# 作业四：混态

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## 作业四：混态

``````###### -- QUANTA -- #######
# Author: Y. Liu, W. Shi  #
# Data: 2022-09-17        #
###########################

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

## 第一步：混态的 Bloch 坐标

``````def getCoordFromRho(Rho):
coord = np.array([0.0000, 0.0000, 1.0000], dtype = float)
#######################################
#todo: complete this function.

#######################################
return coord

def getRhoFromCoord(coord):
Rho = np.array([[1.0000 + 0.0000j, 0.0000 + 0.0000j], [0.0000 + 0.0000j, 0.0000 + 0.0000j]], dtype = complex)
#######################################
#todo: complete this function.

#######################################
return Rho
``````

``````#don't modify the code in this block
n_ck = 3
rho_ck = np.array([[[ 0.5000 + 0.0000j,   0.0000 + 0.0000j],
[ 0.0000 + 0.0000j,   0.5000 + 0.0000j]],
[[ 0.7500 + 0.0000j,   0.2500 - 0.2500j],
[ 0.2500 + 0.2500j,   0.2500 + 0.0000j]],
[[ 0.5000 + 0.0000j,   0.3535 + 0.3535j],
[ 0.3535 - 0.3535j,   0.5000 + 0.0000j]]], dtype = complex)
coord_ck = np.array([[ 0.0000,  0.0000,  0.0000],
[ 0.5000,  0.5000,  0.5000],
[ 0.7071, -0.7071,  0.0000]], dtype = float)

def checkCoord2Rho():
print('Checking converting coordinate to state vector...')
err = [np.sum(abs(getRhoFromCoord(coord_ck[c]) - rho_ck[c])) for c in range(n_ck)]
if np.sum(err) < 0.01:
print('Pass!')
else:

def checkRho2Coord():
print('Checking converting state vector to coordinate...')
err = [np.sum(abs(getCoordFromRho(rho_ck[c]) - coord_ck[c])) for c in range(n_ck)]
if np.sum(err) < 0.01:
print('Pass!')
else:

checkCoord2Rho()
checkRho2Coord()
``````

## 第二步 混态的绘制

``````Rho_to_add = np.array([[[ 0.5000 + 0.0000j,   0.0000 + 0.0000j],
[ 0.0000 + 0.0000j,   0.5000 + 0.0000j]],
[[ 0.7500 + 0.0000j,   0.2500 - 0.2500j],
[ 0.2500 + 0.2500j,   0.2500 + 0.0000j]],
[[ 0.5000 + 0.0000j,   0.3535 + 0.3535j],
[ 0.3535 - 0.3535j,   0.5000 + 0.0000j]]], dtype = complex)

fig = plt.figure(figsize=(6,6))
sphere = Bloch(axes = axes)

######################
#todo:
#converting the 3 states in Rho_to_add into coordinates, and ploting them onto the Bloch sphere

######################
sphere.make_sphere()

#when using CMD to run this script, the Bloch sphere can be shown in an independent view by the following code.
#fig.show()
``````

## 第三步 混态的幺正变换

``````def evo_Mix(U, Rho):
Rho = np.matrix(Rho)
U = np.matrix(U)
Rho_evo = np.matrix([[1.0000, 0.0000], [0.0000, 0.0000]], dtype = complex)
#######################################
#todo: complete this function.

#######################################
return Rho_evo
``````

``````U_ck = np.array([[0.1245 + 0.4403j, -0.5523 + 0.6968j],
[0.2248 + 0.8603j,  0.2914 - 0.3528j]], dtype = complex)

Rho_ck = np.array([[ 0.5000 + 0.0000j,   0.3535 + 0.3535j],
[ 0.3535 - 0.3535j,   0.5000 + 0.0000j]], dtype = complex)

Rho_ans = np.array([[ 0.9015 + 0.0000j,   0.2884 + 0.0741j],
[ 0.2884 - 0.0741j,   0.0984 + 0.0000j]], dtype = complex)

Rho_out = evo_Mix(U_ck, Rho_ck)
err = np.sum(abs(Rho_out - Rho_ans))

if np.sum(err) < 0.01:
print('Pass!')
else:
``````

## 第四步 混态的测量

``````def getBasisState(O):
psi_0 = np.matrix([[1.0000],[0.0000]], dtype = complex)
psi_1 = np.matrix([[0.0000],[1.0000]], dtype = complex)
#######################################
#todo: no difference with the funciton in HW-3. Using that code directly.

#######################################
# order: eigenvalue(psi_0) > eivenvalue(psi_1)
return [psi_0, psi_1]

def Meas_Mix(Rho, O):
m_base = getBasisState(O)
p0 = 0.5000
p1 = 0.5000
#######################################
#todo: modify the following code to complete this function. The initial values are assigned manually

#######################################
return [p0, p1]
``````

``````H_ck = np.matrix([[0.0000 + 0.0000j,  1.0000 + 0.0000j],
[1.0000 + 0.0000j,  0.0000 + 0.0000j]], dtype = complex)
rho_ck = np.matrix([[ 0.9015 + 0.0000j,   0.2884 + 0.0741j],
[ 0.2884 - 0.0741j,   0.0984 + 0.0000j]], dtype = complex)

def checkProb():
print("Check the obtained probabilities...")

p = Meas_Mix(rho_ck, H_ck)

err = abs(p[0] - 0.7883) + abs(p[1] - 0.2116)
if err < 0.01:
print('Pass!')
else:
return

checkProb()
``````

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