Holographic entanglement entropy

At leading semiclassical order, holographic entanglement entropy is the area of a bulk extremal surface anchored on the boundary of the chosen region, divided by $4G_N$.

For a static holographic state and boundary region $A$,where ${\gamma }_A$ is the least-area bulk surface homologous to $A$ and anchored on $\partial A$.

For a round disk of radius $R$ in the vacuum of a holographic conformal field theory on flat space, the static Poincare-AdS Ryu--Takayanagi surface is the hemisphere $r^2+z^2=R^2$.

The Brown--Henneaux central charge is the central charge of the two-dimensional conformal field theory dual to Einstein gravity on ${\mathrm{AdS}}_3$:

For a two-dimensional conformal field theory on a spatial circle of circumference $L$, with propagation speed $v$ and left/right central charges $c_L,c_R$, the plane-vacuum sector has Casimir energyWhen $c_L=c_R=c$, the unit-radius convention $L=2\pi$, $\hslash =v=1$ gives $E_0=-c/12$. A circle of unit circumference instead gives $E_0=-\pi c/6$ in these units. Specifying radius versus circumference prevents a factor-of-$2\pi$ ambiguity.

The exponential map from a cylinder to the complex plane shifts the stress-energy tensor by its Schwarzian derivative term. Equivalently, the cylinder Hamiltonian is $H=(2\pi \hslash v/L)[L_0+{\bar{L}}_0-(c_L+c_R)/24]$. Setting both plane-vacuum weights to zero yields the formula. Other sectors can have different lowest weights. In AdS/CFT correspondence, the unit-radius result matches the renormalized vacuum energy of global three-dimensional Anti-de Sitter spacetime.^[ref]

A Hubeny–Rangamani–Takayanagi surface is the covariant extremal-area surface homologous to a boundary region. Its area divided by $4G_N$ gives leading semiclassical entanglement entropy in a time-dependent holographic spacetime.

A unitary acting on only one factor of a thermofield double state preserves the spectrum of either reduced density matrix and hence preserves left--right entanglement. A bulk shell created on one boundary can enlarge the late event horizon while the HRT surface remains at the earlier bifurcation surface.

For two copies of a quantum system,Tracing out either copy produces the thermal density operator $e^{-\beta H}/Z$.

The modular Hamiltonian of a density operator is $K=-\log \rho$. For a Gibbs state, $K=\beta H+\log Z$.

Linearizing the Von Neumann entropy around a reference state gives $\delta S=\mathrm{Tr}(\delta \rho K)$. Around a thermal state this becomes $\delta S=\beta \delta ⟨H⟩$.

Holographic mutual information is nonnegative because the disconnected union ${\gamma }_A\cup {\gamma }_B$ is an admissible surface for $A\cup B$, whose minimizing surface can have no larger area.

As two boundary regions separate, the minimizing surface for their union can switch from connected to disconnected. At leading order this produces a continuous entropy with a kink and makes holographic mutual information vanish beyond a critical separation.