NLHV.R

This is an example of how a non-local hidden variables model can easily violate the CHSH inequality, even when all correlations are computed on the same set of hidden variables. Intended for pedagogical purposes.

set.seed(9875)

## For reproducibility.

M <- 10^6  ## Sample size. 

## Use the same, single sample of 'M' realizations of hidden variables for
## all four correlations

## The hidden variable is just a random number t between -1 and 1.  We send t
## to Alice and -t to Bob.

t <- runif(M, -1, 1)

## At both Alice's and Bob's stations the same model is used to determine
## outcomes.  We put this into a function obs(hv), that takes the hidden
## variable as the argument and returns -1 or 1.

obs <- function(hv) {
    s <- sign(hv)
    hv <- abs(hv)
    ## Since the following variable depend on both settings a and b, the model is
    ## blatantly non-local:
    L <- (1 + sum(a * b))/4

    if (hv < L) {
        o <- s
    } else if (hv < 2 * L) {
        o <- -s
    } else if (hv < L + 0.5) {
        o <- -1
    } else {
        o <- 1
    }
    return(o)
}

## Now we compute the four correlations in the CHSH inequality:

alpha <- 0
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

## We generate the list of observations by applying the obs function to each
## element in the list of hidden variables.  First for Alice:
ca <- sapply(t, obs)
## Then for Bob:
cb <- sapply(-t, obs)

E11 <- mean(ca * cb)
E11

## [1] -0.7077

alpha <- 0
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca <- sapply(t, obs)
cb <- sapply(-t, obs)

E12 <- mean(ca * cb)
E12

## [1] 0.7085

alpha <- 90
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca <- sapply(t, obs)
cb <- sapply(-t, obs)

E21 <- mean(ca * cb)
E21

## [1] -0.7077

alpha <- 90
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca <- sapply(t, obs)
cb <- sapply(-t, obs)

E22 <- mean(ca * cb)
E22

## [1] -0.7077

## CHSH expression
-E11 + E12 - E21 - E22

## [1] 2.832