## Quantum Entanglement in Continuous Systems

Define the coupled harmonic oscillators:

``coupled.harm.fcn<- function(x1, x2) sqrt(0.7)*dnorm(x1, mean=0, sd=0.5)*dnorm(x2, mean=0, sd=sqrt(0.15)) + sqrt(0.3)*dnorm(x1, mean=5, sd=0.5)*dnorm(x2, mean=6, sd=sqrt(0.1))``

## Schmidt Decompostions

Then run the Schmidt decompotions:

``modes<- continuous.schmidt.decompose(coupled.harm.fcn, -10, 10, -10, 10, nbx1 = 200, nbx2 = 200)``

Then we retrieve the weights of the Schmidt modes, and plot the first ten of them:

``````data.frame(n=1:10, eigenvalue=lapply(modes[1:10], function(mode) mode\$eigenvalue) %>% unlist) %>%
ggplot(aes(x=n, y=eigenvalue)) + geom_point() + ggtitle('Schmidt weights')``````

Then we can plot the first Schmidt mode for both subsystems:

``````xarray<- seq(-5, 5, 10/50)
data.frame(x=xarray, y1=modes[[1]]\$sys1eigfcn(xarray), y2=modes[[1]]\$sys2eigfcn(xarray)) %>%
ggplot(aes(x=x)) + geom_line(aes(y=y1), col='red') + geom_line(aes(y=y2), col='blue') + xlab('x') + ylab('y') +
ggtitle('Schmidt mode 1')``````

And the second Schmidt modes for both subsystems:

``````xarray<- seq(-10, 10, 10/50)
data.frame(x=xarray, y1=modes[[2]]\$sys1eigfcn(xarray), y2=modes[[2]]\$sys2eigfcn(xarray)) %>%
ggplot(aes(x=x)) + geom_line(aes(y=y1), col='red') + xlab('x') + ylab('y') +
ggtitle('Schmidt mode 2')``````

``````xarray<- seq(-10, 10, 10/50)
data.frame(x=xarray, y1=modes[[2]]\$sys1eigfcn(xarray), y2=modes[[2]]\$sys2eigfcn(xarray)) %>%
ggplot(aes(x=x)) + geom_line(aes(y=y2), col='red') + xlab('x') + ylab('y') +
ggtitle('Schmidt mode 2')``````