# A World without Pythons would be so Simple

Netta Engelhardt  Geoff Penington  and Arvin Shahbazi-Moghaddam

###### Abstract

We show that bulk operators lying between the outermost extremal surface and
the asymptotic boundary admit a simple boundary reconstruction in the
classical limit. This is the converse of the Python’s lunch conjecture, which
proposes that operators with support between the minimal and outermost
(quantum) extremal surfaces – e.g. the interior Hawking partners – are highly
complex. Our procedure for reconstructing this “simple wedge” is based on the
HKLL construction, but uses causal bulk propagation of perturbed boundary
conditions on Lorentzian timefolds to expand the causal wedge as far as the
outermost extremal surface. As a corollary, we establish the Simple Entropy
proposal for the holographic dual of the area of a marginally trapped surface
as well as a similar holographic dual for the outermost extremal surface. We
find that the simple wedge is dual to a particular coarse-grained CFT state,
obtained via averaging over all possible Python’s lunches. An efficient
quantum circuit converts this coarse-grained state into a “simple state” that
is indistinguishable in finite time from a state with a local modular
Hamiltonian. Under certain circumstances, the simple state modular Hamiltonian
generates an exactly local flow; we interpret this result as a holographic
dual of black hole uniqueness.

##  1 Introduction

Recent developments in the black hole information frontier have pointed to a
holographic geometrization of the degrees of freedom of the Hawking radiation
[1, 2, 3, 4, 5, 6]. For an AdS black hole evaporating into a bath, the
“entanglement wedge of the radiation” after the Page time includes a large
part of the black hole interior, bounded by the minimal quantum extremal
surface (QES) [7].

This geometric description of the information naturally accounts for both the
Page curve [8] and the Hayden-Preskill decoding criterion [9]. It also leads
to a geometrical explanation for the expectation of Harlow-Hayden [10] that
decoding Hawking radiation should be exponentially complex. Even though the
interior degrees of freedom lie on the radiation side of the minimal QES and
so lie within the radiation entanglement wedge, they are still hidden behind a
nonminimal QES; in the case of the single-sided black hole, the nonminimal QES
is simply the empty set. The region between the nonminimal and minimal
extremal surfaces was dubbed “the Python’s lunch” in [6], because appropriate
Cauchy slices in the bulk (quantum) geometry have a constriction at each
extremal surface, together with a bulge in the middle (the eponymous “lunch”).
See Fig. 1 for an illustration.

Figure 1: An illustration of the Python’s lunch. On the left, the geometry of
a Cauchy slice featuring the titular python’s lunch between the two
constrictions, the dominant QES $X_{\mathrm{min}}$ and the non-minimal QES
$X$, both of which lie behind the causal surface $C$. On the right, a
spacetime diagram of the same.

The claim of [6] was that any bulk operator with support in the interior of a
Python’s lunch should be exponentially difficult to decode, with an exponent
that is controlled by the size of the bulge and grows as $O(1/G_{N})$ in the
semiclassical limit. The justification for this conjecture was based primarily
on tensor network toy models, where the fastest known protocols for decoding
operators inside a lunch use a Grover-search-based algorithm that takes
exponential time.

An additional important consistency check comes from the quantum focusing
conjecture [11], which is the quantum avatar of classical gravitational
lensing when the null energy condition is violated due to quantum corrections.
Quantum focusing and global hyperbolicity ensure that no causal semiclassical
Lorentzian evolution of the bulk geometry can result in causal communication
from behind a Python’s lunch to the asymptotic boundary, no matter what the
asymptotic boundary conditions are. If such communication were possible,
reconstruction of information from within the lunch could be implemented using
only boundary time evolution with simple sources and the extrapolate
dictionary relating bulk observables at the asymptotic boundary to local
boundary operators. From a boundary perspective, this is a very simple
procedure compared to the exponential complexity claimed to be necessary for
operator reconstruction inside the lunch.

Without input from nonperturbative quantum gravity (such as entanglement wedge
reconstruction), simple reconstruction using only low-complexity, causally-
propagating operators and sources is all that semiclassical Lorentzian gravity
is capable of: i.e. all that semiclassical gravity “sees”. Calculations and
arguments that rely exclusively on semiclassical gravity with no further input
(e.g. Hawking’s original calculation) are thus certainly restricted by the
Python’s lunch proposal to recover no more than the domain of dependence
between the outermost quantum extremal surface – the “appetizer” of the lunch
– and the asymptotic boundary.

So how much does purely semiclassical gravity actually recover? On the one
hand, if the Python’s lunch conjecture is true, it is natural to expect that
simple reconstruction can in fact obtain the entire bulk up to the outermost
extremal surface. This “converse” to the Python’s lunch conjecture is
certainly true in tensor network toy models where anything not in a lunch can
be reconstructed using a simple unitary circuit. Any gap in gravitational
theories between the simply reconstructible region and the start of the lunch
would therefore be somewhat puzzling and demand explanation. On the other
hand, simple reconstruction appears to be little more than a glorified version
of the HKLL procedure [12, 13, 14], which is supposed to recover just the so-
called causal wedge: the region that can simultaneously send and receive
signals from the asymptotic boundary. And generically the causal wedge and the
outermost extremal wedge do not coincide.

To clarify this conundrum, let us first briefly review HKLL here, as it will
be instrumental for our work in this paper. The HKLL procedure is a
reconstruction protocol for bulk matter on a fixed background, in which the
bulk fields (which can include gravitons [15, 16], $\mathcal{O}(1/N)$
corrections [16, 17], interactions [18], and higher spins [16, 19]) are
obtained via a non-standard Cauchy evolution from their boundary counterparts
(related to them via the extrapolate dictionary). Quantitatively,

$\phi(x)=\int dXK(x;X)\mathcal{O}(X),$ (1.1)

where $K(x;X)$ is a smearing function that depends on the spacetime geometry
supported on the set of boundary points spacelike-separated to $x$.111Note
that the validity of this non-standard “rotated” Cauchy problem is far from
well-established (though see [20, 21] for proofs in certain cases). Our
purpose here is not to put HKLL on a firm footing, but rather to show that
reconstruction of the simple wedge is as simple as HKLL. We shall therefore
assume HKLL, but any other simple reconstruction procedure for operators in
the causal wedge would do just as well for our purposes. The sense in which
HKLL is “simple” is evident: from a boundary perspective it consists of
boundary time evolution with local sources turned on. And local Hamiltonian
evolution can be simulated efficiently using a quantum circuit.

The immediate prediction therefore, as expressed in [22], is that HKLL can
reconstruct operators within the causal wedge. We might then expect that the
simply reconstructible region – which we shall henceforth refer to as the
simple wedge – is to be identified with the causal wedge. Since the causal
wedge is always a subset of the outermost quantum extremal wedge [7] and is
generically a proper subset 222As shown in [23], quantum effects can allow the
causal wedge to be outside the outermost extremal wedge when defined using
time evolution couples the asymptotic boundary to an auxiliary system.
However, this is only true if we do not include the auxiliary coupled system
when defining the outermost extremal wedge. When comparing apples to apples by
doing so, one indeed finds that the causal wedge is still contained in the
outermost extremal wedge., this leads to the undesirable no-man’s land between
the simple wedge and the outermost quantum extremal surface.

To see deeper into the bulk, we need to expand the causal wedge via the
addition of simple boundary sources as proposed in [24, 25]. In the very
special case when the gap between the outermost extremal surface and the
causal wedge is Planckian, [26] showed that certain causal unitaries produce
just enough backreaction to maximally expand the causal past or future.
However, in generic spacetimes the gap region is non-empty even in the
classical limit and can in fact be arbitrarily large. It was conjectured in
[24, 25] that it should be possible to fine-tune simple sources in order to
“turn off” any extant focusing and so expand the causal wedge up to an
apparent horizon, all without violating the null energy condition.

A central result of this paper is an explicit, constructive derivation of this
fact – in the limit where the bulk dynamics are classical and with a variety
of matter fields. Furthermore, by evolving backwards and forwards in time
using timefolds with different boundary conditions, one can continue to
iteratively expand the causal wedge, from apparent horizon to apparent
horizon, all the way to the outermost extremal surface. Combining this result
with ordinary HKLL leads to simple reconstructions of arbitrary operators in
the outermost extremal wedge.

It is easy to see how this works in the case of Jackiw-Teitelboim gravity [27,
28] minimally coupled to a (classical) massless free scalar. In this setup,
the matter factorizes into left and right movers; by changing the boundary
conditions, we may “absorb” the right movers and turn off focusing on the
future event horizon; this pushes the future event horizon backwards. We can
then repeat the same procedure for the past horizon by evolving backwards in
time; this will now push the past causal horizon backwards. The shift will
have likely revealed additional left-movers, so the procedure needs to be
iteratively repeated until it converges on a stationary bifurcate horizon.
This is illustrated in Fig. 2. The generalization to higher dimensions is
significantly more technically challenging – rather than removing sources of
focusing entirely, it is more practical to “stretch out” the focusing over the
causal horizon and so dilute its effect – but the essential intuition is the
same.

Figure 2: A caricature of the procedure used to push the causal wedge towards
the appetizer in JT gravity coupled to a classical massless scalar. The
leftmost panel is the original spacetime including left- and right-movers with
reflecting boundary conditions. The causal surface of the right boundary is
$C_{1}$. In the middle panel, the left movers have been turned off, which
causes the future event horizon to shift inwards. The resulting causal surface
is $C_{2}$, which is null-separated from $C_{1}$. The final panel shows that
the right movers have been turned off, which causes the past event horizon to
move inwards, shifting the causal surface to $C_{3}$. This shift reveals new
left-movers in the causal wedge, which will have to be removed in subsequent
zigzags along the past and future event horizons.

The original motivation in [24] for attempting to expand the causal wedge
using simple sources was to understand the holographic dual of the _simple
entropy_ , defined as a maximization of the von Neumann entropy over all CFT
density matrices with the same one-point functions – with arbitrary time-
ordered simple sources turned on after some initial time $t$ – as the original
CFT state. In other words, the simple entropy coarse-grains over all of the
details of the state, except for simple observables that can be measured in
the future of the initial time $t$. It was conjectured in [24, 25] that the
simple entropy is the boundary dual of the _outer entropy_ , a bulk quantity
that coarse-grains over the geometry behind the outermost apparent horizon
null-separated from the boundary at time $t$; it is equal to (one quarter of)
the area of the apparent horizon. As a corollary of the results discussed
above, we prove that this conjecture is indeed true whenever the bulk physics
can be treated classically.

What if we generalize the definition of the simple entropy to allow not just
time-ordered insertions of simple operators, but insertions on arbitrary
timefolds (and at arbitrary time)? In this case, there is no obstacle to
seeing behind apparent horizons. By evolving the state backwards (and
forwards) in time and then turning simple sources, it is possible to causally
alter the spacetime near the apparent horizon, changing its location and
“seeing” degrees of freedom that were originally hidden behind it. As per the
discussion above, the first obstruction that cannot be bypassed in this way is
the outermost extremal surface. Indeed, our results demonstrate that the
simple entropy with arbitrary timefolds allowed is holographically dual to the
area of the outermost extremal surface. Similarly, the density matrix
$\rho_{\mathrm{coarse}}$ whose von Neumann entropy is the simple entropy with
timefolds allowed reconstructs exactly the entire outermost extremal wedge and
no more. In fact, we can actually construct a complete spacetime in which the
outermost extremal wedge is the entire entanglement wedge of one connected
asymptotic boundary obtained using the spacetime doubling procedure of [24,
25]; thus $\rho_{\mathrm{coarse}}$ is the actual CFT state dual to the
canonical purification as proposed in [24, 25] and proven in [29]. See Fig. 3.

Figure 3: The coarse-graining procedure of [24, 25] as applied to the
outermost extremal surface. The spacetime behind the outermost extremal
surface $X$ is discarded and replaced with a CPT conjugate of the outermost
extremal wedge. The rest of the spacetime is generated by standard Cauchy
evolution.

