Entanglement of quantum fluctuations in the inflationary universe

We investigate quantum entanglement of a scalar field in the inflationary universe. By introducing a bipartite system using a lattice model of scalar field, we apply the separability criterion based on the partial transpose operation and numerically calculate the bipartite entanglement between separate spatial regions. We find that the initial entangled state becomes separable or disentangled when the size of the spatial regions exceed the Hubble horizon. This is a necessary condition for the appearance of classicality of the quantum fluctuation. We further investigate the condition for the appearance of the classical distribution function and find that the condition is given by the inequality for the symplectic eigenvalue of the covariance matrix of the scalar field.

Figure 1

A bipartite system in our one-dimensional lattice model.

Figure 2

The dependence of the separation $d$ on $E_N$ for region size $n=4$. The left panel is $E_N$ at $\eta =-10$ and the right panel is $E_N$ at $\eta =-0.9$.

Figure 3

Evolution of symplectic eigenvalues for $d=0$, $n=4$. The upper line represents ${\nu }_{-}$ and the lower line represents ${\tilde{\nu }}_{-}$. The physical condition ${\nu }_{-}>1/2$ is always satisfied.

Figure 4

$E_N(d=0)$ as a function of the conformal time $\eta$ for $n=4$ case. After $\eta ={\eta }_c\approx -1$, $E_N$ becomes zero and the system is separable.

Figure 5

The relation between the critical time ${\eta }_c$ and the region size $n$. The deviation from a linear relation represents the effect of boundary condition.