Holographic proof of the strong subadditivity of entanglement entropy

When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as strong subadditivity. For a field theory this inequality can be stated as follows: given any two regions of space $A$ and $B$, $S(A)+S(B)\ge S(A\cup B)+S(A\cap B)$. Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. We give a simple geometrical proof of strong subadditivity employing this holographic prescription.

Figure 1

A constant-time slice of a spacetime on which a gravity theory lives, and the conformal boundary on which its holographically dual field theory lives. $A$ is a region of the boundary; $m$ is the minimal hypersurface in the bulk ending on $\partial A$; and $r$ is a region of the bulk such that $\partial r=A\cup m$.

Figure 2

Two examples in which the QFT lives on a compact space and the bulk contains a black hole. (Technically, since we are considering an eternal black hole in static coordinates, in each case the full spacetime consists of two copies of the region shown connected by an Einstein-Rosen bridge; this will not affect our discussion.) The boundary is divided into two regions $A$ and $B$. Since $\partial A=\partial B$, if the bulk had trivial $(d-1)$st homology the corresponding minimal hypersurfaces $m_A$ and $m_B$ would be identical, and we would have $S(A)=S(B)$. However, due to the requirement that each hypersurface be homologous to the corresponding boundary region, $m_B$ can either (top) wrap around the other side of the event horizon, or (bottom) separate into two connected components, one being $m_A$ and the other the event horizon. In the latter case we have $S(B)=S(A)+S_{\mathrm{BH}}$, where $S_{\mathrm{BH}}$ is the black hole’s Bekenstein-Hawking entropy; since $S_{\mathrm{full}}=S_{\mathrm{BH}}$, the Araki-Lieb inequality is saturated.

Figure 3

Two overlapping regions $A$ and $B$ of the boundary, with (top) their respective minimal bulk hypersurfaces $m_A$, $m_B$ and bulk regions $r_A$, $r_B$, and (bottom) their minimal hypersurfaces $m_A$ and $m_B$ cut up and rearranged into two new hypersurfaces $m_{A\cup B}$ (the bulk part of the surface of $r_A\cup r_B$) and $m_{A\cap B}$ (the bulk part of the surface of $r_A\cap r_B$). $m_{A\cup B}$ and $m_{A\cap B}$ end on $\partial (A\cup B)$ and $\partial (A\cap B)$ respectively (although they are not necessarily the minimal such hypersurfaces).