In [1]:
# EPR Paradox Example
# A neutral pi meson (pi0) decays into an electron/positron (e-/e+)
# pair.
# pi0 --> e+ + e- (electron-positron pair)
# pi0 has angular momentum l = s = 0
# Align electron and positron detectors in opposite directions.
# | pi0 |
# |e- <------ ------> e+|
# Where hbar = 1, the measurement of the spin in some direction is
# +/-1 with spin state [1 0] (up) or [0 1] (down).
# The Pauli exclusion principle with conserved angular momentum 0
# dictates 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 [1 0], the
# electron must have spin [0 1], or vice versa. There is an equal
# probability to find either state during 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 apparently instantaneous, however.
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
%matplotlib inline
In [2]:
# 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.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.text(0.5,565,"e-",size=20)
plt.text(3.5,565,"e+",size=20)
plt.suptitle("Uncorrelated Spins",fontsize=20)
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.show()
In [3]:
# How many violations of angular momentum would be measured?
# From running several simulations, it's evident a violation in the
# conservation of angular momentum would be measured half of the time.
# We can conclude that the information that the entangled particles are in
# orthogonal spin states is instantaneously agreed 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()
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# 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()
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# 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,
# which is what one would expect from the quantum mechanical
# prediction.
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()
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# The prediction (Bell's inequality) made by assuming a local hidden
# variable is violated by a significant number of the possibile
# bborientations. 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.
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plt.show()