Python’s Lunches in Jackiw-Teitelboim gravity with matter

Dongsu Bak,${}^{\negthinspace a}$ Chanju Kim,${}^{\negthinspace b}$ Sang-Heon
Yi,${}^{\negthinspace c}$ Junggi Yoon${}^{\,d,e,f}$

a) Physics Department & Natural Science Research Institute

University of Seoul, Seoul 02504 KOREA

b) Department of Physics, Ewha Womans University, Seoul 03760 KOREA

c) Center for Quantum Spacetime & Physics Department, Sogang University, Seoul
04107 KOREA

d) Asia Pacific Center for Theoretical Physics, Pohang 37673 KOREA

e) Department of Physics, POSTECH, Pohang 37673 KOREA

f) School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
ABSTRACT

We study Python’s lunch geometries in the two-dimensional Jackiw-Teitelboim
model coupled to a massless scalar field in the semiclassical limit. We show
that all extrema including the minimal quantum extremal surface, bulges and
appetizers lie inside the horizon. We obtain fully back-reacted general bulk
solutions with a massless scalar field, which can be understood as
deformations of black holes. The temperatures of the left/right black holes
become in general different from each other. Moreover, in the presence of both
state and source deformations at the same time, the asymptotic black hole
spacetime is further excited from that of the vacuum solution. We provide
information-theoretic interpretation of deformed geometries including Python’s
lunches, minimal quantum extremal surface and appetizers according to the
entanglement wedge reconstruction hypothesis. By considering the restricted
circuit complexity associated with Python’s lunch geometries and the operator
complexity of the Petz map reconstructing a code space operation, we show that
the observational probability of Python’s lunch degrees of freedom from the
boundary is exponentially suppressed. Thus, any bulk causality violation
effects related with Python’s lunch degrees are suppressed nonperturbatively.

##  1 Introduction

The concept of entropy plays a fundamental role in various fields of
theoretical physics. Especially in black hole physics, the entropy may pave a
door-way to our understanding of quantum gravity. One of the well-established
results in this regard is the Bekenstein-Hawking area formula [1, 2], which
says that the black hole entropy is given by its horizon area divided by $4G$.
Though this formula is regarded as a clue to the holographic nature of
underlying degrees of freedom, it does not reveal their precise microscopic
origin unfortunately. Based on the fact that the horizon works as a one-way
door (not allowing anything to escape from a black hole classically), it has
been proposed that the entropy may be understood as a fine-grained
entanglement entropy between the interior and the exterior regions, divided by
the black-hole horizon [3, 4].

As was argued by Page some time ago [5], the entanglement entropy of black
holes should respect the unitary evolution of quantum mechanics and follow the
so-called Page curve. This is incompatible with the result that the Hawking
radiation is Planckian [6] while any correction to this result should be
exponentially suppressed [7] and so an initially pure state is evolved into a
mixed, as far as there is no remnant in the black hole evaporation process and
the low energy effective field theory approach is valid. This conflict is one
way to explain the information loss paradox in black hole physics, which is
sharply reformulated as the firewall paradox [8]. For old black holes which
have emitted more than half of their mass through the Hawking radiation, a
newly generated Hawking radiation is entangled with both the early radiation
and the behind horizon degrees of freedom, which is claimed to be in
contradiction with the monogamy of the entanglement. Thus this conclusion must
be evaded in some ways.

In this regard, there were rather crucial developments based on the so-called
generalized entropy [1, 3]. One begins with a boundary region ${\cal I}$ and
its complement on a given boundary time slice. Along with a codimension two
bulk surface $P$ that is homologous to ${\cal I}$ and divides a bulk Cauchy
slice ending on the boundary time slice into two regions, the generalized
entropy $S_{P}$ can be defined, which consists of two contributions; One is
given by the area of $P$ (divided by $4G$) and the other by the entanglement
entropy of bulk fields between the two regions divided by the surface $P$.
With this preliminary, the procedure to obtain the minimal quantum extremal
surface (QESmin) goes as follows; First extremize $S_{P}$ spatially on a
Cauchy slice specified in the above111Usually, one picks all local minima of
$S_{P}$ at this first stage. In this note, we instead include all extrema,
which leads to the same QESmin at the end of the day., and then maximize among
all possible choices of Cauchy slices for the given boundary, which leads to a
set of extremal surfaces. Among these extremal surfaces, we choose the global
minimum, which singles out the desired QESmin [9, 10, 11]. This can be thought
of as a generalization of the Ryu-Takayanagi surface [12, 13, 14], which has
been explored in various contexts as a holographic realization of the
entanglement entropy of a boundary region. The generalized entropy in some
examples can be computed explicitly, explains the Page curve, and evades the
firewall through some further identification: entanglement wedge island. It
has been argued under the central dogma that some part of black hole interior
degrees of freedom associated with the island are actually belonging to the
radiation (see for a review [15]); An outside observer acting on the radiation
only may in principle probe the island degrees of freedom directly, which
appears rather bizarre as they are at least shielded by the horizon!

Python’s lunches are closely related to the island wedges just mentioned [16].
In general, among all extremal surfaces, one may have a local maximum placed
between two saddle surfaces. Such a local maximal surface is called a bulge
surface whereas the remaining saddle surfaces except the QESmin are called
appetizers. Python’s lunch geometries are involving at least one bulge and one
appetizer. In this note we would like to understand the nature of Python’s
lunch degrees of freedom. In order to understand them, one may borrow various
concepts and results from quantum information theory and its realization in
the context of the AdS/CFT correspondence. These include the entanglement
wedge reconstruction, quantum error correcting codes, the holographic tensor
network, circuit complexity and so on [17, 18, 19, 20, 21, 22, 23, 24, 25].

In particular, quantum complexity plays an essential role in our understanding
of the Python’s lunch degrees of freedom. It has been known that there are two
versions of complexity. One is the unrestricted version of complexity
associated with an observer who may access the entire quantum system of
interest. For instance, in two-sided eternal black holes, probed by an
observer who may access both boundaries, the unrestricted complexity of the
thermo-field-double (TFD) state under time evolution is holographically
related to the time-dependent expansion of the black hole interior region [26,
27, 28]. In the bulk picture, this complexity may be identified using the
volume/action conjecture. The other is the restricted complexity where an
associated observer can access only part of the underlying quantum system. A
rather well-known example is the computational complexity arising in decoding
information out of Hawking radiation [29, 30]. In this note, we shall be
mainly concerned about the restricted complexity in order to understand the
nature of Python’s lunch degrees of freedom.

In this note we focus on the two-dimensional Jackiw-Teitelboim (JT) model [31,
32] coupled to massless scalar fields in its semiclassical limit. We shall
identify all possible deformations of black hole solutions which may involve
Python’s lunch geometries. It turns out that, in the semiclassical limit, all
extrema including QESmin, bulges and appetizers lie inside the horizon, which
basically follows from the null energy condition (NEC). This deformed two-
sided black hole spacetime is to be divided into left and right wedges that
share QESmin as a separating surface (point in 2d). Under the entanglement
wedge reconstruction hypothesis, all local bulk operations within the
left/right wedge can be reconstructed out of the corresponding operations on
the left/right boundary Hilbert space. For definiteness, let us choose the
right wedge and consider a minimal Python’s lunch lying behind the horizon of
the right side. Within this, there are specific local bulk degrees of freedom
that are genuinely associated with the Python’s lunch ($\alpha$-degrees in
short) and the remaining local bulk degrees of freedom in the region outside
the appetizer ($i$-degrees in short). The combined bulk Hilbert space should
be mapped to the full boundary Hilbert space of the right side. We shall
review the restricted complexity of Python’s lunch geometries that is
exponential in their number of degrees. We shall also review the operational
complexity of the Petz map by which one reconstructs code space operations
acting on the bulk Python’s lunch Hilbert space out of $i$-Hilbert space of a
one-sided boundary. Based on these, we shall argue that observational
probability of Python’s lunch degrees of freedom by some boundary experiment
acting on $i$-Hilbert space is exponentially suppressed. Thus, any bulk
causality violation effects related with Python’s lunch degrees are suppressed
nonperturbatively.

This paper is organized as follows. In section 2 our setup is given with the
emphasis on the two-sided black hole solutions. In section 3 we provide a
generic deformation of a scalar field solving the equation of motion. For
concreteness we consider the massless scalar field only. We also present the
general profiles of the dilaton field. In section 4 it is shown that the
deformation of the dilaton field solution can be understood as the deformation
of black holes. Exploring the effects of the scalar field deformation on the
black holes, we show that the temperatures of two-sided black holes can be
asymmetric between the left and right sides. The end points of the horizon are
also studied and it is shown that the traversable wormholes cannot be formed
by our deformation. We show that QESmin, Python’s lunches, and appetizers
reside behind the horizon in our semiclassical setup. In section 5 we provide
some geometric interpretation of our results including python’s lunch,
appetizer, QESmin, and the entanglement wedge reconstruction hypothesis. In
section 6, based on our study, we give some reviews and comments on the
related aspects of the complexity, the information reconstruction from the
one-sided boundary state, and the interaction among the degrees of freedom
including Python’s lunch degrees of freedom. In the final section, we
summarize our results and provide some future directions.

