In this section we follow the beautiful review \cite{Almheiri:2020cfm}.
The most fundamental result concerning classical black holes is the Bekenstein-Hawking entropy formula \cite{Bekenstein:1972tm,Bekenstein:1973ur}. This states that the entropy $S_{B-H}$ of a classical black hole is proportional to the area $A$ of the event horizon, viz
\begin{eqnarray}
S_{\rm B-H}=\frac{A}{4\hbar G_N}.
\end{eqnarray}
By including the entropy $S_{\rm outside}$ of matter and gravitons in the outside region of the black hole we obtain the Bekenstein-Hawking total entropy
\begin{eqnarray}
S_{\rm gen}=\frac{A}{4\hbar G_N}+S_{\rm outside}.
\end{eqnarray}
For classical black holes the area always increases in time and hence we obtain the second law of thermodynamics. In fact, we obtain the second law of thermodynamics even if we include the outside entropy, viz
\begin{eqnarray}
\Delta S_{\rm gen}\geq 0.
\end{eqnarray}
However, it was shown by Hawking that a quantum black holes evaporate at a temperature $T$ proportional to the surface gravity $\kappa$ of the black hole \cite{Hawking:1974rv,Hawking:1974sw}. More precisely, we have
\begin{eqnarray}T=\frac{\hbar \kappa}{2\pi}.\end{eqnarray}
In other words, surface gravity and the area of the horizon are conjugate variables in general relativity in the same way that temperature and entropy are conjugate variables in thermodynamic.
Thus, an evaporating quantum black hole should be described instead by the quantum von Neumann-Landau entropy defined in terms of the density matrix $\rho$ by the standard formula
\begin{eqnarray}S_{vN}=-{\rm Tr}\rho ln\rho.\end{eqnarray}
This vanishes for a pure state, i.e. for $\rho=|\psi\rangle\langle \psi|$ we have $S_{vN}=0$. Thus, $S_{vN}$ measures the degree of mixing of the quantum state, i.e. our ignorance about the quantum state of the system. This entropy is precisely Shannon's entropy of information.
The Bekenstein-Hawking entropy should be thought of as the coarse-grained Gibbs thermodynamic entropy (measures Boltzmann's number of microscopic states of the black hole). In contrast von Neumann entropy is thought of as the fine-grained entropy of the black hole (measures Bell's quantum entanglement characterizing the quantum state of the black hole). The coarse-grained entropy is obtained from the fine-grained entropy by a maximization procedure over all possible choices of density matrices. Thus, we must have
\begin{eqnarray}S_{vN}\leq S_{\rm B-H}.\end{eqnarray}
The generalized entropy should also be thought of as a coarse-grained thermodynamic entropy.
In contrast to the coarse-grained Bekenstein-Hawking entropy which can only increase in time, if there is no black hole evaporation, the fine-grained von Neumann entropy can both increase or decrease in time after the start of the evaporation process of the black hole.
Summary $1$: An evaporating black hole can be viewed, by an outside observer located at infinity, as a unitary quantum system with a total number of degrees of freedom given by the area term $A/4\hbar G_N$ of the Bekenstein-Hawking entropy. This area term is precisely the logarithm of the dimension of the Hilbert space of quantum states of the black hole.
The most important discovery regarding von Neumann entropy in the past 20 years, which was originally motivated by the gauge/gravity duality and quantum entanglement, is the result that the fine-grained von Neumann entropy of quantum systems coupled to gravitational theories can be computed using the so-called Ryu-Takayanagi formula \cite{Ryu:2006bv,Hubeny:2007xt} which is a quantum generalization of the Bekenstein-Hawking fromula where the horizon is replaced with the so-called quantum extremal surfaces.