An immediate application of our result is then the construction of the CFT
dual to the simple wedge in the final spacetime where the causal and outermost
extremal wedges coincide. That is, this dual can be produced from
$\rho_{\mathrm{coarse}}$ via a set of simple operations, with the dual bulk
result being a two-sided black hole in which the bifurcation surface is
extremal. The significance of this statement is manifold: we prove that the
causal and entanglement wedges coincide if and only if the CFT state has a
local modular Hamiltonian, which shows that finite time measurements cannot
tell that the modular flow generated by the simple state is not local. In
spacetimes with sufficient isotropy, the simple wedge CFT modular flow is in
fact exactly local. This is analogous to a type of no-hair theorem: the set of
holographic black holes with a stationary bifurcation surface is identical to
the highly limited set of states with local modular Hamiltonians in the dual
CFT.

From the perspective of holographic complexity, we may therefore interpret the
absence of a Python’s lunch in the dual theory as the CFT state being related
by a simple circuit to a rather special state with local modular flow (or at
least indistinguishable from local in finite time). The world, it would seem,
is rarely simple; pythons are ubiquitous. An explicit example of how a python
might spring on an unsuspecting holographer in what would prima facie appear
to be a python-less spacetime will be provided in our upcoming work [30].

The paper is structured as follows. In Section 2, we define the outermost
extremal wedge and the simple wedge, and we prove that the former is well-
defined. In Section 3 we showcase our procedure for the simple case of JT
gravity coupled to a massless (classical) scalar. In Section 4, we prepare the
perturbation that moves the causal horizon backwards along a the future event
horizon in higher dimensional gravity (with arbitrary, null energy condition-
satisfying matter), and we prove that the required perturbation satisfies the
constraint equations. Section 5 describes the zigzag portion of the procedure
and completes the proof. We discuss the implications of our results, from the
dual to the area of the outermost extremal surface to the nature of the simple
state, in Section 6. We finish with a discussion of generalizations and other
implications in Section 7.

#### Assumptions and Conventions:

The bulk spacetime $(M,g)$ is assumed to be classical with the dynamics
governed by the Einstein field equation, i.e. we work in the large-$N$ and
large $\lambda$ limit of AdS/CFT except where otherwise stated. We will assume
the AdS analogue of global hyperbolicity [31]. We also assume that the initial
spacetime under consideration is one that satisfies the Null Energy Condition
(NEC):

$\displaystyle T_{ab}k^{a}k^{b}\geq 0$ (1.2)

where $T_{ab}$ is the stress energy tensor and $k^{a}$ is any null vector. We
will demonstrate that our perturbations of the spacetime maintain the NEC. All
other conventions are as in [32] unless otherwise stated.

  * •

We shall use $J^{\pm}$ to refer to the bulk causal future and past and
$I^{\pm}$ to refer to the bulk chronological future and past. Given a closed
achronal set $S$, we use $D[S]$ to denote its domain of dependence, which we
shall take to contain its boundary, as in [32]. $D^{+}[S]$ and $D^{-}[S]$
refer to the future and past components of the domain of dependence.

  * •

Hypersurfaces will refer to codimension-one embedded submanifolds of arbitrary
signature.

  * •

By a “surface” we will always mean an achronal, codimension-two embedded
submanifold which is Cauchy-splitting [33]. Two surfaces $\sigma_{1}$ and
$\sigma_{2}$ are homologous whenever there exists a hypersurface $H$ such that
$\partial H=\sigma_{1}\cup\sigma_{2}$. We will be primarily interested in
surfaces homologous to (partial) Cauchy slices of the asymptotic boundary (CFT
(sub)regions).

  * •

Let $\Sigma$ be a Cauchy slice containing a surface $\sigma$ homologous to a
boundary (sub)region $R$. By definition, $\sigma$ splits $\Sigma$ into two
disjoint components that we will denote $\mathrm{Int}_{\Sigma}[\sigma]$ and
$\mathrm{Out}_{\Sigma}[\sigma]$, where the conformal completion of the latter
contains the boundary subregion $R$. We define $W_{\sigma}\equiv
D[\mathrm{Out}_{\Sigma}[\sigma]]$, the outer wedge of $\sigma$. Similarly, we
define $I_{\sigma}\equiv D[\mathrm{Int}_{\Sigma}[\sigma]]$, the inner wedge of
$\sigma$. See also [25].

  * •

For a smooth surface $\sigma$ homologous to a boundary (sub)region, we denote
by $k^{a}$ and $\ell^{a}$ the unique future-directed orthogonal null vector
fields on the $C^{1}$ subsets of $\sigma$ pointing towards
$\mathrm{Out}_{\Sigma}[\sigma]$ and towards $\mathrm{Int}_{\Sigma}[\sigma]$
respectively.

  * •

We define $\partial^{+}W_{\sigma}=\partial
D^{+}[\mathrm{Out}_{\Sigma}[\sigma]]$ and $\partial^{-}W_{\sigma}=\partial
D^{-}[\mathrm{Out}_{\Sigma}[\sigma]]$. When $\sigma$ is smooth,
$\partial^{+}W_{\sigma}$ and $\partial^{-}W_{\sigma}$ can be constructed by
firing null congruences starting from $k^{a}$ and $\ell^{a}$, terminating the
congruence at caustics and non-local self-intersections [32, 34].

  * •

Given any orthogonal null vector field $k^{a}$ on surface $\sigma$,
$\theta_{(k)}$ denotes the expansion of $\sigma$ along $k^{a}$.333$k^{a}$ is
not uniquely defined in places where $\sigma$ is not $C^{1}$. However, the
expansion still has a definite sign, which can be computed via a limiting
procedure. We will refer to the following types of $\sigma$ based on its
expansions:

    * –

A compact $\sigma$ is trapped if $\theta_{(k)}<0$ and $\theta_{(\ell)}<0$, and
marginally trapped if $\theta_{(\ell)}<0$ and $\theta_{(k)}=0$.

    * –

$\sigma$ is _extremal_ if $\theta_{(k)}=0$ and $\theta_{(\ell)}=0$. By
linearity, $\sigma$ is then stationary under deformations along any direction.

  * •

The future and past causal horizons associated to any boundary spacetime
region $R\subset\mathscr{I}$ are defined as $\partial J^{-}[R]$ and $\partial
J^{+}[R]$ respectively. By convention, we use $k^{a}$ and $\ell^{a}$ to refer
to the generators of the future and past horizons respectively. More
specifically, the future and past event horizons are defined as
$\mathscr{H}^{+}\equiv\partial J^{-}[\mathscr{I}]$ and
$\mathscr{H}^{-}\equiv\partial J^{+}[\mathscr{I}]$. The causal surface can be
defined as $C\equiv\mathscr{H}^{+}\cap\mathscr{H}^{-}$ and the causal wedge is
$\mathscr{W}_{C}\equiv J^{+}[\mathscr{I}]\cap J^{-}[\mathscr{I}]$[35].444Note
that although the term causal wedge is common in the literature, unlike the
entanglement wedge it does not refer to a domain of dependence. An important
result in general relativity – which follows from NEC and cosmic censorship –
is that future causal horizons satisfy an area law: the areas of their cross
sections do not decrease as we move the cross section to the future [36, 37].
In particular, this means that any congruence of null generators on a future
horizon has nonnegative expansion. By time-reversal symmetry, a “reverse” area
law holds for past horizons.

  * •

We define the terminated horizons
$\mathscr{H}^{+}_{C}\equiv\mathscr{H}^{+}\cap J^{-}[\mathscr{I}]$ and
$\mathscr{H}^{-}_{C}\equiv\mathscr{H}^{-}\cap J^{-}[\mathscr{I}]$. These are
natural definition for us since we are interested in perturbations of
$\mathscr{H}^{+}$ and $\mathscr{H}^{-}$ caused by causal boundary sources.

##  2 Which Wedge?

Three bulk regions are under consideration here: the outermost extremal wedge,
the causal wedge, and the simple wedge. We will ultimately argue that the
outermost extremal wedge is in fact the simple wedge, but in order to avoid
subscribing to our own conclusions before we have demonstrated them, we
introduce terminology that distinguishes between the two.

We will argue for the equivalence between the outermost extremal and simple
wedges by showing that simple operations and sources (together with a finite
number of time-folds) are sufficient to shift the causal wedge so that it
comes arbitrarily close to coinciding with the outermost extremal wedge. While
our primary results are for compact extremal surfaces, many of our
intermediate results remain valid for boundary-anchored surfaces. In Sec. 7.1
we will discuss in more detail the extent to which our results apply to the
latter case.

We have already defined the more familiar causal wedge in the introduction.
Let us now give a precise definition of the outermost extremal wedge and the
simple wedge.

Intuitively, the extremal wedge is defined as the analogue of the entanglement
wedge for the outermost extremal surface – be it minimal or not. So before
defining the outermost extremal wedge, we must prove that a unique outermost
extremal surface exists in the first place:

######  Proposition 1.

If there exists more than one extremal surface homologous to a connected
component of $\mathscr{I}$, then exactly one is outermost; i.e. there exists
an extremal surface $X$ contained in the outer wedge $W_{X^{\prime}}$ of all
other extremal surfaces $X^{\prime}$ homologous to $\mathscr{I}$.

We will prove this proposition using a series of three lemmas:

######  Lemma 1.

Given two surfaces $\sigma_{1}$ and $\sigma_{2}$ homologous to $\mathscr{I}$,
$W_{\sigma_{1}}\cap W_{\sigma_{2}}$ is a domain of dependence.

###### Proof.

Let $\Sigma$ be a Cauchy slice containing $\sigma_{1}$. We define a new
surface $\Sigma^{\prime}=\partial((J^{+}[\Sigma]\cup
J^{+}[\sigma_{2}])\cap\overline{I^{-}[\sigma_{2}]})$, where
$\overline{I^{-}[\sigma_{2}]}\equiv M-I^{-}[\sigma_{2}]$. Since every
inextensible timelike curve at some point in its past is outside
$J^{+}[\Sigma]\cup J^{+}[\sigma_{2}]$, but eventually ends up in
$J^{+}[\Sigma]\cap\overline{I^{-}[\sigma_{2}]}$, and since no timelike curve
can exit $(J^{+}[\Sigma]\cup
J^{+}[\sigma_{2}])\cap\overline{I^{-}[\sigma_{2}]})$ after entering it,
$\Sigma^{\prime}$ is Cauchy. Note that, despite appearances, the definition of
$\Sigma^{\prime}$ is invariant under time-reversal symmetry.

We will now show that $W_{\sigma_{1}}\cap W_{\sigma_{2}}$ is the domain of
dependence of $H=\Sigma^{\prime}\cap W_{\sigma_{1}}\cap W_{\sigma_{2}}$. Any
causal curve intersecting $W_{\sigma_{1}}\cap W_{\sigma_{2}}$ needs to
intersect $\Sigma\cap W_{\sigma_{1}}$ either (i) in $W_{\sigma_{2}}-\partial
W_{\sigma_{2}}$ or (ii) outside of it . In case (ii), the causal curve needs
to leave $J^{-}[\sigma_{2}]$ in $W_{\sigma_{1}}$ after intersecting $\Sigma$
or enter $J^{+}[\sigma_{1}]$ in $W_{\sigma_{1}}$ before intersecting $\Sigma$.
Therefore, in both cases i and ii we conclude that the causal curve intersects
$H$.

∎

Let $V$ now be a domain of dependence that intersects $\mathscr{I}$. Then
there must exist an “edge” surface $\sigma$ homologous to $\mathscr{I}$ such
that $V=W_{\sigma}$. More precisely, $\sigma$ can be defined as the set of
points $p\in\partial V$ such that in any small neighborhood of $p$ any
inextensible timelike curve crossing $p$ only intersects $V$ at $p$.

######  Lemma 2.

Let $V=W_{\sigma_{1}}\cap W_{\sigma_{2}}$ and let $\sigma$ be the edge of $V$
as defined above. Then,
$\sigma\subset\sigma_{1}\cup\sigma_{2}\cup(\partial^{+}W_{\sigma_{1}}\cap\partial^{-}W_{\sigma_{2}})\cup(\partial^{-}W_{\sigma_{1}}\cap\partial^{+}W_{\sigma_{2}})$.

###### Proof.

Clearly, the edge is contained in $\partial V\subset\partial
W_{\sigma_{1}}\cup\partial W_{\sigma_{2}}$. Say some point $p$ in the edge of
$V$ were in $\partial^{+}W_{\sigma_{1}}-\sigma_{1}$. Then, every timelike
curve crossing $p$ leaves $W_{\sigma_{1}}$ (and therefore $V$) in $I^{+}(p)$
and enters $W_{\sigma_{1}}$ in $I^{-}(p)$. Furthermore, for $p$ to be in the
edge of $V$ it needs to be in $W_{\sigma_{2}}$, but $I^{-}(p)$ must not
intersect $W_{\sigma_{2}}$. Therefore, $p$ needs to be in
$\partial^{-}W_{\sigma_{2}}$. Together with the time reverse of this argument
and also switching $\sigma_{1}$ and $\sigma_{2}$, we conclude that the edge of
$V$ must be contained in
$\sigma_{1}\cup\sigma_{2}\cup(\partial^{+}W_{\sigma_{1}}\cap\partial^{-}W_{\sigma_{2}})\cup(\partial^{-}W_{\sigma_{1}}\cap\partial^{+}W_{\sigma_{2}})$.
∎

Lastly, we will state a lemma from Sec. 2.2 of [38], without providing the
proof (see also Appendix B of [6] for a similar discussion). The lemma assumes
the existence of a stable maximin surface [31] in any domain of dependence.

######  Lemma 3.

If $\sigma$ is a surface homologous to $\mathscr{I}$ satisfying
$\theta_{(k)}\leq 0$ and $\theta_{(\ell)}\geq 0$, then there exists an
extremal surface $Y\subset W_{\sigma}$ homologous to $\mathscr{I}$.

Let us provide some intuition for Lemma 3. The restricted-maximin prescription
returns a surface in $W_{\sigma}$ which is homologous to $\mathscr{I}$ and is
minimal on some Cauchy slice of $W_{\sigma}$. If $\sigma$ satisfies
$\theta_{(k)}<0$ and $\theta_{(\ell)}>0$, then the maximin surface cannot
intersect $\sigma$, since its area would get smaller under deformations away
from such intersections. In [38], it was further shown that the max property
of the maximin surface prohibits intersections with $\partial
W_{\sigma}-\sigma$. This shows that the maximin surface is in the interior of
$W_{\sigma}$ and thus extremal. Furthermore, it was argued that even when the
inequalities are not strict, there still exists an extremal surface homologous
to $\mathscr{I}$ in $W_{\sigma}$ even though in the surface might lie on
$\partial W_{\sigma}$.

We are now ready to prove Proposition 1:

###### Proof.

In this proof any “surface” will mean a surface that is homologous to
$\mathscr{I}$. Let an extremal surface $X$ be called exposed if there does not
exist any extremal surface $Y\neq X$ such that $Y\subseteq W_{X}$. Let us
first argue that there must always exist at least one exposed surface. Define
a partial ordering on extremal surfaces by declaring $X\geq Y$ if and only if
$W_{X}\subseteq W_{Y}$. Note that exposed surfaces would correspond to maximal
elements with respect to this partial order, while an outermost extremal
surface would be a greatest element. Upper bounds exist for any chain because
monotonicity and boundedness (there are no extremal surfaces near asymptotic
infinity) ensure that any sequence $\\{X_{n}\\}$ of extremal surfaces with
$W_{X_{n+1}}\subseteq W_{X_{n}}$ converges to an extremal surface $X_{\infty}$
with $W_{X_{\infty}}\subseteq W_{X_{n}}$ for any finite $n$. Hence, by Zorn’s
lemma, at least one maximal element, i.e. exposed surface, exists.

Now suppose, by way of contradiction, that there exists an exposed surface
$X_{1}$ that is not outermost, i.e. there exists some other extremal surface
$X_{2}$ such that $W_{X_{1}}\nsubseteq W_{X_{2}}$. Let $V=W_{X_{1}}\cap
W_{X_{2}}$ (by definition then $V\subset W_{X_{1}}$). By Lemma 2 and 3,
$V=W_{\sigma}$ for some surface $\sigma\subseteq X_{1}\cup
X_{2}\cup(\partial^{+}W_{X_{1}}\cap\partial^{-}W_{X_{2}})\cup(\partial^{-}W_{X_{1}}\cap\partial^{+}W_{X_{2}})$.
Let us consider each component of the set to which $\sigma$ belongs
separately. The first two are subsets of extremal surfaces and hence have zero
expansion. By focusing, the null hypersurface $\partial^{+}W_{X_{1}}$ and
$\partial^{+}W_{X_{2}}$ are non-expanding towards the future, while the null
hypersurface $\partial^{-}W_{X_{1}}$ and $\partial^{-}W_{X_{2}}$ are non-
expanding towards the past. Therefore, $\sigma$ satisfies $\theta_{(k)}\leq 0$
and $\theta_{(\ell)}\geq 0$555This $\sigma$ may have kinks in which case the
expansions are technically not well-defined. However, we expect that all of
our results in this section generalizes easily to such surfaces using standard
geometric techniques. Intuitively, by slightly smoothing out the kinks we can
get a new infinitesimally nearby surface with very large expansions of the
desired signs around the kinks., and hence by Lemma 3, there exists an
extremal surface $Y$ contained in $W_{\sigma}=W_{X_{1}}\cap W_{X_{2}}$. The
existence of this surface means that $X_{1}$ is not exposed, giving our
desired contradiction.

∎

As an aside, note that extending the last part of this argument to quantum
extremal surfaces, under assumption of the quantum focusing conjecture
requires a small amount of extra work because the definition of quantum
expansion is nonlocal. However strong subadditivity is enough to ensure that
the quantum expansion of $\sigma$ satisfies the desired inequalities.

It is now easy to define the outermost extremal wedge:

######  Definition 1.

Let $X$ be the outermost extremal surface for a connected component of
$\mathscr{I}$. The outermost extremal wedge $W_{X}$ is the outer wedge of $X$.

Next we would like to define the simple wedge. Conceptually, the simple wedge
is the largest bulk region that can be reconstructed from the near boundary
state of the bulk fields using exclusively the bulk equations of motion.
Consider some boundary state $\rho$ whose bulk dual $(M,g)$ we would like to
reconstruct and evolve it to the far past and future with some Hamiltonian. In
the classical regime, following Eq. (1.1), HKLL prescribes the values of the
bulk fields in $\mathscr{W}_{C}$ from the set of one-point functions of their
corresponding local boundary operators on $\mathscr{I}$. In fact, the bulk
equations of motions are sufficient to reconstruct the maximal Cauchy
development of $\mathscr{W}_{C}$ – which we can denote by $W_{C}$.

Cosmic censorship in general [39, 40] and causal wedge inclusion in particular
[31, 41, 42] guarantees that the causal wedge contains no extremal surfaces in
its interior: $W_{C}\subseteq W_{X}$. It is therefore impossible to
reconstruct the region behind $X$ causally. HKLL alone, however, appears to
prima facie fail at an earlier stage: the non-standard Cauchy evolution
appears to stop short of recovering the gap region between $W_{C}$ and
$W_{X}$, which is generically non-empty.

What if we evolve $\rho$ using a different Hamiltonian? Consider turning on a
set of CFT operators at various times during the evolution. This would
“extract” a new set of one-point functions from $\rho$ and therefore has the
potential to expand the reconstructible region. In keeping with our philosophy
of the simple wedge, we must restrict to sources that have a (semi-)classical
bulk dual. Therefore, following [24, 25], we refer to boundary sources as
simple if the bulk fields that they produce propagate causally into the bulk
from the boundary insertion point – and restrict to such sources henceforth.
The change in time evolution when such simple sources are applied within some
boundary time interval $[t_{i},t_{f}]$ is given by the following time-ordered
operator:

$\displaystyle
E=\mathcal{T}exp\left[-i\int_{t_{i}}^{t_{f}}dt^{\prime}~{}J(t^{\prime})\mathcal{O}_{J}(t^{\prime})\right],$
(2.1)

where $J(t)$ is a simple source and $\mathcal{O}_{J}$ is its corresponding
simple operator. Note that $\mathcal{O}_{J}$ might involve spatial integrals
of local boundary operators $\mathcal{O}(t^{\prime},x^{\prime})$. An example
of a simple operator is a spatial integral of a single-trace operator of the
boundary gauge theory.

Adding $E$ to the evolution, say in a future-directed timefold, changes the
spacetime from $M$ to some $M^{\prime}$. By causality, $M-J^{+}[t_{i}]\subset
M^{\prime}$666Here $t_{i}$ stands for a timeslice of $\mathscr{I}$
geometrically.. In particular, the perturbation to the spacetime is localized
away from the past event horizon $\mathscr{H}^{-}$. However, sources like Eq.
(2.1) will typically change where the new future event horizon
$\mathscr{H}^{+}$ intersects $\mathscr{H}^{-}$. In particular, suppose that we
find simple sources that “expand” the causal wedge, i.e. place the new causal
surface $C^{\prime}$ in the future of $C$ on $\mathscr{H}^{-}$. Said in the
CFT language, the new set of one-point data reconstructs a
$\mathscr{W}^{\prime}_{C}$ that contains $\mathscr{H}^{-}_{C}$. Furthermore,
knowing the bulk equations of motion and the original Hamiltonian, we can
reconstruct the Cauchy development of $\mathscr{H}^{-}_{C^{\prime}}$, a wedge
in the _original_ spacetime that contains $W_{C}$ as a proper subset.

It is natural to define the simple wedge according to the maximal success of
this procedure:

######  Definition 2.

The simple wedge is the maximal bulk region that can be reconstructed from
simple operators acting on the dual CFT state, with the inclusion of simple
sources and timefolds.

Although we have defined the simple wedge in the context of classical field
theory in the bulk, it is important to note that HKLL can reconstruct the
quantum state of the bulk propagating in the causal wedge at each step. For
bulk fields in the $1/N$ expansion, Eq. (1.1) provides the dictionary between
local bulk operators and simple CFT operators, realizing this reconstruction
in the Heisenberg picture.

Finally, we close this section by relating the causal and outermost extremal
wedges. Intuitively, the causal surface should coincide with the extremal
surface if and only if there is no focusing whatsoever on the horizons.
However, because the extremal wedge is defined as a domain of dependence and
the causal wedge is defined in terms of causal horizons, it does not
immediately follow that the two must coincide in the absence of focusing. To
reassure the reader, we prove the following lemma:

######  Lemma 4.

Let $X$ be the outermost extremal surface homologous to one or more connected
components of $\mathscr{I}$. $W_{X}=\mathscr{W}_{C}$ if and only if $X$ is a
bifurcation of stationary horizons (and thus $X=C$).

###### Proof.

If $\mathscr{W}_{C}=W_{X}$, then $\partial\mathscr{W}_{C}=\partial W_{X}$. If
both wedges had been defined in terms of domains of dependence, it would
immediately follow that $C=X$. However, since $\mathscr{W}_{C}$ is not defined
as a domain of dependence, we have to work a little harder. The component of
$\partial\mathscr{W}_{C}$ which is spacelike separated to every point in
Int$[\mathscr{W}_{C}]$ is identical to the component of $W_{X}$ which is
spacelike-separated to every point in Int$[W_{X}]$. By definition of the
causal wedge, the former is $C$. Since $W_{X}$ is generated by the domain of
dependence of a hypersurface $H$ whose boundary in $M$ is $X$, $X$ is exactly
the set of points that are spacelike separated from every point in
Int$[W_{X}]$. This immediately shows that $C=X$. Because $\theta_{(n)}[X]=0$
for all $n^{a}$ in the normal bundle of $X$, we find that
$\theta_{(\ell)}[C]=0=\theta_{(k)}[C]$. By the NCC, a future (past) causal
horizon can only have vanishing expansion on a slice if it has vanishing
expansion everywhere to the future (past) of that slice. So
$\partial\mathscr{W}_{C}$ and subsequently $\partial W_{X}$ is generated by
stationary horizons, and $X$ is a bifurcate stationary horizon.

If $X$ is a bifurcate stationary horizon, then results of [33, 34] immediately
imply that $\partial W_{X}$ is the union of two truncated stationary horizons
$\mathscr{H}^{\pm}$. Since $\mathscr{H}^{+}\cap\partial M=\partial D^{+}$,
$\mathscr{H}^{+}$ is a past-directed null congruence fired from $i^{+}$. By
the theorems of [33], $\partial J^{-}[\mathscr{I}]=\mathscr{H}^{+}$ up to
geodesic intersections. Since $\mathscr{H}^{+}$ is stationary, it has no
intersections, so $\partial J^{-}[\mathscr{I}]=\mathscr{H}^{+}$. Similarly
$\partial J^{+}[\mathscr{I}]=\mathscr{H}^{-}$. Thus $\partial
W_{X}=\partial\mathscr{W}_{C}$ and $W_{X}=\mathscr{W}_{C}$ by the homology
constraint. ∎

Note that this result remains valid for quantum extremal surfaces assuming the
quantum focusing conjecture and a suitable generalization of AdS
hyperbolicity.

##  3 Two Dimensions

Let us illustrate our iterative procedure for removing matter falling across
the past and future causal horizons in a simple toy model of JT gravity
minimally coupled to a massless scalar field $\varphi$ (with no direct
coupling between the dilaton $\Phi$ and $\varphi$). The absence of propagating
degrees of freedom of the gravitational field as well as the factorization of
the bulk matter into left- and right-movers are simplifications that naturally
do not generalize to higher dimensions; nevertheless the procedure itself is
well-illustrated in this setting, which we include for pedagogical reasons.
The additional complications introduced in higher dimensions are resolved in
subsequent sections.

Due to focusing resulting from the scalar field $\varphi$, the bifurcation
surface will generically not be extremal, i.e.

$\partial_{n}\Phi|_{C_{1}}\neq 0$ (3.1)

where $n^{a}$ is some vector normal to the causal surface $C_{1}$ (in
particular, for null $n^{a}$ future-outwards directed, this would be positive;
similarly for a time-reverse). As a consequence of the highly limited number
of degrees of freedom in the problem, the extremality failure can only be a
result of focusing: the future causal horizon will experience focusing due to
the $\varphi$ left-moving modes and the past causal horizon will experience
focusing due to the $\varphi$ right-moving modes. Our procedure instructs us
to first remove the source of focusing of the future horizon by modifying the
boundary conditions of $\varphi$, which we can easily do by implementing
absorbing boundary conditions for the right movers in order to remove all the
left-moving modes. This removes focusing from the future causal horizon, which
pushes the horizon deeper into the bulk. As a consequence, the new causal
surface (which is now marginally trapped) – let us call it $C_{2}$ – is pushed
further along the past event horizon, which remains unmodified by this
procedure. This first step is illustrated in the second panel of Fig. 2.

To turn off focusing of the past horizon, we evolve backwards in time,
imposing boundary conditions that remove the right-movers. The past event
horizon moves backwards, and the new causal surface $C_{3}$ (which is now
marginally anti-trapped) is displaced from $C_{2}$ along the future event
horizon of $C_{2}$. However $C_{3}$ is not necessarily extremal since shifting
the past causal horizon reveals a part of the spacetime that was previously
not included in the causal wedge: in particular, new left-moving modes can now
appear in the causal wedge. This is illustrated in the third panel of Fig. 2.
This piecewise-null zigzag procedure thus shifts the causal surface deeper
into the bulk; we may simply repeat the zigzag iteratively.

In classical gravity, the focusing theorem and cosmic censorship (or strong
asymptotic predictability) together guarantee that no extremal surface is ever
in causal contact with $\mathscr{I}$: so the zigzag procedure can never modify
an extremal surface nor move the causal surface deeper than any extremal
surface. Thus the outermost extremal surface is an upper bound on the success
of the procedure. Our goal, of course, is to show that this upper bound is in
fact attained.

Figure 4: An illustration of the approach to the limit point
$C_{\mathrm{lim}}$, where $C_{A}$ and $C_{B}$ are infinitesimally close to the
limiting point.

Because the success of this procedure is bounded by the outermost extremal
surface and because furthermore the procedure moves the surface monotonically
inwards, a limiting causal surface $C_{\mathrm{lim}}$ exists, and the
corresponding causal wedge does not intersect any extremal surface. We will
now argue that $C_{\mathrm{lim}}$ is in fact extremal. Let $C_{A}$ be a causal
surface obtained via iterative zigzags which is infinitesimally close to
$C_{\mathrm{lim}}$; without loss of generality we may take $C_{A}$ to be in
the marginally trapped portion of the zigzag (i.e. the left-movers had just
been removed). Let $v$, $u$ be the affine parameters along the future and past
event horizons, respectively, in the spacetime in which $C_{A}$ is the causal
surface. By construction, $C_{A}$ has no expansion along the future event
horizon:

$\frac{\partial}{\partial v}\Phi|_{C_{A}}=0,$ (3.2)

and because by assumption it is not identical to $C_{\mathrm{lim}}$,

$\frac{\partial}{\partial u}\Phi|_{C_{A}}<0.$ (3.3)

Let $C_{B}$ be the causal surface obtained by removing the right-movers from
the spacetime where $C_{A}$ is the causal surface. By construction

$\frac{\partial}{\partial u}\Phi|_{C_{B}}=0.$ (3.4)

By the zigzag procedure, $C_{A}$ and $C_{B}$ are null-separated along the
“old” future event horizon: i.e. along the null congruence that is the future
event horizon in the spacetime in which $C_{A}$ is the causal surface. Let
$\delta v$ be the amount of affine parameter separating $C_{A}$ and $C_{B}$.
See Fig. 4 for an illustration. Since the points $C_{A}$ and $C_{B}$ must be
infinitesimally close to one another (since both infinitesimally near
$C_{\mathrm{lim}}$), the spacetime metric in that neighborhood may be
approximated as locally flat instead of AdS2; using $u$ and $v$ as
coordinates:

$ds^{2}=-2dudv.$ (3.5)

In these coordinates, it is trivial to relate $\delta v$ to the change in
$\partial_{u}\Phi$ along $v$. In particular, we may bound it from below:

$\delta
v\geq\frac{\partial_{u}\Phi}{\partial_{v}\partial_{u}\Phi|_{\mathrm{max}}},$
(3.6)

where $\partial_{v}\partial_{u}\Phi|_{\mathrm{max}}$ is the maximum value of
$\partial_{v}\partial_{u}\Phi$ on the $\delta v$ interval. Similarly defining
$\delta u$ as the null separation between $C_{B}$ and the next causal surface
after again removing left movers, we obtain an analogous bound:

$\delta
u\geq\frac{\partial_{v}\Phi}{\partial_{u}\partial_{v}\Phi|_{\mathrm{max}}}.$
(3.7)

Under the assumption that $\partial_{u}\partial_{v}\Phi$ is bounded from above
in this neighborhood (which we generically expect to be true), $\delta v$ and
$\delta u$ approach zero no slower than $\partial_{u}\Phi$ and
$\partial_{v}\Phi$ approach zero: thus $C_{\mathrm{lim}}$ must be extremal.
Because focusing arguments ensure that the causal wedge is always contained in
the outermost extremal wedge, it must be the outermost extremal surface.

##  4 The Perturbation

Our task in higher dimensional gravity is now clear: we must find a
perturbation that removes focusing from the causal horizons (without violating
the null energy condition anywhere in the perturbed spacetime), thus shifting
the causal wedge deeper in. What kind of perturbation $\delta g$ would move
the causal surface towards rather than away from the appetizer? On a heuristic
level, we are looking to open up the bulk lightcones so that more of the bulk
is in causal contact with the asymptotic boundary. In searching for such a
perturbation, we may build on the intuition of the boundary causality
condition [43], which states that the inequality

$\int\delta g_{ab}k^{a}k^{b}\geq 0$ (4.1)

(where the integral is over a complete null geodesic with generator $k^{a}$)
is equivalent to demanding that perturbations $\delta g$ of pure AdS source
focusing (as opposed to defocusing). Here we are looking to do the opposite:
we are looking to undo focusing, so it makes sense to look for a perturbation
that satisfies an opposite inequality, with $\delta g_{kk}<0$ everywhere on
$\mathscr{H}^{+}_{C}$. It is a priori not clear that it is possible to find a
perturbation that simultaneously satisfies this inequality and also results in
a spacetime that solves the Einstein equation. To prove this point, we must
show that the requisite $\delta g$ solves the characteristic constraint
equations on the event horizon.

In this section, we will prove that as long as the causal surface is not
marginally outer trapped – i.e. as long as $\theta_{(k)}[C]\neq 0$, it is
possible to find exactly such a perturbation that (1) satisfies the
characteristic constraint equations on the causal horizon and (2) shifts the
causal surface deeper into the bulk. The procedure is roughly as follows: we
prescribe a $\delta g$ deformation on $\mathscr{H}^{+}$; some elements of this
$\delta g$ resemble the “left stretch” construction [44, 45] involving a
rescaling of the generators of certain achronal null hypersurfaces –
intuitively, this dilutes the infalling content on $\mathscr{H}^{+}$ and in
turn reduces focusing. We then demonstrate that the gravitational constraints
on $\delta g$ along with boundary conditions fix the requisite components of
$\delta g$ in such a way that the perturbed spacetime has a larger causal
wedge. In Sec. 5, we will argue how repeating these perturbations pushes in
the causal surface up to an apparent horizon in a given timefold, and to the
appetizer using several timefolds.

We will call the generators of the future and past event horizons $k^{a}$ and
$\ell^{a}$ respectively. 777These will agree with $k^{a}$ and $\ell^{a}$ on
$C$ whenever $C$ is $C^{1}$. We will extend $\ell^{a}$ to the entire spacetime
by picking a smooth Cauchy foliation $\\{{\cal C}_{\alpha}\\}_{\alpha}$ of
$\mathscr{I}$ and defining $\ell^{a}$ to be the bulk generators of $\partial
I^{+}[{\cal C}_{\alpha}]$. This defines a null foliation of
$J^{+}[\mathscr{I}]$ by past causal horizons; The past event horizon, which
$C$ lies on, is a member of this foliation.

We adopt the coordinate and gauge choices of [25]: first, we introduce double
null coordinates $(u,v)$ 888Calling these double-null coordinates is a slight
abuse of notation, as we will only require $g_{vv}=0$ at the horizon $u=0$.,
where

$k^{a}=\left(\frac{\partial}{\partial v}\right)^{a}\ \ ;\ \
\ell^{a}=\left(\frac{\partial}{\partial u}\right)^{a}.$ (4.2)

Figure 5: On the left: three-dimensional (left) illustration of the past
($v=0$) and future ($u=0$) event horizons, with a slicing of the future event
horizon given by the intersection of $u=0$ with past causal horizons
originating from complete slices of the $\mathscr{I}$. On the right: a
conformal diagram illustrating the same.

In these coordinates, the causal surface is at $u=v=0$, the future event
horizon $\mathscr{H}^{+}$ is at $u=0$ and the past event horizon
$\mathscr{H}^{-}$ is at $v=0$. See Fig. 5. We can further fix the gauge in a
neighborhood of $u=0$ so that the metric there takes the form:

$ds^{2}=-2dudv+g_{vv}dv^{2}+2g_{vi}dvdy^{i}+g_{ij}dy^{i}dy^{j}$ (4.3)

where $i,j$ denote the transverse direction on $\mathscr{H}^{+}$. At $u=0$
exactly, we of course require that $k^{a}=(\partial/\partial v)^{a}$ is null,
so that:

$g_{vv}\rvert_{u=0}=0,$ (4.4)

and we may further fix the gauge:

$g_{vi}\rvert_{u=0}=0,$ (4.5)

but we cannot require these components to be identically zero in a
neighborhood of $u=0$, i.e. the derivatives may not vanish. For instance, the
inaffinity of $k^{a}$ – given by
$\kappa_{(v)}=\frac{1}{2}\partial_{u}g_{vv}\rvert_{u=0}$ – cannot be set to
zero in general since we have independently fixed the $\ell^{a}$ vector field,
and orthogonality to $\ell^{a}$ defines the constant (or affine) $v$ slices on
$\mathscr{H}^{+}$.999The covariant definition of the inaffinity of $k^{a}$ is
$k^{a}\nabla_{a}k^{b}=\kappa_{(k)}k^{b}$.

