11institutetext: Department of Physics, University of California, Santa
Barbara, CA 93106, USA22institutetext: Center for Theoretical Physics and
Department of Physics, University of California, Berkeley, CA 94720, USA

# Constrained HRT Surfaces and their Entropic Interpretation

Xi Dong<EMAIL_ADDRESS>1 Donald Marolf<EMAIL_ADDRESS>1,2 and Pratik Rath
<EMAIL_ADDRESS>
###### Abstract

Consider two boundary subregions $A$ and $B$ that lie in a common boundary
Cauchy surface, and consider also the associated HRT surface $\gamma_{B}$ for
$B$. In that context, the constrained HRT surface $\gamma_{A:B}$ can be
defined as the codimension-2 bulk surface anchored to $A$ that is obtained by
a maximin construction restricted to Cauchy slices containing $\gamma_{B}$. As
a result, $\gamma_{A:B}$ is the union of two pieces, $\gamma^{B}_{A:B}$ and
$\gamma^{\bar{B}}_{A:B}$ lying respectively in the entanglement wedges of $B$
and its complement $\bar{B}$. Unlike the area
${\mathcal{A}}\left(\gamma_{A}\right)$ of the HRT surface $\gamma_{A}$, at
least in the semiclassical limit, the area
${\mathcal{A}}\left(\gamma_{A:B}\right)$ of $\gamma_{A:B}$ commutes with the
area ${\mathcal{A}}\left(\gamma_{B}\right)$ of $\gamma_{B}$. To study the
entropic interpretation of ${\mathcal{A}}\left(\gamma_{A:B}\right)$, we
analyze the Rényi entropies of subregion $A$ in a fixed-area state of
subregion $B$. We use the gravitational path integral to show that the
$n\approx 1$ Rényi entropies are then computed by minimizing
${\mathcal{A}}\left(\gamma_{A}\right)$ over spacetimes defined by a boost
angle conjugate to ${\mathcal{A}}\left(\gamma_{B}\right)$. In the case where
the pieces $\gamma^{B}_{A:B}$ and $\gamma^{\bar{B}}_{A:B}$ intersect at a
constant boost angle, a geometric argument shows that the $n\approx 1$ Rényi
entropy is then given by $\frac{\mathcal{A}(\gamma_{A:B})}{4G}$. We discuss
how the $n\approx 1$ Rényi entropy differs from the von Neumann entropy due to
a lack of commutativity of the $n\to 1$ and $G\to 0$ limits. We also discuss
how the behaviour changes as a function of the width of the fixed-area state.
Our results are relevant to some of the issues associated with attempts to use
standard random tensor networks to describe time dependent geometries.

##  1 Introduction

In the context of the AdS/CFT correspondence Maldacena_1999 , the connection
between entanglement measures in the boundary theory and areas of surfaces in
the bulk theory has been an active area of research. The Ryu-Takayanagi
formula Ryu:2006bv , and its generalization to time dependent spacetimes by
Hubeny-Rangamani-Takayanagi (HRT) Hubeny:2007xt , led the way in establishing
a duality between the von Neumann entropy of boundary subregion $R$ and the
area of an extremal surface $\gamma_{R}$ anchored to $\partial R$. Here and in
the main text below we restrict discussion to the case where the bulk gravity
theory is described by the Einstein-Hilbert action with minimally-coupled
matter. The AdS/CFT dictionary has also been extended to many more
entanglement measures including the Rényi entropy Headrick:2010zt ;
Hung:2011nu ; Hartman:2013mia ; Faulkner:2013yia ; Lewkowycz:2013nqa ;
Dong:2016fnf , entanglement negativity Kudler-Flam:2018qjo ; Kusuki:2019zsp ;
Dong:2021clv  and reflected entropy Dutta:2019gen ; Akers:2021pvd ;
Akers:2022max ; Akers:2022zxr  among others.

An interesting such proposal was made in Ref. Chen:2018rgz , which considered
a surface that we call the constrained HRT surface $\gamma_{A:B}$. This
$\gamma_{A:B}$ is defined when $A$, $B$ are boundary subregions that lie on a
common boundary Cauchy surface. In that case, the codimension-2 surface
$\gamma_{A:B}$ is defined by minimizing the area of spacelike111While the
spacelike qualifier was not explicitly stated in Ref. Chen:2018rgz , it is
implied by the idea that the area of the relevant surfaces defines a partial
order with respect to which one can minimize. surfaces anchored to $\partial
A$ (and appropriately homologous to $A$) that are extremal everywhere except
at possible intersections with the HRT surface $\gamma_{B}$ for subregion $B$;
see Fig. 1. It was also suggested in Ref. Chen:2018rgz  that, at least under
appropriate conditions, no point on $\gamma_{A:B}$ can be timelike separated
from any point on $\gamma_{B}$ (and we will show this below under the
assumptions of Ref. Wall:2012uf ). As a result, $\gamma_{B}$ defines two
pieces of $\gamma_{A:B}$ which we will call $\gamma^{B}_{A:B}$ and
$\gamma^{\bar{B}}_{A:B}$, that respectively lie in the entanglement
wedges222We define the entanglement wedges to be closed, and in particular
include $\gamma_{B}$. of $B$ and its complement $\bar{B}$. Ref. Chen:2018rgz
proposed that when the intersection between $\gamma^{B}_{A:B}$ and
$\gamma^{\bar{B}}_{A:B}$ defines a constant boost angle, the quantity
$\frac{{\mathcal{A}}\left(\gamma_{A:B}\right)}{4G}$ computes the von Neumann
entropy of $A$ minimized over modular flow applied to subregion $B$. In this
paper, we revisit constrained HRT surfaces finding a different (but closely
related) entropic interpretation for their area.

Figure 1: Left: The HRT surface $\gamma_{B}$ (red) for subregion $B$ (marked
with a long orange double arrow), along with a family of spacelike separated
HRT surfaces (light blue) corresponding to nested subregions can be placed on
the same Cauchy slice. The HRT surface $\gamma_{A}$ (solid, green) for
subregion $A$ (marked with a long purple double arrow) does not generally lie
on the same Cauchy slice. The constrained HRT surface
$\gamma_{A:B}=\gamma^{B}_{A:B}\cup\gamma^{\bar{B}}_{A:B}$ (dashed, green) is
extremal everywhere except at the location where it meets $\gamma_{B}$. Right:
A side view of the same configuration.

In Sec. 2, we provide a maximin construction of the constrained HRT surface.
We also note that
$\\{\mathcal{A}\left(\gamma_{A:B}\right),\mathcal{A}\left(\gamma_{B}\right)\\}=0$,
where $\mathcal{A}$ denotes the area of the specified codimension-2 surface
and $\\{f,g\\}$ is the Poisson/Peierls bracket of observables $f$ and $g$.
This demonstrates that the area operators of $\gamma_{A:B}$ and $\gamma_{B}$
commute in the classical limit.333To define an area operator carefully in the
quantum theory, we need to impose a bulk ultraviolet cutoff to suppress
divergences coming from quantum fluctuations.

We then move on to discuss the entropic interpretation of
${\mathcal{A}}\left(\gamma_{A:B}\right)$. As shown in Fig. 1, the tangent
spaces of $\gamma^{B}_{A:B}$ and $\gamma^{\bar{B}}_{A:B}$ may not be
continuous across $\gamma_{B}$. We now characterize this potential
discontinuity by defining a boost angle. The minimization over possible
intersections with $\gamma_{B}$ in the definition of
${\mathcal{A}}\left(\gamma_{A:B}\right)$ implies that the projections of those
tangent spaces onto directions parallel to $\gamma_{B}$ are actually
continuous across $\gamma_{B}$. So at any given intersection point where
$\gamma^{B}_{A:B}$ meets $\gamma^{\bar{B}}_{A:B}$, we only need to consider
the projections of their tangent spaces onto the timelike plane orthogonal to
$\gamma_{B}$. As long as these two projections are nonzero444There are special
situations where these projections vanish, such as when a connected component
of $\gamma_{A:B}$ coincides with a connected component of $\gamma_{B}$. At
these special intersection points, the boost angle is not defined. and thus
are one-dimensional vector spaces, they define a boost angle that relates one
to the other. Like the conjecture of Ref. Chen:2018rgz , our proposal
considers the case where the boost angle is the same at all intersection
points where the boost angle is defined. It then states that, for
$\epsilon=O(G^{0})$ but small and positive, the $n=1+\epsilon$ Rényi entropy
of subregion $A$ in a fixed-area state of subregion $B$ is given by

$S_{n}(A)=\frac{\mathcal{A}(\check{\gamma}_{A:B})}{4G}+O\left(\frac{\epsilon}{G}\right).$
(1)

Here and below the use of a check in denoting a constrained HRT surface (as in
$\check{\gamma}_{A:B}$) indicates that the boost angle between the pieces
$\gamma^{B}_{A:B}$ and $\gamma^{\bar{B}}_{A:B}$ is in fact independent of the
intersection point.555Ref. Chen:2018rgz  instead used a superscript > for the
constant boost-angle case.

In order to obtain the result Eq. (1), we will first discuss the calculation
of entropies in fixed-area states in Sec. 3. In particular, it was argued in
Ref. Kaplan:2022orm  that the area ${\cal A}(\gamma_{B})$ of an HRT-surface
$\gamma_{B}$ generates a flow on the covariant phase space that acts as
boundary-condition-preserving (BCP) kink transformation which, as described in
Ref. Bousso:2020yxi , inserts a relative boost between the entanglement wedges
of $B$ and $\bar{B}$. As will be discussed in detail below, this means that a
fixed-area state for $\gamma_{B}$ must decompose into a superposition of
states with different relative boosts across $\gamma_{B}$. Furthermore, when
the area is sharply-fixed (so as to define a pseudo-eigenstate in the language
of Ref. Dong:2022ilf ), the superposition will explore a large range of such
relative boosts.

