Thus, a qubit is described by a unit vector of a complex vector space of dimension 2.

$|\psi ⟩$ is a ket in Dirac notation; think of a column vector $X=\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]$.

$⟨\psi |$ is the corresponding bra , the dual vector; think of the line vector ${\bar{X}}^T=\left[\begin{array}{cc}\bar{\alpha }& \bar{\beta }\end{array}\right]$.

The notation takes on its full meaning when we note that $$\begin{array}{rl}{\overline{X_1}}^T{X}_2& =\overline{{\alpha }_1}{\alpha }_2+\overline{{\beta }_1}{\beta }_2\\ & =⟨{\psi }_1|{\psi }_2⟩\end{array}$$ is the usual inner product.

If we try to measure a qubit in state $|\psi ⟩$, using a device capable of distinguishing the two orthogonal states $|0⟩$ and $|1⟩$, the result is random! We'll get:$$\begin{array}{rl}& =\{\begin{array}{ll}0& \text{with probability }{\left|\alpha \right|}^2\\ 1& \text{with probability }{\left|\beta \right|}^2.\end{array}\end{array}$$

Moreover, after the measurement, the qubit state becomes: $$\begin{array}{rl}|\psi ⟩& =\{\begin{array}{ll}|0⟩& \text{if we read 0}\\ |1⟩& \text{if we read 1}.\end{array}\end{array}$$ This is another specificity of quantum mechanics: observation irreversibly disturbs the system.

If we let $$\begin{array}{rl}\alpha & ={\rho }_1{e}^{i{\phi }_1}\\ \beta & ={\rho }_2{e}^{i{\phi }_2}\end{array}$$ where ${\rho }_1,{\rho }_2\in {\mathbb{R}}^{+}$,we have: $$\begin{array}{rl}|\psi ⟩& ={\rho }_1{e}^{i{\phi }_1}|0⟩+{\rho }_2{e}^{i{\phi }_2}|1⟩\\ & =e^{i{\phi }_1}({\rho }_1|0⟩+{\rho }_2{e}^{i({\phi }_2-{\phi }_1)}|1⟩)\end{array}$$

Since the factor $e^{i{\phi }_1}$ (global phase) has no physical significance, we can assume ${\phi }_1=0$. So any qubit is uniquely represented by: $$\begin{array}{rl}|\psi ⟩& ={\rho }_1|0⟩+{\rho }_2{e}^{i({\phi }_2-{\phi }_1)}|1⟩\text{ with }{\rho }_1,{\rho }_2\in {\mathbb{R}}^{+}\text{ and }{\rho }_1^2+{\rho }_2^2=1\\ & =cos\frac{\theta }{2}|0⟩+\sin \frac{\theta }{2}{e}^{i\phi }|1⟩\end{array}$$ with $\theta \in [0,\pi ]$ and $\phi \in [0,2\pi [$.

The qubit $|\psi ⟩$ is associated with the point on the unit sphere of ${\mathbb{R}}^3$ with spherical coordinates $\theta ,\phi$; this bijection leads to the representation known as the Bloch sphere: