Post 4 — Homotopies Between Loops

This entry introduces homotopies as continuous deformations between loops on the Bloch sphere, with examples using the accompanying Python code.

A homotopy is a continuous deformation from one loop $\gamma_0$ to another loop $\gamma_1$. Formally, it is a function

\[ H : [0,1] \times [0,1] \to S^2, \]

such that

\[ H(0,t) = \gamma_0(t), \quad H(1,t) = \gamma_1(t). \]

On the Bloch sphere, this means smoothly sliding one path into another without cutting or breaking it.

Because $\pi_1(S^2) = 0$, all loops are contractible: they can be deformed to a point.

We implement homotopies numerically by interpolating each pair of points on two loops with the shortest geodesic on the sphere. For points $p,q \in S^2$, the interpolated point at fraction $s$ is

\[ \gamma_s = \frac{\sin((1-s)\theta)}{\sin \theta} \, p + \frac{\sin(s\theta)}{\sin \theta} \, q, \]

where $\theta = \arccos(p \cdot q)$.

The function homotopy_between_loops constructs these intermediate loops. Example: deforming the meridian loop into the equatorial loop.

import numpy as np
import hottbloch as h
 
# Get canonical loops
states_a, loop_a = h.loop_meridian()
states_b, loop_b = h.loop_equator()
 
# Build homotopy (5 intermediate slices)
s_values = np.linspace(0, 1, 5)
slices = h.homotopy_between_loops(loop_a, loop_b, s_values)
 
print("Number of slices:", len(slices))
print("Shape of one slice:", slices[0].shape)

Output: five slices, each a discretized loop with shape `(401, 3)`.

To generate homotopy plots:

python hottbloch.py --out ./hott_outputs --loop equator --theta 6.283185 --slices 5

This produces a series of images in `./hott_outputs`, each showing a loop at a different interpolation value $s$.

Homotopies reveal that all loops on the Bloch sphere can be continuously deformed into one another, reflecting the trivial fundamental group of $S^2$. In the next post we extend this picture by considering the Hopf fibration and the role of global phase.