For such a pseudo-eigenstate, and assuming replica symmetry a la Lewkowycz-
Maldacena Lewkowycz:2013nqa , we consider the limit of small $\epsilon$ in
which backreaction from the cosmic brane Dong:2016fnf  at the HRT surface can
be neglected. The associated path integral is then dominated by the choice of
boost angle $s$ that minimizes the area of the associated HRT surface
$\gamma_{A}(s)$. In the case where the pieces $\gamma^{B}_{A:B}$,
$\gamma^{\bar{B}}_{A:B}$ of $\gamma_{A:B}$ intersect at constant boost angle,
a maximin argument along the lines of Ref. Wall:2012uf  shows that the
dominant saddle is the one where the HRT surface $\gamma_{A}(s)$ agrees with
the constrained HRT surface $\gamma_{A:B}$, thus establishing Eq. (1).

In the above context, it is important to distinguish the $n\approx 1$ Rényi
entropy from the von Neumann entropy. The two turn out to behave differently,
thus showing that the $n\to 1$ and $G\to 0$ limits do not commute. While the
$n\approx 1$ Rényi entropy is computed by
$\min_{s}\frac{{\mathcal{A}}\left(\gamma_{A}(s)\right)}{4G}$, the von Neumann
entropy is given by the expectation value
$\langle\frac{{\mathcal{A}}\left(\gamma_{A}(s)\right)}{4G}\rangle$, where the
average is performed over $s$ and is weighted by the probability distribution
defined by the gravitational path integral.

In Sec. 4, we discuss the implications of our work for tensor networks.
Moreover, we discuss the potential addition of higher-derivative and quantum
corrections to our analysis.

##  2 Constrained HRT surfaces

Consider two boundary subregions $A$ and $B$ that can be placed on the same
boundary Cauchy slice. If $A\cap B\neq\emptyset$, the corresponding HRT
surfaces $\gamma_{A}$ and $\gamma_{B}$ generally contain portions that are
timelike separated from each other; see again Fig. 1. As discussed in e.g.
Ref. Bao:2018pvs , this suggests that the areas of these surfaces will fail to
commute with each other in a quantum theory of gravity, i.e.,
$\left[\mathcal{A}\left(\gamma_{A}\right),\mathcal{A}\left(\gamma_{B}\right)\right]\neq
0$; see e.g. Ref. Kaplan:2022orm  for explicit semiclassical examples of such
non-vanishing commutators. As a result, we cannot simultaneously project onto
fixed eigenvalues of the two operators. The classical limit of the above
statement is that the Poisson/Peierls bracket between the two observables is
non-vanishing, i.e.,
$\\{\mathcal{A}\left(\gamma_{A}\right),\mathcal{A}\left(\gamma_{B}\right)\\}\neq
0$.

In contrast, Ref. Chen:2018rgz  defined a constrained analogue of an HRT
surface that we will call $\gamma_{A:B}$. This $\gamma_{A:B}$ was defined as
the codimension-2 surface anchored to $\partial A$ (and appropriately
homologous to $A$) that has minimal area among all such surfaces that are
extremal everywhere except at potential intersections with $\gamma_{B}$. It
was also suggested in Ref. Chen:2018rgz  that, at least under appropriate
conditions, no point on $\gamma_{A:B}$ can be timelike separated from any
point on $\gamma_{B}$.

This motivates us to provide a restricted maximin definition (inspired by the
maximin construction of Ref. Wall:2012uf ) for the area ${\cal
A}(\gamma_{A:B})$ wherein we restrict to Cauchy slices that include all of
$\gamma_{B}$ as well as the boundary anchors defined by $A$. In doing so, we
assume as in Ref. Wall:2012uf  that our spacetime satisfies the null energy
condition, and that bulk Cauchy surfaces $\Sigma$ for which $\partial\Sigma$
is a given Cauchy surface of the asymptotically locally AdS boundary can touch
future and past boundaries only at Kasner-like singularities.666If this last
assumption is dropped, then HRT surfaces may fail to exist, or they may fail
to correctly compute the desired entropies. See Ref. Fischetti:2014uxa  for
discussion and proposals for dealing with such situations.

That such Cauchy slices exist is clear from e.g. the maximin construction of
$\gamma_{B}$ given in Ref. Marolf:2019bgj . On any Cauchy slice
$\Sigma_{B}\supset\gamma_{B}$, we first find the minimal area surface anchored
to $\partial A$. We then maximize over the choice of $\Sigma_{B}$ to obtain a
maximin surface $\gamma^{mm}_{A:B}$. We assume as in Ref. Wall:2012uf  that
the associated stability criterion is satisfied. When the relevant Cauchy
surface is both spacelike and smooth, this can be shown using the technical
argument in Sec. 3.5 of Ref. Wall:2012uf , but it remains an assumption more
generally. With this assumption, the surface $\gamma^{mm}_{A:B}$ exists and
can be studied as in Ref. Wall:2012uf .

The usual arguments are then readily applied near points of
$\gamma^{mm}_{A:B}$ located away from $\gamma_{B}$, showing that
$\gamma^{mm}_{A:B}$ is extremal at such points. Let us refer to the class of
all such extremal-away-from-$\gamma_{B}$ surfaces (including those that do not
intersect $\gamma_{B}$ at all) as ${\cal E}_{\gamma_{B}}$, and let us also
recall that $\gamma_{A:B}$ as defined in Ref. Chen:2018rgz  is the minimal-
area surface777In Chen:2018rgz  it was simply assumed that such a minimum
exists. But we will soon see that $\gamma^{mm}_{A:B}$ must be the minimum, so
a minimum does in fact exist. in ${\cal E}_{\gamma_{B}}$. Thus we obtain the
bound

${\cal A}(\gamma_{A:B})\leq{\cal A}(\gamma^{mm}_{A:B}).$ (2)

We can then also see that $\gamma^{mm}_{A:B}$ must minimize the area within
${\cal E}_{\gamma_{B}}$ using the concept of the representative of
$\gamma_{A:B}$ on an arbitrary Cauchy surface containing $\gamma_{B}$, and in
particular using the Cauchy surface $\Sigma$ on which $\gamma^{mm}_{A:B}$ is
minimal. This representative was defined in Sec. 3.1 of Ref. Wall:2012uf  as
the intersection of $\Sigma$ with a null congruence orthogonal to
$\gamma_{A:B}$. The focussing argument of Ref. Wall:2012uf  again shows that
the area of the representative $r(\gamma_{A:B})$ on $\Sigma$ can be no larger
than the area of $\gamma_{A:B}$. And since $\gamma^{mm}_{A:B}$ is minimal on
$\Sigma$, we must also have

${\mathcal{A}}(\gamma^{mm}_{A:B})\leq{\mathcal{A}}(r(\gamma_{A:B}))\leq{\mathcal{A}}(\gamma_{A:B}).$
(3)

Comparing Eq. (2) and Eq. (3) shows that the areas of $\gamma^{mm}_{A:B}$ and
$\gamma_{A:B}$ must coincide. This will generically also imply that
$\gamma^{mm}_{A:B}$ and $\gamma_{A:B}$ define the same surface, though
equality of their areas will be enough for our purposes. In particular, since
this equality holds in all spacetimes that satisfy the equations of motion,
the areas ${\mathcal{A}}(\gamma_{A:B})$ and ${\mathcal{A}}(\gamma^{mm}_{A:B})$
define identical functions on the phase space of classical solutions.
Focussing on the areas in this way allows a clean way to deal with cases where
the minimal surface is not unique. Indeed, in such non-unique cases we are
always free to choose $\gamma_{A:B}=\gamma^{mm}_{A:B}$, which we will do
below.

For later purposes, we also note that our maximin construction yields the
bound

${\mathcal{A}}(\gamma_{A:B})\leq{\mathcal{A}}(\gamma_{A}).$ (4)

This follows directly from the fact that $\gamma_{A}$ can be obtained from the
similar-but-less-restrictive maximin principle stated in Ref. Wall:2012uf . In
particular, since $\gamma_{A}$ is obtained by maximizing the area over a
strictly larger set of Cauchy surfaces, it must be at least as large as our
restricted maximum ${\mathcal{A}}(\gamma_{A:B})$.

The restricted maximin construction above establishes that the surfaces
$\gamma_{B}$ and $\gamma_{A:B}$ lie in a common Cauchy slice. This opens up
the possibility that
$\\{\mathcal{A}(\gamma_{A:B}),\mathcal{A}(\gamma_{B})\\}=0$. To see that this
is indeed the case, we recall from Ref. Kaplan:2022orm  that, in Einstein-
Hilbert gravity, the flow generated by $\mathcal{A}(\gamma_{B})$ is given by a
BCP kink-transformation. This transformation has a simple description on any
Cauchy slice $\Sigma_{B}$ that includes $\gamma_{B}$. While the induced metric
is invariant under the transformation, the extrinsic curvature shifts
according to888Our flow parameter $s$ corresponds to $-s$ in the convention of
Ref. Kaplan:2022orm .

$\displaystyle K_{\perp\perp}$ $\displaystyle\rightarrow K_{\perp\perp}+2\pi
s\,\delta_{\gamma_{B}}.$ (5)

Here $\delta_{\gamma_{B}}$ is a Dirac delta function localized on $\gamma_{B}$
and $\perp$ represents the normalized vector tangent to $\Sigma_{B}$ but
orthogonal to $\gamma_{B}$; see Ref. Kaplan:2022orm  for details. The initial
data on $\Sigma_{B}$ can then be used to obtain the corresponding spacetime by
solving the equations of motion.

