Statevector and its representation in the Bloch sphere

In quantum computing, the state of the qubit is probabilistic. Therefore, we can't represent a quantum coin as either heads or tails. Instead, it's usually a combination of both.

Statevector

A quantum statevector is a vector that shows the probability amplitudes of different basis quantum states comprising the overall state of the quantum object. For instance, if we describe the $\text{HEAD}$ state of the quantum coin as $\vert \text{HEAD}\rangle = \begin{pmatrix}1\\0\end{pmatrix}$and the $\text{TAIL}$ state of the coin as $\vert \text{TAIL}\rangle = \begin{pmatrix}0\\1\end{pmatrix}$, any other state of the coin (which is a superposition of these two) can be defined as:

Note: The probability amplitudes in quantum statevectors do not correspond to classical probability. Instead, they will need to be squared to obtain classical probabilities. For instance, in the statevector mentioned above, the probability of obtaining $\vert\text{HEAD}\rangle$ is $\alpha^2$and the probability of obtaining $\vert\text{TAIL}\rangle$is $\beta^2$.

In quantum computing, the state of a single qubit can be represented using the following statevector:

Bloch sphere

The Bloch sphere is a geometric representation of the state of the qubit. Named after the Nobel-winning physicist Felix Bloch, the Bloch sphere allows the qubit to be depicted as a point on the sphere.

A qubit is represented in the Bloch sphere by using the following formula:

Diagrammatically, a qubit is represented on the Bloch sphere like this:

Therefore, by varying the values of $\theta$and $\phi$, all possible states of a single qubit can be represented on the Bloch sphere.