Consider the observable M that corresponds to a measurement in the sign basis where the measurement value is 3 if the outcome is + and −2 if the outcome is −.
Write M in the standard basis.
If M is the Hamiltonian acting on a qubit which is in the state |ψ(0)⟩=|0⟩ at time 0, what is the state of the qubit at time t? Assume that you are working in units such that ℏ=1. You may use e and t in your answer.
Now you perform an X=(0110) measurement on the qubit at time 0. What is the expected value of your measurement?
What if you perform the above measurement at time t=4π3 instead? What is the expected value of your measurement?
To write M in the standard basis, we'll need to express it in terms of the Pauli matrices σx and σz. In the standard basis, the measurement operator M can be written as:
M = 3|+⟩⟨+| - 2|-⟩⟨-|
Here, |+⟩ and |-⟩ represent the eigenstates of the Pauli σz matrix:
|+⟩ = (1/sqrt(2))(|0⟩ + |1⟩)
|-⟩ = (1/sqrt(2))(|0⟩ - |1⟩)
To find the state of the qubit at time t when the Hamiltonian H = M, we can use the time evolution equation:
|ψ(t)⟩ = e^(-iHt)|ψ(0)⟩
Given that |ψ(0)⟩ = |0⟩, we have to apply the exponential of the Hamiltonian to the initial state. However, since the measurement operator M doesn't commute with the Pauli matrices, the time evolution cannot be written in a simple exponential form.
Moving on to the next question, let's consider the measurement X = (0110) applied at time 0. The measurement's expected value is given by:
⟨X⟩ = ⟨ψ(0)|X|ψ(0)⟩
Plugging in the values, we get:
⟨X⟩ = ⟨0| (0110) |0⟩
Calculating the inner product, we find:
⟨X⟩ = (1/sqrt(2)) * (0*0 + 1*1 + 1*0 + 0*0) * (1/sqrt(2))
Simplifying this expression, we get:
⟨X⟩ = 1/2
Now, let's consider performing the same measurement at time t = 4π/3. To find the expected value, we'll need to compute:
⟨X⟩ = ⟨ψ(t)|X|ψ(t)⟩
First, we need to find the state |ψ(t)⟩ using the time evolution equation. However, without knowing the Hamiltonian H, we cannot determine |ψ(t)| explicitly, and therefore, we cannot compute the expected value of the measurement at that particular time.