In particular, the causal development of the data on $\Sigma_{B}$ on either
side of $\gamma_{B}$ is unaffected by the transformation (since the change in
the initial data on $\Sigma_{B}$ is localized to $\gamma_{B}$). Furthermore,
since $\gamma_{A:B}$ is nowhere timelike-separated from $\gamma_{B}$, away
from $\gamma_{B}$ it must be entirely contained in that causal development. In
certain cases, there can also be a finite contribution to the area of
$\gamma_{A:B}$ from its intersection with $\gamma_{B}$ (see footnote 4), but
this area will be determined by the induced metric on any Cauchy slice that
contains $\gamma_{B}$, and this induced metric is also invariant under the
transformation. It thus follows that our flow cannot change ${\cal
A}(\gamma_{A:B})$, and thus that the Poisson/Peierls bracket
$\\{\mathcal{A}(\gamma_{A:B}),\mathcal{A}(\gamma_{B})\\}$ must vanish. At the
quantum level, the two areas will then form a commuting set of operators at
least at leading order in $\hbar$.

##  3 Entropic Interpretation

We now analyze the entropic interpretation of the constrained HRT surface
$\gamma_{A:B}$. In order to do so, we will consider the calculation of the
Rényi entropy of subregion $A$ in a state where the area of the HRT surface
$\gamma_{B}$ is fixed polynomially sharply (i.e., the width is polynomially
small in $G$). Using the gravitational path integral, we will explain how the
fixed-area state decomposes into a superposition over BCP kink-transformed
states.999At the quantum level we simply define the BCP kink transform to be
the transformation generated by exponentiating the HRT area
${\mathcal{A}}(\gamma_{B})$. In the case when the intersection between
$\gamma_{A:B}$ and $\gamma_{B}$ happens at a constant boost angle, we will
show that $\frac{{\mathcal{A}}(\gamma_{A:B})}{4G}$ computes the $n\approx 1$
Rényi entropy of subregion $A$ in the fixed-area state of subregion $B$. We
will also clarify the difference between the $n\approx 1$ Rényi entropy and
the entanglement entropy, highlighting an order-of-limits issue which
obstructs us from interpreting the area of the constrained HRT surface as the
von Neumann entropy of a subregion in our fixed-area state.

Consider a CFT state $\ket{\psi}$ whose norm is computed by a Euclidean path
integral with smooth sources on a manifold $\mathcal{M}$. The corresponding
bulk state is prepared using a gravitational path integral with the
appropriate smooth boundary conditions, i.e.,

$\bra{\psi}\ket{\psi}=\int_{\partial\mathcal{B}=\mathcal{M}}Dg\,e^{-I[g]}.$
(6)

In the saddle point approximation, Eq. (6) will be dominated by a spacetime
which satisfies the Einstein equations everywhere and which also satisfies the
boundary condition $\partial\mathcal{B}=\mathcal{M}$. To obtain a fixed-area
state for subregion $B$, we simply restrict this state to a small window of
areas; i.e., we construct

$\ket{\psi_{{\mathcal{A}}_{0}}}=\Pi_{{\mathcal{A}}_{0}}\ket{\psi},$ (7)

where $\Pi_{{\mathcal{A}}_{0}}$ is an operator that sharply restricts the
wavefunction to eigenvalues of
${\mathcal{A}}(\gamma_{B})\equiv{\mathcal{A}}_{B}$ around ${\mathcal{A}}_{0}$.

So long as the above window contains a large number of area-eigenvalues, there
will be an approximate notion of a ‘conjugate’ variable to
${\mathcal{A}}_{B}$. Semiclassically, this conjugate is the relative boost
angle $s_{B}$ across $\gamma_{B}$ Kaplan:2022orm ; Dong:2022ilf  (see also
Refs. Jafferis:2015del ; Bousso:2020yxi ; Witten:2021unn ;
Chandrasekaran:2022eqq ), i.e., we have the uncertainty relation

$\Delta\left(\frac{{\mathcal{A}}_{B}}{4G}\right)\Delta s_{B}\gtrsim O(1).$ (8)

For semiclassical states prepared using a (say, Schwinger-Keldysh)
gravitational path integral, we typically have $\Delta{\mathcal{A}}_{B},\Delta
s_{B}=O(\sqrt{G})$. Reducing the variance in ${\mathcal{A}}_{B}$ increases the
variance in $s_{B}$. However, as long as both $\Delta{\mathcal{A}}_{B}$ and
$\Delta s_{B}$ vanish in the $G\to 0$ limit, the limit is still associated
with a well defined classical geometry. In particular, the Lorentzian
spacetimes associated to such squeezed states were discussed in Ref.
Dong:2022ilf .

On the other hand, for $\Delta{\mathcal{A}}_{B}$ of order $G$ or smaller, the
corresponding $\Delta s_{B}$ fail to vanish as $G\to 0$ and the state is not
semiclassical. We will focus on states of this sort, with the restriction that
$\Delta{\mathcal{A}}_{B}$ vanish no faster than polynomially as $G\to 0$. Thus
we still expect the associated density matrix for each region $B,\bar{B}$ to
have an exponentially large number of eigenvalues. Such states were called
pseudo-eigenstates in Ref. Dong:2022ilf . These are precisely the kind of
states where we will be able to provide an entropic interpretation for
$\gamma_{A:B}$.

To be specific, we can choose to define the operator $\Pi_{{\mathcal{A}}_{0}}$
by using a Gaussian window function

$\Pi_{{\mathcal{A}}_{0}}=f(\hat{{\mathcal{A}}}_{B})=N\exp\left[-\frac{(\hat{x}_{B}-x_{0})^{2}}{2\sigma^{2}}-i\hat{x}_{B}s_{0}\right],$
(9)

where $N$ is a normalization factor, $x=\frac{{\mathcal{A}}}{4G}$, and we take
$s_{0}$ independent of $G$. We will correspondingly refer to
$\Pi_{{\mathcal{A}}_{0}}$ as a window operator. Note that we can also write
Eq. (9) in the form

$f(\hat{{\mathcal{A}}}_{B})=N^{\prime}\int
ds\,\tilde{f}(s)\exp\left[-i\hat{x}_{B}s\right],$ (10)

where $\tilde{f}(s)$ is the Fourier transform of $f$ (up to normalization),
i.e.,

$\tilde{f}(s)=\exp\left[-\frac{\sigma^{2}}{2}(s-s_{0})^{2}+ix_{0}(s-s_{0})\right].$
(11)

As described above, the desired pseudo-eigenstates (9) have $\sigma=O(1)$ or
parametrically smaller in $G$. We will choose to focus on
$\sigma=O(G^{\alpha})$ with some $\alpha>0$. As a result, the width of Eq.
(11) is polynomially large in $1/G$. We also see that the parameter $s_{0}$
determines the peak of the probability distribution for the boost angle.

Figure 2: The boundary-condition-preserving (BCP) kink-transformation modifies
the Cauchy data ${\cal C}$ on a bulk Cauchy slice $\Sigma$ to new data ${\cal
C}_{s}$ by adding a delta function in the codimension-1 extrinsic curvature
component $K_{\perp\perp}$, where as shown $x_{\perp}$ represents the
intrinsic coordinate that measures proper distance on $\Sigma$ in the
direction orthogonal to the HRT surface $\gamma_{B}$. This essentially results
in a shift in the relative boost $s$ across $\gamma_{R}$. The qualifier BCP
emphasizes that the slice $\Sigma$ with the new data ${\cal C}_{s}$ remains
glued to the asymptotic boundary regions $B$ and $\bar{B}$ in precisely the
same way as for $\Sigma$ with the original data ${\cal C}$.

Since the state $\ket{\psi}$ is defined by a gravitational path integral with
smooth boundary-sources, in the semiclassical limit it will be associated with
some spacetime geometry $g_{\psi}$. When we apply Eq. (10) to $\ket{\psi}$, we
will obtain an integral over states
$\ket{\psi(s)}:=e^{-i\hat{x}_{B}s}\ket{\psi}$. These states will also be
semiclassical and will be associated with spacetime geometries $g_{\psi(s)}$
obtained by applying the boundary-condition-preserving (BCP) kink
transformation to $g_{\psi}$ as summarized in Fig. 2. This can be seen, for
example, from the fact that the insertion of $e^{-i\hat{x}_{B}s}$ into the
gravitational path integral is equivalent to including a cosmic brane that
classically sources a Lorentz-signature conical defect. As a result,
Schwinger-Keldysh path integrals that compute correlators in the state
$\ket{\psi(s)}$ can be approximated by saddles that agree with the saddle for
$\ket{\psi}$ near the Euclidean boundaries, but which contain new Lorentz-
signature conical defects at $\gamma_{B}$. A Schwinger-Keldysh path integral
will in fact contain two copies of $\gamma_{B}$, and the desired correlators
are computed by inserting operators in the region between the resulting
conical singularities; see Fig. 3. As a result, at the semiclassical level,
those correlators probe the spacetime $g_{\psi(s)}$ constructed by applying
the BCP kink transformation to $g_{\psi}$ as claimed above.

Figure 3: Left: A saddle for the (Euclidean or complex) path integral for
correlators in $\ket{\psi}$. The blue line indicates a surface with initial
data that defines the associated Lorentz-signature solution and which contains
$\gamma_{B}$. The same Euclidean saddle computes the norm of any
$|\psi(s)\rangle$, since they are related by unitary operators. Right: A
Schwinger-Keldysh contour that attaches a piece of Lorentz-signature spacetime
at the blue surface in fact yields two copies of $\gamma_{B}$. For any fixed
value of $s$, in the semiclassical approximation, the path integral for
correlators in $\ket{\psi(s)}$ inserts the operator $e^{-i\hat{x}_{B}s}$ at
both copies, enacting a relative boost of rapidity $s$ between the initial
data on the half of the blue slice on the closer side of $\gamma_{B}$ and the
data on the farther side. The Lorentzian region (light blue) is then a BCP
kink-transform of the corresponding region for the original $\ket{\psi}$.

The Lorentzian spacetimes $g_{\psi(s)}$ are well understood Bousso:2020yxi .
In Einstein gravity, the BCP kink transformed initial data results in Weyl
curvature delta function shocks along the lightcones of the HRT surface but
the rest of the spacetime is expected to be smooth.

