Chapter 4 in-class problems - Solutions

Using what you learned in Lab, answer questions 4.7, 4.8, 4.10, 4.11, 4.12, and 4.13



In [1]:

    
from numpy import sin,cos,sqrt,pi
from qutip import *

Remember, these states are represented in the HV basis:



In [2]:

    
H = Qobj([[1],[0]])
V = Qobj([[0],[1]])
P45 = Qobj([[1/sqrt(2)],[1/sqrt(2)]])
M45 = Qobj([[1/sqrt(2)],[-1/sqrt(2)]])
R = Qobj([[1/sqrt(2)],[-1j/sqrt(2)]])
L = Qobj([[1/sqrt(2)],[1j/sqrt(2)]])

The sim_transform function creates the matrix $\bar{\mathbf{S}}$ that can convert from one basis to another. As an example, it will create the tranform matrix to convert from HV to ±45 if you run:

Shv45 = sim_transform(H,V,P45,M45)    #  creates the matrix Shv45

Then you can convert a ket from HV to ±45 by applying the Shv45 matrix:

Shv45*H    #  will convert H from the HV basis to the ±45 basis

To convert operators, you have to sandwich the operator between $\bar{\mathbf{S}}$ and $\bar{\mathbf{S}}^\dagger$:

Shv45*Ph*Shv45.dag()     #  converts Ph from HV basis to the ±45 basis.



In [3]:

    
def sim_transform(o_basis1, o_basis2, n_basis1, n_basis2):
    a = n_basis1.dag()*o_basis1
    b = n_basis1.dag()*o_basis2
    c = n_basis2.dag()*o_basis1
    d = n_basis2.dag()*o_basis2
    return Qobj([[a.data[0,0],b.data[0,0]],[c.data[0,0],d.data[0,0]]])

4.11: Express $\hat{R}_p(\theta)$ in ±45 basis



In [4]:

    
def Rp(theta):
    return Qobj([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]).tidyup()



In [5]:

    
Shv45 = sim_transform(H,V,P45,M45)

4.12:



In [6]:

    
Rp45 = Shv45*Rp(pi/4)*Shv45.dag()



In [7]:

    
Rp45*Shv45*P45 == Shv45*V   # convert P45 to the ±45 basis









    Out[7]:





True



In [8]:

    
Rp45* Qobj([[1],[0]])









    Out[8]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.707\\-0.707\\\end{array}\right)\end{equation*}



In [9]:

    
ShvLR = sim_transform(H,V,L,R)



In [10]:

    
ShvLR*Rp(pi/4)*ShvLR.dag()









    Out[10]:





Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.707-0.707j) & 0.0\\0.0 & (0.707+0.707j)\\\end{array}\right)\end{equation*}



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