Entanglement and impropriety

We describe a classical model of quantum entanglement corresponding to spontaneous parametric downconversion. Specifically, we consider the interaction of classical stochastic vacuum modes from the zero-point field and a high intensity pump with a nonlinear optical medium. The vacuum modes are treated as proper complex Gaussian random variables and are the sole source of randomness in the model. Phase matching conditions mimic the quantum mechanical energy and momentum conservation laws and result in outgoing classical fields of maximum intensity. The resulting transformation resembles the Bogoliubuv transformation for multi-mode squeezed light, albeit with operators replaced by random variables, and yields a vector of improper complex Gaussian random variables. Using an amplitude threshold detection and post-selection measurement scheme, the degree of impropriety is found to correspond to the degree of entanglement in the corresponding quantum description.