Let the \reftext{def:qubit-state-space-2026a}{qubit state space} be as in that definition, with computational basis vectors $e_{1}$ and $e_{2}$, \reftext{def:standard-inner-product-cn-2026a}{standard inner product} $\langle\cdot,\cdot\rangle$ and \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$. Let $u=(u_{1},u_{2})$ be an element of $\mathbb{C}^{2}$, and let $|u_{k}|$ denote the \reftext{def:complex-modulus-2026a}{modulus} of $u_{k}$. Then the following hold.

\textbf{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 \reftext{def:unit-vector-2026a}{unit vectors}.

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

\textbf{3. (Expansion in the computational basis)} $u=u_{1}e_{1}+u_{2}e_{2}$, where the operations are those of the \reftext{def:complex-coordinate-space-cn-2026a}{complex coordinate space}; and if $\alpha,\beta$ are complex numbers with $u=\alpha e_{1}+\beta e_{2}$, then $\alpha=u_{1}$ and $\beta=u_{2}$.

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