Strong subadditivity and the covariant holographic entanglement entropy formula

TL;DR

The work investigates whether the covariant holographic entanglement entropy formula (HRT) respects strong subadditivity (SSA) in time-dependent holographic states by analyzing planar AdS$_3$-Vaidya spacetimes. The authors compute boundary interval entropies via spacelike geodesics, framing SSA through the concavity and monotonicity of the length function $L(l)$ and examining the role of the null energy condition (NEC). They find SSA holds for positive-energy shells (NEC satisfied) but can be violated for negative-energy shells, with non-concave $L(l)$ and negative $I_2$ tracing the breakdown. These results provide strong, concrete evidence for the SSA validity of HRT in dynamical settings and highlight NEC as a potential key constraint guiding a general covariant SSA proof and its implications for the area–entropy connection in gravity.

Abstract

Headrick and Takayanagi showed that the Ryu-Takayanagi holographic entanglement entropy formula generally obeys the strong subadditivity (SSA) inequality, a fundamental property of entropy. However, the Ryu-Takayanagi formula only applies when the bulk spacetime is static. It is not known whether the covariant generalization proposed by Hubeny, Rangamani, and Takayanagi (HRT) also obeys SSA. We investigate this question in three-dimensional AdS-Vaidya spacetimes, finding that SSA is obeyed as long as the bulk spacetime satisfies the null energy condition. This provides strong support for the validity of the HRT formula.

Figures (15)

• Figure 1: Penrose diagram for a thin-shell positive energy Vaidya space. The 45-degree red line is the infalling null shell. The vertical blue line is the asymptotic AdS boundary, and the black curve on the top is $r=0$ singularity. The dashed 45-degree lines are BTZ's horizons and their extension back to AdS region. The dotted curves indicate constant radius tracks. A series of coordinate transformations are imposed on BTZ region, to match the coordinates in AdS along the mass-shell. The asymptotic AdS boundary of BTZ spacetime is intentionally set to be vertical. However the black curved singularity $r=0$ is in general not able to keep horizontal straight simultaneously.
• Figure 2: Several geodesics(green curves) are shown in the figure. Any symmetric geodesic with the symmetric point in the shaded region falls to the singularity, while all the others will eventually approach the asymptotic AdS boundary. The constant time (orange) curve through the crossing point of the horizon and mass-shell is drawn for comparison. The second geodesic from top is interesting, that it passes through behind horizon region and reaches asymptotic AdS boundary. Some numeric detail is $v_0=-0.4$, and $p_x=\{0.5,0.65,0.75,0.95\}$ respectively for the geodesics from top to bottom.
• Figure 3: $L(l_x)$ for pure spacelike interval at $t_b=\{0.5,1.1,2\}$ on the boundary, respectively from bottom to top. The black curves are $L(l_x)$ functions for BTZ and pure AdS, from top to bottom respectively.
• Figure 4: With $(l_A,l_B,l_C)=(0.5,1,0.5)$, we move the pure spacelike intervals on boundary from AdS to BTZ. $I_2$ is always positive. The green part is for AdS, and the red part is for the case that all the relevant geodesics are in BTZ space.
• Figure 5: A series of geodesics ending on AdS boundary at $t_b=\{0.5, 1, 1.3, 2, 2.5, 3, 5\}$ from top to bottom respectively. Red curves are calculated from $r_c>1$, the blue ones are from $\frac{1}{2}<r_c^2<1$, and the brown come from $r_c^2<\frac{1}{2}$. Clearly the middle three with $t_b=\{1.3,2,2.5\}$ are combination from two cases. The top and bottom black curves are given for BTZ and pure $AdS_3$ respectively.