Now note that the following becomes good approximations as θ→0:
sinθ≈θ
sin2(π4−θ)≈12−c⋅θ
(where c is some constant.) Use these approximations to estimate the probability of success of the two measurements as x→0 (as a function of x and c).
Measurement I?
Measurement II?
To estimate the probability of success of the two measurements as x→0, we can use the given approximations. Let's evaluate each measurement separately.
Measurement I:
The problem does not provide any specific information about Measurement I. If you can provide additional details or assumptions about Measurement I, I can help you estimate its probability of success using the given approximations.
Measurement II:
The given approximation is sin(π/4 - θ) ≈ 1/2 - c * θ, where c is some constant.
To estimate the probability of success for Measurement II as x→0, we need to express the probability in terms of x and c. However, the given information provides θ as the variable, not x. We need to establish a relationship between θ and x.
Since θ is approaching 0, we can assume that x is very small compared to π/4. Therefore, we can express x in terms of θ as x = π/4 - θ.
Substituting this expression into the approximation, we get:
sin(π/4 - θ) ≈ 1/2 - c * θ
sin(x) ≈ 1/2 - c * (π/4 - x)
Now, we can estimate the probability of success by treating sin(x) as the probability and solving for x:
sin(x) = 1/2 - c * (π/4 - x)
Keep in mind that this is an approximation, and the accuracy of the estimation depends on the value of the constant c and how small x is.