mirror of
https://asciireactor.com/otho/phy-521.git
synced 2024-11-24 14:05:05 +00:00
208 lines
6.4 KiB
Python
208 lines
6.4 KiB
Python
# 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.
|
|
|
|
|
|
|
|
|