##  2 Two-dimensional dilaton gravity

Our basic setup is a 2d dilaton gravity known as the JT model coupled to a
matter field [31, 32, 33], whose action is given by

$\displaystyle I=I_{top}+{1\over 16\pi
G}\int_{M}d^{2}x\sqrt{-g}\,\phi\left(R+\frac{2}{\ell^{2}}\right)+I_{M}(g,\chi)\,,$
(2.1)

where

$\displaystyle I_{top}$ $\displaystyle={\phi_{0}\over 16\pi
G}\int_{M}d^{2}x\sqrt{-g}R\,,$ $\displaystyle I_{M}$
$\displaystyle=-\frac{1}{2}\int_{M}d^{2}x\sqrt{-g}\left(g^{ab}\nabla_{a}\chi\nabla_{b}\chi+m^{2}\chi^{2}\right)\,.$
(2.2)

For a well-defined variational principle, we need to add a surface term

$I_{surf}={1\over 8\pi G}\int_{\partial M}\sqrt{-\gamma}\,(\phi_{0}+\phi)\,K$
(2.3)

to the above where $\gamma_{ij}$ and $K$ denote the induced metric and the
extrinsic curvature on the boundary $\partial M$ which is taken to be
timelike.

The equations of motion read

$\displaystyle R+\frac{2}{\ell^{2}}=0\,,$ (2.4)
$\displaystyle\nabla^{2}\chi-m^{2}\chi=0\,,$ (2.5)
$\displaystyle\nabla_{a}\nabla_{b}\phi-
g_{ab}\nabla^{2}\phi+\frac{1}{\ell^{2}}g_{ab}\phi=-8\pi GT_{ab}\,,$ (2.6)

where

$T_{ab}=\nabla_{a}\chi\nabla_{b}\chi-\frac{1}{2}g_{ab}\left(g^{cd}\nabla_{c}\chi\nabla_{d}\chi+m^{2}\chi^{2}\right)\,.$
(2.7)

The global AdS2 space corresponds to the metric solution to the equations of
motion, whose form is given by

$ds^{2}=\frac{\ell^{2}}{\cos^{2}\mu}\left(-d\tau^{2}+d\mu^{2}\right)\,,$ (2.8)

where the spatial coordinate $\mu$ is ranged over
$[-\frac{\pi}{2},\frac{\pi}{2}]$. The dilaton field configuration for the most
general vacuum black hole solution reads

$\phi=\phi_{BH}(L,b,\tau_{B})\equiv\bar{\phi}\,L\,\,\frac{(b+b^{-1})\cos(\tau-\tau_{B})-(b-b^{-1})\sin\mu}{2\cos\mu}\,.$
(2.9)

By the coordinate transformation

$\displaystyle\frac{r}{L}$
$\displaystyle=\frac{(b+b^{-1})\cos(\tau-\tau_{B})-(b-b^{-1})\sin\mu}{2\cos\mu}\,,$
$\displaystyle\tanh\frac{tL}{\ell^{2}}$
$\displaystyle=\frac{2\sin(\tau-\tau_{B})}{(b+b^{-1})\sin\mu-(b-b^{-1})\cos(\tau-\tau_{B})}\,,$
(2.10)

one is led to the corresponding AdS black hole metric

$ds^{2}=-\frac{r^{2}-L^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-L^{2}}dr^{2}\,,$
(2.11)

with

$\phi=\bar{\phi}\,r\,.$ (2.12)

In general, this black hole metric describes only the Rindler wedge of two-
sided AdS black holes. The location of singularity is defined by the curve
$\Phi^{2}\equiv\phi_{0}+\phi=0$ in the above dilaton field, and $\Phi^{2}$
might be viewed as characterizing the size of transverse space from the
viewpoint of dimensional reduction from higher dimensions222For example, the
dimensional reduction of 4d near-extremal black holes has been investigated in
[34], which leads to our JT gravity with matter fields. This reduction is well
controlled once $\phi|_{\rm cutoff}\ll\phi_{0}$ [35, 34] where $\phi|_{\rm
cutoff}$ denotes the value of the dilaton at asymptotic cut-off trajectories
introduced below. Of course, for the well-defined nearly AdS2 dynamics, we
also require $\bar{\phi}\,\ell\ll\phi|_{\rm cutoff}$. With these conditions,
our solutions below can be consistently embedded into the 4d theory.  [33]. In
this left/right symmetric two-sided black hole, one can see that the Hawking
temperature, the entropy and energy are given by

$T=\frac{1}{2\pi}\frac{L}{\ell^{2}}\,,\qquad S=S_{0}+{\cal C}T\,,\qquad
E=\frac{1}{2}{\cal C}T^{2}\,,$ (2.13)

where $S_{0}=\frac{\phi_{0}}{4G}$ and ${\cal
C}=\frac{\pi\bar{\phi}\ell^{2}}{2G}$ denote the ground state entropy and the
specific heat, respectively. In general, these physical quantities may be
different for left/right Rindler wedges in deformed two-sided black holes. In
the next sections we consider general deformations with a scalar field
starting from the left-right symmetric black hole configuration with
$\tau_{B}=0$ and $b=1$. We find that the above left-right thermodynamic
quantities indeed become different from each other in general with a
nontrivial matter field profile.

Before going ahead, let us recall a simple left-right asymmetrically deformed
configuration with a massless scalar, known as the eternal Janus deformation
[36, 37]. In this case, the black hole temperature and the dilaton field
configuration are left-right symmetric under the exchange
$\mu\leftrightarrow-\mu$, but the matter field configuration is not left-right
symmetric. This simple Janus solution is given by

$\chi=\gamma(\mu-\kappa_{0})\,,\qquad\phi=\bar{\phi}L_{0}\frac{\cos\tau}{\cos\mu}-4\pi
G\gamma^{2}(1+\mu\tan\mu)\,,$ (2.14)

The Penrose diagram of this configuration with $\phi_{0}$ contribution is
given in Figure 1.

Figure 1: The Penrose diagram for the Janus deformation is depicted.

Now, we would like to present the matter field solution to the equations of
motion in (2.5), which is given by

$\chi=\sum^{\infty}_{n=0}c_{n}\,{\cal N}_{n}\cos^{\cal D}\mu\,C_{n}^{\cal
D}(\sin\mu)\,e^{-i(n+{\cal D})\tau}+{\rm c.c.}\ ,$ (2.15)

where

${\cal N}_{n}=2^{{\cal D}-1}\Gamma({\cal
D}){\textstyle\sqrt{\frac{\Gamma(n+1)}{\pi\Gamma(n+2{\cal D})}}}\,,$ (2.16)

and $C^{\cal D}_{n}(x)$ denotes the Gegenbauer polynomial. Here, the parameter
${\cal D}$ is related to the mass of the scalar field $\chi$ as

${\cal D}=\Delta_{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+4m^{2}}\right)\,.$ (2.17)

The AdS/CFT dictionary tells us that the bulk matter field $\chi$ with mass
$m$, when $m^{2}\geq 0$, is dual to a scalar primary operator $O_{\Delta}(t)$,
where its dimension $\Delta$ is given by the larger one of $\Delta_{\pm}$ as

$\Delta=\Delta_{+}=\frac{1}{2}\left(1+\sqrt{1+4m^{2}}\right)\,,$ (2.18)

The deformation by the scalar field $\chi$ with ${\cal D}=\Delta_{+}$
corresponds to a state deformation of the dual field theory, while the other
deformation with ${\cal D}=\Delta_{-}$ does to a source deformation. When
scalar field mass squared is negative but above the Breitenlohner-Freedman
bound [38], which is given by $0>m^{2}>-1/4$ in our case, both possibilities
of operator dimensions may be realized as

$\displaystyle\Delta=\Delta_{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+4m^{2}}\right)\,.$
(2.19)

In this paper, we focus on the bulk classical configuration by ignoring the
loop corrections. This limited consideration may be justified by keeping the
Newton constant very small and by introducing a controllable number of
identical scalar fields, say $K$, without self-interactions [39]. With the
combination $KG$ fixed, one finds that any loop corrections are suppressed as
$\mathcal{O}(1/K^{n})$ where $n$ is counting the number of loops involved.

##  3 General deformations with a massless scalar field

From now on, we shall focus on turning on a massless scalar field that is dual
to a scalar operator of dimension $\Delta=1$. In this section we are mainly
interested in fully back-reacted bulk solutions with the massless scalar
field. Based on these solutions, we shall identify rather general Python’s
lunch geometries that are arising if certain parameter conditions are met. For
this, we note that the metric solution to the equation (2.4) remains the same
while the bulk scalar equation (2.5) (with $m^{2}=0$) may be solved explicitly
by

$\displaystyle\chi=\chi_{v}+\chi_{s}\ ,$ (3.1)

where

$\chi_{v}=\sum^{\infty}_{n=1}a_{n}\sin n\big{(}\mu+{\frac{\pi}{2}}\big{)}\cos
n(\tau-\tau^{v}_{n})\ ,$ (3.2)

and

$\chi_{s}=b_{0}+\sum^{\infty}_{n=1}b_{n}\cos
n\big{(}\mu+{\frac{\pi}{2}}\big{)}\cos n(\tau-\tau^{s}_{n})\ .$ (3.3)

This describes a full-set of classical solutions of the scalar equation in the
ambient AdS2 space with the global coordinates. The bulk solution with
$\chi=\chi_{v}$ corresponds to the state deformation of the left/right
boundary quantum system without deforming our boundary theories333One may
specify the well-known correspondence between JT gravity and SYK system. But
such detailed specification does not play much role in our description of this
paper.; This leads to a nontrivial vev of the dimension-one operator $O$ in
the boundary theory. On the other hand, the bulk solution with $\chi=\chi_{s}$
corresponds to the source deformation of boundary Lagrangian $L_{0}$ by
$L_{0}+s_{L/R}(u)O(u)$ where $u$ denotes the time coordinate of our left/right
boundary system going upward direction444Note that the left-right values of
vev and source are not independent since our undeformed bulk configuration
corresponds to the TFD state in the boundary..

