Metric Topology and the Origins of Quantum Correlation
by Joy Christian
Einstein Centre for Local-Realistic Physics, 15 Thackley End, Oxford OX2 6LB, United Kingdom
Publication date: 12 August 2015
# For a theoretical background, please see http://arxiv.org/abs/1405.2355
# and http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=188#p5129Angles = seq(from = 0, to = 360, by = 7.2) * 2 * pi/360
K = length(Angles) # The total number of angles between 0 and 2pi
corrs = matrix(nrow = K, ncol = K, data = 0) # Container for correlations
Js = matrix(nrow = K, ncol = K, data = 0) # Container for "zero" events
Ns = matrix(nrow = K, ncol = K, data = 0) # Container for non-zero events
Ls = matrix(nrow = K, ncol = K, data = 0) # Container for initial states# A canvas of auxiliary pre-ensemble (analogous to pre-reduced phase space):
M = 10^4 # Size of the pre-ensemble. Next one can try 10^5, or even 10^6
r = runif(M, 0, 2*pi) # M uniformly distributed numbers between 0 and 2pi
z = runif(M, -1, +1) # M uniformly distributed numbers between -1 and +1
h = sqrt(1 - z^2)
x = h * cos(r)
y = h * sin(r)
e = rbind(x, y, z) # A 3xM matrix, with M columns of e-vectors representing
# the x, y, z coordinates of uniformly distributed points on an S^2. In what
# follows these M number of pre-states will define an auxiliary pre-ensemble# Defining the metrical and topological structures on S^3:
# So far we have set the variables in R^3. We now construct a metric on S^3
s = runif(M, 0, pi) # Initial states of the spins are the pairs (e, s) within S^3
f = -1 + (2/sqrt(1 + ((3 * s)/pi))) # For details see the paper arXiv:1405.2355
g = function(u,v,s){ifelse(abs(colSums(u*v)) > f, colSums(u*v), 0)}
# g(u,v,s) is non-vanishing only if |u.v| > f(s) ; g(u,v,s) --> "dot" as s --> pi
# Defines an inner product on S^3, thus changing the space from R^3 to S^3
# colSums(u * v) = u.v = the standard inner product between u and v in R^3
# u and v are orthogonal to each other in S^3 when abs(colSums(u * v)) < f
# The metric g(u,v,s) reduces to the standard Euclidean metric in R^3 for f --> 0# Computing the "quantum" correlations:
for (i in 1:K) {
alpha = Angles[i]
a = c(cos(alpha), sin(alpha), 0) # Measurement direction 'a'
for (j in 1:K) {
beta = Angles[j]
b = c(cos(beta), sin(beta), 0) # Measurement direction 'b'
A = +sign(g(a,e,s)) # Alice's measurement results A(a, e, s) = +/-1
B = -sign(g(b,e,s)) # Bob's measurement results B(b, e, s) = -/+1
N = length((A*B)[A & B]) # Number of all possible events observed in S^3
corrs[i,j] = sum(A*B)/N # Product moment correlation coefficient E(a, b)
# ----------------------------------------------------------------------
# Metric g(u,v,s) forbids the results (0,+), (0,-), (+,0), (-,0), & (0,0)
# Probabilities for (0,+), (0,-), (+,0), (-,0), and (0,0) vanish exactly
Cuu = length((A*B)[A > 0 & B > 0]) # Coincidence count of (+,+) events
Cdd = length((A*B)[A < 0 & B < 0]) # Coincidence count of (-,-) events
Cud = length((A*B)[A > 0 & B < 0]) # Coincidence count of (+,-) events
Cdu = length((A*B)[A < 0 & B > 0]) # Coincidence count of (-,+) events
# corrs[i,j] = (Cuu + Cdd - Cud - Cdu) / (Cuu + Cdd + Cud + Cdu) # = -a.b
# -----------------------------------------------------------------------
# There are no "0 outcomes" within S^3: Cou = Cod = Cuo = Cdo = Coo = 0
J = length((A*B)[g(a,e,s) & A == 0 & g(b,e,s) & B == 0]) # Number of null events
J = length((A*B)[g(a,e,s) & A == 0 | g(b,e,s) & B == 0]) # Number of null events
Js[i,j] = J # Total number of "vanishing" or "zero" events "observed" within S^3
Ns[i,j] = N # The total number of events simultaeously observed by Alice and Bob
# -------------------------------------------------------------------------------
# corrs[i] = sum(A)/length(A) # Verifies < A > = 0 for all 'a' regardless of 'b'
# corrs[j] = sum(B)/length(B) # Verifies < B > = 0 for all 'b' regardless of 'a'
# --------------------------------------------------------------------------
# Computing the same correlations with the standard Euclidean metric in R^3
C = sign(g(a,e,s))
D = sign(g(b,e,s))
o = x[C & D]
p = y[C & D]
q = z[C & D]
w = rbind(o,p,q) # N vectors in R^3 representing the initial states within S^3
a.w = colSums(a*w) # Standard inner product in R^3 between the vectors a and w
b.w = colSums(b*w) # Standard inner product in R^3 between the vectors b and w
# corrs[i,j] = sum(-sign(a.w)*sign(b.w))/N # Another way of calculating E(a, b)
L = length(w)/3 # The total number of initial states (e,s) emitted within S^3
Ls[i,j] = L # Total number of initial particles (e,s) emitted from the source
}
}par(mar = c(0, 0, 2, 0))
persp(x = Angles, y = Angles, main = "The strong correlations predicted by the 3-sphere model", z = corrs, zlim = c(-1, 1), col = "lightblue1", theta = 135, phi = 30, scale = FALSE, xlab = "alpha", ylab = "beta")
par(mar = c(0, 0, 2, 0))
persp(x = Angles, y = Angles, main = "Ratio N / L in S^3 = Number N of simultaneous events / Number L of initial states (e, s); from 0 to 1", z = Ns/Ls, zlim = c(0, 1), col = "bisque", theta = 135, phi = 30, scale = FALSE, xlab = "alpha", ylab = "beta", zlab = "N / L")
par(mar = c(0, 0, 2, 0))
persp(x = Angles, y = Angles, main = "Ratio J / N within S^3 = Number J of vanishing or zero events / Number N of all possible events; from 0 to 1", z = Js/Ns, zlim = c(0, 1), col = "darkseagreen1", theta = 135, phi = 30, scale = FALSE, xlab = "alpha", ylab = "beta", zlab = "J / N")
The last plot can be used to verify that there are no zero outcomes regardless of the metric defining the physical space by the parameter “s”.
