## Comparison of CHSH and J for recent Bell experiments, 
## together with optimally noise-reduced versions of both.
## Theory: https://pub.math.leidenuniv.nl/~gillrd/Peking/Peking_4.pdf

## In short: assume four multinomial samples, 
## estimate covariance matrix of estimated relative frequencies, 
## use sample deviations from no-signalling to optimally reduce 
## the noise in the estimate of Bell's S or Eberhard's J

## AKA: generalized least squares

############# DELFT #############

## The basic data, four 2x2 tables

table11 <- matrix(c(23,  3,  4, 23), 
    2, 2, byrow = TRUE,
    dimnames = list(Alice = c("+", "-"), Bob = c("+", "-")))
table12 <- matrix(c(33, 11,  5, 30), 
    2, 2, byrow = TRUE,
    dimnames = list(Alice = c("+", "-"), Bob = c("+", "-")))
table21 <- matrix(c(22, 10,  6, 24), 
    2, 2, byrow = TRUE,
    dimnames = list(Alice = c("+", "-"), Bob = c("+", "-")))    
table22 <- matrix(c(4, 20, 21,  6), 
    2, 2, byrow = TRUE,
    dimnames = list(Alice = c("+", "-"), Bob = c("+", "-")))

table11
##      Bob
## Alice  +  -
##     + 23  3
##     -  4 23
table12
##      Bob
## Alice  +  -
##     + 33 11
##     -  5 30
table21
##      Bob
## Alice  +  -
##     + 22 10
##     -  6 24
table22
##      Bob
## Alice  +  -
##     +  4 20
##     - 21  6
## Check of the total number of trials

# "The number of valid trials is N = 245"
sum(table11) + sum(table12) + sum(table21) + sum(table22)
## [1] 245
## The same data now in one 4x4 table

tables <- cbind(as.vector(t(table11)), as.vector(t(table12)), as.vector(t(table21)), as.vector(t(table22)))
dimnames(tables) = list(outcomes = c("++", "+-", "-+", "--"), 
                      settings = c(11, 12, 21, 22))
tables
##         settings
## outcomes 11 12 21 22
##       ++ 23 33 22  4
##       +-  3 11 10 20
##       -+  4  5  6 21
##       -- 23 30 24  6
## The total number of trials for each setting pair

Ns <- apply(tables, 2, sum)
Ns
## 11 12 21 22 
## 53 79 62 51
## observed relative frequencies, one 4x4 matrix

rawProbsMat <- tables / outer(rep(1,4), Ns)
rawProbsMat
##         settings
## outcomes         11         12         21         22
##       ++ 0.43396226 0.41772152 0.35483871 0.07843137
##       +- 0.05660377 0.13924051 0.16129032 0.39215686
##       -+ 0.07547170 0.06329114 0.09677419 0.41176471
##       -- 0.43396226 0.37974684 0.38709677 0.11764706
## Convert the relative frequencies to one vector of length 16

VecNames <- as.vector(t(outer(colnames(rawProbsMat), rownames(rawProbsMat), paste, sep = "")))
rawProbsVec <- as.vector(rawProbsMat)
names(rawProbsVec) <- VecNames

VecNames
##  [1] "11++" "11+-" "11-+" "11--" "12++" "12+-" "12-+" "12--" "21++" "21+-"
## [11] "21-+" "21--" "22++" "22+-" "22-+" "22--"
rawProbsVec
##       11++       11+-       11-+       11--       12++       12+- 
## 0.43396226 0.05660377 0.07547170 0.43396226 0.41772152 0.13924051 
##       12-+       12--       21++       21+-       21-+       21-- 
## 0.06329114 0.37974684 0.35483871 0.16129032 0.09677419 0.38709677 
##       22++       22+-       22-+       22-- 
## 0.07843137 0.39215686 0.41176471 0.11764706
## Building up the 4 no-signalling constraints, combined in one 16 x 4 matrix "NS"

Aplus <- c(1, 1, 0, 0)
Aminus <- - Aplus
Bplus <- c(1, 0, 1, 0)
Bminus <- - Bplus
zero <- c(0, 0, 0, 0)
NSa1 <- c(Aplus, Aminus, zero, zero)
NSa2 <- c(zero, zero, Aplus, Aminus)
NSb1 <- c(Bplus, zero, Bminus, zero)
NSb2 <- c(zero, Bplus, zero, Bminus)
NS <- cbind(NSa1 = NSa1, NSa2 = NSa2, NSb1 = NSb1, NSb2 = NSb2)
rownames(NS) <- VecNames
NS
##      NSa1 NSa2 NSb1 NSb2
## 11++    1    0    1    0
## 11+-    1    0    0    0
## 11-+    0    0    1    0
## 11--    0    0    0    0
## 12++   -1    0    0    1
## 12+-   -1    0    0    0
## 12-+    0    0    0    1
## 12--    0    0    0    0
## 21++    0    1   -1    0
## 21+-    0    1    0    0
## 21-+    0    0   -1    0
## 21--    0    0    0    0
## 22++    0   -1    0   -1
## 22+-    0   -1    0    0
## 22-+    0    0    0   -1
## 22--    0    0    0    0
## Build the 16x16 estimated covariance matrix of the 16 observed relative frequencies

cov11 <- diag(rawProbsMat[ , "11"]) - outer(rawProbsMat[ , "11"], rawProbsMat[ , "11"])
cov12 <- diag(rawProbsMat[ , "12"]) - outer(rawProbsMat[ , "12"], rawProbsMat[ , "12"])
cov21 <- diag(rawProbsMat[ , "21"]) - outer(rawProbsMat[ , "21"], rawProbsMat[ , "21"])
cov22 <- diag(rawProbsMat[ , "22"]) - outer(rawProbsMat[ , "22"], rawProbsMat[ , "22"])