Using Eq. (10), the fixed-area state can be written as

$\displaystyle\ket{\psi_{{\mathcal{A}}_{0}}}$ $\displaystyle=N^{\prime}\int
ds\tilde{f}(s)\ket{\psi(s)}.$ (12)

Correlation functions in the future/past of $\gamma_{B}$ can be computed by
the expression

$\bra{\psi_{{\mathcal{A}}_{0}}}O\ket{\psi_{{\mathcal{A}}_{0}}}=|N^{\prime}|^{2}\int
ds\,ds^{\prime}\,\tilde{f}(s)\,\tilde{f}^{*}(s^{\prime})\,\langle\psi(s^{\prime})|O|\psi(s)\rangle.$
(13)

It is convenient to rewrite the flow parameters as
$\bar{s}=\frac{s+s^{\prime}}{2}$ and $\Delta=\frac{s-s^{\prime}}{2}$. The
overlap $\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}$ may be
evaluated using a Euclidean gravitational path integral, and for a given
$\bar{s}$ the Cauchy-Schwarz inequality makes clear that the overlap is
maximized at $\Delta=0$. And since the overlap is computed by a semiclassical
gravitational path integral, away from $\Delta=0$ it will decay as
$\exp\left[-\frac{\\#\Delta^{2}}{G}\right]$.101010To the extent that we may
replace area-flow by modular flow we may write
$|\bra{\psi(\bar{s}+\Delta)}\ket{\psi(\bar{s}-\Delta)}|^{2}=\left|\tr\big(e^{-(1+2i\Delta)K_{B}^{\psi}}\big{missing})\right|^{2}$
and note that this quantity can be thought of as a natural generalization of
the spectral form factor to modular flow. Such quantities are similarly
expected to decay exponentially with $\Delta^{2}$. In particular,
semiclassical states separated by $\Delta\gg O(\sqrt{G})$ will be
approximately orthogonal. For the interested reader, Appendix A provides an
explicit computation illustrating the analogue of this result in pure JT
gravity, and which shows that the exponential decay is associated with complex
saddle points. See especially the discussion around Eq. (33).

Since $\tilde{f}(\bar{s}-\Delta),\tilde{f}^{*}(\bar{s}+\Delta)$ vary slowly
with $\Delta$, the integral over $\Delta$ will be dominated by a saddle very
close to $\Delta=0$. Setting $\Delta=0$ in Eq. (13) defines a diagonal
approximation that yields a simple picture for the bulk state as a statistical
mixture of the states $\ket{\psi(s)}$. Note that if $\tilde{f}(s)$ was sharply
peaked at $s=s_{0}$, then Eq. (13) would instead be well approximated by a
single spacetime as expected for a semiclassical squeezed state.

We can now consider the reduced density matrix on subregion $A$ in the state
$\ket{\psi_{{\mathcal{A}}_{0}}}$. It is given by

$\rho_{A,{\mathcal{A}}_{0}}=|N^{\prime}|^{2}\int
ds\,ds^{\prime}\tilde{f}(s)\tilde{f}^{*}(s^{\prime})\tr_{\bar{A}}\left(\ket{\psi(s)}\bra{\psi(s^{\prime})}\right).$
(14)

In terms of $\\{\bar{s},\Delta\\}$, the kernel
$\tilde{f}(s)\tilde{f}^{*}(s^{\prime})=\exp\left[-\sigma^{2}\left((\bar{s}-s_{0})^{2}+\Delta^{2}\right)+2ix_{0}\Delta\right]$,
and its magnitude decays slowly in both the $\bar{s}$ and $\Delta$ directions.

We can now take this density matrix and try to compute its $n$th moments for
integer $n$ using the gravitational path integral. For $n=1$, we have

$\tr\left(\rho_{A,{\mathcal{A}}_{0}}\right)=|N^{\prime}|^{2}\int
2d\bar{s}\,d\Delta\,\tilde{f}(\bar{s}+\Delta)\tilde{f}^{*}(\bar{s}-\Delta)\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}.$
(15)

As discussed above, the magnitude
$|\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}|$ of the overlap
$\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}$ is strongly localized
at $\Delta=0$. This suggests that the integral over $\Delta$ can be evaluated
in the saddle point approximation, though we must also take into account that
there will be a rapidly-varying phase set by the most-likely value $x_{\psi}$
of $x_{B}=\frac{{\mathcal{A}}(\gamma_{B})}{4G}$ in the original state
$|\psi\rangle$. The factor
$\tilde{f}(\bar{s}+\Delta)\tilde{f}^{*}(\bar{s}-\Delta)$ also contributes a
rapidly-varying phase that is of the opposite sign, and which is set by the
parameter $x_{0}$ in Eq. (9). For the moment, we will simply choose $x_{0}$ so
that these phases cancel. This is equivalent to taking the window function
$f({\mathcal{A}}(\gamma_{B}))$ to be peaked at the area of the $\gamma_{B}$ in
the classical geometry $g_{\psi}$ associated with $|\psi\rangle$.

On the other hand, the $\bar{s}$ direction is a soft mode over which we must
still integrate. For integer $n>1$, to the extent that we can ignore any
possible breaking of replica symmetry, we can use the Lewkowycz-Maldacena
method Lewkowycz:2013nqa  for every choice of $\\{\bar{s},\Delta\\}$ to reduce
the problem to the computation of the action of a solution with a cosmic brane
of tension $T_{n}=\frac{n-1}{4nG}$ in the gravitational path integral
satisfying the same boundary conditions as the calculation for $n=1$. This
yields111111One might expect Eq. (16) to integrate over $n$ copies of
$\bar{s},\Delta$ but in the saddle point approximation it is sufficient to
integrate over one copy as in Eq. (16) at the cost of neglecting subleading
$O\left(\log G\right)$ contributions to the Rényi entropy.

$\tr\left(\rho_{A,{\mathcal{A}}_{0}}^{n}\right)\approx|N^{\prime}|^{2n}\int
2d\bar{s}\,d\Delta\,\tilde{f}(\bar{s}+\Delta)^{n}\tilde{f}^{*}(\bar{s}-\Delta)^{n}\bra{\psi(\bar{s}-\Delta)}O_{n}\ket{\psi(\bar{s}+\Delta)}^{n},$
(16)

where $O_{n}$ represents the operator inserting the cosmic brane of tension
$T_{n}$. We will be interested in computing Eq. (16) for $n\approx 1$ where we
can neglect the backreaction caused by the cosmic brane, giving us

$\tr\left(\rho_{A,{\mathcal{A}}_{0}}^{n}\right)\approx|N^{\prime}|^{2n}\int
2d\bar{s}\,d\Delta\,\tilde{f}(\bar{s}+\Delta)^{n}\tilde{f}^{*}(\bar{s}-\Delta)^{n}e^{-(n-1)\frac{{\mathcal{A}}(\gamma_{A}(\bar{s},\Delta))}{4G}}\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}^{n},$
(17)

where $\gamma_{A}(\bar{s},\Delta)$ is the HRT surface for $A$ in the saddle
corresponding to fixed $\bar{s},\Delta$ (which, as discussed above, will be
complex for $\Delta\neq 0$). There will again be a sharp decay in the $\Delta$
direction due to the overlap
$\bra{\psi(\bar{s}-\Delta)}\ket{\psi(\bar{s}+\Delta)}^{n}$ while the factor
$\tilde{f}(\bar{s}+\Delta)^{n}\tilde{f}^{*}(\bar{s}-\Delta)^{n}$ has a slowly-
varying magnitude. Thus we may again use saddle point methods to set
$\Delta=0$, leaving us only with the integral over $\bar{s}$. We will write
$\gamma_{A}(\bar{s},\Delta=0)$ simply as $\gamma_{A}(\bar{s})$ below.

When $n=1+\epsilon$ with $\epsilon=O(1)$, we can use a further saddle point
approximation to evaluate Eq. (17) since the soft mode is then sharply
localized by the contribution ${\mathcal{A}}(\gamma_{A}(\bar{s}))$ from the
cosmic brane action. In particular, since $1/\sigma$ is large we will assume
that ${\mathcal{A}}(\gamma_{A}(\bar{s}))$ is minimized at some
$\bar{s}_{\text{min}}$ such that $|\bar{s}_{\text{min}}-s_{0}|\ll 1/\sigma$ .
We may thus write

$\tr\left(\rho_{A,{\mathcal{A}}_{0}}^{n}\right)\approx|N^{\prime}|^{2n}\tilde{f}(\bar{s}_{\text{min}})^{n}\tilde{f}^{*}(\bar{s}_{\text{min}})^{n}e^{-(n-1)\frac{{\mathcal{A}}(\gamma_{A}(\bar{s}_{\text{min}}))}{4G}}.$
(18)

Note that this is a general result relating the Rényi entropy of
$\rho_{A,{\mathcal{A}}_{0}}$ to the area of
$\gamma_{A}(\bar{s}_{\text{min}})$.

We now connect the above result to the constrained HRT surface $\gamma_{A:B}$.
To do so, we specialize to the case $\gamma_{A:B}=\check{\gamma}_{A:B}$ where
$\gamma^{B}_{A:B}$ intersects $\gamma^{\bar{B}}_{A:B}$ at a constant boost
angle $\check{s}$. We then claim that $\bar{s}_{\text{min}}=-\check{s}$, and
that the area of $\gamma_{A}(\bar{s}_{\text{min}})$ is precisely that of
$\check{\gamma}_{A:B}$. Furthermore, in a given classical geometry the
rapidity $\check{s}$ will be independent of $G$. As a result, since $1/\sigma$
is parametrically large, we will indeed find $|\bar{s}_{\text{min}}-s_{0}|\ll
1/\sigma$.