In our JT model, the metric remains to be AdS2 with an arbitrary matter
perturbation, as mentioned previously. For the dilaton field, we shall
consider deformations from the vacuum solution

$\phi_{bg}=\,\bar{\phi}L_{0}\,\frac{\cos\tau}{\cos\mu}\,,$ (3.4)

which describes a two-sided black hole with temperature
$T_{0}=\frac{1}{2\pi}\negthinspace\frac{L_{0}}{\ell^{2}}$. When the matter
field is turned on, the two-sided black hole will be perturbed in general
left-right asymmetrically as we shall analyze in detail in later sections.

Considering first the state perturbation with $\chi=\chi_{v}$, the dilaton
solution becomes

$\phi=\phi_{bg}+8\pi
G\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}a_{m}a_{n}\phi^{v}_{m,n}\ ,$ (3.5)

where

$\displaystyle\phi^{v}_{n,n}$
$\displaystyle=\frac{(-1)^{n}\,n}{8(4n^{2}-1)}\left(2n{\cos 2n\mu}+\sin
2n\mu\tan\mu\right)\cos 2n(\tau-\tau^{v}_{n})-\frac{n^{2}}{4}(1+\mu\tan\mu)\
,$ (3.6)

and

$\displaystyle\phi^{v}_{n,n+1}=\phi^{v}_{n+1,n}$
$\displaystyle=\frac{(-1)^{n+1}\sec\mu}{16(2n+1)}\Big{[}(n+1){\sin
2n\mu}+n\sin(2n+2)\mu\Big{]}\cos[(2n+1)\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}-{\negthinspace}n\tau^{v}_{n}]$
$\displaystyle\ \ \
-\frac{n(n+1)}{{8}}(\sin\mu+\mu\sec\mu)\cos[\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}+{\negthinspace}n\tau^{v}_{n}]\
,$ (3.7)

with $n=1,2,\cdots$. The remaining off-diagonal contribution
$\phi^{v}_{m,n}\,\,(m\neq n,n\pm 1)$ is given by

$\displaystyle\phi^{v}_{m,n}=$
$\displaystyle\frac{mn}{8\cos\mu}\left[\cos[n(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{n}){\negthinspace}-{\negthinspace}m(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{m})]\textstyle{\left(\frac{\sin(n-m+1)(\mu+\frac{\pi}{2})}{(n-m+1)(n-m)}-\frac{\sin(n-m-1)(\mu+\frac{\pi}{2})}{(n-m-1)(n-m)}\right)}\right.$
$\displaystyle\left.+\cos[n(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{n}){\negthinspace}+{\negthinspace}m(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{m})]\textstyle{\left(\frac{\sin(n+m+1)(\mu+\frac{\pi}{2})}{(n+m+1)(n+m)}-\frac{\sin(n+m-1)(\mu+\frac{\pi}{2})}{(n+m-1)(n+m)}\right)}\right]\,,$
(3.8)

which satisfies $\phi^{v}_{m,n}=\phi^{v}_{n,m}$. The terms in (3.6) and (3)
can be obtained by taking $\lim_{m\rightarrow n}\phi^{v}_{m,n}$ and
$\lim_{m\rightarrow n+1}\phi^{v}_{m,n}$ up to some homogeneous terms.

For the sake of an illustration, we here present the gravity solution with
$\chi=a_{1}\chi^{v}_{1}+a_{2}\chi^{v}_{2}+a_{3}\chi^{v}_{3}$ where
$\chi_{n}^{v}=\sin n(\mu+{\frac{\pi}{2}})\cos n(\tau-\tau^{v}_{n})$. The
corresponding dilaton solution in this case becomes

$\phi=\phi_{bg}+8\pi
G\left(a_{1}^{2}\phi^{v}_{1,1}+a_{2}^{2}\phi^{v}_{2,2}+a_{3}^{2}\phi^{v}_{3,3}+2a_{1}a_{2}\phi^{v}_{1,2}+2a_{1}a_{3}\phi^{v}_{1,3}+2a_{2}a_{3}\phi^{v}_{2,3}\right)\,,$
(3.9)

where the three diagonal terms are given by

$\displaystyle\phi^{v}_{1,1}\,$
$\displaystyle=-\left[\frac{1}{12}\cos^{2}\mu\cos
2(\tau-\tau^{v}_{1})+\frac{1}{4}(1+\mu\tan\mu)\right]\,,$
$\displaystyle\phi^{v}_{2,2}\,$ $\displaystyle=-\left[\frac{1}{15}(2-3\cos
2\mu)\cos^{2}\mu\cos 4(\tau-\tau^{v}_{2})+(1+\mu\tan\mu)\right]\,,$ (3.10)
$\displaystyle\phi^{v}_{3,3}\,$
$\displaystyle=-\frac{3}{140}\Bigl{[}\cos^{2}\mu(9-16\cos 2\mu+10\cos
4\mu)\cos 6(\tau-\tau^{v}_{3})+105(1+\mu\tan\mu)\Bigr{]}\ .$

The remaining off-diagonal terms can be organized as

$\displaystyle\phi^{v}_{1,2}$
$\displaystyle=\negthinspace\frac{1}{6}\cos^{2}\mu\sin\mu\cos(3\tau\negthinspace-\negthinspace\tau^{v}_{1}\negthinspace-\negthinspace
2\tau^{v}_{2})\negthinspace-\negthinspace\frac{1}{4}\left(\frac{\mu}{\cos\mu}\negthinspace+\negthinspace\sin\mu\right)\cos(\tau\negthinspace+\negthinspace\tau^{v}_{1}\negthinspace-\negthinspace
2\tau^{v}_{2})\ ,$ $\displaystyle\phi^{v}_{2,3}\,$
$\displaystyle=-\frac{1}{80}\Bigl{[}(60\cos(\tau+2\tau^{v}_{2}-3\tau^{v}_{3})\Bigl{(}\frac{\mu}{\cos\mu}+\sin\mu\Bigr{)}$
$\displaystyle\ \
-\cos(5\tau-2\tau^{v}_{2}-3\tau^{v}_{3})8\cos^{2}\mu(-1+4\cos
2\mu)\sin\mu)\Bigr{]}\,,$ $\displaystyle\phi^{v}_{1,3}\,$
$\displaystyle=-\frac{1}{20}\cos^{2}\mu\Bigl{[}5\cos(2\tau+\tau^{v}_{1}-3\tau^{v}_{3})+(2-3\cos
2\mu)\cos(4\tau-\tau^{v}_{1}-3\tau^{v}_{3})\Bigr{]}\,.$

Note that the left and right boundaries of our AdS2 spacetime are located
respectively at $\mu=\pm\pi/2$. In these asymptotic regions, the dilaton
solution behaves as

$\phi\rightarrow\bar{\phi}L_{0}\frac{Q(\tau)}{\cos\mu}+O(\cos\mu)\,.$ (3.11)

Indeed one may explicitly check this behavior for
$\phi^{v}_{1,1},\,\phi^{v}_{2,2},\,\phi^{v}_{3,3}$ and
$\phi^{v}_{1,2},\,\phi^{v}_{2,3}$ with nonvanishing contributions to
$Q(\tau)$. The remaining term $\phi^{v}_{1,3}\rightarrow O(\cos^{2}\mu)$ as we
approach the left/right boundaries does not contribute to $Q(\tau)$. This
trend continues with the general mode-solution $\phi^{v}_{m,n}$ for arbitrary
$m$ and $n$. Namely, the $Q(\tau)$ contributions from $\phi^{v}_{n,n}$ and
$\phi^{v}_{n,n+1}(=\phi^{v}_{n,n+1})$ are nonvanishing. On the other hand,
$\phi^{v}_{m,n}(m\neq n,n\pm 1)\rightarrow O(\cos^{2}\mu)$ as
$\mu\rightarrow\pm\pi/2$ and, consequently, has no contribution to $Q(\tau)$.

We here also present a general solution for the source deformation with
$\chi=\chi_{s}$. The corresponding dilaton solution can be obtained as

$\phi=\phi_{bg}+8\pi
G\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}b_{m}b_{n}\phi^{s}_{m,n}\ ,$ (3.12)

where

$\displaystyle\phi^{s}_{n,n}$
$\displaystyle=\frac{(-1)^{n+1}\,n}{8(4n^{2}-1)}\left(2n{\cos 2n\mu}+\sin
2n\mu\tan\mu\right)\cos 2n(\tau-\tau^{s}_{n})-\frac{n^{2}}{4}(1+\mu\tan\mu)\
,$ (3.13)

and

$\displaystyle\phi^{s}_{n,n+1}=\phi^{s}_{n+1,n}$
$\displaystyle=\frac{(-1)^{n}\sec\mu}{16(2n+1)}\Big{[}(n+1){\sin
2n\mu}+n\sin(2n+2)\mu\Big{]}\cos[(2n+1)\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{s}_{n+1}{\negthinspace}-{\negthinspace}n\tau^{s}_{n}]$
$\displaystyle\ \ \
-\frac{n(n+1)}{{8}}(\sin\mu+\mu\sec\mu)\cos[\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{s}_{n+1}{\negthinspace}+{\negthinspace}n\tau^{s}_{n}]\
,$ (3.14)

with $n=1,2,\cdots$. The remaining off-diagonal contribution
$\phi^{s}_{m,n}\,\,(m\neq n,n\pm 1)$ is given by

$\displaystyle\phi^{s}_{m,n}=$
$\displaystyle\frac{mn}{8\cos\mu}\left[\cos[n(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{n}){\negthinspace}-{\negthinspace}m(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{m})]\textstyle{\left(\frac{\sin(n-m+1)(\mu+\frac{\pi}{2})}{(n-m+1)(n-m)}-\frac{\sin(n-m-1)(\mu+\frac{\pi}{2})}{(n-m-1)(n-m)}\right)}\right.$
$\displaystyle\left.-\cos[n(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{n}){\negthinspace}+{\negthinspace}m(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{m})]\textstyle{\left(\frac{\sin(n+m+1)(\mu+\frac{\pi}{2})}{(n+m+1)(n+m)}-\frac{\sin(n+m-1)(\mu+\frac{\pi}{2})}{(n+m-1)(n+m)}\right)}\right]\
,$ (3.15)

which satisfies $\phi^{s}_{m,n}=\phi^{s}_{n,m}$. The contributions in (3.13)
and (3) can be obtained by taking $\lim_{m\rightarrow n}\phi^{s}_{m,n}$ and
$\lim_{m\rightarrow n+1}\phi^{s}_{m,n}$ up to some homogeneous terms.

