The Bloch Sphere

Now that we know something about complex numbers in quantum mechanics, we need a representation different from our customary two-dimensional state space diagram. The angle of the state vector in a two-dimensional state space diagram shows only the different state amplitudes of the superposition state; it does not show information about the relative phases we discussed earlier.

The most common representation that includes the relative phase is called the Bloch sphere, named after Felix Bloch, a Nobel Laureate in physics, and a pioneer in the field of nuclear magnetic resonance. Those techniques eventually led to magnetic resonance imaging (MRI), one of the most important medical imaging techniques available to us today. Bloch was concerned with the nuclei of atoms and their magnetic and spin properties. In fact, the nucleus of a hydrogen atom (a proton) is an example of a qubit because it has spin-$½$. If you read more about qubits and their quantum states, you will eventually come across this pictorial representation: the Bloch sphere.