We have a qubit in the state |ψ>= √3/2 |0>+ 1/2 |1>, which we want to measure in the {Cos θ |0> + Sin θ |1>, Sin θ |0> - Cos θ |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 find the value of θ that makes the two possible outcomes equiprobable, we need to calculate the probabilities of measuring each outcome and set them equal to each other.
The probabilities of measuring each outcome can be calculated using the formula:
P(0) = |<0|ψ>|²
P(1) = |<1|ψ>|²
Let's calculate these probabilities step by step.
First, let's expand the state |ψ> in the computational basis:
|ψ> = √3/2 |0> + 1/2 |1>
Now, let's write the basis states in terms of the given basis:
|0> = Cos θ |0> + Sin θ |1>
|1> = Sin θ |0> - Cos θ |1>
To find the probabilities, we need to calculate the absolute squares of the inner products between the states.
Calculating <0|ψ>:
<0|ψ> = <0| (√3/2 |0> + 1/2 |1>)
= (√3/2) <0|0> + (1/2) <0|1>
= √3/2
Calculating <1|ψ>:
<1|ψ> = <1| (√3/2 |0> + 1/2 |1>)
= (√3/2) <1|0> + (1/2) <1|1>
= 1/2
Now, let's calculate the probabilities.
P(0) = |<0|ψ>|²
= (|√3/2|)²
= 3/4
P(1) = |<1|ψ>|²
= (|1/2|)²
= 1/4
To make the two outcomes equiprobable, we need to set P(0) equal to P(1) and solve for θ.
3/4 = 1/4
Since these two probabilities are not equal, there is no value of θ that would make the two outcomes equiprobable in this case.