7.5 The Bloch sphere

We describe the state of a qubit by a vector

a |0⟩ + b |1⟩ = r₁ e^{ϕ₁ i} |0⟩ + r₂ e^{ϕ₂ i} |1⟩

in C² with r₁ and r₂ non-negative numbers in R.

The magnitudes r₁ and r₂ are related by r₁² + r₂² = 1. This is a mathematical condition. We saw in section 7.3 ₂ − ϕ₁ that is significant and not the individual phases ϕ₁ and ϕ₂. This is a physical condition and it also means we can take a to be real.

We also saw that we could represent a quantum state as

|ψ⟩ = cos(θ/2) |0⟩ + sin(θ/2) e^{ϕ i} |1⟩.

We do this via a non-linear projection and a change of coordinates, and get a point on the surface of the Bloch sphere.

The two angles have the ranges 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. θ is measured from the positive z axis and ϕ from the positive x axis in the xy...