作业三：测量的Bloch球表示「答案」

Bloch-HW-3.ipynb

第一步：测量算符的测量基

from qutip import *
from scipy.linalg import *
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
import matplotlib.pyplot as plt


def getBasisState(O):
    # 初始化态矢量
    psi_0 = np.matrix([[1.0000], [0.0000]], dtype=complex)
    psi_1 = np.matrix([[0.0000], [1.0000]], dtype=complex)

    # 对角化算符
    eig_v, eig_k = eig(O)
    eig_v = np.real(eig_v)

    # 对角化后的基矢量
    m_base = [np.matrix(eig_k[:, i]).T for i in range(len(eig_v))]

    # 对角化后基矢量的顺序
    order = np.argsort(eig_v)[::-1]
    m_base = [m_base[i] for i in order]

    return m_base


def Meas(psi, O):
    m_base = getBasisState(O)

    # 计算投影概率
    p0 = (abs(m_base[0].H * psi) ** 2)[0, 0]
    p1 = (abs(m_base[1].H * psi) ** 2)[0, 0]

    return [p0, p1]

第二步：测量的 Bloch 球表示

未连线

def plotLine(point1, point2, config):
    plt.plot([point1[1], point2[1]], [-point1[0], -point2[0]], [point1[2], point2[2]], config)


fig = plt.figure(figsize=(6, 6))
axes = Axes3D(fig, auto_add_to_figure=False)
fig.add_axes(axes)
sphere = Bloch(axes=axes)

# coordinates of given basis states
coord0 = np.array([0.0995, 0.0000, 0.9950], dtype=float)
coord1 = np.array([-0.0995, -0.0000, -0.9950], dtype=float)
psi_ex_coord = np.array([0.9518, 0.0000, 0.3068], dtype=float)

# probabilities are:
p = [0.3, 0.7]
coord_projection = p[0] * coord1 + p[1] * coord0

sphere.add_points(coord0)
sphere.add_points(coord1)
sphere.add_points(psi_ex_coord)
sphere.add_points(coord_projection)

# measurement operator
H = np.matrix([[2 ** (-0.5), 2 ** (-0.5)], [2 ** (-0.5), -2 ** (-0.5)]], dtype=complex)
psi = np.matrix([[3 ** (-0.5)], [(2 / 3) ** (0.5) * 1j]], dtype=complex)

# add the coordinates of the projection of psi onto the measurement basis
meas_probs = Meas(psi, H)
psi_proj_coords = meas_probs[0] * coord0 + meas_probs[1] * coord1
sphere.add_points(psi_proj_coords)

sphere.make_sphere()

# plot the lines connecting the points
plotLine(coord_projection, psi_ex_coord, '-k')
plotLine(coord_projection, coord0, '-b')
plotLine(coord_projection, coord1, '-r')
# plot the line corresponding to the probabilities of measurement
plotLine(psi_ex_coord, psi_proj_coords, '-g')

plt.show()

连线版本

from qutip import *
from scipy.linalg import *
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
import matplotlib.pyplot as plt


def getBasisState(O):
    # 初始化态矢量
    psi_0 = np.matrix([[1.0000], [0.0000]], dtype=complex)
    psi_1 = np.matrix([[0.0000], [1.0000]], dtype=complex)

    # 对角化算符
    eig_v, eig_k = eig(O)
    eig_v = np.real(eig_v)

    # 对角化后的基矢量
    m_base = [np.matrix(eig_k[:, i]).T for i in range(len(eig_v))]

    # 对角化后基矢量的顺序
    order = np.argsort(eig_v)[::-1]
    m_base = [m_base[i] for i in order]

    return m_base


def Meas(psi, O):
    m_base = getBasisState(O)

    # 计算投影概率
    p0 = (abs(m_base[0].H * psi) ** 2)[0, 0]
    p1 = (abs(m_base[1].H * psi) ** 2)[0, 0]

    return [p0, p1]


def plotLine(point1, point2, config):
    xs = [point1[1], point2[1]]
    ys = [-point1[0], -point2[0]]
    zs = [point1[2], point2[2]]
    axes.plot(xs, ys, zs, config)

fig = plt.figure(figsize=(6, 6))
axes = Axes3D(fig, auto_add_to_figure=False)
fig.add_axes(axes)
sphere = Bloch(axes=axes)

# coordinates of given basis states
coord0 = np.array([0.0995, 0.0000, 0.9950], dtype=float)
coord1 = np.array([-0.0995, -0.0000, -0.9950], dtype=float)
psi_ex_coord = np.array([0.9518, 0.0000, 0.3068], dtype=float)

# probabilities are:
p = [0.3, 0.7]
coord_projection = p[0] * coord1 + p[1] * coord0

sphere.add_points(coord0)
sphere.add_points(coord1)
sphere.add_points(psi_ex_coord)
sphere.add_points(coord_projection)

# measurement operator
H = np.matrix([[2 ** (-0.5), 2 ** (-0.5)], [2 ** (-0.5), -2 ** (-0.5)]], dtype=complex)
psi = np.matrix([[3 ** (-0.5)], [(2 / 3) ** (0.5) * 1j]], dtype=complex)

# add the coordinates of the projection of psi onto the measurement basis
meas_probs = Meas(psi, H)
psi_proj_coords = meas_probs[0] * coord0 + meas_probs[1] * coord1
sphere.add_points(psi_proj_coords)

sphere.make_sphere()

# plot the lines connecting the points
plotLine(coord_projection, psi_ex_coord, '-k')
plotLine(coord_projection, coord0, '-b')
plotLine(coord_projection, coord1, '-r')
# plot the line corresponding to the probabilities of measurement
plotLine(psi_ex_coord, psi_proj_coords, '-g')

plt.show()

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