Author: Kwan-yuet Ho, Ph.D.
Date: 2018-03-28

# Entanglement in Coupled Harmonics (Simple Two-Body System)

#### Kwan-yuet Ho, Ph.D.

#### March 28, 2018

## Quantum Entanglement in Continuous Systems

Define the coupled harmonic oscillators:

    coupled.harm.fcn<- function(x1,x2) exp(-((0.5*(x1+x2))**2))*exp(-(x1-x2)**2)*sqrt(2./pi)

## Schmidt Decompostions

Then run the Schmidt decompotions:

    modes<- continuous.schmidt.decompose(coupled.harm.fcn, -10, 10, -10, 10)

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(-5, 5, 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') + geom_line(aes(y=y2), col='blue') + xlab('x') + ylab('y') +
      ggtitle('Schmidt mode 2')