Lab 4 - Measurements

First our standard definitions:

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

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)]])

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 $\hat{\mathcal{P}}_{HV}$ operator¶

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

Q: What is the expectation value $\langle \hat{\mathcal{P}}_{HV}\rangle$ 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.¶

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

psi.dag()*Phv*psi

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

psi.dag()*Phv*Phv*psi

1- (-0.6)**2

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:

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?¶

data.mean()

data.var()

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