In this formulation a codimension-2 surface $X$ of area $A(X)$ is used to compute the generalized entropy given by the formula
\begin{eqnarray}S_{\rm gen}(X)&=&\frac{A(X)}{4\hbar G_N}+S_{\rm semi-cl}(\Sigma_X).\end{eqnarray}
$\Sigma_X$ is the region bounded by a cutoff surface (an arbitrarily chosen surface demarcating the black hole system and separating it from its Hawking radiation) and the codimension-two surface $X$. The entropy $S_{\rm semi-cl}(\Sigma_X)$ is then the von Neumann entropy of the quantum fields of matter and gravitons which are propagating on the classical geometry of the region $\Sigma_X$ (semi-classical approximation). This fine-grained quantum entropy is therefore computed using the density matrix $\rho_{\Sigma_X}$ of the region $\Sigma_X$, viz
\begin{eqnarray}S_{\rm semi-cl}(\Sigma_X)&=&S_{\rm vN}(\rho_{\Sigma_X}).\end{eqnarray}
The codimension-2 surface $X$ is a surface which has two dimensions less than the embedding spacetime and it is chosen in such a way that the generalized entropy is minimized in the spatial direction but maximized in the temporal direction. In other words, this surface is in fact an extremal surface, i.e. the generalized entropy takes an extremal value on this so-called ''quantum extremal surface''. This extremal value of the generalized entropy is precisely the fine-grained von Neumann entropy of the black hole system, viz
\begin{eqnarray}S_{b-h}&=&{\rm min}_X\{{\rm ext}_X\big[S_{\rm gen}[X]\big]\}\nonumber\\&=&{\rm min}_X\{{\rm ext}_X\bigg[\frac{A(X)}{4\hbar G_N}+S_{\rm semi-cl}(\Sigma_X)\bigg]\}.\end{eqnarray}
The quantum extremal surface can be shown to lie behind the event horizon. There are two different surfaces which dominate respectively the early and late times of the evaporation process. At early times the quantum extremal surface is found to be the vanishing surface, i.e. a trivial surface of zero size. At late times the quantum extremal surface is found to be a non-vanishing surface which lies just behind the event horizon.
The so-called ''entanglement wedge'' of this fine-grained entropy is the region bounded between the cutoff surface and the quantum extremal surface, i.e. it includes only a portion of the interior of the black hole. The degrees of freedom of the black hole in this region describe the geometry up to the extremal surface.
In more detail, at very early times there is only the vanishing quantum extremal surface. Thus, at these very early times, the entropy of the area term is zero. Furthermore, since no Hawking modes has enough time to escape the black hole region at these very early times, the semi-classical von Neumann entropy of the matter enclosed by the cutoff and the vanishing surfaces is zero because there is no quantum entanglement.
At early times when some Hawking radiation has the chance to escape the black hole region the semi-classical fine-grained von Neumann entropy of the matter modes enclosed by the cutoff and the vanishing surfaces becomes non-zero due to the quantum entanglement between these inner modes and the outer modes of the Hawking radiation. This entropy increases as the black hole evaporates and thus as more Hawking modes escape or more matter modes accumulate. This increasing semi-classical entropy dominates the generalized entropy as the entropy of the area term always vanishes for the vanishing surface. In other words, the generalized entropy of the vanishing quantum extremal surface is increasing in time.
However, at some early time another quantum extremal surface appears. This surface is non-vanishing and time-dependent and lies close to the event horizon. This surface can be found as follows. At time $t$ on the cutoff surface (which determines how much Hawking radiation has escaped) we must go backward an amount of time of the order of the so-called srcambling time $r_s\ln S_{B-H}$ and then shoot a light ray towards the event horizon. It is near the intersection point behind the event horizon where the non-vanishing surface is found. The entropy of the area term is now non-zero given precisely by the entropy of the black hole which is decreasing in time as the area of the black hole is constantly shrinking due to the evaporation process. The semi-classical von Neumann entropy is clearly negligible compared to the Bekenstein-Hawking entropy . In other words, the generalized entropy of the non-vanishing quantum extremal surface is decreasing in time.
In summary, the generalized entropy of the vanishing surface is dominated by the area term and is increasing while the generalized entropy of the non-vanishing surface is dominated by the semi-classical entropy and is decreasing. Thus, the vanishing surface gives the minimum at early times whereas the non-vanishing surface gives the minimum at late times and as a consequence the generalized entropy goes through a maximum at some time called the Page time. In other words, the generalized entropy follows the so-called Page curve where a phase transition at the Page time occurs between the vanishing and non-vanishing surfaces.
This behavior can be seen more explicitly as follows. For a generic surface $X$ (which starts on the horizon) the area term is always decreasing as we move the surface inward along a null direction whereas the semi-classical entropy starts by decreasing (since the included interior modes purify the exterior modes inside the black hole region) and then becomes increasing (since at this stage the interior modes are entangled with the Hawking modes outside the black hole region). It is in this regime where the area term is decreasing while the semi-classical entropy is increasing and thus their derivatives can be balanced, i.e. the switch or phase transition from the vanishing surface to the non-vanishing surface occurs.
