7.5 The Bloch sphere

We describe the state of a qubit by a vector Bloch sphere

in C² with r¹ and r² nonnegative 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 that it’s the relative phase of φ₂ – φ₁ that is significant and not the individual phases φ₁ and φ₂. This is a physical condition and means we can take a to be real.

We also saw that we could represent a quantum state as

We do this via a nonlinear projection and a change of coordinates and get a point on the surface of the Bloch sphere, shown in Figure 7.7.

Figure 7.7: 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-plane.

The nonlinear...