

In [4]:

    
from numpy import sqrt
from qutip import *



In [5]:

    
H = Qobj([[1],[0]])
V = Qobj([[0],[1]])

Example 8.A.1:



In [6]:

    
psi = 1/sqrt(2)*tensor(H,H) + 1/sqrt(2)*tensor(V,V)
psi









    Out[6]:





Quantum object: dims = [[2, 2], [1, 1]], shape = [4, 1], type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.707\\0.0\\0.0\\0.707\\\end{array}\right)\end{equation*}



In [7]:

    
rho_ent = psi*psi.dag()



In [8]:

    
(rho_ent*rho_ent).tr()









    Out[8]:





0.9999999999999996



In [9]:

    
rho_mix = 0.5*tensor(H,H)*tensor(H,H).dag() + 0.5*tensor(V,V)*tensor(V,V).dag()



In [10]:

    
(rho_mix*rho_mix).tr()









    Out[10]:





0.5

Example 8.A.2

Remember the 45 states:



In [11]:

    
P45 = Qobj([[1/sqrt(2)],[1/sqrt(2)]])
M45 = Qobj([[1/sqrt(2)],[-1/sqrt(2)]])

Create the projection operator for $|+45,+45\rangle$



In [12]:

    
Proj_4545 = tensor(P45,P45)*tensor(P45,P45).dag()



In [13]:

    
(Proj_4545*rho_mix).tr()









    Out[13]:





(0.2499999999999999+0j)

Create projection operator for $|+45\rangle_i$



In [14]:

    
Proj_45i = tensor(qeye(2),P45)*tensor(qeye(2),P45).dag()



In [15]:

    
(Proj_45i*rho_mix).tr()









    Out[15]:





(0.4999999999999999+0j)



In [16]:

    
0.25/0.5









    Out[16]:





0.5

Extend the example for the pure (superposition) state



In [17]:

    
(Proj_4545*rho_ent).tr() / (Proj_45i*rho_ent).tr()









    Out[17]:





(0.9999999999999998+0j)

The photons are entanlged and therefore show perfect correlation even in the +45 measurements.