We note that the dilaton solution is independent of $b_{0}$ since its
contribution to the energy momentum tensor vanishes. Hence we shall set
$b_{0}$ to zero for simplicity in our presentation. Also one finds that the
general structure of $\phi^{s}_{m,n}$ is similar to that of $\phi^{v}_{m,n}$
which is from the state deformation. Indeed one finds that the relation

$\phi^{s}_{m,n}|_{\tau^{s}_{k}=\tau^{v}_{k}+\frac{\pi}{2k}}=\phi^{v}_{m,n}$
(3.16)

works with $k=1,2,\cdots$. Hence in our study of Python’s lunch geometries in
Sec. 5, we will focus on the case of the state deformation (or the source
deformation alternatively) unless both are turned on at the same time.

Finally one may also consider the full general deformation with
$\chi=\chi_{v}+\chi_{s}$. Then the corresponding dilaton solution may be
obtained as

$\phi=\phi_{bg}+8\pi
G\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}\Bigl{[}a_{m}a_{n}\phi^{v}_{m,n}+b_{m}b_{n}\phi^{s}_{m,n}+2a_{m}b_{n}\phi^{c}_{m,n}\Bigr{]}\
,$ (3.17)

where

$\displaystyle\phi^{c}_{n,n}$
$\displaystyle=\frac{(-1)^{n}\,n}{8(4n^{2}-1)}\left(\cos 2n\mu\tan\mu-2n{\sin
2n\mu}\right)\cos n(2\tau-\tau^{v}_{n}-\tau^{s}_{n})$
$\displaystyle+\frac{n^{2}}{4}\tau\tan\mu\sin n(\tau^{v}_{n}-\tau^{s}_{n})\ ,$
(3.18)

and

$\displaystyle\phi^{c}_{n,n+1}=\frac{(-1)^{n+1}\sec\mu}{16(2n+1)}\Big{[}(n{\negthinspace}+{\negthinspace}1){\cos
2n\mu}{\negthinspace}+{\negthinspace}n\cos(2n{\negthinspace}+{\negthinspace}2)\mu\Big{]}\cos[(2n{\negthinspace}+{\negthinspace}1)\tau{\negthinspace}-{\negthinspace}n\tau^{v}_{n}{\negthinspace}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{s}_{n{\negthinspace}+{\negthinspace}1}]$
$\displaystyle\ \
-\frac{n(n+1)}{8}\Bigl{[}\cos\mu\cos[\tau{\negthinspace}+{\negthinspace}n\tau^{v}_{n}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{s}_{n{\negthinspace}+{\negthinspace}1}{\negthinspace}]-\tau\sec\mu\sin[\tau{\negthinspace}+{\negthinspace}n\tau^{v}_{n}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{s}_{n{\negthinspace}+{\negthinspace}1}{\negthinspace}]\Bigr{]}\
,$ (3.19)
$\displaystyle\phi^{c}_{n+1,n}=\frac{(-1)^{n+1}\sec\mu}{16(2n+1)}\Big{[}(n{\negthinspace}+{\negthinspace}1){\cos
2n\mu}{\negthinspace}+{\negthinspace}n\cos(2n{\negthinspace}+{\negthinspace}2)\mu\Big{]}\cos[(2n{\negthinspace}+{\negthinspace}1)\tau{\negthinspace}-{\negthinspace}n\tau^{s}_{n}{\negthinspace}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{v}_{n{\negthinspace}+{\negthinspace}1}]$
$\displaystyle\ \
+\frac{n(n+1)}{8}\Bigl{[}\cos\mu\cos[\tau{\negthinspace}+{\negthinspace}n\tau^{s}_{n}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{v}_{n{\negthinspace}+{\negthinspace}1}{\negthinspace}]-\tau\sec\mu\sin[\tau{\negthinspace}+{\negthinspace}n\tau^{s}_{n}-{\negthinspace}(n{\negthinspace}+{\negthinspace}1)\tau^{v}_{n{\negthinspace}+{\negthinspace}1}{\negthinspace}]\Bigr{]}\
,$ (3.20)

with $n=1,2,\cdots$.

The remaining term $\phi^{c}_{m,n}\,\,(m\neq n,n\pm 1)$ is given by

$\displaystyle\phi^{c}_{m,n}$
$\displaystyle=\frac{mn}{8\cos\mu}\left[\cos[{\negthinspace}m(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{m}){\negthinspace}-{\negthinspace}n(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{n}){\negthinspace}]\textstyle{\left(\frac{\cos(m-n-1)(\mu+\frac{\pi}{2})}{(m-n-1)(m-n)}-\frac{\cos(m-n+1)(\mu+\frac{\pi}{2})}{(m-n+1)(m-n)}\right)}\right.$
$\displaystyle\left.+\cos[m(\tau{\negthinspace}-{\negthinspace}\tau^{v}_{m}){\negthinspace}+{\negthinspace}n(\tau{\negthinspace}-{\negthinspace}\tau^{s}_{n})]\textstyle{\left(\frac{\cos(m+n+1)(\mu+\frac{\pi}{2})}{(m+n+1)(m+n)}-\frac{\cos(m+n-1)(\mu+\frac{\pi}{2})}{(m+n-1)(m+n)}\right)}\right]\,.$
(3.21)

Again the contributions in (3) and (3) can be obtained by taking
$\lim_{m\rightarrow n}\phi^{c}_{m,n}$, $\lim_{m\rightarrow
n}\phi^{c}_{m,n+1}$and $\lim_{m\rightarrow n+1}\phi^{c}_{m,n}$ up to some
homogeneous terms.

In the following sections, we shall analyze the asymptotic form of the above
solutions in detail. The resulting left/right black holes are deformed in
general. For instance, the Hawking temperatures of the left/right black holes
become in general different from each other. Especially when both state and
source deformations are present at the same time, the asymptotic black hole
spacetime is further excited rather nontrivially.

Finally one may also consider chiral deformations for which either $T_{++}=0$
and $T_{--}=0$. (In our classical case with the massless scalar field,
$T_{-+}=T_{+-}=0$ automatically.) This type of deformations is realized by
choosing $a_{n}=\pm\,b_{n}$ and $\tau_{n}^{s}=\tau_{n}^{v}+\frac{\pi}{2n}$.
Then the scalar field becomes

$\chi_{\pm}=\sum^{\infty}_{n=1}b_{n}\sin\big{(}\tau\pm\mu\pm\frac{\pi}{2}-\tau^{v}_{n}\big{)}\,,$
(3.22)

which gives us $T_{\pm\pm}=(\partial_{\pm}\chi_{\pm})^{2}$ with
$T_{\mp\mp}=0$, respectively. Here, we introduced $x^{\pm}=\tau\pm\mu$. In
these coordinates, the equation of motion for the dilaton involving
$T_{\mp\mp}$ becomes

$-\sec^{2}\mu\,\partial_{\mp}(\cos^{2}\mu\,\partial_{\mp}\phi)=T_{\mp\mp}\ .$
(3.23)

Thus one may show that the dilaton field can be written in the form of

$\phi=f_{\pm}(x^{\pm})\tan\mu+g_{\pm}(x^{\pm})\ ,$ (3.24)

for $\chi=\chi_{\pm}(x^{\pm})$, respectively.

##  4 Deformations of two-sided black holes

In this section, we shall analyze the resulting deformation of the two-sided
black hole spacetime including the horizon structure and the late-time
temperature of the black hole system.

Physical properties of the deformed system can be obtained by studying the
asymptotic behavior of the dilaton solution. We shall see that, when only one
kind of deformation of the two in (3.1) is turned on, the asymptotic behavior
of the dilaton reduces to the form of the vacuum solution (2.9) with deformed
constants. In general the Hawking temperatures of the left/right black holes
changes differently from each other. Although there can be many nontrivial
Python’s lunches developed in the bulk, the boundary dynamics will be that of
simply deformed black holes with corresponding modifications in the cut-off
trajectory and does not directly capture the effect of the Python’s lunches.
In the presence of both the state deformation and the source deformation, the
asymptotic behavior of the dilaton turns out to be more complicated than the
vacuum solution (2.9). This would be then reflected in the boundary dynamics
as a new term in the boundary action as well as more complicated trajectories.

