Qubit State Space

Let $\mathbb{C}^{2}$ be the complex coordinate space with $n=2$, which is a complex vector space by The Complex Coordinate Space is a Complex Vector Space, equipped with the standard inner product, which is an inner product on it by The Standard Inner Product Makes the Complex Coordinate Space an Inner Product Space.

The qubit state space is this complex inner product space. Its computational basis vectors are the elements

$$e_{1}=(1,0),\qquad e_{2}=(0,1)$$

of $\mathbb{C}^{2}$, also written $|0\rangle$ and $|1\rangle$ respectively. A qubit state vector is a unit vector of the qubit state space.