## The More You Know: Bloch Sphere

The Bloch Sphere is a useful geometrical representation of the state of a Qubit or any other two level quantum system. As discussed in this article an arbitrary pure state $|\psi\rangle$ of a Qubit can be expressed in the computational basis as

$|\psi\rangle=\alpha|0\rangle + \beta|1\rangle$,

where $\alpha$ and $\beta$ are complex number such that $|\alpha|^2+|\beta|^2=1$. Since an overall phase has no observable effects, $\alpha$ or $\beta$ can be chosen  as real numbers and $|\psi\rangle$ can written as

$|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+\exp^{i\phi}\sin\frac{\theta}{2}|1\rangle$,

where $\theta$ and $\phi$ $\in \mathbb{R}$. The parameters $0\leq\theta\leq\pi$ and $0\leq\phi\leq2\pi$ define a point on the $3D$ sphere called the Bloch Sphere (Fig. 1).

Fig. 1: Bloch Sphere representation of the State of a Qubit.

The points in the surface of the sphere represent pure states of the qubit, with $|0\rangle$ as the North pole and $|1\rangle$ as the South pole, and the equator states of the form $|\psi\rangle=|0\rangle+\exp^{i\phi}|1\rangle$. Additionally any mixed state of a qubit can be represented as a point inside the sphere, with maximally mixed states at the center of the Bloch Sphere.

This representation is useful in Quantum Computation to visualize the effect of single qubit gates.