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Lab 4 - Measurements

First our standard definitions:

import matplotlib.pyplot as plt
from numpy import sqrt,pi,cos,sin,arange,random,exp
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,b],[c,d]])
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 - (eO)**2)

Q: Define the P^HV\hat{\mathcal{P}}_{HV} operator

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

Q: What is the expectation value ⟨P^HV⟩\langle \hat{\mathcal{P}}_{HV}\rangle for state ∣ψ⟩=15∣H⟩+25∣V⟩|\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 PHV\mathcal{P}_{HV}?

psi.dag()*Phv*Phv*psi
1.0 - (-0.6)**2

Ex: Use the random function to generate a mock data set for the state ∣ψ⟩|\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=20,p=[0.2,0.8])
data.mean()
data.var()

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

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 P^C\hat{\mathcal{P}}_{C}

# 5.11
P_45 = P45*P45.dag() - M45*M45.dag()
# 5.12
P_c = L*L.dag() - R*R.dag()
# 5.13
(Phv*P_45 - P_45*Phv) == 2j * P_c
# 5.14
(Phv*P_c - P_c*Phv) == -2j * P_45
# 5.15
P_p45 = P45*P45.dag()
P_v = V*V.dag()
P_p45*P_v - P_v*P_p45

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

psi = 1/sqrt(3)*H + sqrt(2/3.0)*exp(1j*pi/3.0)*V
psi.norm()
Delta(psi,P_45)*Delta(psi,Phv)
1/2j*(psi.dag()*(Phv*P_45 - P_45*Phv)*psi)
Quantum Mechanics with QuTiP
Lab 4 - Measurements
Quantum Mechanics with QuTiP
Two-particle systems