Suppose we have a qubit in the state |ψ⟩=12|0⟩+3√2|1⟩. If we measure this qubit in |u⟩=3√2|0⟩+12|1⟩,|u⊥⟩=−12|0⟩+3√2|1⟩ basis, what is the probability that the outcome is u

To find the probability of measuring a qubit in a particular basis state, we need to calculate the inner product (also known as the dot product) between the qubit's current state and the basis state we are interested in. In this case, the basis state we are interested in is |u⟩.

The inner product between two qubits is given by:

⟨u|ψ⟩

Let's plug in the values and calculate it step by step:

|ψ⟩ = 1/2 |0⟩ + 3√2 |1⟩
|u⟩ = 3√2 |0⟩ + 1/2 |1⟩

Taking the conjugate of |u⟩:

|u⟩* = (3√2)* |0⟩ + (1/2)* |1⟩
= 3√2* |0⟩ + 1/2 |1⟩

Multiplying the corresponding coefficients of |u⟩* and |u⟩:

⟨u|ψ⟩ = (3√2* * 1/2) ⟨0|0⟩ + (1/2 * 3√2) ⟨1|0⟩
= (3√2* * 1/2) * 1 + (1/2 * 3√2) * 0
= 3√2* * 1/2
= 3√2/2

The absolute value of the inner product squared gives us the probability:

P(u) = |⟨u|ψ⟩|²
= |3√2/2|²
= (3√2/2)²
= 3/2

So, the probability of measuring the qubit in the basis state |u⟩ is 3/2. However, probabilities should always be between 0 and 1. Thus, this result is not physically valid and suggests that there might be an error in the given quantum states or basis states.