Suppose we have a qubit in the state |ψ>. We know that if we measure it in the standard basis, the probability of getting a 0 is 7/9. Now, if we instead �first apply a Hadamard gate and then measure the resulting qubit H|ψ> in the sign basis, what is the probability of getting a +?
7/9
2/9 is correct
To find the probability of getting a "+" when measuring the resulting qubit (H|ψ>) in the sign basis, we need to first apply the Hadamard gate (H gate) to the initial state |ψ>, and then calculate the probability of getting the state |+>.
Let's assume the initial qubit state |ψ> can be represented as a linear combination of the basis states |0> and |1>:
|ψ> = α|0> + β|1>
After applying the H gate to |ψ>, we get:
H|ψ> = (Hα|0> + Hβ|1>)
The H gate can be represented as:
H = 1/√2 * [1 1]
[1 -1]
Applying this gate to |0> and |1>, we get:
H|0> = 1/√2 * (|0> + |1>)
H|1> = 1/√2 * (|0> - |1>)
Substituting these values in H|ψ>:
H|ψ> = 1/√2 * (α|0> + α|1> + β|0> - β|1>)
= 1/√2 * [(α + β)|0> + (α - β)|1>]
To calculate the probability of getting "+" when measuring in the sign basis, we need to compute the squared magnitude of the coefficient of the |+> state in H|ψ>.
The |+> state can be represented as:
|+> = 1/√2 * (|0> + e^(iπ/2) |1>)
= 1/√2 * (|0> + i |1>)
Comparing this with H|ψ>:
|+> = 1/√2 * [(α + β)|0> + (α - β)|1>]
We can see that the coefficient of |+> in H|ψ> is (α + β).
Therefore, the probability of getting "+" when measuring H|ψ> in the sign basis is:
P(+) = |(α + β)|^2.
Given that the probability of getting 0 when measured in the standard basis is 7/9, we have:
P(0) = |α|^2 = 7/9
Using this information, we can rewrite the probability of getting "+" as:
P(+) = |(α + β)|^2 = |α|^2 + |β|^2 + 2 * Re(α*β*)
Since |α|^2 + |β|^2 = 1 (as the state is normalized), we have:
P(+) = 1 + 2 * Re(α*β*)
Substituting |α|^2 = 7/9, we get:
P(+) = 1 + 2 * Re(α*β*) = 1 + 2 * Re(√(7/9) * β*)
Unfortunately, without having the specific details of the state |ψ>, we cannot determine the exact value of β and calculate the probability of getting "+".
To find the probability of getting a "+" when measuring the qubit after applying a Hadamard gate, we need to understand the effect of the Hadamard gate on the state of the qubit.
The Hadamard gate, denoted as H, transforms the basis states |0⟩ and |1⟩ into superpositions. It can be represented by the following matrix:
H = 1/sqrt(2) * [[1, 1], [1, -1]]
Given that we have a qubit initially in the state |ψ⟩, we can represent it as a linear combination of the basis states |0⟩ and |1⟩:
|ψ⟩ = α|0⟩ + β|1⟩
The coefficients α and β are complex numbers. The condition that the probability of getting a 0 is 7/9 means that |α|^2 = 7/9.
To find the state H|ψ⟩ after applying the Hadamard gate, we simply multiply the Hadamard matrix with the state vector:
H|ψ⟩ = 1/sqrt(2) * [[1, 1], [1, -1]] * (α|0⟩ + β|1⟩)
= (1/sqrt(2))(α|0⟩ + β|1⟩ + α|0⟩ - β|1⟩)
= 1/sqrt(2) * (2α|0⟩)
= sqrt(2)α|0⟩
Now, let's find the probability of obtaining a "+" when measuring H|ψ⟩ in the sign basis. The sign basis consists of the states |+⟩ and |-⟩, which are defined as:
|+⟩ = 1/sqrt(2) * (|0⟩ + |1⟩)
|-⟩ = 1/sqrt(2) * (|0⟩ - |1⟩)
To calculate the probability of obtaining a "+" (|+⟩), we need to find the coefficient of |+⟩ when expressing H|ψ⟩ in the sign basis.
H|ψ⟩ = sqrt(2)α|0⟩
= sqrt(2)α * (1/sqrt(2)) * (|+⟩ + |-⟩) [Expanding |0⟩ in the sign basis]
= α * (|+⟩ + |-⟩)
The coefficient of |+⟩ is α, which is the same as the coefficient of |0⟩ in the original state |ψ⟩. Therefore, the probability of obtaining a "+" when measuring the qubit after applying the Hadamard gate is |α|^2 = (7/9).
In conclusion, the probability of getting a "+" is 7/9.