Suppose we are given a single qubit which is either in the state |u> = cos(pi/8) |0> + sin(pi/8) |1> or |v> = cos(3pi/8) |0> + sin(3pi/8) |1>. We measure this qubit in the standard basis and guess u if the outcome is 0 and guess v if the outcome is 1.
(a) What is the probability that you guess right?
(b) Is there a measurement which is correct more often?
Yes
No
b- no
a) 0.853
a) What quantum state do you have to input in order to get output |00⟩ ?
b) What quantum state do you have to input in order to get output |11⟩ ?
(c) What quantum state do you have to input in order to get output 1/2�ã(|00⟩+|11⟩)?
Helppp
u are all cheaters on the exam and will be reported.
lolol at Anonymous
:D im glad someone has a sense of humour..
If the first qubit is in the state 2/sqrt(5)|0> - 1/sqrt(5)|1> and the second qubit is in the state 1/sqrt(2)|0> - i/sqrt(2)|1>, what is the state of the composite system?
To find the probability of guessing right in this scenario, we need to calculate the probability of getting the outcome 0 when measuring the given qubit in the standard basis.
Let's start with the state |u> = cos(pi/8) |0> + sin(pi/8) |1>. To measure this qubit in the standard basis, we need to find the probability amplitude for the outcome 0.
The probability amplitude for getting the outcome 0 is given by the coefficient of |0> in the state |u>. In this case, the coefficient is cos(pi/8), which means the probability amplitude is cos(pi/8). To find the probability of getting the outcome 0, we square the probability amplitude:
Probability of outcome 0 for state |u> = |cos(pi/8)|^2 = cos^2(pi/8).
Similarly, for the state |v> = cos(3pi/8) |0> + sin(3pi/8) |1>, the probability of getting the outcome 0 is given by the coefficient of |0> in the state |v|. In this case, the coefficient is cos(3pi/8), so the probability amplitude is cos(3pi/8). Squaring the probability amplitude gives us:
Probability of outcome 0 for state |v> = |cos(3pi/8)|^2 = cos^2(3pi/8).
Now, to find the probability of guessing right, we need to add up the probabilities of getting the outcome 0 for both states and then take the average:
Probability of guessing right = (Probability of outcome 0 for state |u> + Probability of outcome 0 for state |v>)/2
= (cos^2(pi/8) + cos^2(3pi/8))/2.
This gives us the answer to part (a).
To address part (b), we can compare the probabilities obtained in part (a) for guessing right to see if there is a measurement that is correct more often. If one of the probabilities is higher than the other, then we can say that there is a measurement that is correct more often. If the probabilities are equal, then there is no measurement that is correct more often.
So, to determine if there is a measurement that is correct more often, we compare the probabilities:
Probability of guessing right for state |u> = cos^2(pi/8)
Probability of guessing right for state |v> = cos^2(3pi/8)
By comparing these probabilities, we can determine if one is greater than the other or if they are equal.