The extrinsic curvature tensors of constant-$v$ slices are simple in this
gauge

$\displaystyle{B_{(v)}}_{ij}$
$\displaystyle=\frac{1}{2}\partial_{v}g_{ij}\rvert_{u=0},$ (4.6a)
$\displaystyle{B_{(u)}}_{ij}$
$\displaystyle=\frac{1}{2}\partial_{u}g_{ij}\rvert_{u=0},$ (4.6b)
$\displaystyle\chi_{i}$
$\displaystyle=\frac{1}{2}\partial_{u}g_{vi}\rvert_{u=0}.$ (4.6c)

where ${B_{(v)}}_{ij}$ and ${B_{(u)}}_{ij}$ denote the null extrinsic
curvatures of constant $v$ slices and $\chi_{i}$ denotes the their twist.
Since $\mathscr{H}^{+}$ is achronal, we can specify a new solution to the
Einstein equation via a perturbative deformation of the metric on it, so long
as the null constraint equations are satisfied. In particular, we consider the
following perturbation on $\mathscr{H}^{+}_{C}(u=0,v\geq 0)$:

$ds^{2}=-2dudv+(g_{vv}+\delta g_{vv})dv^{2}+2(g_{vi}+\delta
g_{vi})dvdy^{i}+(g_{ij}+\delta g_{ij})dy^{i}dy^{j}.$ (4.7)

The perturbation components $\delta g_{vv}$, $\delta g_{vi}$, $\delta g_{ij}$
and their first $u$ derivatives are initial data that we can freely specify on
the characteristic hypersurface $u=0$ so long as this data satisfies the null
constraint equations101010The null constraint equations are the equations of
motion containing at most one $u$ derivative of the metric perturbations.:

$\displaystyle\delta G_{vv}+\Lambda\delta g_{vv}=8\pi G\delta T_{vv}$ (4.8a)
$\displaystyle\delta G_{uv}=8\pi G\delta T_{uv}$ (4.8b) $\displaystyle\delta
G_{vi}+\Lambda\delta g_{vi}=8\pi G\delta T_{vi}$ (4.8c)
$\displaystyle\hat{\delta G_{ij}}+\Lambda\hat{\delta g_{ij}}=8\pi G\hat{\delta
T_{ij}}$ (4.8d)

where $\delta G_{ab}$ denotes the linearized perturbation of the Einstein
tensor, $\Lambda$ is the cosmological constant, and the hat in the last
equation denotes the traceless part. Note that the corresponding deformations
of the stress energy tensor must be sourced by a perturbative modification to
the matter fields that itself satisfies the fields’ equations of motion on the
background geometry.

Thus we will need to prescribe $\delta g$ as well as a matter source for it.
The latter is accomplished by a perturbative “stretch”. A nonperturbative
stretch is an exponential rescaling [44, 45]:

$\displaystyle g^{\prime}_{ij}(u=0,v,y)=g_{ij}(u=0,ve^{-s},y)$ (4.9a)
$\displaystyle T^{\prime}_{vv}(u=0,v,y)=e^{-2s}T_{vv}(u=0,ve^{-s},y)$ (4.9b)
$\displaystyle\kappa^{\prime}_{(v)}(u=0,v,y)=e^{-s}\kappa_{(v)}(u=0,ve^{-s},y)$
(4.9c)

where prime denotes the transformed quantities. Our matter source will be
obtained in the perturbative limit of this, by setting $e^{-s}=1-\epsilon$,
where $\epsilon\sim\mathcal{O}(\delta g)$ is the parameter controlling the
expansion. Our choice for the full perturbation on $v\geq 0$, metric and
matter, is then:

$\displaystyle\delta g_{vi}(u=0,v,y)=0$ (4.10a)
$\displaystyle\partial_{u}\delta
g_{vv}(u,v,y)\rvert_{u=0}=2(1-\epsilon)\kappa_{(v)}(0,v(1-\epsilon),y)-2\kappa_{(v)}(u=0,v,y)$
(4.10b) $\displaystyle\delta
g_{ij}(u=0,v,y)=g_{ij}(u=0,v(1-\epsilon),y)-g_{ij}(u=0,v,y)$ (4.10c)
$\displaystyle\delta
T_{vv}(u=0,v,y)=(1-\epsilon)^{2}T_{vv}(u=0,v(1-\epsilon),y)-T_{vv}(u=0,v(1-\epsilon),y)$
(4.10d)

where in this linearized analysis we will only need to keep track of first
order terms in $\epsilon$ and $\delta g$ (i.e. we will drop all terms of order
$\epsilon\delta g$). Note that $\delta g_{vv}$, $\partial_{u}\delta g_{ij}$,
and $\partial_{u}\delta g_{vi}$ are allowed to be non-zero. We will see that
their values are constrained subject to the above restrictions.

Before we move forward with the analysis, we need to ask whether we can always
obtain the stress tensor profile in Eq. (4.10d). This question is difficult to
answer in broad generality. Therefore, from now on we restrict our matter
sector to consist of a minimally coupled complex scalar field theory $\phi$
(with an arbitrary potential) coupled to some Maxwell field (or consider
either separately), with Lagrangian density

$\displaystyle\mathcal{L}_{\text{matter}}=-\frac{1}{4}g^{ac}g^{bd}F_{ab}F_{cd}-g^{ab}\bar{\nabla}_{a}\phi\bar{\nabla}_{b}\phi^{*}-V(|\phi|^{2})$
(4.11)

where $F_{ab}$ is the field strength and $\bar{\nabla}$ denotes the covariant
derivative with respect to the vector potential $A^{a}$. Then

$T_{vv}=2\bar{\nabla}_{v}\phi\bar{\nabla}_{v}\phi^{*}+F_{iv}F^{i}_{v},$ (4.12)

Since both $\phi$ and $A_{v}$ are free initial data in the characteristic
problem, we can simply generate the desired transformation by setting
$\phi^{\prime}(v,y)=\phi(ve^{-s},y)$ and
$A^{\prime}_{v}=e^{-s}A_{v}(ve^{-s},y)$, and
$A_{i}^{\prime}=A_{i}(ve^{-s},y)$.

We now proceed to prove that our choice of perturbation solves the null
constraint equations with $\delta g_{vv}<0$ everywhere on
$\mathscr{H}^{+}_{C}$. Because the unperturbed spacetime satisfies the NEC and
no new matter terms are introduced by the perturbation, the perturbed
spacetime will likewise satisfy the NEC. For pedagogical clarity, we will
focus on the more illuminating $vv$-constraint here and relegate the remaining
constraint equations to Appendix A. Our analysis here will be twofold: we will
first analyze the constraint (4.8a) in the absence of $\delta g_{vv}$,
separately compute the contribution of $\delta g_{vv}$, and sum the two
together; this is possible so long as we work in the linearized regime.

By the Raychaudhuri equation, $R_{vv}$ depends only on the geometry of the
$u=0$ hypersurface:

$R_{vv}=-\partial_{v}\theta_{(v)}-{B_{(v)}}_{ij}{B_{(v)}}^{ij}-\kappa_{(v)}\theta_{(v)}.$
(4.13)

Therefore in the absence of $\delta g_{vv}$ (i.e. implementing only the
stretch):

$\displaystyle
R^{\prime}_{vv}(u=0,v,y)=e^{-2s}R_{vv}(u=0,ve^{-s},y)-(1-e^{-s})\theta_{(v)}(u=0,v=0,y)\delta(v)$
$\displaystyle=T^{\prime}_{vv}(u=0,v,y)-(1-e^{-s})\theta_{(v)}(u=0,v=0,y)\delta(v)$
(4.14)

where the delta function term results from the discontinuity in
$\theta^{\prime}_{(v)}$ across $v=0$. So
$R^{\prime}_{vv}-T^{\prime}_{vv}=-(1-e^{-s})\theta_{(v)}[C]\delta(v)$. We now
take the same perturbative limit of this transformation and re-introduce
$\delta g_{vv}$:

$\displaystyle-\frac{1}{2}\theta_{(u)}\partial_{v}\delta
g_{vv}-\frac{1}{2}\nabla_{\perp}^{2}\delta g_{vv}+\chi^{i}\partial_{i}\delta
g_{vv}+$
$\displaystyle\left(\nabla_{\perp}.\chi-\partial_{v}\theta_{(u)}-{B_{(v)}}_{ij}{B_{(u)}}^{ij}+8\pi
G(-T_{uv}-\mathcal{L}_{\text{matter}}+F_{uv}^{2})\right)\delta g_{vv}$
$\displaystyle-\epsilon\theta_{(v)}[C]\delta(v)=0$ (4.15)

where all of the quantities multiplying $\delta g_{vv}$ and its derivatives
are background quantities. We offer an alternative derivation of Eq. (4) in
Appendix B by implementing the “stretch” using an inaffinity shock [44, 45],
which directly induces the delta function term in Eq. (4).

Since by construction, we are only perturbing the data on
$\mathscr{H}^{+}_{C}$, $\delta g_{vv}(u=0,v=0^{-},y)=0$, so the delta function
term in Eq. (4) enforces a jump in $\delta g_{vv}$:

$\delta
g_{vv}(u=0,v=0^{+},y)=2\epsilon\frac{\theta_{(v)}[C]}{\theta_{(u)}[C]}\leq 0$
(4.16)

where the sign comes from the fact that $\theta_{(v)}[C]\geq 0$ and
$\theta_{(u)}[C]<0$.111111In a non-generic spacetime, it is possible that
$\theta_{(u)}[C]=0$, which raises potential concerns about the divergence in
Eq. (4.16). We can avoid this issue by shifting the earliest location of the
perturbation to some arbitrarily small $v=\delta>0$ instead of $v=0$. This new
cut lies on a past causal horizon that reaches $\mathscr{I}$ and therefore
cannot be stationary. In fact, originating the perturbation at $v=\delta>0$
makes sense physically since by sending $t_{i}$ to arbitrarily small values in
the source Eq. (2.1) we can affect the region arbitrarily close to $C$, but
not $C$ itself. With this subtlety in mind, we pick $v=0$ as the origin of the
perturbation in the main text because we can get arbitrarily close to $C$.
Note that by assumption $\theta_{(v)}[C]>0$, and so $\delta g_{v}v<0$, at
least in a subset of $C$. This implies that the curve generated by
$(\partial_{v})^{a}$ is nowhere spacelike and at least timelike on a subset of
$C$. In order to open up the lightcone and move the causal surface deeper into
the bulk, it would be sufficient if $\delta g_{vv}\leq 0$ everywhere on
$\mathscr{H}^{+}_{C}$, not just at $C$.

We will now demonstrate that if $\delta g_{vv}(u=0,v=0^{+},y)\leq 0$, then
$\delta g_{vv}(u=0,v,y)\leq 0$ for all $v>0$, by analyzing the constraint that
$\delta g_{vv}$ satisfies on $v>0$:

$\displaystyle-\frac{1}{2}\theta_{(u)}\partial_{v}\delta g_{vv}$
$\displaystyle-\frac{1}{2}\nabla_{\perp}^{2}\delta
g_{vv}+\chi^{i}\partial_{i}\delta g_{vv}+$
$\displaystyle\left(\nabla_{\perp}.\chi-\partial_{v}\theta_{(u)}-{B_{(v)}}_{ij}{B_{(u)}}^{ij}+8\pi
G(-T_{uv}-\mathcal{L}_{\text{matter}}+F_{uv}^{2})\right)\delta g_{vv}=0$
(4.17)

which we may view as an “evolution” equation for $\delta g_{vv}$ on $u=0$ from
which we can derive $\delta g_{vv}$ on $\mathscr{H}^{+}_{C}$ from its value at
$C$.

