Constraining gravity using entanglement in AdS/CFT

Abstract

We investigate constraints imposed by entanglement on gravity in the context of holography. First, by demanding that relative entropy is positive and using the Ryu-Takayanagi entropy functional, we find certain constraints at a nonlinear level for the dual gravity. Second, by considering Gauss-Bonnet gravity, we show that for a class of small perturbations around the vacuum state, the positivity of the two point function of the field theory stress tensor guarantees the positivity of the relative entropy. Further, if we impose that the entangling surface closes off smoothly in the bulk interior, we find restrictions on the coupling constant in Gauss-Bonnet gravity. We also give an example of an anisotropic excited state in an unstable phase with broken conformal invariance which leads to a negative relative entropy.

Figures (3)

• Figure 1: (colour online) For $d>2$ we get the allowed $n_1,n_2$ region to be the blue triangle above for a generic stress tensor. The region above the blue solid line and below the blue dashed and dotted lines are allowed from the relative entropy positivity. For $d\rightarrow \infty$ the region collapses to a line $0\leq n_1\leq 1$ indicated in green. The Einstein value $(n_1,n_2)=(\frac{1}{2},-\frac{1}{8(d-1)})$ is shown by the black dot. The region below the solid red line and above the dashed and dotted red lines are allowed by the null energy condition. By turning on a generic component of the stress tensor only the Einstein value is picked out. By switching off certain components of the stress tensor, various bands bounded by the solid, dashed and dotted lines are picked out.
• Figure 2: Negative of the function $f(x_2')$ is plotted which is a positive valued function
• Figure 3: Comparison between the various constraints on the GB coupling. The length of the line represents the range of allowed $\lambda$.