Qubits

Representation

A qubit, like a bit, can take on one of two values when measured.

It is represented by two complex numbers, usually in a $2 \times 1$ matrix, often called a ket.

Basis

$ |0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \qquad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $

$ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle $

As matrices

$ |\psi\rangle = \alpha \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0 \\ 1 \end{bmatrix} $

$ |\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} $

$ |\alpha|^2 + |\beta|^2 = 1 $

Probability Amplitudes

$ |\alpha|^2 + |\beta|^2 = 1 $

$ \alpha = \alpha_0 + \alpha_1 i \qquad \beta = \beta_0 + \beta_1 i \qquad \alpha_0,\alpha_1,\beta_0,\beta_1 \in \mathbb{R} $

$ |\alpha| = \sqrt{\alpha_0^2 + \alpha_1^2} $

$ |\alpha|^2 = \alpha_0^2 + \alpha_1^2 $

Note:

$ \alpha^* = \alpha_0 - \alpha_1 i $

$ \alpha \alpha^* = (\alpha_0 + \alpha_1 i)(\alpha_0 - \alpha_1 i) = \alpha_0^2 + \alpha_1^2 = |\alpha|^2 $

$ |\alpha|^2 = \alpha \alpha^* = \alpha^* \alpha $

Examples

$ |\psi_1\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} \qquad \left|\frac{1}{\sqrt{2}}\right|^2 = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} $

$ |\psi_2\rangle = \begin{bmatrix} \frac{i}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} \qquad \left|\frac{i}{\sqrt{2}}\right|^2 = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} $

$ |\psi_3\rangle = \begin{bmatrix} \frac{-i}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix} \qquad \left|\frac{-i}{\sqrt{2}}\right|^2 = \left|\frac{-1}{\sqrt{2}}\right|^2 = \frac{1}{2} $