It is not too difficult to see why $\delta g_{vv}(u=0,v,y)\leq 0$ for all
$v>0$ starting from $\delta g_{vv}(u=0,v=0^{+},y)\leq 0$. Suppose $\delta
g_{vv}>0$ at some value of $v$. Then, assuming that all quantities in (4) are
continuous, there must exist a “last” constant-$v$ slice $\sigma$ on which
$\delta g_{vv}\leq 0$ everywhere. By continuity, there exists a point
$p\in\sigma$ where $\delta g_{vv}|_{p}=0$ (and then immediately becomes
positive for larger $v$). By construction, we must have $\partial_{i}\delta
g_{vv}|_{p}=0$ and $\nabla_{\perp}^{2}\delta g_{vv}|_{p}\leq 0$. But by (4)
this implies that $\partial_{v}\delta g_{vv}|_{p}\leq 0$, and so $\delta
g_{vv}$ cannot become positive.

This reasoning may seem a bit fast, but it can be made more rigorous (and free
of simplifying assumptions) using standard techniques. The operator
$\mathscr{L}$ acting on $\delta g_{vv}$ in (4) is parabolic whenever
$\theta_{(u)}<0$; it thus satisfies the weak comparison principle for
parabolic operators, which states that if $f$ and $h$ are functions satisfying
$\mathscr{L}f\leq 0$ and $\mathscr{L}h\geq 0$ everywhere in the interior of
the parabolic domain, and $f\leq h$ on the boundary of the parabolic domain,
then $f\leq h$ everywhere in the parabolic domain. Setting $f=\delta g_{vv}$
and $h\equiv 0$, we immediately find that $\mathscr{L}f=0$ and
$\mathscr{L}h=0$, so the weak comparison principle yields the desired
conclusion: $\delta g_{vv}\leq 0$ everywhere on $\mathscr{H}^{+}_{C}$.
Technically, the weak comparison principle is usually stated for domains in
$\mathbb{R}^{n}$, fortunately, it follows as a fairly direct consequence of
the maximum principle for elliptic operators, which does hold for more general
manifolds [46]. The functions $f$ and $h$ need only be of Sobolev type
$W^{1,2}_{0}$; that is, only their local weak derivatives are required to
exist [47], which is sufficient for our purposes. In fact, a version of the
maximum principle for elliptical operators on “rough” null hypersurfaces
(including caustics and non-local intersections specifically on event
horizons) was proved in [48].

To make sure that our $\delta g_{vv}$ solution exists, we need to also satisfy
the other constraints (4.8b), (4.8c), and (4.8d). This is easy to do because
they are “evolution” equations for $\partial_{u}\delta g_{ij}$ and
$\partial_{u}\delta g_{vi}$ on $\mathscr{H}^{+}_{C}$ which we can solve no
matter what $\delta g_{vv}$ is. We relegate this discussion to appendix A.

Let us now discuss possible subtleties in our construction due to caustics.
Since caustic lines will generically be a measure zero subset of
$\mathscr{H}^{+}$ [49, 37, 50], we believe that they do not pose a fundamental
obstruction to our procedure. Caustic lines can intersect $C$, at which point
$C$ will generically be kinked. At the location of the kink, a chunk of
transverse directions, associated to the generators that emanate from the
caustic line, needs to get inserted in the transverse domain on which we place
our boundary data for Eq. (4). However, so long as this data satisfies $\delta
g_{vv}\leq 0$, Eq. (4) still guarantee $\delta g_{vv}\leq 0$ everywhere on
$\mathscr{H}^{+}_{C}$. In fact, since these new generators do not extend to
the past of the caustics by definition, we expect to have even more freedom in
specifying this boundary data because we do not have to worry about how this
boundary is glued to some past hypersurface.

Let us offer an alternative argument to further ameliorate caustic-related
worries. As $\mathscr{H}^{+}_{C}$ settles down to Kerr-Newman, there exists an
earliest cross section $\mu_{\text{earliest}}$ lying on a past horizon with no
caustics in its future. By setting $\mu_{\text{earliest}}$ as the origin of
our perturbation (the new $v=0$), we can avoid caustics altogether.
Furthermore, each perturbation should make the portion of the horizon to the
future of $\mu_{\text{earliest}}$ more stationary, pushing the new
$\mu_{\text{earliest}}$ further to the past eventually approaching
$\mathscr{H}^{-}$.

Lastly, it is important to show that the perturbation has not shifted the
$u=0$ surface to the point where it is no longer close to the event horizon –
especially in the asymptotic region $v\to\infty$. This is simplest to do if we
assume that the _background_ horizon settles down to a stationary spherically
symmetric configuration at some finite affine parameter, though we expect
proper falloffs to hold more generally. The equation for $\delta g_{vv}$ then
simplifies to:

$\displaystyle-\frac{1}{2}\theta_{(u)}\partial_{v}\delta g_{vv}$
$\displaystyle-\frac{1}{2}\nabla_{\perp}^{2}\delta
g_{vv}-\partial_{v}\theta_{(u)}~{}\delta g_{vv}=0$ (4.18)

On $\mathscr{H}^{+}$, we have $\theta_{(u)}\sim v$ asymptotically. We can then
solve for the asymptotic behavior of $\delta g_{vv}$ from Eq. (4.18):

$\delta g_{vv}\sim v^{-2}$ (4.19)

Therefore $u=0$ asymptotes to a stationary null hypersurface after the
perturbation, so it naturally lines up with the new causal horizon in the
$v\to+\infty$ limit. We find that our proposed perturbation results in a
spacetime that solves the Einstein equation and in which the causal horizon is
pushed deeper into the bulk unless $\theta_{(k)}[C]=0$.

##  5 Zigzag

We will now use the above perturbation to show that the causal surface $C$ can
be moved arbitrarily close to the outermost extremal surface $X$, the
appetizer of the lunch, using simple sources only. This requires us to show
that (1) the perturbation analyzed in the previous section can be engineered
from simple sources on the boundary, (2) the perturbation can be iteratively
repeated both in the past and future, resulting in the approach of $C$ to $X$,
without incurring high complexity, and (3) that this procedure does not change
the geometry of the lunch, nor does the causal surface breach the lunch region
in the process. We will begin our discussion by assuming for
simplicity121212No pun intended. that the causal surface and the appetizer
have the same topology; topological differences between the two surfaces are
discussed at the end.

To show (1), we simply employ our assumption of the validity of HKLL,
discussed in Sec. 1, and evolve the data on
$\mathscr{H}^{+}_{C}\cup\mathscr{H}^{-}_{C}$ “sideways” towards $\mathscr{I}$
to find appropriate boundary conditions at $\mathscr{I}$, which will be
smeared local sources.131313We are aware that some mathematician somewhere
must be apoplectic with rage after reading this paragraph. Since we are
physicists, we will nevertheless persevere under the usual assumption in
holography that HKLL does in fact work out. Concerned readers who do not a
priori wish to subscribe to HKLL may take comfort in the following
interpretation of our results: we show that reconstruction of the simple wedge
is no more complex than that of the causal wedge. This sideways evolution was
also used in [25] to prove the simple entropy conjecture in the case where the
horizon was only perturbatively non-stationary.

We will shortly demonstrate (2) in detail. The process is similar in spirit to
the zigzag process in Sec. 3, but instead of removing infalling chiral modes,
we apply our perturbation in Sec. 4. First, we will discuss the consequences
of repeated iterations of our perturbation on the future horizon and then add
timefolds into our procedure.

Let us begin by a comparison between the perturbed and unperturbed causal
surfaces, denoted $C^{\prime}$ and $C$ respectively, as a result of one
instance of our perturbation on $\mathscr{H}^{+}_{C}$. Since the perturbation
is localized away from the past event horizon $\mathscr{H}^{-}$, it is
expedient to compare the relative location of $C$ with that of $C^{\prime}$
using their position on the past event horizon: both are slices of
$\mathscr{H}^{-}$. Note that in the perturbed geometry no special role is
played by $C$. As shown in Sec. 4, the perturbation guarantees that
$C^{\prime}$ is “inwards” (i.e. at larger $u$) compared with $C$ so long as
$\theta_{(k)}[C]\neq 0$. Nothing stops us from then repeating the perturbation
above on the _new_ future horizon. As long as some point on the causal surface
satisfies $\theta_{(k)}>0$, the perturbation pushes the causal wedge further
inwards.

The only obstruction in the construction above occurs if $\theta_{(k)}$
vanishes identically on the causal surface. Thus it is clear that the inwards
shift of the causal surface obtained via simple sources limits to an outermost
marginally outer trapped surface $\mu$. (We can define a rigorous notion of
the causal surface approaching arbitrarily close to a surface with
$\theta_{(k)}=0$ by picking an affine parameterization on $\mathscr{H}^{-}$
and defining proximity of the two surfaces in terms of the maximal elapsed
affine parameter between them.)

Let us provide intuition for the existence of $\mu$. Say on $\mathscr{H}^{-}$
we can identify two cuts $\mu_{1}$ and $\mu_{2}$ such that $\mu_{2}$ encloses
$\mu_{1}$ and $\theta_{(k)}[\mu_{1}]\leq 0$, $\theta_{(k)}[\mu_{2}]\geq 0$.
Then we expect an outermost marginally outer trapped surface $\mu$ in-between
$\mu_{1}$ and $\mu_{2}$.141414This is proved when $\mu_{1}$ and $\mu_{2}$
satisfy $\theta_{(k)}[\mu_{1}]<0$, $\theta_{(k)}[\mu_{2}]>0$ and lie on smooth
spacelike slices in $D=4$ in [51], but since we can approach $\mathscr{H}^{-}$
with such spacelike slices we expect the same to hold for $\mathscr{H}^{-}$,
and the technology used in the extant proof is expected to generalize to
higher dimensions. On $\mathscr{H}^{-}$, $C$ plays the role of $\mu_{2}$. For
$\mu_{1}$, we can pick $\partial J^{+}[X]\cap\mathscr{H}^{-}$ – whenever it is
a full cross section of $\mathscr{H}^{-}$ – which satisfies $\theta_{(k)}\leq
0$ by the focusing theorem. However, note that $\partial
J^{+}[X]\cap\mathscr{H}^{-}$ might be empty if $\mathscr{H}^{-}$ falls into a
singularity before intersecting $\partial J^{+}[X]$. Even so, at least for
Kasner-like singularities, we can find cross sections of $\mathscr{H}^{-}$ in
a neighborhood of the singularity which are trapped [31]. Note also that
generically, our choices for $\mu_{1}$ and $\mu_{2}$ satisfy
$\theta_{(k)}[\mu_{1}]<0$, $\theta_{(k)}[\mu_{2}]>0$. We will then have
candidates for both $\mu_{1}$ and $\mu_{2}$, so $\mu$ exists.

Prima facie the procedure at this point appears to have failed! The causal
surface will generically stop well away from null separation with $X$, and
even further away from coincidence with $X$. However, this is only true on the
particular timefold under consideration. To proceed to close the gap further,
we reverse the arrow of time. We can then repeat the procedure above in time
reverse to shrink the discrepancy between the causal wedge and $W_{X}$. We
iterate this procedure via forward and reverse timefolds; each step brings the
causal surface and $X$ closer. Just like for the JT gravity case in Section 3,
the causal surface should limit to the outermost extremal surface after
sufficiently many timefolds. Importantly, since the bulk physics involved is
entirely classical, the number of timefolds required for the causal surface to
approach the outermost extremal surface, within a given precision, should be
independent of the Planck scale. This means that the complexity of the process
cannot diverge in the classical limit $N\to\infty$. We therefore conclude that
for $X$ and $C$ of identical topology, the simple wedge and the outermost
extremal wedge coincide.

Finally, we address (3): throughout this construction, the geometry of the
lunch is left unaltered: the perturbation is localized in the causal
complement of the lunch, guaranteeing that the lunch remains undisturbed.