By employing the assumption of unitarity we know that the black hole system and the Hawking radiation system should be described by a pure state. This means that the fine-grained entropy of the Hawking radiation should be equal to the fine-grained entropy of the black hole which must always be less than the Bekenstein-Hawking entropy, viz
\begin{eqnarray}S_{\rm rad}={\rm S}_{\rm b-h}\leq S_{B-H}\end{eqnarray}
As we have discussed, the fine-grained entropy region (entanglement wedge) of the black hole system is bounded between the cutoff surface and the quantum extremal surface $X$.
Naively, the fine-grained entropy region (entanglement wedge) of the Hawking radiation system should be located beyond the cutoff surface where Hawking radiation has escaped and where gravity can be neglected and spacetime is flat. However, and as it turns out, the entanglement wedge of the Hawking radiation is a disconnected surface which contains, in addition to the region beyond the cutoff surface, the region inside the black hole behind the quantum extremal surface $X$.
The degrees of freedom in the Hawking radiation are obviously entangled with the degrees of freedom in the interior of the black hole. Thus, the fine-grained von Neumann entropy of the emitted Hawking radiation which has escaped beyond the cutoff surface increases steadily in the early stages of the evaporation. As we accumulate more radiation the entropy keeps rising until it reaches in value the Bekenstein-Hawking entropy which defines the maximum number of degrees of freedom contained originally in the black hole, i.e. it defines the maximum number of degrees of freedom which the radiation can be entangled with. The time at which the fine-grained entropy of the radiation ceases increasing and starts decreasing is precisely the Page time and it is the time when the quantum extremal surface $X$ changes or jumps from the trivial or vanishing surface to the non-trivial quantum extremal surface which lies just behind the horizon.
The fine-grained von Neumann entropy of the radiation contains no area term and the semi-classical entropy of the region $\Sigma_{\rm rad}$ beyond the cutoff surface is always increasing. The area can be increased and the semi-classical entropy $S_{\rm semi-cl}(\Sigma_{\rm rad})$ can be decreased if we modify the region where we compute the entropy, i.e. the entanglement wedge of the radiation in such a way that it becomes a disconnected region (the area term becomes thus increasing) which contains entangled matter in far away parts (the semi-classical entropy becomes increasing).
The only obvious solution to achieve this outcome is to modify the entanglement wedge of the radiation in such a way that it includes a portion of the interior of the black hole. More precisely, the region $\Sigma_{\rm rad}$ is replaced with the disconnected region $\Sigma_{X}^{'}=\Sigma_{\rm rad}+\Sigma_{\rm island}$ where $\Sigma_{\rm island}$ represents the portion of the black hole interior which must be included in the entanglement wedge of the radiation. This region $\Sigma_{\rm island}$ is also known as an ''island" and it is centered around the origin. It appears a time $r_s\log S_{B-H}$ (scrambling time) after the formation of the black hole. The boundary of the island, i.e. the boundary of the disconnected region $\Sigma_{X}^{'}$ is precisely given by the quantum extremal surface $X$ and thus the area term is given as before by $A(X)/4\hbar G_N$.
The fine-grained von Neumann entropy of the radiation is then given explicitly by
\begin{eqnarray}S_{\rm rad}&=&{\rm min}_X\{{\rm ext}_X\big[S_{\rm gen}[X]\big]\}\nonumber\\&=&{\rm min}_X\{{\rm ext}_X\bigg[\frac{A(X)}{4\hbar G_N}+S_{\rm semi-cl}(\Sigma_{\rm rad}+\Sigma_{\rm island})\bigg]\}.\end{eqnarray}
This is the von Neumann entropy of the exact quantum state of the radiation computed using only the state of the radiation in the semi-classical approximation. This island formula can be derived from the gravitational path integral. Also, it is not difficult to show that the von Neumann entropies of the Hawing radiation and of the black hole system are identically equal. Indeed, we have $S_{\rm rad}=S_{\rm b-h}$ since the area term is the same in both cases and by using the properties of quantum entanglement we have $S_{\rm semi-cl}(\Sigma_{\rm rad}+\Sigma_{\rm island})=S_{\rm semi-cl}(\Sigma_X)$.
We can check that the von Neumann entropy of the radiation as defined in terms of the island by the above formula will decrease at late times. In fact, this entropy follows exactly the Page curve. This von Neumann entropy is the minimum of the vanishing and non-vanishing islands contributions. The increasing segment for early times corresponds to the no-island contribution whereas the decreasing segment for late times corresponds to the with-island contribution.
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