Entanglement of a coarse grained quantum field in the expanding universe

We investigate the entanglement of a quantum field in the expanding universe. By introducing a bipartite system using a coarse-grained 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 after the spatial separation of two points exceed the Hubble horizon. This provides the necessary conditions for the appearance of classicality of the quantum fluctuation. We also investigate the condition of classicality that the quantum field can be treated as the classical stochastic variables.

Figure 1

The function $x_0(c)$. Each line corresponds to $x_{01}$, $x_{02}$, $x_{03}$.

Figure 2

The relation between the spatial distance $r$ and the logarithmic negativity $E_N$ [log plot of $E_N(r)$]. Each line corresponds to the different value of the mass $m/k_0=0$ (blue), 10 (red), 20 (yellow), 40 (green) (top to bottom). The inset is the same plot in the small $r$ region and shows exponential decay of $E_N$. The decay rate depends on the mass $m$ in the small $r$ region.

Figure 3

Spatial dependence of the logarithmic negativity at $N=10$ for the massless field. The distance $r_{\mathrm{phys}}$ is in the unit of $H_0^{-1}$. $E_N$ goes to zero at $r_{\mathrm{phys}}=r_{\mathrm{separable}}$. Each line corresponds to $p=100$ (blue), 10 (red), 5 (yellow), 3 (green) (left to right).

Figure 4

The value of the symplectic eigenvalue $({\tilde{\nu }}_{-}{)}^2$ in the $(r_{\mathrm{phys}},N)$ space for the massless scalar field in the Universe with power law expansion. The distance $r_{\mathrm{phys}}$ is in the unit of $H_0^{-1}$. Each panel corresponds to the different expansion rate $p=100$, 10, 5, 3 from the top left to the down right. The contour lines $({\tilde{\nu }}_{-}{)}^2=0.25,1.0,1.5$ are shown. The dark green region corresponds to $({\tilde{\nu }}_{-}{)}^2<1/4$ and the system is entangled in this region. The light green region corresponds to $1/4<({\tilde{\nu }}_{-}{)}^2<1$ and the system is separable but the classicality condition is not satisfied. The pink region corresponds to $1<({\tilde{\nu }}_{-}{)}^2$. The blue solid line is $r_{\mathrm{phys}}=H^{-1}$ (horizon scale) and the black solid lines represent different comoving scales.

Figure 5

Spatial dependence of the logarithmic negativity at $N=10$ for the massive scalar field in the de Sitter spacetime. The distance $r_{\mathrm{phys}}$ is in the unit of $H_0^{-1}$. Each line corresponds to $m^2/H_0^2=0$ (blue), $1/4$ (red), $1/2$ (yellow), $3/4$ (green), 1 (blue) (left to right).

Figure 6

The $(r_{\mathrm{phys}},N)$ dependence of the symplectic eigenvalue $({\tilde{\nu }}_{-}{)}^2$ for the massive scalar field in the de Sitter spacetime. The distance $r_{\mathrm{phys}}$ is in units of $H_0^{-1}$. Each panel show $({\tilde{\nu }}_{-}{)}^2$ for $m^2/H_0^2=0$, $1/4$, $1/2$, 1 from the top left to the down right. The contour lines $({\tilde{\nu }}_{-}{)}^2=0.25,1.0,1.5$ are shown. The dark green region corresponds to $({\tilde{\nu }}_{-}{)}^2<1/4$ and the system is entangled. The light green region corresponds to $({\tilde{\nu }}_{-}{)}^2>1/4$ and the system is separable. The pink region satisfies the condition of the classicality $({\tilde{\nu }}_{-}{)}^2>1$. The black solid lines represent the different comoving scales. The line $({\tilde{\nu }}_{-}{)}^2=1/4$ deviates from the horizon scale $H_0^{-1}$ (the blue solid line) as the mass increases.

Figure 7

The dependence of the comoving scale $\delta$ of the symplectic eigenvalues for the massive scalar field in the de Sitter universe. The left panel shows $({\nu }_{-}{)}^2$ (the upper line) and $({\tilde{\nu }}_{-}{)}^2$ (the lower line) at $N=2$. The right panel is at $N=5$. The mass of the scalar field is $m^2/H_0^2=1/4$. At any time, ${\tilde{\nu }}_{-}>1/2$ for $1/a<\delta \le 1$.