HW1 - Problem 3

Problem 3: Qubit States

Consider a qubit in the following state:

\(\left|\psi\right> := \frac{i}{\sqrt{2}} \left|0\right> - \frac{1}{\sqrt{2}} \left|1\right>\)

  1. Verify that \(\left|\psi\right>\) is indeed a quantum state.
  2. If we measure \(\left|\psi\right>\), what is the probability that we observe \(\left|0\right>\)?
  3. Suppose we did observe \(\left| 0 \right>\). What is the state of the qubit after the measurement?
  4. Find a state that is orthogonal to \(\left|\psi\right>\).

Let’s try one more time with another quantum state! Practice makes perfect.

Consider the following quantum state

\(\left|\psi\right> := \left( \frac{\sqrt{3}}{4} + i \frac{1}{4} \right) \left|0\right> + \left(- \frac{\sqrt{3}}{4} + i \frac{3}{4}\right) \left|1\right>.\)

  1. Verify that \(\ket{\psi}\) is a quantum state.
  2. What is the probability that we measure \(\ket{0}\) if we measure in the standard basis? What is the state after the measurement?
  3. Find a state orthogonal to \(\ket{\psi}\).
  4. Consider the state \(e^{i \pi / 4}\ket{\psi}\). That is, multiply the two amplitudes by \(e^{i \pi / 4}\). What is the probability that we measure \(\ket{0}\) if we measure in the standard basis? \end{enumerate}