To see first the causal structure of the solution (3.12), let us study its
asymptotic behavior as $\mu\rightarrow\pm\frac{\pi}{2}$. Note that, as
$\mu\rightarrow\pm\frac{\pi}{2}$,

$\displaystyle\phi^{v}_{n,n}\ $
$\displaystyle=-\frac{n^{2}}{4}(1+\mu\tan\mu)+{\cal O}(\cos^{2}\mu)\ ,$
$\displaystyle\phi^{v}_{n,n+1}$
$\displaystyle=\phi^{v}_{n+1,n}=-\frac{n(n+1)}{8}(\sin\mu+\mu\sec\mu)\cos[\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}+{\negthinspace}n\tau^{v}_{n}]+{\cal
O}(\cos^{2}\mu)\ ,$

while all the remaining of $\phi^{v}_{m,n}$ are ${\cal O}(\cos^{2}\mu)$. We
shall also introduce $\alpha_{n}$ by the rescaling
$\alpha_{n}=na_{n}\sqrt{\frac{\pi^{2}G}{\bar{\phi}L_{0}}}$. The dilaton in
asymptotic regions can then be written in the form of (3.11) with

$Q(\tau)=\cos\tau-\sum^{\infty}_{n=1}\alpha_{n}^{2}+(\pm)\sum^{\infty}_{n=1}\alpha_{n}\alpha_{n+1}\cos[\tau{\negthinspace}-{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}+{\negthinspace}n\tau^{v}_{n}]\
.$ (4.1)

Here and below the upper/lower sign inside parenthesis is for the left/right
boundary region respectively.

As was noted previously [35], the reparameterization can be fixed by the
prescription of the dilaton cutoff

$\phi|_{\rm cutoff}=\bar{\phi}\frac{\ell}{\epsilon}\ ,$ (4.2)

treating $\phi$ as a radial coordinate in the asymptotic region. This will fix
the asymptotic left/right cutoff trajectory $\mu^{L/R}_{c}(\tau)$. For
simplicity below, we shall frequently drop the index $L$ and $R$ since the
left and right asymptotic regions can be treated in the same way. Through the
asymptotic form of the dilaton field in (3.11), the trajectory is then given
by

$\cos\mu_{c}=\frac{L_{0}}{\ell}Q(\tau)\epsilon\ .$ (4.3)

The reparameterization on this trajectory enables us to express $\tau$ in
terms of the boundary time ${u}$. Namely, from the prescription in the metric
in (2.8), we have

$ds^{2}|_{\rm
cutoff}\left(=-\frac{\ell^{2}d\tau^{2}}{\cos^{2}\mu_{c}}\right)=-\frac{du^{2}}{\epsilon^{2}}\
.$ (4.4)

This leads to

$du=d\tau\frac{\ell^{2}}{L_{0}Q(\tau)}\ .$ (4.5)

We arrange $Q(\tau)$ as in the form of

$Q(\tau)=\sqrt{A^{2}+B^{2}}(\cos(\tau-\tau_{B})-q)\ ,$ (4.6)

where

$\displaystyle A\,$
$\displaystyle=1+(\pm)\sum^{\infty}_{n=1}\alpha_{n}\alpha_{n+1}\cos[{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}-{\negthinspace}n\tau^{v}_{n}]\
,$ $\displaystyle B\,$
$\displaystyle=(\pm)\sum^{\infty}_{n=1}\alpha_{n}\alpha_{n+1}\sin[{\negthinspace}(n+1)\tau^{v}_{n+1}{\negthinspace}-{\negthinspace}n\tau^{v}_{n}]\
,$ $\displaystyle q\ $
$\displaystyle=\frac{\sum^{\infty}_{n=1}\alpha^{2}_{n}}{\sqrt{A^{2}+B^{2}}}\
,$ $\displaystyle\tau_{B}$ $\displaystyle=\arctan\frac{B}{A}.$ (4.7)

For the asymptotic region to be well-defined, $q<1$ and $A>0$ is required.
Then the reparameterization is solved by

$\tanh\frac{L}{\ell^{2}}\frac{u}{2}=\sqrt{\frac{1+q}{1-q}}\tan\frac{\tau-\tau_{B}}{2}\
,$ (4.8)

where $L=L_{0}\sqrt{A^{2}+B^{2}}\sqrt{1-q^{2}}$. Since $u$ is the boundary
time coordinate of the cutoff trajectory in the Rindler wedge in (2.11), the
late-time temperature of the deformed system becomes

$T=T_{0}\sqrt{A^{2}+B^{2}}\sqrt{1-q^{2}}\ .$ (4.9)

One further find that the future/past infinity of the Rindler wedge is at

$\tau_{f/p}=\pm 2\arctan\sqrt{\frac{1-q}{1+q}}+\tau_{B}\ ,$ (4.10)

which can also be obtained from $Q(\tau)=0$. Then the L/R future horizons are
described by straight lines

$(\mp)\mu=(\tau-\tau^{L/R}_{f})+\frac{\pi}{2}\ ,$ (4.11)

while the L/R past horizons are by

$(\pm)\mu=(\tau-\tau^{L/R}_{p})-\frac{\pi}{2}\ .$ (4.12)

The corresponding Penrose diagram is depicted in Figure 2.

Figure 2: Penrose diagram of deformed two-sided black hole spacetime.

Note that in this figure the end points of the future/past horizon
$\tau_{f/p}$ are shifted from $\pm\frac{\pi}{2}$ to values with
$|\tau_{f/p}|\,\leq\,\frac{\pi}{2}$. This can be seen from

$\cos\tau_{f/p}=\frac{1}{A^{2}+B^{2}}[A\alpha\mp{\rm
sign}(B)\sqrt{A^{2}\alpha^{2}-(A^{2}+B^{2})(\alpha^{2}-B^{2})}]\ ,$ (4.13)

where $\alpha\equiv\sum^{\infty}_{n=1}\alpha^{2}_{n}$. Since

$\displaystyle\alpha\pm B$
$\displaystyle=\frac{1}{2}\alpha_{1}^{2}+\frac{1}{2}\sum^{\infty}_{n=1}\left\\{\alpha_{n}^{2}+\alpha_{n+1}^{2}\pm(\pm)2\alpha_{n}\alpha_{n+1}\sin[(n+1)\tau^{v}_{n+1}-n\tau^{v}_{n}]\right\\}\
,$ $\displaystyle\geq 0\ ,$ (4.14)

the numerator of (4.13) is positive semidefinite, which proves
$|\tau_{f/p}|\,\leq\,\frac{\pi}{2}$.

In case of the source deformations where the dilaton is given by (3.17), the
functional form of each mode $\phi_{m,n}^{s}$ is essentially the same as that
of the state deformation $\phi^{v}_{m,n}$ and hence we get the same results.
However, for the full general deformations (3.1) including both state as well
as source deformations, we obtain rather different behaviors due to the cross
terms $\phi_{m,n}^{c}$ in (3.17). To see this, let us first consider the
simplest case where both the state and the source deformations are turned on,

$\chi=a_{1}\cos\mu\cos(\tau-\tau^{v}_{1})-b_{1}\sin\mu\cos(\tau-\tau^{s}_{1})\
,$ (4.15)

with the corresponding dilation solution

$\phi=\phi_{bg}+8\pi
G(a_{1}^{2}\phi^{v}_{1,1}+b_{1}^{2}\phi^{s}_{1,1}+2a_{1}b_{1}\phi^{c}_{1,1})\
.$ (4.16)

As $\mu\rightarrow(\mp)\frac{\pi}{2}$, $\phi$ behaves as in (3.11) with

$Q(\tau)=\cos\tau-\alpha_{1}^{2}-\beta_{1}^{2}+(\mp)2\alpha_{1}\beta_{1}\left[\frac{1}{3\pi}\cos(2\tau-\tau^{v}_{1}-\tau_{1}^{s})+\frac{2\tau}{\pi}\sin(\tau^{v}_{1}-\tau_{1}^{s})\right]\
,$ (4.17)

where $\beta_{n}=nb_{n}\sqrt{\frac{\pi^{2}G}{\bar{\phi}L_{0}}}$. Note that the
cross term in $Q(\tau)$ contains a linear term in $\tau$ which cannot be
obtained as a function of $u$ in a closed form. For small deformations, one
can still compute future/past infinities at which $Q(\tau)$ vanishes. The
results are

$\tau_{f/p}=\pm\frac{\pi}{2}+\delta\tau_{f/p}\ ,$ (4.18)

where

$\displaystyle\delta\tau_{f}$
$\displaystyle=-\alpha_{1}^{2}-\beta_{1}^{2}+(\mp)2\alpha_{1}\beta_{1}\left[\sin(\tau^{v}_{1}-\tau_{1}^{s})-\frac{1}{3\pi}\cos(\tau^{v}_{1}+\tau_{1}^{s})\right]+\cdots\
,$ $\displaystyle\delta\tau_{p}$
$\displaystyle=\alpha_{1}^{2}+\beta_{1}^{2}+(\mp)2\alpha_{1}\beta_{1}\left[\sin(\tau^{v}_{1}-\tau_{1}^{s})+\frac{1}{3\pi}\cos(\tau^{v}_{1}+\tau_{1}^{s})\right]+\cdots\
.$ (4.19)

Here the ellipses are quartic order terms in $\alpha_{1}$ and $\beta_{1}$ and
the upper/lower sign refers to the left/right boundary respectively as before.
Note that $\delta\tau_{f/p}$ is not negative/positive definite. In other
words, $|\tau_{f/p}|$ can be larger than $\frac{\pi}{2}$ when both the state
and the source are deformed together, in contrast with the previous cases
where only one kind of deformation is turned on. Nevertheless, for this
particular deformation with the lowest modes, the difference of the future
infinity and the past infinity is still less than or equal to $\pi$, i.e.,

$\displaystyle\tau^{L/R}_{f}-\tau^{L/R}_{p}$
$\displaystyle=\pi-2(\alpha_{1}^{2}+\beta_{1}^{2})+(\pm)\frac{4}{3\pi}\alpha_{1}\beta_{1}\cos(\tau^{v}_{1}+\tau_{1}^{s})+\cdots$
$\displaystyle\leq\pi\ .$ (4.20)

Then the difference becomes smaller by the perturbation, although the overall
spacetime is slightly shifted either upward or downward.

