Bloch Sphere Coordinates

The Bloch Sphere

Every pure single-qubit state can be represented as a point on the surface of the Bloch sphere — a unit sphere in 3D space.

Parametrization

Any normalized qubit state can be written as:

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

where:

• $\theta \in [0, \pi]$ is the polar angle (latitude from the north pole)
• $\phi \in [0, 2\pi)$ is the azimuthal angle (longitude)

Key Points on the Bloch Sphere

Extracting Angles from a State

Given a normalized state $[\alpha, \beta]$:

$\theta = 2\arccos(|\alpha|)$

$\phi = \arg(\beta) - \arg(\alpha) \pmod{2\pi}$

When $|\alpha| \approx 0$ (i.e., the state is near $|1\rangle$), $\alpha$ has no well-defined argument, so we set $\phi = 0$ by convention.

Implementation

import cmath, math

def bloch_angles(state):
    alpha = complex(state[0])
    beta = complex(state[1])
    theta = 2 * math.acos(min(abs(alpha), 1.0))
    if abs(alpha) < 1e-10:
        phi = 0.0
    else:
        phi = (cmath.phase(beta) - cmath.phase(alpha)) % (2 * math.pi)
    return (theta, phi)

Why the Bloch Sphere Matters

• Gate actions become geometric rotations: X is a $\pi$-rotation around the X axis, H is a $\pi$-rotation around the $(X+Z)/\sqrt{2}$ axis.
• Decoherence contracts the Bloch sphere inward — mixed states live inside the sphere.
• Visualization makes it easy to reason about the effect of gate sequences.

Your Task

Implement bloch_angles(state) that returns (theta, phi) for a normalized qubit state.