To show this, we first prove that ${\mathcal{A}}(\check{\gamma}_{A:B})$ gives
the area of the HRT surface $\gamma_{A}(-\check{s})$ in the geometry
$g_{\psi(-\check{s})}$. This $g_{\psi(-\check{s})}$ is by definition the
result of applying a BCP-kink transformation with boost parameter $-\check{s}$
to $g_{\psi}$. Since this does not change either the entanglement wedge of $B$
or that of $\bar{B}$, the spacetime $g_{\psi(-\check{s})}$ contains extremal
surfaces corresponding to both $\gamma^{B}_{A:B}$ and $\gamma^{\bar{B}}_{A:B}$
that are anchored both to the boundary and to a common locus on $\gamma_{B}$.
Furthermore, due to the relative boost by $-\check{s}$, their tangents are now
continuous across $\gamma_{B}$. And since extremal surfaces satisfy a second
order partial differential equations, it follows that they in fact join
smoothly across $\gamma_{B}$ to form a surface that is everywhere extremal.
This surface also satisfies the same homology constraint as
$\gamma_{A}(-\check{s})$ and, indeed, it must be the minimal such surface on
some Cauchy slice $\Sigma$ that contains $\gamma_{B}$ (since our maximin
construction ensured that this was the case for the original $\gamma_{A:B}$).
As a result, the representative argument of Sec. 3.1 of Ref. Wall:2012uf
again shows that, in a spacetime satisfying the null energy condition, there
can be no smaller extremal surface that is both anchored and homologous to
$A$. It now follows immediately from Eq. (4) that the choice
$\bar{s}=-\check{s}$ minimizes the area of the surfaces $\gamma_{A}(\bar{s})$
as claimed, and thus that

$S_{1+\epsilon}(\rho_{A,{\mathcal{A}}_{0}})\approx\frac{{\mathcal{A}}(\check{\gamma}_{A:B})}{4G},$
(19)

where we have ignored logarithmic corrections arising from the prefactors in
Eq. (18).

Thus far we have considered only the case where the window function
$f({\mathcal{A}}(\gamma_{B}))$ is peaked at the area of the surface
$\gamma_{B}$ in the classical geometry $g_{\psi}$ associated with the state
$|\psi\rangle$. However, the case of more general window functions is now
straightforward. We may simply choose the operator $\Pi_{{\mathcal{A}}_{0}}$
to be a product of two separate (commuting) window operators. We take the
first to be of the form (9) with $s_{0}=0$, some choice of $x_{0}$, and a
width $\sigma_{1}$ of order $G^{-\alpha_{1}}$ for $1/2>\alpha_{1}>0$. Acting
with the associated $\Pi_{{\mathcal{A}}_{0},1}$ thus produces what Ref.
Dong:2022ilf  called a squeezed fixed-area-state
$|\psi_{1}\rangle:=\Pi_{{\mathcal{A}}_{0},1}|\psi\rangle$ with area
${\mathcal{A}}_{0}$. We then take the second window operator
$\Pi_{{\mathcal{A}}_{0},2}$ to again be of the form (9), with the same
$x_{0}$, but now with arbitrary $s_{0}$ (that is independent of $G$) and with
a width $\sigma_{2}$ of order $G^{\alpha_{2}}$ for some $\alpha_{2}>0$. Note
that the product
$\Pi_{{\mathcal{A}}_{0}}=\Pi_{{\mathcal{A}}_{0},2}\Pi_{{\mathcal{A}}_{0},1}$
is again of the form (9) with the above values of $x$, $s_{0}$ and with a
width slightly smaller than $\sigma_{2}.$ Thus this final operator
$\Pi_{{\mathcal{A}}_{0}}$ prepares the desired pseudo-eigenstate.

The point of this two-step construction is that the squeezed state
$|\psi_{1}\rangle$ is still semiclassical and, in particular, it is still
well-approximated by a single classical geometry of the form described in Ref.
Dong:2022ilf . We emphasize that this geometry solves the usual classical
equations of motion without additional sources. In particular, while the
Euclidean (or complex) saddle associated with a path integral definition of
this state will have a conical singularity at $\gamma_{B}$, there is no
conical singularity in the Lorentzian solution. While the Lorentzian solution
will generally contain power law divergences near the null congruences
orthogonal to $\gamma_{B}$, these satisfy the equations of motion; see Ref.
Dong:2022ilf  for details.

Since $|\psi_{1}\rangle$ is semiclassical, and since it is now peaked at a
value of ${\mathcal{A}}(\gamma_{B})$ that is very close to $4Gx_{0}$, we can
analyze the pseudo-eigenstate
$\Pi_{{\mathcal{A}}_{0}}|\psi\rangle=\Pi_{{\mathcal{A}}_{0},2}|\psi_{1}\rangle$
by repeating the argument given above involving Eqs. (13)–(19) replacing the
state $|\psi\rangle$ by $|\psi_{1}\rangle$ and $\Pi_{{\mathcal{A}}_{0}}$ by
$\Pi_{{\mathcal{A}}_{0},2}$. The addition of power law divergences does not
significantly affect the discussion. In particular, since we are focused here
on Einstein-Hilbert gravity, the areas of codimension-2 surfaces remain finite
and well-defined (see Ref. Dong:2022ilf ). And while the most likely value
$x_{1}$ of $\hat{x}_{B}$ in $|\psi_{1}\rangle$ will differ slightly from
$x_{0}$, one may check that the difference is sufficiently small that setting
$\Delta=0$ in Eq. (17) remains a good approximation to the saddle-point
result.121212Or, alternatively, one can adjust the peak of the window function
for $\Pi_{{\mathcal{A}}_{0},1}$ so as to place the peak of $|\psi_{1}\rangle$
at the desired value $x_{0}$. We thus again arrive at the desired result (19).

###  3.1 Density matrix interpretation

We will now interpret the above results in terms of the structure of the
reduced density matrix (14) on subregion $A$. In doing so, we will see that
much of the above story is reproduced by applying standard bounds on Rényi and
von Neumann entropies.

We will take as input two lessons from the discussion above. The first lesson
is that the density matrix (14) is nearly diagonal in $s$. This fact is in
strong accord with the expectation that in the bulk semiclassical limit, $s$
describes a boost angle at $\gamma_{B}$. As a result, in this limit it can be
measured locally in both the entanglement wedge of $A$ and in that of
$\bar{A}$. The second lesson, however, is that this diagonality is only
approximate. Indeed, it arises only in the semiclassical limit. As a result,
the quantum state $\ket{\psi}$ has an $O(\sqrt{G})$ width in the boost angle
$s_{B}$. This width is then shared by each $|\psi(s)\rangle$, so that the
$s,s^{\prime}$ terms in the density matrix (14) for subregion $A$ superpose
coherently over any $s$-interval with width smaller than $O(\sqrt{G})$. We
therefore expect that Eq. (14) can be approximated by choosing a discrete set
${\cal S}$ of $\bar{s}$-values and considering the state
$\tilde{\rho}_{A,{\mathcal{A}}_{0}}$ below:

$\rho_{A,{\mathcal{A}}_{0}}\approx\tilde{\rho}_{A,{\mathcal{A}}_{0}}:=\sum_{\bar{s}\in{\cal
S}}c(\bar{s})|\tilde{f}(\bar{s})|^{2}\tilde{\rho}_{A,\bar{s}},$ (20)

where $\tilde{\rho}_{A,\bar{s}}$ is a reduced density matrix on subregion $A$
obtained from Eq. (14) by inserting an additional Gaussian window function
$e^{-\left[(s-\bar{s})^{2}+(s^{\prime}-\bar{s})^{2}\right]/2\alpha^{2}}$ of
width $\alpha\sim\sqrt{G}$ and normalizing the result.131313In Eq. (20) we did
not include contributions from slightly off-diagonal terms (i.e., from $s$,
$s^{\prime}$ that are within a few standard deviations of each other).
Including these would contribute a subleading $O\left(\log G\right)$ term to
the Rényi entropy, similar to what was explained in footnote 11. The factor
$c(\bar{s})$ is a slowly-varying positive function defined by a combination of
the factor involved in the above renormalization of $\tilde{\rho}_{A,\bar{s}}$
and the local density of $\bar{s}$ values in ${\cal S}$. What is important for
us is that $c(\bar{s})$ will again be polynomial in $G$. The coefficients
$c(\bar{s})|\tilde{f}(\bar{s})|^{2}$ play the role of a probability
distribution over the boost parameter $\bar{s}$.

The ansatz (20) describes the fixed-area reduced density matrix as a mixture
of reduced density matrices on subregion $A$ associated with different values
of the flow parameter $\bar{s}$. Note that we have not required the density
matrices $\tilde{\rho}_{A,\bar{s}}$ to be orthogonal to each other in any way.
Non-zero overlaps are allowed, and we will account for such overlaps in our
analysis below. We expect the approximation (20) to be accurate in the limit
where the set ${\cal S}$ becomes dense, though for later use we will assume
that the number of points in ${\cal S}$ is taken to be at most polynomially
large in $1/G$.

The width $\alpha\sim O(\sqrt{G})$ is optimal since making it much sharper or
wider would typically reduce the width of either ${\mathcal{A}}_{B}$ or
$s_{B}$. Since neither of these quantities commutes with
${\mathcal{A}}(\gamma_{A})$, either case would make it difficult to have a
semiclassical formula for entropies of $A$.

We can now use Eq. (20) to compute Rényi entropies of $A$. In particular, we
have the inequality for a mixture nielsen2001majorization

$\frac{1}{1-n}\log(\sum_{\bar{s}}c(\bar{s})|\tilde{f}({\bar{s}})|^{2}e^{-(n-1)S_{n}(\tilde{\rho}_{A,{\bar{s}}})})\leq
S_{n}(\tilde{\rho}_{A,{\mathcal{A}}_{0}})\leq\frac{1}{1-n}\log(\sum_{\bar{s}}c(\bar{s})^{n}|\tilde{f}({\bar{s}})|^{2n}e^{-(n-1)S_{n}(\tilde{\rho}_{A,\bar{s}})}),$
(21)

which follows from Schur concavity of the Rényi entropy. We emphasize that the
bounds hold certainly for the density matrix
$\tilde{\rho}_{A,{\mathcal{A}}_{0}}$, but that additional so-far-uncontrolled
errors can enter if we wish to interpret Eq. (21) as placing bounds on the
original density matrix $\rho_{A,{\mathcal{A}}_{0}}$. Thus the current
discussion represents an interpretation of our earlier results, in which the
approximation of $\rho_{A,{\mathcal{A}}_{0}}$ by Eq. (20) will be justified a
posteriori by using Eq. (21) to reproduce Eq. (19).

