two_by_two_manifold.py

import numpy as np
import numpy.linalg as LA


def flag_distance(mat1, mat2):
    n_columns = mat1.shape[1]
    running_metric = 0.0
    for i in range(n_columns):
        running_metric = LA.norm(np.outer(mat1[:,i], mat1[:,i].conj()) - np.outer(mat2[:,i], mat2[:,i].conj()), 'fro')**2
    return np.sqrt(running_metric)

def gen_2by2_unitary_matrix(a):
    ma = np.abs(a)
    theta = np.angle(a)
    cos = np.cos
    sin = np.sin
    return np.array([[cos(ma), np.exp(1j * theta) * np.sin(ma)],
                     [-np.exp(-1j * theta) * np.sin(ma), cos(ma)]])

a1 = np.random.rand() * np.pi / 2 * np.exp(1j * np.random.rand() * np.pi * 2)
a2 = np.random.rand() * np.pi / 2 * np.exp(1j * np.random.rand() * np.pi * 2)
A1 = gen_2by2_unitary_matrix(a1)
A2 = gen_2by2_unitary_matrix(a2)
print(flag_distance(A1, A2))
A11 = gen_2by2_unitary_matrix(np.abs(a1))
A21 = gen_2by2_unitary_matrix(np.abs(a2))
print(flag_distance(A11, A21))