For Wednesday, October 30th, 2024

  1. Happy Almost Hallowe’en!
  2. There is a test on Wednesday.  Check out the “test” tab for some practice problems & review materials.  Check discord for a review session schedule.
  3. Today we did two qubit systems, so here are a few problems in that direction.
  4. Consider the state $\frac{1}{\sqrt{2}}|01\rangle + \frac{1}{\sqrt{2}}|10\rangle$.
    1. What is the probability of measuring a 0 in the first bit?
    2. What is the probability of measuring a 0 in the second bit?
    3. What is the probability of measuring a 0 in both bits?  (I.e. after measuring both bits, what’s the probability you obtained 00?)
  5. For each of the following, determine if it is entangled or not:
    1. $|111\rangle$
    2. $\frac{1}{\sqrt{2}}|10\rangle + \frac{1}{\sqrt{2}}|11\rangle$
    3. $\frac{1}{\sqrt{2}}|110\rangle + \frac{1}{\sqrt{2}}|111\rangle$
    4. $\frac{1}{\sqrt{2}}|010\rangle + \frac{1}{\sqrt{2}}|111\rangle$
  6. In the state  $\frac{1}{\sqrt{2}}|010\rangle + \frac{1}{\sqrt{2}}|111\rangle$, what is the probability of measuring a 0 in the first qubit?  middle qubit?  last qubit?
  7. Measure the following states in the first qubit and report the possible outcome bits, outcome states and probabilities.  Don’t forget to normalize.
    1. the Bell state
    2. $\frac{1}{\sqrt{14}}|01\rangle + \frac{2i}{\sqrt{14}}|10\rangle +  \frac{3}{\sqrt{14}}|11 \rangle$