What about the case where the topologies are different? This could for example
be the case in a time-symmetric and spherically symmetric null shell collapse
in AdS where $C$ is a sphere and $X=\varnothing$ [52]. If such topology
difference exists, we would expect it to present itself between $C$ and the
outermost marginally trapped surface on $\mathscr{H}^{-}$ in a timefold (or
several timefolds) of our procedure above. Furthermore, because each iteration
of our $\mathscr{H}^{+}_{C}$ perturbation pushes the causal surface inwards on
$\mathscr{H}^{-}$ a bit, at some point the jump from $C$ to $C^{\prime}$ would
have to involve a topology change. As there is nothing explicit in our
construction that constrains the topology of $C^{\prime}$ according to that of
$C$, we do not see a fundamental obstruction against such topology changes
arising from our iterative procedure. A rigorous treatment of such cases might
be interesting but is left to future work.

##  6 Simplicity Killed the Python

Having established that the simple wedge is in fact reconstructible using
exclusively simple experiments, we now explore the implications of our results
beyond the converse to the Python’s lunch: what is the dual to the area of the
outermost extremal surface? What is the field theory interpretation of our
results, and in particular what is the “simple state” dual to the simple
wedge?

###  6.1 Time-Folded Simple Entropy

As noted in the introduction, the simple entropy of [24, 25] is a coarse-
graining over high complexity measurements conducted after a fixed boundary
time $t_{\mathrm{bdy}}$ on a single timefold:

$S_{\mathrm{simple}}[t_{\mathrm{bdy}},\rho_{\mathrm{bdy}}]=\max_{\rho\in{\cal
B}}S_{\mathrm{vN}}[\rho]$ (6.1)

where $\rho_{\mathrm{bdy}}$ is the actual state of the CFT, $t_{\mathrm{bdy}}$
is a choice of boundary time slice, and ${\cal B}$ is the set of all CFT
states (density matrices) that have the same one-point functions as
$\rho_{\mathrm{bdy}}$ under any simple sources turned on after the time
$t_{\mathrm{bdy}}$ (and with some very late time cutoff to avoid recurrences).
That is, ${\cal B}$ consists of the set of CFT states $\rho$ such that

$\left\langle
E\mathcal{O}E^{\dagger}\right\rangle_{\rho_{\mathrm{bdy}}}=\left\langle
E\mathcal{O}E^{\dagger}\right\rangle_{\rho},$ (6.2)

for all possible $E$ defined as in Eq. 2.1. The simple entropy at a given
boundary time is thus a coarse-graining over high complexity data that
preserves all of the simple data to the future (or past) of that time.

With these restrictions to a particular subset of boundary time and a fixed
timefold, the simple entropy was proposed as a dual to the outer entropy,
which is a bulk-defined quantity that coarse-grains over the exterior of an
apparent horizon (a surface which is by definition always marginally
trapped151515A refinement of an apparent horizon that was named a “minimar
surface”. Apparent horizons are generically minimar surfaces.). The outer
entropy coarse-grains over all possible spacetimes that look identical outside
of a given apparent horizon to find the spacetime with the largest HRT
surface, and thus the largest von Neumann entropy in the CFT:

$S_{\mathrm{outer}}[\mu]=\max\limits_{{\cal
X}}\frac{\mathrm{Area}[X]}{4G\hbar}=\max\limits_{\rho\in{\cal
H}}S_{\mathrm{vN}}[\rho]=\frac{\mathrm{Area}[\mu]}{4G\hbar},$ (6.3)

where ${\mathcal{X}}$ consists of the HRT surfaces of all classical
holographic spacetimes containing $O_{W}[\mu]$, and ${\cal H}$ is the
corresponding set of CFT states; the final equality is proved in [24, 25].
This is done by discarding the spacetime behind $\mu$, constructing a
spacetime with an HRT surface $X_{\mu}$ whose area is identical to the area of
$\mu$, and then CPT conjugating the spacetime around $X_{\mu}$. By
construction, $O_{W}[\mu]$ is left unaltered.

The proposal that the simple and outer entropies are identical says that there
is a particular definition of black hole entropy which is a consequence of
coarse-graining over the highly complex physics that we expect describes the
interior:

$S_{\mathrm{outer}}[\mu(t_{\mathrm{bdy}})]=S_{\mathrm{simple}}[t_{\mathrm{bdy}}]$
(6.4)

where $t_{\mathrm{bdy}}=\partial J^{-}[\mu]\cap\mathscr{I}$. Our construction
in Section 4 establishes this conjecture for apparent horizons: in a given
timefold, it is possible to push the event horizon all the way up to
$\mu(t_{\mathrm{bdy}})$ without accessing any high complexity data for
$t>t_{\mathrm{bdy}}$.

Our construction is of course more general, as it applies to timefolds.
Extending the simple entropy proposal to include timefolds immediately yields
the holographic dual to the area of the outermost extremal surface $X$:

$\frac{\mathrm{Area}[X]}{4G\hbar}=S_{\mathrm{outer}}[X]=S_{\mathrm{simple}}$
(6.5)

where $S_{\mathrm{simple}}$ is obtained from
$S_{\mathrm{simple}}[t_{\mathrm{bdy}}]$ by taking
$t_{\mathrm{bdy}}\rightarrow-\infty$ and including arbitrary timefolds. The
inclusion of timefolds removes the need for a reference apparent horizon, and
the coarse-grained spacetime (in which the outermost extremal surface is in
fact the HRT surface) is obtained by CPT-conjugating around the outermost
extremal surface $X$; see Fig. 3. Crucially, note that the coarse-graining
procedure leaves the outermost extremal wedge untouched and coarse-grains only
over the lunch. Standard entanglement wedge reconstruction via quantum error
correction [53, 54, 55] applies to reconstruction of the outermost extremal
wedge, since in the coarse-grained spacetime obtained by CPT conjugation, the
outer wedge of $X$ is exactly the entanglement wedge. Since we will argue
below that the coarse-grained spacetime has a simple modular Hamiltonian,
entanglement wedge reconstruction using, for example, modular flow as in [56]
should be much simpler when based on the coarse-grained state rather than the
original state. This is consistent with the simplicity of reconstructing the
outermost extremal wedge.

###  6.2 The Simple State

So far we have introduced two manipulations that can be done to our original
spacetime. The first was the zigzag procedure, introduced in Section 5 which
makes the causal wedge coincide with (or become arbitrarily close to) the
outermost extremal wedge. We say that the resulting spacetime is ‘exposed’
because everything in the simple wedge can be directly seen by the
boundary.161616Note that this definition is unrelated to our definition of an
‘exposed surface’ in the proof of Proposition 1. The second is the coarse-
graining procedure introduced above where we CPT conjugate about the outermost
extremal surface and thereby create a state where the outermost extremal wedge
coincides with the entanglement wedge. If we apply _both_ manipulations to the
spacetime, we can produce a spacetime where all three wedges (approximately)
coincide. This is illustrated in Fig. 6.

Figure 6: A diagram illustrating the relationships between the different
states: the original CFT state $\rho$, which may be either coarse-grained to
obtain $\rho_{\mathrm{coarse}}$ by forgetting about high complexity operators,
or it may be “exposed” by acting on it with simple operators. The two
operations modify causally independent and non-intersecting portions of the
spacetime, so they commute: after obtaining the coarse-grained state in which
$X$ is the HRT surface, we may perform our zigzag procedure to push the causal
surface up to $X$ and obtain the simple state in which all three wedges
coincide. Or, after obtaining the exposed state in which the causal and
outermost extremal wedge coincide, but the entanglement wedge properly
contains both, we may coarse-grain to obtain the same simple state.

That is, given any holographic CFT state $\rho$ dual to some entanglement
wedge (which will likely have a Python’s lunch), there exists a
$\rho_{\mathrm{coarse}}$ which is indistinguishable from $\rho$ via simple
experiments and has no Python’s lunch. Executing our procedure zigzag on this
coarse-grained state yields the coarse-grained exposed spacetime, in which the
causal, simple, and entanglement wedges all coincide or come arbitrarily close
to coinciding. The dual to this is described by the state obtained via the
zigzag procedure together with the set of simple operators that remains from
the final timefold. We shall refer to this history that includes the the
remaining simple operators the simple history, and in a slight abuse of
notation we will denote this entire history of states as
$\rho_{\mathrm{simple}}$.

What is the CFT interpretation of $\rho_{\mathrm{simple}}$? Below we prove
that the causal and entanglement wedges coincide exactly if and only if the
state dual to the entanglement wedge has a geometric modular flow. The
immediate implication is that in the case where the zigzag procedure gives an
exact coincidence, the simple state has an exactly local modular Hamiltonian,
and rather than being a history it is in fact a single state. Note that this
is suggestive of a CFT dual to a gravitational no-hair theorem. To be more
precise: this result suggests that the set of stationary holographic black
holes is to be identified with the high limited set of states with exactly
local modular flow. If coincidence between the causal surface and the
appetizer is asymptotic rather than exact, then we come to the conclusion that
the modular flow generated by the simple state is very close to local in the
sense that only operators with support in the asymptotically shrinking region
between the causal and entanglement wedges are able to definitively tell that
the two are not identical. Since that region translates (for simple operators)
to access to arbitrarily late or early times, we find that finite-time simple
measurements are unable to tell that the modular flow generated by
$\rho_{\mathrm{simple}}$ at each stage is not local.

The secondary implication is that it is possible to take any holographic state
$\rho$ and, via a series of simple operations, render its modular flow
(nearly) indistinguishable from a geometric flow via any simple experiments.
If the appetizer has sufficient symmetry, then the statement should be true
exactly. Let us now prove our theorem, which we do in broad generality for
boundary subregions.

######  Theorem 1.

Let $\mathscr{W}_{C}[R]$ denote the causal wedge of a boundary subregion $R$,
and let $W_{E}[R]$ denote the entanglement wedge of the boundary subregion.
$\mathscr{W}_{C}[R]=W_{E}[R]$ if and only if the boundary modular Hamiltonian
on $R$ generates a geometric flow with respect to a boundary Killing vector
field on $R$.

###### Proof.

Assume that the boundary modular Hamiltonian generates a geometric flow with
respect to some Killing vector field $\xi^{I}$ on $\partial M$ (here $I$ is a
boundary spacetime index). Under modular flow, a local operator is mapped to
another local operator:

$\mathcal{O}(x,s)=\rho_{R}^{-is/2\pi}\mathcal{O}(x)\rho_{R}^{is/2\pi}=\mathcal{O}(x_{\xi}(s))$
(6.6)

where $x_{\xi}(s)$ is the boost along $\xi^{I}$ of $x\in D[R]$. Via [56],
$W_{E}[R]$ can be reconstructed by smearing the modular flow of local
operators over $D[R]$. Since in this case modular flow is an automorphism on
the space of local operators in $D[R]$, we can reconstruct all operators in
$W_{E}[R]$ by smearing local operators:

$\Phi(x)=\int\limits_{D[R]}f(X|x)\mathcal{O}(X)dx,$ (6.7)

where $f(X|x)$ is a smearing function. If there is a gap between $W_{E}[R]$
and $\mathscr{W}_{C}[R]$, then operators that are localized to the gap should
commute with all local operators on the boundary (within our code subspace)
via the extrapolate dictionary (note that this only works in the large-$N$
limit where we don’t have to worry about gravitational dressing). However this
is inconsistent with the equation above; so $x\in\mathscr{W}_{C}[R]$. But this
argument holds for all local operators: there exist no local operators in the
gap between $W_{X}[R]$ and $\mathscr{W}_{C}[R]$. In the large-$N$ limit
(without backreaction), this means that there simply is no gap between the two
wedges: $W_{X}[R]=\mathscr{W}_{C}[R]$.

To prove the other direction, we consider the proof of [56] for the zero-mode
formula of entanglement wedge reconstruction (appendix B.1 of [56]). Starting
with equation B.71, it is shown that the nonlocality of modular flow on
$\partial W_{E}[R]$ is due to the change in the instrinsic metric of spatial
slices of $\partial W_{E}[R]$ (in particular, the loss of ultralocality). When
$\partial W_{E}[R]$ is stationary, the metric on codimension-two slices does
not change with evolution along the congruence. This means that the modular
flow on $\partial W_{E}[R]$ is local, which in turn implies that the boundary
modular flow is local as well. ∎

##  7 Discussion

Our primary technical result in this article is the proof of the converse to
the Python’s lunch proposal in the strict large-$N$ limit: operators that lie
outside of a Python’s lunch are simply reconstructible in the dual CFT, and
moreover this reconstruction only relies on the bulk dynamics in the large-$N$
limit, manifestly respecting the causal structure of the background metric.

