-
-
-
-
-
-
-
-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()
-
-
-
-
-
-
-
-
-
-
-
-Out[3]:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-In [4]:
-
-
-
-
-
-
-# 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()
-
-
-
-
-
-
-
-
-
-
-
-Out[4]:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-In [5]:
-
-
-
-
-
-
-# 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()
-
-
-
-
-
-
-
-
-
-
-
-Out[5]:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-In [6]:
-
-
-
-
-
-
-# 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.
-
-
-
-
-
-In [7]:
-
-
-
-
-
-
-plt.show()
-