From (4), one can also compare shifted values in other boundaries. Suppose for
instance that $\delta\tau_{f}^{R}\geq 0$. (The other case
$\delta\tau_{f}^{L}\geq 0$ can be treated in a similar manner.) Assuming
$\alpha_{1}\beta_{1}\geq 0$, this happens only when

$\sin(\tau^{v}_{1}-\tau_{1}^{2})-\frac{1}{3\pi}\cos(\tau^{v}_{1}+\tau_{1}^{s})\geq
1\ .$ (4.21)

Then it is not difficult to see that the following relation holds

$\delta\tau_{f}^{L}\leq
0\leq\delta\tau_{f}^{R}\leq\delta\tau_{p}^{L}\leq\delta\tau_{p}^{R}\ .$ (4.22)

Therefore although the position $\tau_{f}^{R}$ of the future infinity at the
right boundary is shifted slightly upward, the other three positions are also
moved enough to compensate for such shift. In particular,
$\delta\tau_{f}^{L/R}\leq\delta\tau_{p}^{R/L}$ implies that the causal
structure is not of the type of traversable wormholes, as it should be for
classical perturbations.

Having seen what happens when both the state and the source deformations are
turned on, let us now consider the most general deformation (3.1). Denote

$Q(\tau)=\cos\tau+Q_{1}^{L/R}(\tau)\ ,$ (4.23)

where $Q_{1}^{L/R}(\tau)$ is given by

$Q_{1}^{L/R}=\sum_{m,n}(\alpha_{m}\alpha_{n}\varphi^{v,L/R}_{m,n}+\beta_{m}\beta_{n}\varphi_{m,n}^{s,L/R}+2\alpha_{m}\beta_{n}\varphi_{m,n}^{c,L/R})\
,$ (4.24)

with

$\varphi_{m,n}^{L/R}(\tau)\equiv\left.\frac{8\cos\mu}{\pi
mn}\phi_{m,n}(\tau,\mu)\right|_{\mu\rightarrow\mp\frac{\pi}{2}}\ .$ (4.25)

Then

$\tau_{f}^{R}-\tau_{p}^{R}=\pi+Q_{1}^{R}(\tfrac{\pi}{2})+Q_{1}^{R}(-\tfrac{\pi}{2})+\cdots\
,$ (4.26)

where ellipsis is of quartic or higher order in $\alpha_{n}$’s and
$\beta_{n}$’s. Nonvanishing contributions from the cross term
$\varphi_{n,m}^{c,R}$ are

$\displaystyle\varphi_{n,n}^{c,R}(\tfrac{\pi}{2})+\varphi_{n,n}^{c,R}(-\tfrac{\pi}{2})\
\ $ $\displaystyle=\frac{2(-1)^{n}}{\pi
n(4n^{2}-1)}\cos[n(\tau^{v}_{n}+\tau_{n}^{s})],$
$\displaystyle\varphi_{n,n+1}^{c,R}(\tfrac{\pi}{2})+\varphi_{n,n+1}^{c,R}(-\tfrac{\pi}{2})$
$\displaystyle=\cos[n\tau^{v}_{n}-(n+1)\tau_{n+1}^{s}],$
$\displaystyle\varphi_{n+1,n}^{c,R}(\tfrac{\pi}{2})+\varphi_{n+1,n}^{c,R}(-\tfrac{\pi}{2})$
$\displaystyle=-\cos[n\tau^{s}_{n}-(n+1)\tau_{n+1}^{v}],$ (4.27)

and for $m\neq n$ and $m-n$ even,

$\varphi_{m,n}^{c,R}(\tfrac{\pi}{2})+\varphi_{m,n}^{c,R}(-\tfrac{\pi}{2})=(-1)^{\frac{m-n}{2}+1}\left\\{\negthinspace\negthinspace\frac{4\cos(m\tau^{v}_{m}-n\tau_{n}^{s})}{\pi(m\negthinspace-\negthinspace
n)[(m\negthinspace-\negthinspace n)^{2}\negthinspace-\negthinspace
1]}-\frac{4\cos(m\tau^{v}_{m}+n\tau_{n}^{s})}{\pi(m\negthinspace+\negthinspace
n)[(m\negthinspace+\negthinspace n)^{2}\negthinspace-\negthinspace
1]}\right\\}.$ (4.28)

Collecting terms from $\varphi^{v}_{m,n}$, $\varphi_{m,n}^{s}$ and
$\varphi_{m,n}^{c}$, we have

$\displaystyle Q_{1}^{R}(\tfrac{\pi}{2})+Q_{1}^{R}(-\tfrac{\pi}{2})$ (4.29)
$\displaystyle=$
$\displaystyle-2\sum_{n}\left\\{\alpha_{n}^{2}+\beta_{n}^{2}\right\\}+2\hskip-8.5359pt\sum_{m-n\
\textrm{even}}\hskip-8.5359pt\alpha_{m}\beta_{n}\,[\varphi_{m,n}^{c,R}(\tfrac{\pi}{2})+\varphi_{m,n}^{c,R}(-\tfrac{\pi}{2})]\
$
$\displaystyle+2\sum_{n}\left\\{\alpha_{n}\beta_{n+1}[\varphi_{n,n+1}^{c,R}(\tfrac{\pi}{2})+\varphi_{n,n+1}^{c,R}(-\tfrac{\pi}{2})]+\alpha_{n+1}\beta_{n}[\varphi_{n+1,n}^{c,R}(\tfrac{\pi}{2})+\varphi_{n+1,n}^{c,R}(-\tfrac{\pi}{2})]\right\\}\,.$

We can explicitly diagonalize (4.29) for some finite number of deformations.
If $\alpha_{n}=\beta_{n}=0$ for $n\geq 8$, we find that (4.29) is negative
semidefinite. If higher modes are present, however, it turns out that (4.29)
can be positive because of the cross terms. In other words, if high enough
modes of both state and source deformations are turned on at the same time,
the difference between the future infinity and the past infinity in one side
can be larger than $\pi$. Since $|\tau_{f/p}|\leq\frac{\pi}{2}$ in the
presence of only one type of deformations as seen above, this result is due to
the interaction of both deformations for which the asymptotic black hole
spacetime is nontrivially excited from the vacuum solution. Also, the
trajectories of the boundary dynamics would be deformed accordingly.

On the other hand, the difference of the future infinity of one side and the
past infinity of the other side must still be less than or equal to $\pi$ for
arbitrary deformations, since traversable wormholes should not be formed.
Indeed, we find that

$\tau_{f}^{R}-\tau_{p}^{L}=\pi+Q_{1}^{R}(\tfrac{\pi}{2})+Q_{1}^{L}(-\tfrac{\pi}{2})+\cdots\
,$ (4.30)

where $Q_{1}^{R}(\tfrac{\pi}{2})+Q_{1}^{L}(-\tfrac{\pi}{2})$ can be written in
a manifestly negative semidefinite form,

$\displaystyle Q_{1}^{R}$
$\displaystyle(\tfrac{\pi}{2})+Q_{1}^{L}(-\tfrac{\pi}{2})$ (4.31)
$\displaystyle=$
$\displaystyle-(\alpha_{1}\sin\tau^{v}_{1}-\beta_{1}\cos\tau^{s}_{1})^{2}-(\alpha_{1}\cos\tau^{v}_{1}+\beta_{1}\sin\tau^{s}_{1})^{2}$
$\displaystyle-\sum_{n}\left\\{[\alpha_{n}\sin n\tau^{v}_{n}-\beta_{n}\cos
n\tau^{s}_{n}-\alpha_{n+1}\cos(n+1)\tau^{v}_{n+1}-\beta_{n+1}\sin(n+1)\tau^{s}_{n+1}]^{2}\right.$
$\displaystyle\phantom{-\sum_{n}\
}\left.+[\alpha_{n+1}\sin(n+1)\tau^{v}_{n+1}-\beta_{n+1}\cos(n+1)\tau^{s}_{n+1}+\alpha_{n}\cos
n\tau^{v}_{n}+\beta_{n}\sin n\tau^{s}_{n}]^{2}\right\\}.$

Since the general dilaton solution (3.17) is an infinite series of
trigonometric functions with arbitrary coefficients, one may wonder if the
extremum points of $\phi$ can exist anywhere in the spacetime. We shall argue,
however, that classically they should reside behind the horizon thanks to the
NEC. Let us focus on the right Rindler wedge for definiteness. In terms of the
affine parameter for a fixed $x^{-}=x^{-}_{0}$,

$\lambda=\tan\left(\frac{x^{+}-x^{-}_{0}}{2}\right),$ (4.32)

the equation of motion (3.23) in null coordinates can be written as

$\frac{d^{2}\phi}{d\lambda^{2}}=-4\cos^{4}\left(\frac{x^{+}-x^{-}_{0}}{2}\right)~{}T_{++}=-T_{\lambda\lambda}\,,$
(4.33)

which is the two-dimensional analogue of the Raychaudhuri equation. Here,
$\frac{d}{d\lambda}\phi$ plays the role of the expansion of the null geodesic
congruence. Now, the positive NEC ($T_{\lambda\lambda}>0$) implies

$\frac{d^{2}\phi}{d\lambda^{2}}<0\,,$ (4.34)

which shows us that null rays from the extremal point of $\phi$, defined by
$\partial_{+}\phi=\partial_{-}\phi=0$, should be met with a singularity within
a finite affine parameter and so the extremal points should reside behind the
horizon. Note that this is nothing but a two-dimensional version of the famous
singularity theorem or the focusing argument [40, 41, 42].

