5. The Qubit: The Bloch Sphere

The Bloch Sphere

The Bloch Sphere is not a physical object; it is the essential geometric visualization tool for the state space of a single qubit ($\mathbb{C}^2$). It ties together everything you’ve learned about normalization, superposition, and phase.

1. The Mapping

A state vector is a 2-dimensional complex vector, but the Bloch Sphere allows us to map it uniquely onto the surface of a 3-dimensional real sphere (the unit sphere).

• Poles (The Classical Basis): The North Pole is the state $|0\rangle$. The South Pole is the state $|1\rangle$.
• Surface (Superposition): Every single point on the surface represents a valid pure superposition state $|\psi\rangle$.
• Vector Length (Normalization): The vector from the center to any point on the surface has a length of 1, satisfying the normalization condition $\langle \psi | \psi \rangle = 1$.

2. The Spherical Coordinates

The position of any pure state $|\psi\rangle$ on the sphere is defined by two real angles, $\theta$ (polar) and $\phi$ (azimuthal), which directly map to probability and phase:

$$|\psi\rangle = \cos(\frac{\theta}{2})|0\rangle + e^{i\phi}\sin(\frac{\theta}{2})|1\rangle$$

• Angle $\theta$ (Polar, $0 \le \theta \le \pi$): This angle, measured from the Z-axis (North Pole), controls the probability bias. $$P(|0\rangle) = \cos^2(\theta/2)$$ $$P(|1\rangle) = \sin^2(\theta/2)$$
• Angle $\phi$ (Azimuthal, $0 \le \phi < 2\pi$): This angle, measured from the X-axis, controls the crucial relative phase $e^{i\phi}$ between the $|0\rangle$ and $|1\rangle$ components.

3. Gates as Rotations

Unitary gates (Postulate 2) are represented as rotations of the state vector on the sphere.

• The Pauli-X gate is a $180^\circ$ rotation about the X-axis.
• The Pauli-Z gate is a rotation about the Z-axis (changing $\phi$).
• The Hadamard gate is a rotation that maps the Z-axis states ($|0\rangle, |1\rangle$) to the X-axis states ($|+\rangle, |-\rangle$).

Your Task: Mapping the Superposition

The balanced superposition state $|+\rangle$ is created by applying the Hadamard gate to $|0\rangle$. Its vector form is:

$$|+\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$$

Find the corresponding spherical coordinates $(\theta, \phi)$ for this state.

• Determine $\theta$: What value for $\theta$ makes $\cos(\theta/2) = 1/\sqrt{2}$?
• Determine $\phi$: Since the coefficient of $|1\rangle$ is real, what must the phase factor $e^{i\phi}$ equal?