HW1 - Problem 2

Problem 1: Different Bases

For the following states, if they are written in the standard basis, rewrite them in the Hadamard basis. If they are written in the Hadamard basis, rewrite them in the standard basis.

  1. \(\left|\psi_1\right> = -\left|0\right>\)
  2. \(\left|\psi_2\right> = \frac{i}{\sqrt{2}}\left|0\right> + \frac{1 - i}{2}\left|1\right>\)
  3. \(\left|\psi_3\right> = \frac{1}{\sqrt{3}}\left|+\right> - \frac{1+i}{\sqrt{3}}\left|-\right>\)
  4. \(\left|\psi_4\right> = \frac{1}{\sqrt{2}}\left|+\right> - \frac{1}{\sqrt{2}} \left|-\right>\)

Recall the \(\{\left|i\right>, \left|-i\right>\}\) basis where

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

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

Write the following states in the \(\{\left|i\right>, \left|-i\right>\}\) basis, that is as a weighted sum of \(\left|i\right>\) and \(\left|-i\right>\).

  1. \(\left|\psi_1\right> = \left|0\right>\)
  2. \(\left|\psi_2\right> = \left|1\right>\)
  3. \(\left|\psi_3\right> = \left|+\right>\)
  4. \(\left|\psi_4\right> = \left|-\right>\)
  5. \(\left|\psi_5\right> = \frac{\sqrt{3}}{2}\left|0\right> - \frac{1}{2}\left|1\right>\)
  6. \(\left|\psi_6\right> = \frac{\sqrt{3}}{2}\left|+\right> + \frac{1}{2}\left|-\right>\)