Rewrite 13√|0⟩+2√3√|1⟩ in the sign basis.
(1+sqrt(2))/sqrt(6)|+> + (1-sqrt(2))/sqrt(6)|->
We have a qubit in the state |ϕ⟩=3√2|0⟩+12|1⟩, which we want to measure in the {cosθ|0⟩+sinθ|1⟩,sinθ|0⟩−cosθ|1⟩} basis. In order for the two possible outcomes to be equiprobable, what should be the value of θ in degrees? (Answer between 0 and 90.)
To rewrite the expression 13√|0⟩+2√3√|1⟩ in the sign basis, we need to convert the given expression from the computational basis {|0⟩, |1⟩} to the sign basis {|+⟩, |−⟩}.
The sign basis is related to the computational basis by the following transformation:
|+⟩ = 1/√2 * (|0⟩ + |1⟩)
|−⟩ = 1/√2 * (|0⟩ - |1⟩)
To apply this transformation, we substitute the values of |0⟩ and |1⟩ into the expressions for |+⟩ and |−⟩:
|+⟩ = 1/√2 * (|0⟩ + |1⟩)
= 1/√2 * (1√|0⟩ + 0√|1⟩)
= 1/√2 * 1√|0⟩ + 1/√2 * 0√|1⟩
= 1/√2 √|0⟩
|−⟩ = 1/√2 * (|0⟩ - |1⟩)
= 1/√2 * (1√|0⟩ - 0√|1⟩)
= 1/√2 * 1√|0⟩ - 1/√2 * 0√|1⟩
= 1/√2 √|0⟩
So, the expression 13√|0⟩+2√3√|1⟩ in the sign basis becomes:
13√|0⟩+2√3√|1⟩
= 13√(1/√2 √|0⟩) + 2√3(1/√2 √|0⟩)
= 13(1/√2)√|+⟩ + 2√3(1/√2)√|−⟩
= (13/√2)√|+⟩ + (√3/√2)√|−⟩
Therefore, the expression in the sign basis is (13/√2)√|+⟩ + (√3/√2)√|−⟩.