Since each individual density matrix $\tilde{\rho}_{A,\bar{s}}$ has
$O(\sqrt{G})$ fluctuations, it is natural to expect that the associated von
Neumann entropy can be analyzed a la Lewkowycz-Maldacena Lewkowycz:2013nqa ,
and thus that for $n=1+\epsilon$, the Rényi entropies are computed by a cosmic
brane prescription and that backreaction from the brane gives only an
$O(\frac{\epsilon}{G})$ correction. Thus we write

$S_{n}(\tilde{\rho}_{A,\bar{s}})=\frac{{\mathcal{A}}(\gamma_{A,\bar{s}})}{4G}+O\left(\frac{\epsilon}{G}\right),$
(22)

where $\gamma_{A,\bar{s}}$ is the HRT surface in the classical geometry
$g_{\psi(\bar{s})}$ corresponding to $\ket{\psi(\bar{s})}$. The sums in Eq.
(21) involve a slowly varying factor (from
$c(\bar{s})|\tilde{f}(\bar{s})|^{2}$) multiplied by a factor
$\exp\left[-\frac{\epsilon\gamma_{A,\bar{s}}}{4G}\right]$ whose variation is
parametrically more rapid. Thus, the sum is dominated by the term with the
smallest area which, as discussed before, is given by
${\mathcal{A}}(\check{\gamma}_{A:B})$ in the case where $\gamma^{B}_{A:B}$,
$\gamma^{\bar{B}}_{A:B}$ meet at a constant boost angle (which we now assume).

Approximating the sums in Eq. (21) by integrals and using the fact that
$\tilde{f}(\bar{s})$ is Gaussian, we find that the LHS and RHS differ by two
kinds of terms: an $O\left(\frac{\sigma^{2}}{\epsilon}\right)$ difference
arising from the shift in the peak of the Gaussian and $O(\log G)$ corrections
from the prefactors. Thus, putting everything together we have

$\displaystyle S_{1+\epsilon}(\rho_{A,{\mathcal{A}}_{0}})$
$\displaystyle=\frac{{\mathcal{A}}(\check{\gamma}_{A:B})}{4G}+O\left(\frac{\epsilon}{G}\right)+O\left(\frac{\sigma^{2}}{\epsilon}\right)+O\left(\log
G\right).$ (23)

In particular, we see that while the errors are small for
$\epsilon\gg\sigma^{2}$, the errors become large in the strict $\epsilon\to 0$
limit required to compute the von Neumann entropy. This non-commutativity of
the $n\to 1$ and $G\to 0$ limits is analogous to that described in
Akers:2020pmf  associated with computations of entropy near an HRT-phase
transition.

To compute the von Neumann entropy we can use the analogous inequality
nielsen2001quantum

$\sum_{\bar{s}}c(\bar{s})|\tilde{f}(\bar{s})|^{2}S(\tilde{\rho}_{A,\bar{s}})\leq
S(\tilde{\rho}_{A,{\mathcal{A}}_{0}})\leq\sum_{\bar{s}}c(\bar{s})|\tilde{f}(\bar{s})|^{2}S(\tilde{\rho}_{A,\bar{s}})-\sum_{\bar{s}}c(\bar{s})|\tilde{f}(\bar{s})|^{2}\log\left[c(\bar{s})|\tilde{f}(\bar{s})|^{2}\right].$
(24)

The LHS and RHS differ by a Shannon term which is bounded above by $\log k$
where $k$ is the number of relevant terms in the sum. For our case, we can
truncate the sum to the region $|\bar{s}-s_{0}|\leq\frac{1}{\sigma}$. Since
the size of the set ${\cal S}$ of $\bar{s}$ values used in the sum in Eq. (20)
was taken to be at most polynomial in $1/G$, this Shannon term is at most
$O(\log G)$. Note that, since $\sigma$ is at most polynomial in $1/G$, our
restriction on the size of ${\cal S}$ still allows an arbitrary polynomially-
large density of $\bar{s}$ values. Thus, we have

$S(\rho_{A,{\mathcal{A}}_{0}})=\sum_{\bar{s}}c(\bar{s})|\tilde{f}(\bar{s})|^{2}S(\tilde{\rho}_{A,\bar{s}})+O(\log
G).$ (25)

Since we expect $S(\tilde{\rho}_{A,\bar{s}})$ to be given by an HRT-area as
described above, the result (25) can be interpreted as the expectation value
of this area (up to $O(\log G)$ corrections). This can be thought of as the
application of the HRT formula to a superposition of spacetimes with a widely
spread probability distribution Almheiri:2016blp ; Harlow:2016vwg ;
Akers:2018fow .

To summarize, the key point in the above calculation is that the state (7) has
a soft mode associated with the boost angle across the HRT surface. When
evaluating the $n\approx 1$ Rényi entropy, the soft mode is stabilized by the
dependence of ${\mathcal{A}}(\gamma_{A}(\bar{s}))/4G$ on this boost angle. As
a result, the integral can be evaluated using a saddle point approximation at
the smallest value of this area. However, for the entanglement entropy, we
need to perform the full integral over the soft mode. The final answer is thus
not given by the area of a single surface in a single geometry.

A final important point is the dependence of the above calculation on the area
width $\sigma$ and the parameter $s_{0}$. In calculating the $n\approx 1$
Rényi entropy we have assumed that $\bar{s}=-\check{s}$ is within a few
standard deviations from the maximum $s_{0}$ in the probability distribution
for $s$. This is a natural assumption in the limit where the distribution in
$s$ becomes broad. However, if we choose the original width $\sigma$ of
${\mathcal{A}}_{B}/4G$ to be larger than $O(1)$, then the width in $s$ is
small and, in particular, is likely to be small compared with the separation
between $s_{0}$ and $-\check{s}$. In that case we are considering squeezed
states for which case the Gaussian distribution over $\bar{s}$ remains sharp
in the semiclassical limit and we instead obtain the usual HRT formula for the
spacetime associated with the boost $\bar{s}=s_{0}$. On the other hand, if we
choose the width $\sigma$ to be non-perturbatively small in $G$ then, even
though the distribution in $s$ will in some sense be broad enough to access
the regime near $-\check{s}$, the fact that we may then be sensitive to the
discreteness of area-eigenvalues implies that there can be no good notion of a
conjugate variable $s$. In addition, the $O(\log G)$ error terms in Eq. (25)
become very large and it is difficult to claim any precise result for the
entropies. (Note that a similar comment applies to the logarithmic corrections
to Eq. (19) from the neglected prefactors in Eq. (18).) This is natural since
such a regime is far from being semiclassical. What is interesting, then, is
that there is an intermediate regime where $\sigma$ is polynomially small in
$G$ that yields a broad superposition over spacetimes with different
$\bar{s}$-values for which the $n\approx 1$ Rényi entropies can still be
obtained from a semiclassical computation in a single spacetime. This is in
agreement with the expectation of Ref. Almheiri:2016blp  that the area
operator behaves linearly for a small superposition over spacetimes.

##  4 Discussion

The above work considered the constrained HRT surfaces defined in Ref.
Chen:2018rgz  and found an entropic interpretation for their areas in
situations where their failure to be smooth is defined by a single constant
boost angle. This is closely related to the results of Ref. Chen:2018rgz
which interpreted modular flow as a disentangler. We made use of the fact that
a fixed-area state defined by a boundary subregion $B$ automatically explores
a wide range of BCP kink transformations. The action of modular flow on a
semiclassical geometry is well approximated by the BCP kink transformation
jlms . Thus, a fixed-area state effectively explores a wide range of modular
flows applied to the original semiclassical state from which the fixed-area
state was obtained. Consequently, computations of the $n\approx 1$ Rényi
entropy for another boundary subregion $A$ are dominated by values of the flow
parameter that minimize the area ${\mathcal{A}}(\gamma_{A})$ of the HRT
surface $\gamma_{A}$. Interestingly, this differs from the behavior of the von
Neumann entropy $S(A)$, which is one of the qualitative differences between
our entropic interpretation and that of Ref. Chen:2018rgz .

As a small technical comment, our introduction of a possible non-trivial
oscillatory phase $\exp{-is_{0}{\mathcal{A}}_{B}/4G}$ in the window operator
$\Pi_{{\mathcal{A}}_{0}}$ used to construct the fixed-area state (7)
generalizes the class of fixed-area states that have been explicitly described
in the past. In particular, in the analysis of Lorentzian geometries for
squeezed states (so that $\sigma$ is no smaller than $O(1)$) performed in Ref.
Dong:2022ilf , our $s_{0}$ was implicitly set to zero. However, including the
effect of $s_{0}$ was not difficult, especially since it was noted in Sec. 3
that this phase constitutes a source for a Lorentz-signature conical defect as
described in Fig. 3.

We will now comment on various possible corrections to Eq. (19). In the case
where the classical geometry $g_{\psi}$ of $|\psi\rangle$ is smooth and we
apply a window operator $\Pi_{{\mathcal{A}}_{0}}$ with ${\mathcal{A}}_{0}$
equal to the area ${\mathcal{A}}(\gamma_{B})$ in $g_{\psi}$, higher derivative
corrections to the bulk Einstein-Hilbert action will be straightforward to
include if the corrected geometric entropies continue to generate BCP kink
transformations. As will be discussed in Ref. geom , this is the case for some
class of higher derivative theories. Further issues appear at general values
of ${\mathcal{A}}_{0}$, as the associated power law divergences become more
singular in the presence of higher-derivative corrections. It is not clear how
to accommodate this in our calculation, though it is plausible that an
analogous result would follow from a regulated calculation in which one smears
out the fixed-area surface over a codimension-zero region. In that context,
there should be no divergences.

