The Computational Basis and the State Vectors of a Qubit

Let the qubit state space be as in that definition, with computational basis vectors $e_{1}$ and $e_{2}$, standard inner product $\langle\cdot,\cdot\rangle$ and induced norm $\lVert\cdot\rVert$. Let $u=(u_{1},u_{2})$ be an element of $\mathbb{C}^{2}$, and let $|u_{k}|$ denote the modulus of $u_{k}$. Then the following hold.

1. (Orthonormality) $\langle e_{1},e_{1}\rangle=\langle e_{2},e_{2}\rangle=1$ and $\langle e_{1},e_{2}\rangle=\langle e_{2},e_{1}\rangle=0$. In particular $e_{1}$ and $e_{2}$ are unit vectors.

2. (Components as inner products) $\langle e_{1},u\rangle=u_{1}$ and $\langle e_{2},u\rangle=u_{2}$.

3. (Expansion in the computational basis) $u=u_{1}e_{1}+u_{2}e_{2}$, where the operations are those of the complex coordinate space; and if $\alpha,\beta$ are complex numbers with $u=\alpha e_{1}+\beta e_{2}$, then $\alpha=u_{1}$ and $\beta=u_{2}$.

4. (State vectors) $\lVert u\rVert^{2}=|u_{1}|^{2}+|u_{2}|^{2}$, and $u$ is a qubit state vector if and only if $|u_{1}|^{2}+|u_{2}|^{2}=1$.