Lab 4 - Measurements

First our standard definitions:



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



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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]]])

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



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









    Out[4]:





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.



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



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psi.dag()*Phv*psi









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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 $\mathscr{P}_{HV}$?



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psi.dag()*Phv*Phv*psi









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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*}



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1- (-0.6)**2









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0.64

Example: 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:



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data = random.choice([1, -1],size=1000000,p=[0.2,0.8])

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?



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data.mean()









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-0.59854600000000002



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data.var()









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0.64174268588399952

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 $\hat{\mathscr{P}}_{C}$



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



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P_45









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Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 1.000\\1.000 & 0.0\\\end{array}\right)\end{equation*}



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



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P_c









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Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -1.000j\\1.000j & 0.0\\\end{array}\right)\end{equation*}



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