Hi, not sure your reasoning is correct. First when you write
P = |alpha><alpha| + |alpha_p><alpha_p| well this is not a measurement projector here and its just the identity matrix: the sum of two orthogonal projectors in a 2-dim space. There are two projectors in this situation P_1 = |alpha><alpha| and P_2 = |alpha_p> <alpha_2 |.
Here is a simpler way of looking at things. When Bob receives |0> from Alice the prob to get y_i=0 is the prob that the state is projected to |alpha> after Bob's measurement. There are two ways to compute this depending how you apply Born's rule:
prob = |<final state | initial state >|^2 = |<alpha | 0>|^2 = (cos alpha)^2
simply !
Your expression |<alpha|P|0>|^2 is meaningless... you are confusing with the fact that
prob = |<alpha | 0>|^2 = <0|alpha> <alpha | 0> = <0| P_1|0>
which is another way of writing the Born rule.
Hope all this clarifies.
N.M
Your expression |<alpha|P|0>|^2 is meaningless... you are confusing with the fact that
prob = |<alpha | 0>|^2 = <0|alpha> <alpha | 0> = <0| P_1|0>
which is another way of writing the Born rule.
Hope all this clarifies.
N.M