Cov <- matrix(0, 16, 16)
rownames(Cov) <- VecNames
colnames(Cov) <- VecNames
Cov[1:4, 1:4] <- cov11/Ns["11"]
Cov[5:8, 5:8] <- cov12/Ns["12"]
Cov[9:12, 9:12] <- cov21/Ns["21"]
Cov[13:16, 13:16] <- cov22/Ns["22"]

## The vector "S" is used to compute the CHSH statistic "CHSH"
## The sum of the first three sample correlations minus the fourth

S <- c(c(1, -1, -1 ,1), c(1, -1, -1 , 1), c(1, -1, -1, 1), - c(1, -1, -1, 1))
names(S) <- VecNames
CHSH <- sum(S * rawProbsVec)
CHSH
## [1] 2.4225
## Compute the estimated variance of the CHSH statistic, 
## its estimated covariances with the observed deviations from no-signalling,
## and the 4x4 estimated covariance matrix of those deviations.
## We'll later also need the inverse of the latter.

varS <- t(S) %*% Cov %*% S
covNN <- t(NS) %*% Cov %*% NS
covSN <- t(S) %*% Cov %*% NS
covNS <- t(covSN)

InvCovNN <- solve(covNN)

## Estimated variance of the CHSH statistic, 
## and estimated variance of the optimally "noise reduced" CHSH statistic.

varCHSH <- varS
varCHSHopt <- varS - covSN %*% InvCovNN %*% covNS

## The variance, and the improvement as ratio of standard deviations

varS
##            [,1]
## [1,] 0.04154528
sqrt(varCHSH / varCHSHopt)
##          [,1]
## [1,] 1.009601
## The coefficients of the noise reduced CHSH statistic and the resulting improved estimate

Sopt <- S - covSN %*% InvCovNN %*% t(NS)
Sopt
##          11++      11+-       11-+ 11--     12++       12+-       12-+
## [1,] 1.016769 -1.235284 -0.7479471    1 1.240163 -0.7647162 -0.9951203
##      12--      21++       21+-      21-+ 21--      22++      22+-
## [1,]    1 0.9676932 -0.7802539 -1.252053    1 -1.224626 0.7802539
##           22-+ 22--
## [1,] 0.9951203   -1
CHSHopt <- sum(Sopt * rawProbsVec)
CHSHopt
## [1] 2.462658
## p-values assuming approximate normality for testing CHSH inequality

pnorm((CHSH - 2)/ sqrt(varCHSH), lower.tail = FALSE)
##           [,1]
## [1,] 0.0190936
pnorm((CHSHopt - 2)/ sqrt(varCHSHopt), lower.tail = FALSE)
##            [,1]
## [1,] 0.01096277
## Now we repeat for the Eberhard J statistic
## First, the coefficients in the vector "J"
## and the observed value of the statistic

J <- c(c(1, 0, 0 ,0), c(0, -1, 0 ,0), c(0, 0, -1, 0), c(-1, 0, 0, 0))
names(J) <- VecNames
sum(J * rawProbsVec)
## [1] 0.1195162
## Next, its estimated variance and resulting p-value

varJ <- t(J) %*% Cov %*% J
sum(J * rawProbsVec) / sqrt(varJ)
##          [,1]
## [1,] 1.261291
pnorm(sum(J * rawProbsVec) / sqrt(varJ), lower.tail = FALSE)
##          [,1]
## [1,] 0.103602
## The covariances between J and the observed deviations from no-signaling
## The variance of the usual estimate of J and of the improved estimate of J
## The improvement as a ration of standard deviations

covJN <- t(J) %*% Cov %*% NS
covNJ <- t(covJN)
varJopt <- varJ - covJN %*% InvCovNN %*% covNJ

varJ
##             [,1]
## [1,] 0.008978894
sqrt(varJ / varJopt)
##          [,1]
## [1,] 1.877415
## The coefficients of an improved estimataor of Eberhard's J

Jopt <- J - covJN %*% InvCovNN %*% t(NS)
Jopt
##             11++       11+-       11-+ 11--       12++       12+-
## [1,] 0.004192268 -0.5588209 -0.4369868    0 0.06004087 -0.4411791
##            12-+ 12--         21++       21+-       21-+ 21--        22++
## [1,] -0.4987801    0 -0.008076702 -0.4450635 -0.5630132    0 -0.05615644
##           22+-      22-+ 22--
## [1,] 0.4450635 0.4987801    0
## Observed estimate of J, and improved estimate of J

sum(J * rawProbsVec)
## [1] 0.1195162
sum(Jopt * rawProbsVec)
## [1] 0.1156646
## p-values based on J and on improved J
## Note that the p-value based on improved J is the same as that of improved CHSH

pnorm(sum(J * rawProbsVec) / sqrt(varJ), lower.tail = FALSE)
##          [,1]
## [1,] 0.103602
pnorm(sum(Jopt * rawProbsVec) / sqrt(varJopt), lower.tail = FALSE)
##            [,1]
## [1,] 0.01096277
## The p-value of the optimized J got worse, because, apparently,
## the estimate and the optimized estimate of J are somewhat different!
## Our procedure for CHSH did lead to small improvement
## It seems that deviation from no-signalling is quite large!