In attempting to include quantum corrections, a natural extension of our
proposal would be to replace fixed-area states by states where we restrict the
boundary modular Hamiltonian to a small window of eigenvalues. Such a
restricted state can then be expressed as a superposition over modular flowed
states. However, the precise bulk dual of such states is not well understood
since restricting the bulk modular Hamiltonian would require non-local sources
acting on the entire entanglement wedge, unlike the localized conical defect
source for fixed-area states. It would be interesting to understand this
better in the future as well.

We would also like to discuss the limitations of our analysis. We were able to
provide an entropic interpretation for $\gamma_{A:B}$ only when it intersects
$\gamma_{B}$ at a constant boost angle. As pointed out in Ref. Chen:2018rgz ,
this case is not generic. Even in cases with two boundary dimensions, it can
fail when the region $A$ contains multiple disconnected intervals. It would be
interesting to find a generalization of fixed-area states that would allow one
to explore position-dependent boosts, thus opening up the possibility of a
more general entropic interpretation for constrained HRT surfaces.

Our results also have implications for attempts to understand the random
tensor networks of Ref. Hayden:2016cfa  as models of general holographic
states. Such networks have a flat Rényi spectrum for each boundary subregion.
For any fixed subregion $B$, it is thus natural to associate such networks
with bulk states that have a fixed area for $\gamma_{B}$ Akers:2018fow ;
Dong:2018seb . In gravity, due to a lack of commutativity of
${\mathcal{A}}(\gamma_{B})$ with ${\mathcal{A}}(\gamma_{A})$ for other
boundary subregions $A$, it is difficult to fix many such areas simultaneously
Bao:2018pvs . One might therefore try to proceed by studying alternative
operators designed to better commute with ${\mathcal{A}}(\gamma_{B})$; see
also related discussion in Ref. examples . But we see here that doing so with
${\mathcal{A}}(\gamma_{A:B})$ can lead to new issues, such as a failure for
the $n\rightarrow 1$ and $G\rightarrow 0$ limits to commute, with the result
that the new operator ${\mathcal{A}}(\gamma_{A:B})$ fails to represent a von
Neumann entropy after fixing the area of $\gamma_{B}$.

Finally, we note that the methods used in this paper can also be applied to
geodesics in arbitrary dimensions that compute correlation functions in fixed-
area states. For instance, consider the microcanonical thermofield double
state Marolf:2018ldl , where the energy has been fixed to some polynomially
small width in $G$. The left-right two-point correlation function of heavy
operators in this state would be similarly computed by a constrained geodesic
intersecting the bifurcate horizon by arguments similar to those made here.
This comment is closely related to the ideas of Ref. Faulkner:2018faa .

###### Acknowledgements.

We would like to thank Chris Akers, Luca Iliesiu, Geoff Penington and Arvin
Shahbazi-Moghaddam for useful discussions. PR is supported in part by a grant
from the Simons Foundation, and by funds from UCSB. This material is based
upon work supported by the Air Force Office of Scientific Research under award
number FA9550-19-1-0360. This work was supported in part by the Berkeley
Center for Theoretical Physics; by the Department of Energy, Office of
Science, Office of High Energy Physics under QuantISED Award DE-SC0019380 and
under contract DE-AC02-05CH11231. PR was supported by the National Science
Foundation under Award Number 2112880.

##  Appendix A JT model

In this appendix, we consider pure JT gravity. This theory was canonically
quantized in Ref. Harlow:2018tqv . Equivalently, it can be described by the
gravitational path integral restricted to the disk topology Yang:2018gdb . The
phase space variables can be chosen to be $\phi_{h}$, the value of the dilaton
at the horizon, and the conjugate variable $s$ which corresponds to the boost
across the horizon. Another useful phase space variable is the renormalized
geodesic length $L$ between the two boundaries.

We will treat $\phi_{h}$ below analogously to the HRT area of $B$ discussed in
the main text. We will also take $L$ to be an analogue of the HRT area of $A$.
Of course, $L$ has no entropic interpretation in 2D gravity but it may still
serve as a useful model for higher dimensional contexts where codimension-2
surfaces with an entropic interpretation can reach from boundary to boundary.

First we can define a fixed-area state using a Gaussian window function as

$\ket{\phi_{0}}=\int
d\phi_{h}\,e^{-\frac{(\phi_{h}-\phi_{0})^{2}}{2\sigma^{2}}}\ket{\phi_{h}}.$
(26)

The analogue of our main-text computation of Rényi entropies for subregion $A$
is then to consider a correlation function of the form

$\displaystyle S_{n}(A)$
$\displaystyle=\bra{\phi_{0}}e^{-(n-1)\frac{L}{4\beta}}\ket{\phi_{0}}$ (27)
$\displaystyle=\int
d\phi_{h}\,d\phi_{h}^{\prime}\,dL\,e^{-(n-1)\frac{L}{4\beta}}e^{-\frac{(\phi_{h}-\phi_{0})^{2}}{2\sigma^{2}}}e^{-\frac{(\phi_{h}^{\prime}-\phi_{0})^{2}}{2\sigma^{2}}}\psi_{\phi_{h}}(L)\,\psi^{*}_{\phi_{h}^{\prime}}(L),$
(28)

where we have used the fact that $\beta$ plays the role of $G$ in JT gravity
and where $\psi_{\phi_{h}}(L)=\bra{L}\ket{\phi_{h}}$ is the wavefunction of a
dilaton eigenstate in the $L$ basis. We have also set $\phi_{b}=1$.

The wavefunctions were written down explicitly in Ref. Harlow:2018tqv  and are
given by Bessel functions. However, for our purposes, it will be sufficient to
mention only a few properties of $\psi_{\phi_{h}}(L)$. The first is that
$\psi_{\phi_{h}}(L)$ decays rapidly below the classical turning point $L_{c}$
which corresponds to the length of the geodesic on the time symmetric surface
in the associated classical geometry. Thus, it can be taken to vanish for
$L<L_{c}$. The second is that for $L>L_{c}$ this $\psi_{\phi_{h}}(L)$ is well-
approximated by $e^{-2i\phi_{h}L}+Re^{2i\phi_{h}L}$, where $R$ is a pure
phase. This simple form is associated with the fact that the conjugate
variable to $\phi_{h}$ can be described by a relative boost $s$ between the
left and right wedges, and also with the fact that (unless $s$ is very small),
the length $L$ increases linearly with $|s|$. In particular, the fact that
there are two branches stems from the fact that relative boosts of $\pm s$
lead to the same value of $L$.

We will first perform the integrals over $\phi_{h}$ and $\phi_{h}^{\prime}$.
Using the saddle point approximation, one finds that we are left with

$\bra{\phi_{0}}e^{-(n-1)L/4\beta}\ket{\phi_{0}}\approx\int_{L_{c}}^{\infty}dL\,|I(L)|^{2}e^{-(n-1)L/4\beta},$
(29)

where $L_{c}=-2\log\left(\phi_{h}\right)$ is the smallest renormalized length
within the classically allowed region and $I(L)$ is given by

$I(L)\approx
e^{-2L^{2}\sigma^{2}}\left(e^{-2iL\phi_{0}}+Re^{2iL\phi_{0}}\right).$ (30)

Thus, we see that we have a wide Gaussian that decays with width
$\frac{1}{\sigma}$. Finally, the term $e^{-(n-1)\frac{L}{4\beta}}$ for any
$n-1=O(1)$ creates a pressure to pick up the smallest value of $L$ within the
classically allowed region. This is precisely the geodesic at the time
symmetric slice which is the analogue of the constrained HRT surface. On the
other hand, we could also have evaluated the analog of the entanglement
entropy $\bra{\phi_{0}}\frac{L}{4\beta}\ket{\phi_{0}}$ which would be given by
the average value of $L$, instead of the length of a single geodesic.

In order to make a further connection with our calculation of interest, we can
re-express the fixed-area state in terms of a superposition over BCP kink-
transformed states. The terms arising in the computation of the Rényi entropy
then take the form $\bra{\psi(s)}e^{-\bar{\epsilon}L}\ket{\psi(s^{\prime})}$
where $\bar{\epsilon}=\frac{n-1}{4\beta}$. Using the thermofield double state
for $\ket{\psi}$, one of the terms (obtained from incoming waves) takes the
form

$\displaystyle\bra{\psi(s_{1})}e^{-\bar{\epsilon}L}\ket{\psi(s_{2})}$
$\displaystyle\supset$ $\displaystyle\int
d\phi_{1}\,d\phi_{2}\,dL\,\exp[2\pi\phi_{1}-\frac{\beta\phi_{1}^{2}}{2}+2\pi\phi_{2}-\frac{\beta\phi_{2}^{2}}{2}-2i\phi_{1}L+2i\phi_{2}L$
(31) $\displaystyle+$ $\displaystyle
i\phi_{1}s_{1}-i\phi_{2}s_{2}-\bar{\epsilon}L\Bigr{]},$ (32)

where we have used the form of the Hartle-Hawking state. Evaluating the
integrals in the saddle point approximation, we have

$\displaystyle\bra{\psi(s_{1})}e^{-\bar{\epsilon}L}\ket{\psi(s_{2})}\supset\exp\left[\frac{4\pi^{2}}{\beta}-\frac{1}{4}(s_{1}+s_{2})\bar{\epsilon}+\frac{\beta\bar{\epsilon}^{2}}{16}-\frac{(s_{1}-s_{2})^{2}}{4\beta}+\frac{2\pi
i(s_{1}-s_{2})}{\beta}\right],$ (33)

where we have used the complex saddle at
$\phi_{1}=\frac{2\pi}{\beta}+i\left(\frac{s_{1}-s_{2}}{2\beta}+\frac{\bar{\epsilon}}{4}\right)$,
$\phi_{2}=\frac{2\pi}{\beta}+i\left(\frac{s_{1}-s_{2}}{2\beta}-\frac{\bar{\epsilon}}{4}\right)$
and $L=\frac{s_{1}+s_{2}}{4}-\frac{\beta\bar{\epsilon}}{8}$. It is easy to see
that Eq. (33) vanishes for $|s_{1}-s_{2}|\gg O(\sqrt{\beta})$ as claimed. As
discussed previously, $\beta$ plays the role of $G$ in this theory. Moreover,
the imaginary part in the exponent leads to wild oscillations when
$\Delta:=\frac{s_{2}-s_{1}}{2}\neq 0$. Note that a similar term with the
opposite phase arises in the computation of the Rényi entropy as well, and the
overall integral is purely real. We also see that there is a pressure to
decrease the value of $\bar{s}=\frac{s_{1}+s_{2}}{2}$ until the classical
turning point $L_{c}$. The Rényi entropy is then obtained by the integral over
$s_{1},s_{2}$ with a wide Gaussian kernel. In effect, we have shown that this
calculation is dominated by the diagonal terms where $\Delta=0$ and at the
value $\bar{s}=0$ corresponding to the constrained HRT surface.

