A Visual Exploration of the Qubit State

Two-dimensional quantum system

The qubit is a two-dimensional quantum system. Each dimension is denoted by a standard basis vector:

$|0\rangle = \begin{bmatrix}1\\0\end{bmatrix}$, in Python [1, 0] and

$|1\rangle = \begin{bmatrix}0\\1\end{bmatrix}$, in Python [0, 1].

The superposition of both dimensions represents the state of the qubit. This is the qubit state vector $|\psi\rangle$ (“psi”).

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

In Python, $|\psi\rangle$ is the array [alpha, beta].

However, $|\psi\rangle$ must be normalized by: $\alpha^2 + \beta^2 = 1$

Although normalizing the qubit state vector is not a difficult task, doing the math over and over again can be tedious.

Let’s explore an easier way to do this. First, we’ll look at a graphical representation of the qubit state $|\psi\rangle$ in the following figure “2-dimensional qubit system”.

In this representation, both dimensions reside at the vertical axis but in opposite directions. The system’s top and bottom correspond to the standard basis vectors $|0\rangle$ and $|1\rangle$, respectively.