Suppose that a qubit is in the state |\phi\rangle=a|0\rangle+\sqrt{1-a^2}|1\rangle where a\in[-1,1]. If we first perform a standard basis measurement on this qubit and then perform a |u\rangle,|u^\perp\rangle-basis measurement where |u\rangle=b|0\rangle+\sqrt{1-b^2}|1\rangle for some b\in[-1,1], what is the probability that the outcome of the second measurement is u, in terms of a and b?

Explicitly indicate multiplication with a * symbol.

To find the probability of the outcome of the second measurement being |u⟩, we need to consider the possible outcomes of the first measurement.

When performing a standard basis measurement on the qubit in the state |ϕ⟩, we obtain one of the two possible outcomes: |0⟩ or |1⟩. Let's consider these two cases separately.

Case 1: The outcome of the first measurement is |0⟩
In this case, after the first measurement, the qubit collapses to the state |0⟩. To calculate the probability of the outcome of the second measurement being |u⟩, we need to determine the overlap between the state |u⟩ and the state |0⟩.

The overlap between two quantum states can be calculated using the inner product. So, we need to compute ⟨u|0⟩, where |u⟩=b|0⟩+√(1−b²)|1⟩.

⟨u|0⟩ = ⟨0|u⟩* = (b*⟨0|0⟩) + (√(1−b²)*⟨0|1⟩)
Since ⟨0|0⟩ = 1 and ⟨0|1⟩ = 0 (orthogonality of basis states), we can simplify the expression.

⟨u|0⟩ = b

Therefore, for this case, the probability of the outcome of the second measurement being |u⟩ is |b|² = b².

Case 2: The outcome of the first measurement is |1⟩
In this case, after the first measurement, the qubit collapses to the state |1⟩. We follow the same steps as in Case 1 to calculate the probability of the outcome of the second measurement being |u⟩.

The overlap between |u⟩ and |1⟩ is given by:

⟨u|1⟩ = ⟨0|u⟩*⟨0|1⟩ + ⟨1|u⟩*⟨1|1⟩
Since ⟨0|1⟩ = 0 and ⟨1|1⟩ = 1, the expression simplifies to:

⟨u|1⟩ = (√(1−b²)*⟨1|1⟩) = √(1−b²)

Therefore, for this case, the probability of the outcome of the second measurement being |u⟩ is |√(1−b²)|² = 1−b².

Now, to find the overall probability, we need to consider the probabilities of both cases occurring and sum them up.

The probability of the outcome being |0⟩ in the first measurement is |a|² = a².
The probability of the outcome being |1⟩ in the first measurement is |1-a²|.

So, the overall probability of the outcome of the second measurement being |u⟩ can be calculated as:
P(u) = a² * b² + |1-a²| * (1-b²)

Therefore, the probability that the outcome of the second measurement is |u⟩, in terms of a and b, is a² * b² + |1-a²| * (1-b²).