Bloch Sphere

A general pure state of a Qubit is

∣ψ⟩=cos⁡θ2∣0⟩+eiϕsin⁡θ2∣1⟩θ∈[0,π]ϕ∈[0,2π]\left\lvert \psi \right\rangle=\cos\frac{\theta}{2}\left\lvert 0 \right\rangle+e^{i\phi}\sin\frac{\theta}{2}\left\lvert 1 \right\rangle\quad \theta\in[0,\pi]\quad \phi\in[0,2\pi]

Where the orthogonal states are antipodal (North pole is ∣0⟩\left\lvert 0 \right\rangle and South pole is ∣1⟩\left\lvert 1 \right\rangle). The equator contains equal superpositions like ∣+⟩=12(∣0⟩+∣1⟩)\left\lvert + \right\rangle=\frac{1}{\sqrt{2}}(\left\lvert 0 \right\rangle+\left\lvert 1 \right\rangle) and ∣−⟩=12(∣0⟩−∣1⟩)\left\lvert - \right\rangle=\frac{1}{\sqrt{2}}(\left\lvert 0 \right\rangle-\left\lvert 1 \right\rangle).

A state can be on the inside of a Bloch sphere which means its a mixed state. If a state is on the surface of the Bloch sphere, it is a pure state.

A rotation by angle θ\theta around the axis n^\hat{n} is given by

Rn^(θ)=e−iθ n^⋅σ⃗/2=cos⁡(θ2)I−isin⁡(θ2)(nxσx+nyσy+nzσz)R_{\hat{n}}(\theta) = e^{-i\theta\,\hat{n}\cdot\vec{\sigma}/2} = \cos\left(\tfrac{\theta}{2}\right)I - i\sin\left(\tfrac{\theta}{2}\right)(n_x\sigma_x + n_y\sigma_y + n_z\sigma_z)

Where

Rx(θ)=(cos⁡θ2−isin⁡θ2−isin⁡θ2cos⁡θ2)R_x(\theta) = \begin{pmatrix} \cos\tfrac{\theta}{2} & -i\sin\tfrac{\theta}{2} \\ -i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{pmatrix} Ry(θ)=(cos⁡θ2−sin⁡θ2sin⁡θ2cos⁡θ2)R_y(\theta) = \begin{pmatrix} \cos\tfrac{\theta}{2} & -\sin\tfrac{\theta}{2} \\ \sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{pmatrix} Rz(θ)=(e−iθ/200eiθ/2)R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}