Emerging entanglement on network histories

We show that quantum fields confined to Lorentzian histories of freely falling networks in Minkowski spacetime probe entanglement properties of vacuum fluctuations that extend unrestricted across spacetime regions. Albeit instantaneous field configurations are localized on one-dimensional edges, angular momentum emerges on these network histories and establish the celebrated area scaling of entanglement entropy.

Figure 1

Shown is a portion of spacetime deconstructed in accordance with global hyperbolicity. Depicted is an (initial) hypersurface labeled by the time ${\tau }_{\mathrm{in}}$ assigned to the instantaneous events supported on ${\mathrm{\Sigma }}_{{\tau }_{\mathrm{in}}}$ by freely falling observers. On this hypersurface an ideal network resides. As the network is freely falling to ${\mathrm{\Sigma }}_{{\tau }_{\mathrm{fi}}}$ it traces out a piecewise smooth two-dimensional Lorentzian history to which the dynamical degrees of freedom are confined. As an example, in gray is shown the Lorentzian history traced out by the evolution of two edges of the network.

Figure 2

Shown are the mutual information $I$ of quantum fluctuations confined to a single edge in a low-temperature thermal state (data points) and in the ground state at zero temperature (our prediction) as a function of the subsystem size $l$ in fractions of the edge length $L$.

Figure 3

Spatial section of a three-dimensional mesh graph embedded in Minkowski spacetime. The disk of radius $R$ is traced out and the crossing points of this circle and the graph are indicated with dots. The single edge shown in red, the simple loop in blue, and star graph with three edges depicted in green represent basic building blocks of this graph. Moreover, a subgraph consisting of this star graph and two additional edges (dashed) is shown as an example for a loop with a single edge attached.

Figure 4

Entanglement entropy of quantum fluctuations confined to the basic building blocks of the mesh graph shown in Fig. 3 in terms of the radius of an entangling sphere. In particular, the entanglement entropy is given for the red edge and the blue loop equipped with a finite resolution structure consisting of $N=300$ locations for two different choices of $\mu a$. The radius of the entangling sphere shown in Fig. 3 is indicated with a vertical dashed line.

Figure 5

Entanglement entropy of vacuum fluctuations confined on a network idealized as a minimal star graph with three edges joined at a single vertex (referring to Fig. 3) in terms of the entangling sphere radius for two different values of $\mu a$. For comparison the entanglement entropy of vacuum fluctuations confined to a single edge with Dirichlet boundary conditions at both its endpoints is shown in dashed lines for both cases.

Figure 6

The upper plot shows the entanglement entropy $S_{\text{grid}}$ of quantum fields in the ground state confined to a three-dimensional 11x11x11 regular grid graph with ${11}^3$ vertices and universal length $L=7a$ of the edges in terms of a minimal distance scale $a$ implied by an assumed finite resolution structure associated with some experiment. $S_{\text{grid}}$ is shown in terms of the area $A$ of the entangling sphere. Data points correspond to numerical experiments for two different values of $\mu a$. For $\mu a=1$ (displayed in green color), the analytical approximation $S(R)$ (green solid line) described in the main text is in good agreement with the data points (from numerical experiments). For the case $\mu a={10}^{-3}$ (black data points) there is no analytical approximation for the given long-distance scale $L$. For convenience, a fit $S_{\mathrm{fit}}$ (black dashed lines) of the data from the numerical experiments has been added. The lower plot shows the quality of the agreement using $\mathrm{\Delta }S=\delta S-1$ with $\delta S=S(R)/S_{\text{grid}}$ for $\mu a=1$ and $\delta S=S_{\mathrm{fit}}/S_{\text{grid}}$ for $\mu a={10}^{-3}$.

Figure 7

The upper plot shows the entanglement entropy $S_{\text{grid}}$ of quantum fields in the ground state confined to a three-dimensional 17x17x17 regular grid graph with ${17}^3$ vertices and universal length $L=4a$ of the edges in terms of a minimal distance scale $a$ implied by an assumed finite resolution structure associated with some experiment. $S_{\text{grid}}$ is shown in terms of the area $A$ of the entangling cube. Data points correspond to numerical experiments for two different values of $\mu a$. For $\mu a=1$ (displayed in green color) the analytical approximation $S(R)$ is shown using a green solid line. A fit $S_{\mathrm{fit}}$ of the data for $\mu a={10}^{-3}$ is depicted using a dashed black line. For comparison, the fit to the entanglement entropy for an entangling sphere introduced in Fig. 6 is shown in red. The lower plot shows the quality of the agreement using $\mathrm{\Delta }S=\delta S-1$ with $\delta S=S(R)/S_{\text{grid}}$ for $\mu a=1$ and $\delta S=S_{\mathrm{fit}}/S_{\text{grid}}$ for $\mu a={10}^{-3}$.

Figure 8

Entanglement entropy of degrees of freedom confined to a single edge that pierces the entangling surface at its midpoint as a function of $\mu a$. The length of the edge allows for 500 oscillator locations equidistant from each other with separation $a$. This guarantees $\mu a\gg 0.002$ in the displayed domain $\mu a\in (0.05,1.5)$ so that finite-size effects are not affecting the entanglement entropy evaluated at the midpoint of the edge. The red curve shows the numerical value obtained with the code [17], while the blue and green curve, respectively, show $S^{\text{plat}}$ and $S^{(3)}$ given in the main text. The background shading indicates the domain of validity of quantum field theory.