Bloch Sphere

Visualize qubits on a Bloch sphere.

For a one-qubit system, a general pure state $∣ ψ ⟩$ can be written as a linear combination of $∣ 0 ⟩$ and $∣ 1 ⟩$ :

∣ ψ ⟩ = cos (\frac{θ}{2}) ∣ 0 ⟩ + e^{i ϕ} sin (\frac{θ}{2}) ∣ 1 ⟩,

where $0 \leq ϕ < 2 π$ and $0 \leq θ < π$ . When interpreted as the polar and azimuthal angles in spherical polar coordinates, the angles $ϕ$ and $θ$ uniquely identify a point on a unit sphere called the Bloch sphere. With the one-to-one correspondence between the points on the Bloch sphere and the pure state space of a one-qubit system, the Bloch sphere can be used to visualize any pure state of a one-qubit system.

Bloch Sphere provides an interactive way to visualize the pure one-qubit states on the Bloch sphere by manipulating the $ϕ$ and $θ$ angles as well as pseudo-rotating the sphere by clicking/touching and dragging.