In case of the chiral deformations (3.22), the situation is rather simple. For
definiteness, let us choose the upper sign in (3.22) which corresponds to
$T_{--}=0$. Then there are only ingoing/outgoing modes in the right/left sides
and we expect that the future/past horizons of the right/left black holes
would be deformed to outwards/inwards so that they do not meet at one point.
On the other hand, the other side of the horizons, namely the past/future
horizons of the right/left black holes would be still connected as a single
straight line. See Figure 3. This can easily be seen from the general form of
the dilaton solution (3.24). Note that at the horizon we would have
$\frac{d\phi}{d\lambda}\Big{|}_{H}\underset{\lambda=\pm\infty}{=}0$, where
$\lambda$ is the relevant affine parameter. It implies the equation
$f_{+}(x^{+})=0$ which is independent of $x^{-}$. Then in the spacetime under
consideration where the horizon is unique at each asymptotic region, the
solution $x^{+}=x^{+}_{H}$ should define a unique straight-line throughout the
spacetime. In other words, it represents the past/future horizons on the
right/left sides connected as a single straight line as expected.

Figure 3: Penrose diagram of the two-sided black hole spacetime deformed by
the antichiral matter $\chi=\chi(x^{+})$.

We now show that the extremal point should be unique and lie on the horizon
for chiral deformations. First, it is evident that $\partial_{-}\phi=0$
results in $f_{+}(x^{+})=0$ which is nothing but the defining equation of the
horizon $x^{+}=x^{+}_{H}$. The other condition $\partial_{+}\phi=0$ for the
extremal point gives

$\partial_{+}\phi=f_{+}^{\prime}(x^{+})\tan\mu+g_{+}^{\prime}(x^{+})=0\ ,$
(4.35)

which has a unique solution on the horizon $x^{+}=x^{+}_{H}$,

$x^{-}=x^{+}_{H}+2\tan^{-1}\left(\frac{g_{+}^{\prime}(x^{+}_{H})}{f_{+}^{\prime}(x^{+}_{H})}\right)\
,$ (4.36)

which completes the proof. Note in particular that there cannot be a Python’s
lunch at least classically for chiral deformations.

##  5 Python’s lunch geometries

In this section, we discuss Python’s lunch geometries formed by the general
deformations. Let us first consider the Janus deformation (2.14) to identify a
Python’s lunch in the simplest setup. Since $\cos\tau$ is the only
$\tau$-dependent term in the dilaton configuration (2.14), we should consider
the $\tau=0$ slice to find any extremum point. At $\tau=0$, the dilaton field
can be written as

$\phi=\bar{\phi}L_{0}\left[\frac{1}{\cos\mu}-\frac{\gamma^{2}}{2}(1+\mu\tan\mu)\right]\,,$
(5.1)

where we set $\frac{8\pi G}{\bar{\phi}L_{0}}\equiv 1$ in this section. Then as
$\mu\rightarrow\frac{\pi}{2}$,

$\phi=\frac{1-\frac{\pi}{4}\gamma^{2}}{\frac{\pi}{2}-\mu}+O\left(\frac{\pi}{2}-\mu\right),$
(5.2)

which requires $\gamma^{2}<\frac{4}{\pi}$. For $\mu\simeq 0$, $\phi$ becomes

$\phi=\bar{\phi}L_{0}\left[\frac{1}{2}(1-\delta)-\frac{\delta}{2}\mu^{2}+\frac{1}{24}(1-4\delta)\mu^{4}\right]+O(\mu^{6}),$
(5.3)

where $\delta=\gamma^{2}-1$. It is clear that, for small positive $\delta$,
$\mu=0$ is a maximum point of $\phi$. The QESmin is the minimum point of
$\phi$ which occurs at $\mu_{q}\simeq\sqrt{6\delta}(1+O(\delta))$ with

$\phi(\mu_{q})=\bar{\phi}L_{0}\left[\frac{1}{2}(1-\delta)-\frac{3}{2}\delta^{2}\right]+O(\delta^{3}).$
(5.4)

Therefore, for $1<\gamma^{2}<\frac{4}{\pi}$, a bulge is formed at $\mu=0$ and
QESmin is located at $\mu_{q}\simeq\sqrt{6\delta}$. The difference of the two
dilaton values is

$\phi(0)-\phi(\mu_{q})=\frac{3}{2}\bar{\phi}L_{0}\delta^{2}+O(\delta^{3}).$
(5.5)

We draw the dilaton field under the Janus deformation in Figure 4.

Figure 4: 3D plot of dilaton field for Janus deformation with
$\gamma^{2}=6/5$.

Now we look into more general deformations analyzed in section 3. For the
lowest-mode state deformation $\chi=a_{1}\chi_{1}^{v}$, the dilaton solution
is given by (3.9) and (3). Choosing $\tau_{1}^{v}=\frac{\pi}{2}$ and $a_{1}$
such that $\frac{3}{2}<a_{1}^{2}<\frac{8}{\pi}$, the dilaton is positive
definite and has a maximum at $\mu=\tau=0$ and hence a Python’s lunch geometry
is formed around the origin, as depicted in Figure 5. For $a_{1}^{2}\simeq
3/2$, the QESmin occurs at $\mu\simeq 0$ while, as $a_{1}$ increases, its
position moves towards $\mu=\frac{\pi}{2}$.

Figure 5: Various plots of dilaton field with $\tau=0$ and
$\tau_{n}=\frac{\pi}{2n}$. (a) For $\chi=a_{1}\chi_{1}^{v}$ with
$a_{1}=0.9\sqrt{8/\pi}$. (b) For $\chi=a_{1}\chi_{1}^{v}+a_{2}\chi_{2}^{v}$
with $a_{1}=0.9\sqrt{8/\pi}$ and $a_{2}=-0.02\sqrt{8/\pi}$. (c) For
$\chi=a_{1}\chi_{1}^{v}+a_{2}\chi_{2}^{v}+a_{3}\chi_{3}^{v}$ with
$a_{1}=0.094\sqrt{8/\pi}$, $a_{2}=0.94\sqrt{8/\pi}$ and
$a_{3}=-0.094\sqrt{8/\pi}$ . (d) For
$\chi=a_{151}\chi_{151}^{v}+a_{200}\chi_{200}^{v}$ with
$a_{1}=0.0316\sqrt{8/\pi}$, $a_{2}^{2}=0.8888\sqrt{8/\pi}$.

So far, we have considered left-right symmetric configurations. The QESmin in
this case is located at two places, one in $\mu>0$ region and the other in
$\mu<0$ region, respectively, since the dilaton fields are even functions of
$\mu$. If we turn on several modes at the same time, the dilaton has cross
terms which can be odd in $\mu$ as seen in (3). Then the degeneracy is broken
and we have only one QESmin at either left or right region. This is
illustrated in Figure 5, where the first and the second modes are turned on at
the same time, $\chi=a_{1}\chi_{1}^{v}+a_{2}\chi_{2}^{v}$, and the dilaton has
a cross term $\phi^{v}_{1,2}$ in (3) which is an odd function of $\mu$. In the
figure, there are three extrema which can be identified as the QESmin, a bulge
and an appetizer, respectively from the left.

Geometries with two Python’s lunches can be obtained by turning on three
lowest modes (3.9). See Figure 5. By now, it should be clear that one can
generate very complicated geometries by suitably turning on higher modes.
Figure 5 is an example of this kind, where two modes
$\chi=a_{m}\chi_{m}^{v}+a_{n}\chi_{n}^{v}$ are turned on with $m=151$ and
$n=200$. In these figures, phases $\tau_{n}$’s are set to be
$\tau_{n}=\frac{\pi}{2n}$ so that the dilaton configuration is symmetric under
the time reversal $\tau\rightarrow-\tau$ which guarantees that $\tau=0$ slice
has extrema. One can consider, however, more general configurations by
choosing other values for $\tau_{n}$.

Figure 6: A dilaton field with python’s lunch geometries and the corresponding
Penrose diagram.

In Figure 6, we draw a typical dilaton configuration and the corresponding
Penrose diagram of deformed two-sided black hole spacetime. There is a unique
QESmin at the true minimum of the dilaton. At the local maxima, we have three
bulges denoted by $\mathcal{P}_{1}^{L}$, $\mathcal{P}_{1}^{R}$ and
$\mathcal{P}_{2}^{R}$, respectively. There are also three appetizers denoted
by $\mathcal{A}_{1}^{L}$, $\mathcal{A}_{1}^{R}$ and $\mathcal{A}_{2}^{R}$,
respectively, and they correspond to local minima. Superscript L and R refer
to the relative positions with respect to the QESmin. In the Penrose diagram,
horizons are drawn in black lines. Note that all the extrema are behind the
horizons as argued in section 4. The spacetime is divided into the left and
the right wedges with QESmin as a separating point, as drawn in red lines.
Then, according to the entanglement wedge reconstruction hypothesis, all local
bulk operations in each wedge should be able to be reconstructed in the
corresponding boundaries as operations on each boundary Hilbert space. In the
left wedge having an appetizer $\mathcal{A}_{1}^{L}$ and a bulge
$\mathcal{P}_{1}^{L}$, the local bulk degrees of freedom in the leftmost wedge
outside $\mathcal{A}_{1}^{L}$ should admit a simple boundary reconstruction
based on the HKLL reconstruction [44, 45], including those between the horizon
and the appetizer [46]. The degrees of freedom associated with Python’s lunch
with $\mathcal{P}_{1}^{L}$ is discussed in section 6. In the right wedge,
there are two Python’s lunch geometries with bulges $\mathcal{P}_{1}^{R}$ and
$\mathcal{P}_{2}^{R}$ respectively, which are separated by the appetizer
$\mathcal{A}_{1}^{R}$. The local bulk degrees of freedom in the right wedge
are then associated with one of the Python’s lunches or the rightmost wedge
outside the appetizer $\mathcal{A}_{2}^{R}$. Note that the generalized entropy
of the first bulge $\mathcal{P}_{1}^{R}$ is larger than that of the second
bulge $\mathcal{P}_{2}^{R}$.

