Lab 4 - Measurements

First our standard definitions:



In [72]:

    
import matplotlib.pyplot as plt
from numpy import sqrt,pi,cos,sin,arange,random,exp
from qutip import *



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)]])



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]]])



In [4]:

    
def Delta(state, op):
    """Calculate std. dev. of an observable in a given state"""
    eO2 = state.dag()*op*op*state
    eO = state.dag()*op*state
    return sqrt(eO2.data[0,0] - (eO.data[0,0])**2)

Q: Define the $\hat{\mathscr{P}}_{HV}$ operator



In [5]:

    
Phv = H*H.dag() - V*V.dag()
Phv









    Out[5]:





Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0\\0.0 & -1.0\\\end{array}\right)\end{equation*}

Q: What is the expectation value $\langle \hat{\mathscr{P}}_{HV}\rangle$ for state $|\psi\rangle = \frac{1}{\sqrt{5}}|H\rangle + \frac{2}{\sqrt{5}}|V\rangle$? Interpret this result given the amplitudes in the state.



In [6]:

    
psi = 1/sqrt(5)*H + 2/sqrt(5)*V



In [7]:

    
psi.dag()*Phv*psi









    Out[7]:





Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}-0.600\\\end{array}\right)\end{equation*}

Q: What is the variance of $\mathscr{P}_{HV}$?



In [8]:

    
psi.dag()*Phv*Phv*psi









    Out[8]:





Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.000\\\end{array}\right)\end{equation*}



In [9]:

    
1.0 - (-0.6)**2









    Out[9]:





0.64

Ex: Use the random function to generate a mock data set for the state $|\psi\rangle$.

random.choice([1,-1],size=10,p=[0.2,0.8])

gives a list of 10 numbers, either 1 or -1 with the associated probability p:



In [43]:

    
data = random.choice([1, -1],size=20,p=[0.2,0.8])



In [44]:

    
data.mean()









    Out[44]:





-0.69999999999999996



In [45]:

    
data.var()









    Out[45]:





0.51000000000000001

Q: Verify the mean and variance of the mock data set match your QM predictions. How big does the set need to be for you to get ±5% agreement?



In [62]:

    
data = random.choice([1, -1],size=10000,p=[0.2,0.8])



In [63]:

    
data.mean()









    Out[63]:





-0.59840000000000004



In [64]:

    
data.var()









    Out[64]:





0.64191744000000017

10,000 does pretty well for getting to the predictions. "There is no substitute for an adequate sample size."

Q: Answer problems 5.11, 5.12, 5.13, 5.14, 5.17, 5.18, 5.19 from the textbook. These are an opportunity to practice with a new operator $\hat{\mathscr{P}}_{C}$



In [65]:

    
# 5.11
P_45 = P45*P45.dag() - M45*M45.dag()



In [66]:

    
#5.12
P_c = L*L.dag() - R*R.dag()



In [67]:

    
#5.13
(Phv*P_45 - P_45*Phv) == 2j * P_c









    Out[67]:





True



In [68]:

    
#5.14
(Phv*P_c - P_c*Phv) == -2j * P_45









    Out[68]:





True



In [69]:

    
#5.15
P_p45 = P45*P45.dag()
P_v = V*V.dag()



In [70]:

    
P_p45*P_v - P_v*P_p45









    Out[70]:





Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.500\\-0.500 & 0.0\\\end{array}\right)\end{equation*}

5.19 - this one is tricky, but is easier with our custom function



In [73]:

    
psi = 1/sqrt(3)*H + sqrt(2/3.0)*exp(1j*pi/3.0)*V
psi.norm()









    Out[73]:





1.0



In [74]:

    
Delta(psi,P_45)*Delta(psi,Phv)









    Out[74]:





(0.83147941928309788+0j)



In [75]:

    
1/2j*(psi.dag()*(Phv*P_45 - P_45*Phv)*psi).data[0,0]









    Out[75]:





(0.81649658092772592+0j)



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