In this approximation, one encounters another kind of “interference” term
where we use the part of the wavefunction containing the reflected wave for
one of the terms. We do not expect a saddle of this kind in general. In the
approximation we worked in, this term is given by

$\displaystyle\bra{\psi(s_{1})}e^{-\bar{\epsilon}L}\ket{\psi(s_{2})}$
$\displaystyle\supset$ $\displaystyle\int
d\phi_{1}\,d\phi_{2}\,dL\,R\exp[2\pi\phi_{1}-\frac{\beta\phi_{1}^{2}}{2}+2\pi\phi_{2}-\frac{\beta\phi_{2}^{2}}{2}$
(34) $\displaystyle+$ $\displaystyle
2i\phi_{1}L+2i\phi_{2}L+i\phi_{1}s_{1}-i\phi_{2}s_{2}-\bar{\epsilon}L\Bigr{]}.$
(35)

One can check that the naive saddle for this integral is located far from the
original contour, so that it is outside the regime of validity of our
approximation. Instead, a careful analysis involving the exact wavefunctions
$\psi_{\phi_{h}}(L)$ would be required to determine if the original contour
can be deformed so as to access this saddle in a useful way.

However, we will follow a different route since we are only interested in
working perturbatively around $\bar{\epsilon}=0$. At $\bar{\epsilon}=0$, one
can do the exact calculation in Eq. (31) without introducing a complete set of
$L$ eigenstates. One can then check that we obtain precisely the result of Eq.
(33) in the saddle point approximation at $\bar{\epsilon}=0$. This shows that
at $\bar{\epsilon}=0$, the interference term (34) does not contribute. Since
we are working perturbatively in $\bar{\epsilon}$, this continues to be true
at higher orders in $\bar{\epsilon}$. In summary, we have explicitly
demonstrated the diagonal approximation in JT gravity.

## References

  * (1) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113 [hep-th/9711200].
  * (2) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602 [hep-th/0603001].
  * (3) V.E. Hubeny, M. Rangamani and T. Takayanagi, _A Covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062 [0705.0016].
  * (4) M. Headrick, _Entanglement Renyi entropies in holographic theories_ , _Phys. Rev. D_ 82 (2010) 126010 [1006.0047].
  * (5) L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, _Holographic Calculations of Renyi Entropy_ , _JHEP_ 12 (2011) 047 [1110.1084].
  * (6) T. Hartman, _Entanglement Entropy at Large Central Charge_ , 1303.6955.
  * (7) T. Faulkner, _The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT_ , 1303.7221.
  * (8) A. Lewkowycz and J. Maldacena, _Generalized gravitational entropy_ , _JHEP_ 08 (2013) 090 [1304.4926].
  * (9) X. Dong, _The Gravity Dual of Renyi Entropy_ , _Nature Commun._ 7 (2016) 12472 [1601.06788].
  * (10) J. Kudler-Flam and S. Ryu, _Entanglement negativity and minimal entanglement wedge cross sections in holographic theories_ , _Phys. Rev. D_ 99 (2019) 106014 [1808.00446].
  * (11) Y. Kusuki, J. Kudler-Flam and S. Ryu, _Derivation of holographic negativity in AdS 3/CFT2_, _Phys. Rev. Lett._ 123 (2019) 131603 [1907.07824].
  * (12) X. Dong, X.-L. Qi and M. Walter, _Holographic entanglement negativity and replica symmetry breaking_ , _JHEP_ 06 (2021) 024 [2101.11029].
  * (13) S. Dutta and T. Faulkner, _A canonical purification for the entanglement wedge cross-section_ , _JHEP_ 03 (2021) 178 [1905.00577].
  * (14) C. Akers, T. Faulkner, S. Lin and P. Rath, _Reflected entropy in random tensor networks_ , _JHEP_ 05 (2022) 162 [2112.09122].
  * (15) C. Akers, T. Faulkner, S. Lin and P. Rath, _The Page curve for reflected entropy_ , _JHEP_ 06 (2022) 089 [2201.11730].
  * (16) C. Akers, T. Faulkner, S. Lin and P. Rath, _Reflected entropy in random tensor networks. Part II. A topological index from canonical purification_ , _JHEP_ 01 (2023) 067 [2210.15006].
  * (17) Y. Chen, X. Dong, A. Lewkowycz and X.-L. Qi, _Modular Flow as a Disentangler_ , _JHEP_ 12 (2018) 083 [1806.09622].
  * (18) A.C. Wall, _Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy_ , _Class. Quant. Grav._ 31 (2014) 225007 [1211.3494].
  * (19) M. Kaplan and D. Marolf, _The action of HRT-areas as operators in semiclassical gravity_ , _JHEP_ 08 (2022) 102 [2203.04270].
  * (20) R. Bousso, V. Chandrasekaran, P. Rath and A. Shahbazi-Moghaddam, _Gravity dual of Connes cocycle flow_ , _Phys. Rev. D_ 102 (2020) 066008 [2007.00230].
  * (21) X. Dong, D. Marolf, P. Rath, A. Tajdini and Z. Wang, _The spacetime geometry of fixed-area states in gravitational systems_ , _JHEP_ 08 (2022) 158 [2203.04973].
  * (22) N. Bao, G. Penington, J. Sorce and A.C. Wall, _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_ , _JHEP_ 11 (2019) 069 [1812.01171].
  * (23) S. Fischetti, D. Marolf and A.C. Wall, _A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors_ , _Class. Quant. Grav._ 32 (2015) 065011 [1409.6754].
  * (24) D. Marolf, A.C. Wall and Z. Wang, _Restricted Maximin surfaces and HRT in generic black hole spacetimes_ , _JHEP_ 05 (2019) 127 [1901.03879].
  * (25) D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, _Relative entropy equals bulk relative entropy_ , _JHEP_ 06 (2016) 004 [1512.06431].
  * (26) E. Witten, _Gravity and the crossed product_ , _JHEP_ 10 (2022) 008 [2112.12828].
  * (27) V. Chandrasekaran, G. Penington and E. Witten, _Large N algebras and generalized entropy_ , 2209.10454.
  * (28) M.A. Nielsen and G. Vidal, _Majorization and the interconversion of bipartite states._ , _Quantum Inf. Comput._ 1 (2001) 76.
  * (29) C. Akers and G. Penington, _Leading order corrections to the quantum extremal surface prescription_ , _JHEP_ 04 (2021) 062 [2008.03319].
  * (30) M.A. Nielsen and I.L. Chuang, _Quantum computation and quantum information_ , _Phys. Today_ 54 (2001) 60.
  * (31) A. Almheiri, X. Dong and B. Swingle, _Linearity of Holographic Entanglement Entropy_ , _JHEP_ 02 (2017) 074 [1606.04537].
  * (32) D. Harlow, _The Ryu–Takayanagi Formula from Quantum Error Correction_ , _Commun. Math. Phys._ 354 (2017) 865 [1607.03901].
  * (33) C. Akers and P. Rath, _Holographic Renyi Entropy from Quantum Error Correction_ , _JHEP_ 05 (2019) 052 [1811.05171].
  * (34) X. Dong, D. Marolf and P. Rath, _The JLMS Formula, Modular Flow and the Area Operator_ , to appear.
  * (35) X. Dong, D. Marolf and P. Rath, _Geometric entropies and their geometric flow_ , to appear.
  * (36) P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, _Holographic duality from random tensor networks_ , _JHEP_ 11 (2016) 009 [1601.01694].
  * (37) X. Dong, D. Harlow and D. Marolf, _Flat entanglement spectra in fixed-area states of quantum gravity_ , _JHEP_ 10 (2019) 240 [1811.05382].
  * (38) J. Held, M. Kaplan, D. Marolf and J.-q. Wu, _Link-area commutators in AdS 3 Area-networks_, to appear.
  * (39) D. Marolf, _Microcanonical Path Integrals and the Holography of small Black Hole Interiors_ , _JHEP_ 09 (2018) 114 [1808.00394].
  * (40) T. Faulkner, M. Li and H. Wang, _A modular toolkit for bulk reconstruction_ , _JHEP_ 04 (2019) 119 [1806.10560].
  * (41) D. Harlow and D. Jafferis, _The Factorization Problem in Jackiw-Teitelboim Gravity_ , _JHEP_ 02 (2020) 177 [1804.01081].
  * (42) Z. Yang, _The Quantum Gravity Dynamics of Near Extremal Black Holes_ , _JHEP_ 05 (2019) 205 [1809.08647].