We emphasize that bulk reconstructions that are causal in this sense cannot
work for the interior of the Python’s lunch because no causal horizon can
intersect the lunch. The CFT encoding of the Python’s lunch appears to involve
highly non-local quantum gravity effects. An example of such non-local
dynamics is the ER=EPR conjecture [57, 58] which asserts that the entanglement
between an evaporating black hole and its Hawking radiation after the Page
time must allow complicated operations on the distant radiation to change the
state behind the lunch, drastically violating the naive causal structure
dictated by the background metric. It has been speculated that wormhole-like
“corrections” to the background geometry connecting the radiation to the black
hole interior could explain the “true” causal structure not captured by the
background metric. It is natural to speculate that similar dynamics are at
play in the Python’s lunch encoding into the boundary.

We will now discuss various generalizations of our main results:

###  7.1 Boundary-Anchored Surfaces

We have focused here primarily on compact surfaces, but we may pose similar
questions for boundary subregions: given the state $\rho_{R}$ on a boundary
subregion $R$, how complex is the reconstruction of operators behind the event
horizon but within the outermost extremal wedge? This requires a treatment of
surfaces with a boundary-anchored component rather than surfaces whose
components are all compact. Most of our results generalize almost immediately
to the boundary-anchored case: Lemma 3 [38] makes no reference the topology of
surfaces (beyond the homology constraint); similarly for the proofs of Lemmas
1 and 2. The perturbed initial data prescription at the causal surface also
carries over mutatis mutandis. As noted in Section 4, the weak comparison
principle operates on the basis of the maximum principle for elliptic
operators. The latter does indeed apply to bounded domains in general, and to
boundary-anchored hypersurfaces in AdS particular (see [59] for a discussion
in the context of AdS/CFT). The main potential source of difficulty is the
falloff: both the causal surface and the outermost extremal surface approach
the asymptotic boundary, but not with the same tangent space; the asymptotic
falloff of $\delta g$ must approach zero sufficiently fast so as to not spoil
the asymptotics while bridging the gap between the two surfaces. We expect
that the appropriate falloff conditions can be satisfied, but this remains a
subject for future work.

###  7.2 Robustness under Quantum Corrections

Fundamentally, the classical calculations done in this paper are only
interesting as an approximation to the fully quantum dynamics that actually
describe the bulk in AdS/CFT. Do our arguments extend to the semiclassical
setting where the background spacetime is still treated classically, but with
quantum fields propagating on it? Do they generalize to the regime where
perturbative corrections to the geometry, suppressed by powers $G\hbar$, are
allowed to contribute? A number of important assumptions break down in this
case: Raychaudhuri’s equation is still valid, but the null energy condition
will not generally hold, and so light rays emanating from a classical extremal
surface can defocus. Fortunately, the quantum focusing conjecture (QFC) states
that the generalized entropy of null congruences emanating from QESs is always
subject to focusing. In a semiclassical or perturbatively quantum bulk, the
appetizer is the outermost quantum, not classical, extremal surface; the QFC
ensures that the outermost quantum extremal wedge always contains the causal
wedge.171717Apparent counterexamples are only possible when the time evolution
includes interaction with an auxiliary system; once you include these
auxiliary interacting modes when defining quantum extremality then the
apparent causality issues go away. The question is whether we can still expand
the causal wedge using appropriate sources and timefolds in order to bridge
the gap between the causal surface and the appetizer.

This is a much harder question than the classical question discussed here: the
class of allowed QFT states is simply much harder to classify and use than
classical field theory states. However, in particularly simple examples, for
instance where the causal wedge is approximately Rindler-like, a quantum
version of the “left stretch” appears to be well defined and gives exactly the
right change in energy to reduce focusing and remove the perturbatively small,
i.e. $O(1/N)$, distance between the causal and outermost extremal wedges [26].
It is feasible that in the limit of many zigzags, the causal wedge would
approach a Rindler-like region, at which point it becomes possible to apply
the bulk unitaries discussed in [26] to eliminate the remaining small gap. We
leave a detailed study of this to future work.

###  7.3 Asymptotically Flat Spacetimes

In asymptotically flat spacetimes, asymptotic infinity is lightlike rather
than timelike. However, there do not seem to be any major obstructions to
adapting the results of this paper to that setting. Instead of the timefolds
of the asymptotically AdS problem, in asymptotically flat space one would
presumably evolve forwards along future null infinity, then backwards with
different boundary conditions that remove focusing at the past event horizon,
in order to produce a state where the causal surface is very close to a past
apparent horizon. Then one would evolve backwards and forwards along past null
infinity in order to produce a state where the causal surface is very close to
a future apparent horizon. At each step, the causal wedge increases in size.
After sufficiently many such timefolds, the causal surface should approach the
outermost extremal surface, as in the asymptotically AdS case. The
interpretation of our results in the asymptotically flat case is naturally
obfuscated by relatively inchoate status of flat holography. We may speculate
that extremal surfaces are important more generally for defining entropy in
gravity; it is also possible that a similar notion of a Python’s lunch applies
beyond AdS holography. We do not subscribe to any particular interpretation –
here we simply note that the technical aspects of this work are likely not
restricted to AdS.

###  7.4 Complexity Censorship

Let us finish with a few comments on cosmic censorship, a prima facie
unrelated conjecture about classical General Relativity. (Weak) cosmic
censorship [39] is essentially the statement that high curvature physics lies
behind event horizons. One of its landmark consequences is that trapped
surfaces lie behind event horizons, and that consequently marginally trapped
and in particular extremal surfaces lie on or behind event horizons [37]. It
is clear from the above discussion that any violation of cosmic censorship
would be quite problematic for the Python’s lunch picture: if the nonminimal
extremal surface could lie outside of the event horizon (or, in the quantum
case, could communicate with $\mathscr{I}$), then operators behind would lie
properly within the causal wedge and would thus be reconstructible by HKLL
despite being exponentially complex. The Python’s lunch proposal thus appears
to depend heavily on the validity of cosmic censorship – which is known to be
violated in AdS [60, 61, 62]. As matters currently stand, violations of cosmic
censorship notwithstanding, it is possible to prove that the holographic
entanglement entropy prescription guarantees that trapped surfaces must lie
behind event horizons [42]. We could however have proven the same statement
from holographic complexity: marginally trapped surfaces (and therefore,
trapped surfaces also, by the reasoning of [42]) must lie behind event
horizons, for if they did not, operators behind the Python’s lunch could be
reconstructed in a simple procedure. This suggests that in AdS/CFT, aspects of
cosmic censorship may be reformulated as “complexity censorship”: that high
complexity physics must be causally hidden and thus unable to causally
communicate to $\mathscr{I}$.

## Acknowledgments

It is a pleasure to thank S. Alexakis, R. Bousso, S. Fischetti, L. Susskind
for helpful discussions. NE is supported by NSF grant no. PHY-2011905, by the
U.S. Department of Energy under grant no. DE-SC0012567 (High Energy Theory
research), and by funds from the MIT physics department. GP is supported by
the UC Berkeley physics department, the Simons Foundation through the ”It from
Qubit” program, the Department of Energy via the GeoFlow consortium and also
acknowledges support from a J. Robert Oppenheimer Visiting Professorship. ASM
is supported by the National Science Foundation under Award Number 2014215.

##  Appendix A uv and vi constraints

Here we write down the general structure of the perturbative constraints
(4.8b), (4.8c), (4.8d):

$\displaystyle\frac{1}{2}g^{ij}\partial_{v}\partial_{u}\delta
g_{ij}=(\hat{L}_{1})^{ab}\partial_{u}\delta g_{ab}+(\hat{L}_{2})^{ab}\delta
g_{ab}+8\pi G\delta T_{uv}$ (A.1a)
$\displaystyle\partial_{v}\partial_{u}\delta
g_{vi}=(\hat{L}_{1}^{\prime})^{ab}_{i}\partial_{u}\delta
g_{ab}+(\hat{L}_{2}^{\prime})^{ab}_{i}\delta g_{ab}+8\pi G\delta T_{vi}$
(A.1b) $\displaystyle\partial_{v}\partial_{u}\hat{\delta
g_{ij}}=(\hat{L}_{1}^{\prime\prime})^{ab}_{ij}\partial_{u}\delta
g_{ab}+(\hat{L}_{2}^{\prime\prime})^{ab}_{ij}\delta g_{ab}+8\pi G\hat{\delta
T_{ij}}$ (A.1c)

where $\hat{L_{n}}$, $\hat{L^{\prime}_{n}}$, and $\hat{L^{\prime\prime}_{n}}$
represent linear differential operators that depend only on the background
metric and do not have $u$ derivatives. The only significance of Eqs. (A.1a),
(A.1b), and (A.1c) for us is that they are consistent with the $\delta g_{vv}$
solution to Eq. (4). Plugging in the $\delta g_{vv}$ solution reduces Eqs.
(A.1a), (A.1b), and (A.1c) to coupled linear differential equations involving
$\partial_{u}\delta g_{ij}\rvert_{u=0}$ and $\partial_{u}\delta
g_{vi}\rvert_{u=0}$ which are first order in $\partial_{v}$. Note that the
initial conditions at $v=0$ are
$\partial_{u}g_{ij}(u=0,v=0^{-},y)=\partial_{u}g_{vi}(u=0,v=0^{-},y)=0$, as
enforced by the perturbation being localized away from $\mathscr{H}^{-}$.

##  Appendix B Alternative derivation of the vv constraint

Here we provide an alternative derivation for the same perturbation discussed
in Sec. 4. Instead of the transformations (4.9a), (4.9b), and (4.9c) on
$\mathscr{H}^{+}_{C}$, we can equivalently insert an inaffinity shock at $v=0$
[44, 45]:

$\displaystyle\kappa_{(v)}=(1-e^{-s})\delta(v)$ (B.1)

and take the $(1-e^{-s})\sim\epsilon$ limit. In addition, we want to introduce
the following $\delta g_{ab}$ transformation:

$\displaystyle ds^{2}=-2dudv+(g_{vv}+\delta g_{vv})dv^{2}+2(g_{vi}+\delta
g_{vi})dvdy^{i}+(g_{ij}+\delta g_{ij})dy^{i}dy^{j},$ (B.2)

with

$\displaystyle\delta g_{vi}\rvert_{u=0}$ $\displaystyle=0$ (B.3)
$\displaystyle\delta g_{ij}\rvert_{u=0}$ $\displaystyle=0$ (B.4)
$\displaystyle\partial_{u}\delta g_{vv}\rvert_{u=0}$ $\displaystyle=0$ (B.5)

We need to apply the combined perturbations in Eqs. (B.1) and (B.2) to the
$vv$ constraint:

$\displaystyle\delta G_{vv}+\Lambda\delta g_{vv}=8\pi G\delta T_{vv}$ (B.6)

The only contribution in $\delta G_{vv}$ from Eq. (B.1) is through the
$\kappa_{(v)}\theta_{(v)}$ term in Eq. (4.13). Summing this up with the
contribution from the contribution from $\delta g_{ab}$ of Eq. (B.2), we get:

$\displaystyle-\frac{1}{2}\theta_{(u)}\partial_{v}\delta
g_{vv}-\frac{1}{2}\nabla_{\perp}^{2}\delta g_{vv}+\chi^{i}\partial_{i}\delta
g_{vv}+$
$\displaystyle\left(\nabla_{\perp}.\chi-\partial_{v}\theta_{(u)}-{B_{(v)}}_{ij}{B_{(u)}}^{ij}+8\pi
G(-T_{uv}-\mathcal{L}_{\text{matter}}+F_{uv}^{2})\right)\delta g_{vv}$
$\displaystyle-\epsilon\theta_{(v)}[C]\delta(v)=0$ (B.7)

Similarly, the uv and vi constraints could be analyzed resulting in Eqs.
(A.1a) and (A.1b).

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