# Both CHSH and Clauser-Horne / Eberhard inequalities are violated in the 3-sphere model:alpha <- 0 * pi/180
beta <- 45 * pi/180
a <- c(cos(alpha), sin(alpha), 0) # Measurement direction 'a'
b = c(cos(beta), sin(beta), 0) # Measurement direction 'b'
A = +sign(g(a,e,s)) # Alice's measurement results A(a, e, s) = +/-1
B = -sign(g(b,e,s)) # Bob's measurement results B(b, e, s) = -/+1
C_0_45 = length((A*B)[A > 0 & B > 0]) # Coincidence count of (+,+) events
N_0_45 = length((A*B)[A & B]) # Coincidence count of all possible events
E_0_45 <- sum(A * B)/N_0_45
alpha <- 0 * pi/180
beta <- 135 * pi/180
a <- c(cos(alpha), sin(alpha), 0) # Measurement direction 'a'
b = c(cos(beta), sin(beta), 0) # Measurement direction 'b'
A = +sign(g(a,e,s)) # Alice's measurement results A(a, e, s) = +/-1
B = -sign(g(b,e,s)) # Bob's measurement results B(b, e, s) = -/+1
C_0_135 = length((A*B)[A > 0 & B > 0]) # Coincidence count of (+,+) events
N_0_135 = length((A*B)[A & B]) # Coincidence count of all possible events
E_0_135 <- sum(A * B)/N_0_135
alpha <- 90 * pi/180
beta <- 45 * pi/180
a <- c(cos(alpha), sin(alpha), 0) # Measurement direction 'a'
b = c(cos(beta), sin(beta), 0) # Measurement direction 'b'
A = +sign(g(a,e,s)) # Alice's measurement results A(a, e, s) = +/-1
B = -sign(g(b,e,s)) # Bob's measurement results B(b, e, s) = -/+1
C_90_45 = length((A*B)[A > 0 & B > 0]) # Coincidence count of (+,+) events
N_90_45 = length((A*B)[A & B]) # Coincidence count of all possible events
E_90_45 <- sum(A * B)/N_90_45
alpha <- 90 * pi/180
beta <- 135 * pi/180
a <- c(cos(alpha), sin(alpha), 0) # Measurement direction 'a'
b = c(cos(beta), sin(beta), 0) # Measurement direction 'b'
A = +sign(g(a,e,s)) # Alice's measurement results A(a, e, s) = +/-1
B = -sign(g(b,e,s)) # Bob's measurement results B(b, e, s) = -/+1
C_90_135 = length((A*B)[A > 0 & B > 0]) # Coincidence count of (+,+) events
N_90_135 = length((A*B)[A & B]) # Coincidence count of all possible events
E_90_135 <- sum(A * B)/N_90_135abs(E_0_45 - E_0_135 + E_90_45 + E_90_135) # The absolute bound is supposed to be 2## [1] 2.851039# Quantum mechanics and the 3-sphere model predict 2.8284 (up to statistical error)R_0_45 = C_0_45 / N_0_45 # The rate of detecting (+,+) coincidence at (0, 45)
R_0_135 = C_0_135 / N_0_135 # The rate of detecting (+,+) coincidence at (0, 135)
R_90_45 = C_90_45 / N_90_45 # The rate of detecting (+,+) coincidence at (90, 45)
R_90_135 = C_90_135 / N_90_135 # The rate of detecting (+,+) coincidence at (90, 135)
R_a = 1/2 # Since < A > = 0 for all 'a' regardless of 'b', as verified above
R_b = 1/2 # Since < B > = 0 for all 'b' regardless of 'a', as verified above
(R_0_45 - R_0_135 + R_90_45 + R_90_135 - R_a - R_b) # The lower bound is supposed to be -1## [1] -1.21093# Quantum mechanics and the 3-sphere model predict -1.2071 (up to statistical error)