituting these relations into the original vector, we get:
13√|0⟩+2√3√|1⟩ = 13√(1/√2)(|+⟩ + |-⟩) + 2√3√(1/√2)(|+⟩ - |-⟩)
= (13/√2)(√|+⟩ + √|-⟩) + (√3√/√2)(√|+⟩ - √|-⟩)
= (13/√2)√(|+⟩ + |-⟩) + (√3√/√2)√(|+⟩ - |-⟩)
Thus, the vector 13√|0⟩+2√3√|1⟩ in the sign basis is (13/√2)√(|+⟩ + |-⟩) + (√3√/√2)√(|+⟩ - |-⟩).