Post 5 — The Hopf Fibration and Global Phase

This entry introduces the Hopf fibration, which lifts Bloch sphere paths into SU(2), and shows how to visualize the accumulated global phase with the accompanying Python code.

Quantum states are rays: multiplying a state by a global phase $e^{i\phi}$ does not change its physical predictions. Thus, the true state space is complex projective space $\mathbb{C}P^1$, which is topologically the Bloch sphere $S^2$.

The Hopf fibration describes this structure:

\[ S^3 \longrightarrow S^2, \]

with fiber $S^1$ corresponding to global phase. Here $S^3$ is the unit sphere in $\mathbb{C}^2$, i.e. normalized qubit states.

As a state evolves around a closed loop on $S^2$, the lift in $S^3$ accumulates a total phase. Part of this is dynamical, depending on time evolution, and part is geometric (the Berry phase).

Tracking overlaps $\langle \psi_k | \psi_{k+1} \rangle$ along a discretized path gives the incremental phase changes.

The function accumulated_global_phase computes the cumulative phase along a state path. Example: the equatorial loop.

import numpy as np
import hottbloch as h
 
# Build equatorial loop
states, loop_points = h.loop_equator()
 
# Compute accumulated phase
phases = h.accumulated_global_phase(states)
 
print("First few phase values:", phases[:5])
print("Final accumulated phase:", phases[-1])

The final accumulated phase approximates the Berry phase up to integer multiples of $2\pi$.

To plot the phase accumulation along the loop:

python hottbloch.py --out ./hott_outputs --loop equator --theta 6.283185 --hopf

This generates a plot of the global phase vs. step index and saves it in `./hott_outputs`.

The Hopf fibration explains the hidden structure behind the Bloch sphere: each point on $S^2$ corresponds to an entire circle of quantum states differing only by global phase. By tracking accumulated global phase, we connect the geometric Berry phase to its origin in the fiber structure of $S^3 \to S^2$.