HW1 - Problem 4

Problem 4: Linear algebra with Qubits

1. Which of the following matrices are stochastic and/or unitary?

\(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \; B = \begin{bmatrix} 0 & 1/3 \\ 1 & 2/3 \end{bmatrix}, \; C = \begin{bmatrix} 2 & -1 \\ 1/2 & 1/2 \end{bmatrix}, \; D = \frac{1}{\sqrt{2}}\begin{bmatrix} i & 1 \\ 1 & -i \end{bmatrix}\)

2. Define the two qubit states \(\left|\phi\right>\) and \(\left|\psi\right>\) as:

\(\left|\phi\right> = \frac{\left|0\right> + 3i\left|1\right>}{\sqrt{10}} \text{ and } \left|\psi\right> = \frac{2\left|0\right> - i\left|1\right>}{\sqrt{5}}.\)

What is \(\left<\phi | \psi\right>\)?

What about \(\left<\psi | \phi\right>\)

3. A qubit state should be normalized, meaning the norm should be 1: \(|| \left|\phi\right> || = 1\). The following state \(\left|\phi\right>\) is not normalized. Find a constant \(N\) such that \(\frac{1}{N}\left|\phi\right>\) is normalized.

\(\left|\phi\right> = 3\left|0\right> - i\left|1\right>\)

4. Define

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

Prove that the pair of states \(\left|i\right>\), \(\left|-i\right>\) form an orthonormal basis.