phy-521/presentation/simulation.py

# EPR Paradox Example

# A pion (pi0) decays into an electron/positron (e-/e+) pair.

# pi0 --> e+ + e- (electron-positron pair)
# pi0 has angular momentum l = 0

# Align electron and positron detectors in opposite directions.

# |            pi0            |
# |e- <------       ------> e+|

# Where hbar = 1, the measurement of the spin in some direction is either
# [1 0] or [0 1].

# Pauli exclusion principle with conserved angular momentum l=0 says this 
# system must be in the singlet state 
# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].

# In this state, if the positron is measured to have spin up, the electron
# must have spin down, or vice versa. There is an equal probability to
# measure either spin by the first measurement.

# This view is consistent with the realist view. The realist view could hold
# that the electron and position had those angular momenta from creation.

# EPR assumes influences cannot propagate faster than the speed of light.
# "Wave function collapse" is instantaneous.


import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches



# If information about the measurement of the wave function propagated
# at a finite speed, the particles could conceivably be measured such
# that both are equally likely to hold either spin up or spin down.

# What would happen if the measurements were uncorrelated?

plt.ylim([400,600])
plt.xlim([-1,5])
plt.xticks([0.125,0.85,3.125,3.85],["down","up","down","up"])
plt.tick_params(axis='both',labelsize=15)
plt.hist(np.random.randint(0,2,1000),bins=4)
plt.hist(np.random.randint(3,5,1000),bins=4)
elepatch = mpatches.Patch(color='blue', label='e-')
pospatch = mpatches.Patch(color='orange', label='e+')
plt.legend(handles=[elepatch,pospatch])
plt.figure()



# On average, how many violations of angular momentum would be measured?

# Clearly, a violation in angular momentum would be measured half of the time.
# We can conclude that the information that the entangled particles are in
# the opposite spin states of eachother is instantaneously known once
# a measurement is made.

violations = 0

for trial in range(0,1000):
    elespin = np.random.randint(0,2)
    posspin = np.random.randint(0,2)
    if elespin == 0:
        elespin = np.matrix('0 1')
    else:
        elespin = np.matrix('1 0')
    if posspin == 0:
        posspin = np.matrix('0 1')
    else:
        posspin = np.matrix('1 0')
    
    elespin.transpose()
    posspin.transpose()
    chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')
    
    if chi_squared == 0:
        violations = violations + 1

zeroes = np.zeros(violations,dtype=int)
ones = np.full((1000-violations),1,dtype=int)
result = np.concatenate((ones,zeroes))

plt.ylim([400,600])
plt.xlim([-1,2])
plt.xticks([0.125,0.85],["violation","adherence"])
plt.tick_params(axis='both',labelsize=15)
plt.suptitle("Conservation Violations",fontsize=20)
plt.hist([result],bins=4)
plt.figure()
        




# Bell's Experiment took this a step further, to rule out locality completely.
# Establish the detectors to "float" such that they measure the components of
# the spins of the electron and positron along a unit vector a or b, with 
# angles phi_a and phi_b, respectively. Compute a product P of the spins in 
# units of hbar/2. This will give +/-1. 

# /            pi0            \
# /e- <------       ------> e+\

# QM predicts P(a,b) = -a dot b, the expectation value of the product of
# the spins.

# In 1964, Bell derived the Bell inequality for a local hidden variable
# theory:   abs(P(a,b) - P(a,c)) <= 1 + P(b,c)

# For any local hidden variable theory, the Bell inequality must hold. It
# introduces a third unit vector c, which is any other unit vector than
# a or b.

# Does the quantum mechanical prediction violate the Bell inequality? Testing
# several detector configurations in a plane, systematically from 0 to pi,
# we determine whether the QM prediction is consistent with a local hidden
# variable theory.

violations = 0
trials = 0
for step_a in range(0,6):
    for step_b in range (0,6):
        phi_a = step_a/6*np.pi
        phi_b = step_b/6*np.pi
        phi_c = phi_a - 0.5*phi_b
        P_ab = -1 * np.cos(phi_a - phi_b)
        P_ac = -1 * np.cos(phi_a - phi_c)
        P_bc = -1 * np.cos(phi_b - phi_c)
        
        bell_lhs = np.abs(P_ab - P_ac)
        bell_rhs = 1 + P_bc
        
        if bell_lhs > bell_rhs:
            violations = violations + 1
            
        trials = trials + 1
                
zeroes = np.zeros(violations,dtype=int)
ones = np.full((trials-violations),1,dtype=int)
result = np.concatenate((ones,zeroes))

plt.ylim([trials/2-10,trials/2+10])
plt.xlim([-1,2])
plt.xticks([0.125,0.85],["violation","adherence"])
plt.tick_params(axis='both',labelsize=15)
plt.suptitle("Hidden Locality Violations",fontsize=20)
plt.hist([result],bins=4)
plt.figure()




# It is seen that the QM prediction disagrees with a local hidden variable
# theory in a significant number of configurations.

# On average, for random orientations between 0 and pi, how often? After running several trials,
# it appears to be about half of the time.

violations = 0
trials = 0
for rand_a in range(0,10):
    for rand_b in range (0,10):
        phi_a = np.random.rand(1)*np.pi
        phi_b = np.random.rand(1)*np.pi
        phi_c = phi_a - 0.5*phi_b
        P_ab = -1 * np.cos(phi_a - phi_b)
        P_ac = -1 * np.cos(phi_a - phi_c)
        P_bc = -1 * np.cos(phi_b - phi_c)
        
        bell_lhs = np.abs(P_ab - P_ac)
        bell_rhs = 1 + P_bc
        
        if bell_lhs > bell_rhs:
            violations = violations + 1
            
        trials = trials + 1

zeroes = np.zeros(violations,dtype=int)
ones = np.full((trials-violations),1,dtype=int)
result = np.concatenate((ones,zeroes))

plt.ylim([trials/2-10,trials/2+10])
plt.xlim([-1,2])
plt.xticks([0.125,0.85],["violation","adherence"])
plt.tick_params(axis='both',labelsize=15)
plt.suptitle("Hidden Locality violations",fontsize=20)
plt.hist([result],bins=4)
plt.figure()



# The prediction (Bell's inequality) made by assuming a local hidden
# variable is violated by a significant number of the possibile orientations.
# This simulation cannot determine which theory is correct, but the QM
# prediction has been confirmed through experiment. No hidden local variable
# holds actionable information about the state. Entangled states retain
# their entanglement in a non-local nature.