As we have demonstrated above, extrema of the dilaton can change depending on
matter perturbations. Suppose that with a suitable perturbation, the dilaton
value at, for example, $\mathcal{A}_{1}^{R}$ is lowered and eventually gets
smaller than the dilaton value at the current QESmin. Then the position of the
QESmin would make a transition from the current QESmin to
$\mathcal{A}_{1}^{R}$. Accordingly, the degrees of freedom associated with
$\mathcal{P}_{1}^{R}$ would belong to the left wedge of the new QESmin. In
this case, those degrees would no longer be reconstructed from the right
boundary according to the entanglement wedge hypothesis. Instead, it would be
reconstructed from the left boundary.

##  6 Complexity, reconstruction and ghostly interactions

Figure 7:  We draw the tensor network diagram of the Python’s lunch geometry.
In this diagram, the first and the last vertical legs in each column are
identified. Each circle represent a perfect tensor with six legs. The minimal
dotted cut-line represents the QESmin, the dotted line labeled by ${\cal
P}_{1}$ the bulge and the one labeled by ${\cal A}_{1}$ the appetizer. The
isometry $T_{1}$ maps from ${\cal H}_{Q}$ to ${\cal H}_{{\cal P}_{1}}$. The
orange colored dots between the QESmin and the bulge ${\cal P}_{1}$ represent
ancilla qubits required in order to make the map $T_{1}$ be unitary. The map
from ${\cal H}_{{\cal P}_{1}}$ to ${\cal H}_{{\cal A}_{1}}$ can be given by an
adjoint of isometry $T_{2}$. This requires a post-selection and the black dots
represent the post-selected qubits.

In this section, we would like to first briefly review the restricted circuit
complexity associated with Python’s lunch geometries [16, 42]. For
definiteness, we limit our attention to the right wedge that lies on the right
side of the true QESmin as illustrated in Figure 7 and take, for simplicity, a
basic lunch with one bulge ${\cal P}_{1}$ and one appetizer ${\cal
A}_{1}$555At each point, the associated entropy in the classical regime is
given by the formula $S(P)=\frac{\phi(P)}{4G}+S_{0}$ in our JT model.. We
consider an isometry map $T$ from a state $|\psi\rangle_{Q}$ $(\in{\cal
H}_{Q}$ with basis $\\{|I\rangle\\})$ defined on the QESmin to the appetizer
state $|\psi\rangle_{{\cal A}_{1}}$ $(\in{\cal H}_{{\cal A}_{1}}$ with basis
$\\{|i\rangle\\})$. This requires $S_{{\cal A}_{1}}\geq S_{Q}$ which follows
from the fact that the QESmin is the minimum of all extremal surfaces. The
bulge Hilbert space ${\cal H}_{{\cal P}_{1}}$ can be written as ${\cal
H}_{\alpha}\otimes{\cal H}_{{\cal A}_{1}}$ where ${\cal H}_{\alpha}$ with
basis $\\{|\alpha\rangle\\}$ is associated with the Python’s lunch degrees of
freedom. The number $n$ of its associated qubits is identified as

$n=\log_{2}|\alpha|\ ,$ (6.1)

where $|\alpha|$ is the dimension of the Hilbert space ${\cal H}_{\alpha}$
with $\log|\alpha|=\Delta S^{\cal P}_{11}=S_{{\cal P}_{1}}-S_{{\cal A}_{1}}$.
The map $T$ is realized as follows. In the tensor network picture of the bulk
[47] illustrated in Figure 7, one has an isometry $T_{1}$ from ${\cal H}_{Q}$
to ${\cal H}_{{\cal P}_{1}}$ but the map from ${\cal H}_{{\cal P}_{1}}$ to
${\cal H}_{{\cal A}_{1}}$ cannot be isometric. Instead it is given by the
adjoint of an isometry $T_{2}$ and hence constructing the map
$T=T^{\dagger}_{2}T_{1}$ becomes nontrivial. Namely adding $m$ ancilla qubits
with $m=\log_{2}\big{(}|\alpha||i|/|I|\big{)}$, one may represent the map by

$T|\psi\rangle_{Q}=\langle 0|^{\otimes n}U|0\rangle^{\otimes
m}|\psi\rangle_{Q}\ .$ (6.2)

In Figure 7, the orange colored dots between the QESmin and the bulge ${\cal
P}_{1}$ represent the ancilla qubits. The black dots the represent required
post-selected qubits, which are identified as the Python’s lunch degrees of
freedom in the above. Since the postselection is nonunitary, the trick is to
use the unitary operations of Grover search algorithm [48, 49] whose the
required number of simple operation is given by the order of
$2^{\frac{n}{2}}$. This leads to the circuit complexity of the Python’s lunch

$C\sim 2^{\frac{n}{2}}=e^{\frac{1}{2}\Delta S^{\cal P}_{11}}\ ,$ (6.3)

as was shown in [16].

For a general Python’s lunch, we again consider a general Python’s lunch
geometry in the right wedge which involves bulges ${\cal P}_{i}$ and
appetizers ${\cal A}_{j}$ with $i,j=1,2,\cdots,M$. In [42], it was shown that
the associated circuit complexity is given by

$C\sim e^{\frac{1}{2}{\rm max}_{i\leq j}\Delta S^{\cal P}_{ij}}\ ,$ (6.4)

where $\Delta S^{\cal P}_{ij}=S_{{\cal P}_{i}}-S_{{\cal A}_{j}}$ defined only
for $i\leq j$. We shall call ${\rm max}_{i\leq j}\,\Delta S^{\cal P}_{ij}$ as
$\Delta S_{\cal P}$ and the relevant bulge and appetizer of this maximum as
maximum bulge and appetizer respectively.

Next we review the Petz map reconstruction of code space operation from the
boundary theory and its operational complexity in the presence of the Python’s
lunch degrees of freedom [50]. For this purpose, we consider a code space
state of ${\cal H}_{a}$ (with a basis $\\{|a\rangle\\})$ that is embedded into
the full Hilbert space of ${\cal H}_{\alpha}\otimes{\cal H}_{i}$ on the right
multiplied with the Hilbert space ${\cal H}_{I}$ on the left where the left
and the right side are entangled through the QESmin. Specifically this
embedding defines an isometry $V$ given by

$|a\rangle\longrightarrow V|a\rangle=|\psi\rangle_{RL}\ ,$ (6.5)

where

$|\psi\rangle_{RL}=\frac{1}{\sqrt{|I|}}\sum_{I}U|aII_{0}\rangle_{R}|I\rangle_{L}\
.$ (6.6)

This requires some clarifications; First the unitary operator $U$ is
introduced in such a way that

$U|aII_{0}\rangle_{R}=\sum_{\alpha i}|\alpha i\rangle\,U^{\alpha i}_{aII_{0}}\
,$ (6.7)

where $|I_{0}\rangle$ represents a specific ancilla qubit state and their
Hilbert space basis will be denoted by $\\{|I^{\prime}_{0}\rangle\\}$. The
number of total ancilla qubits is determined such that

$|a||I||I^{\prime}_{0}|=|\alpha||i|\ .$ (6.8)

Again the $\alpha$-qubits represent the Python’s lunch degrees of freedom,
whose number of qubits is given by (6.1) as before. On the other hand, the
$i$-qubits are for the degrees of freedom with relatively low-complexity,
which lie on and outside the outermost appetizer extremal surface. We take
their Hilbert space dimension to be large enough such that

$|i|>|a||\alpha||I|\ .$ (6.9)

The $I$-qubits represent the QESmin degrees of freedom, which are responsible
for the left-right entanglement in the state $|\psi\rangle_{RL}$. We also
introduce a further separation of the ancilla qubits into two parts as
indicated by $I_{0}=I_{10}I_{20}$ and their basis by
$\\{|I^{\prime}_{10}I^{\prime}_{20}\rangle\\}$ with the relation
$|i|=|\alpha||a||I||I^{\prime}_{10}|$. With help of this setup, we define $U$
fully by

$U|aII^{\prime}_{0}\rangle_{R}=\sum_{\alpha i}|\alpha i\rangle\,U^{\alpha
i}_{aII^{\prime}_{0}}\ ,$ (6.10)

without the restriction of the specific ancilla qubit state. Now, the operator
$U^{\dagger}$ is completely defined as a unitary operator acting on ${\cal
H}_{\alpha}\otimes{\cal H}_{i}$. The Python’s lunch state
$|\psi\rangle_{\alpha}$ is not available from the viewpoint of a right-side
observer who can perform only relatively low-complexity operations. Thus from
the viewpoint of this observer, one is tracing over the ${\cal
H}_{\alpha}\otimes{\cal H}_{I_{L}}$ space in the construction of the Petz map
below.

Finally when the unitary matrix $U$ is assumed to be random enough [50], the
operator $\tilde{U}$ defined by

$\tilde{U}|\alpha aII^{\prime}_{10}I_{20}\rangle\,(\equiv|e_{\alpha
aII^{\prime}_{10}}\rangle)=\sum_{i}|i\rangle\,U^{\alpha
i}_{aII^{\prime}_{10}I_{20}}\sqrt{|\alpha|}$ (6.11)

becomes a unitary operator that is acting on ${\cal H}_{\alpha
aII^{\prime}_{10}}\cong{\cal H}_{i}$ ignoring any negligibly small corrections
in all practical senses. Namely the set $\\{|e_{\alpha
aII^{\prime}_{10}}\rangle\\}$ forms an orthonormal basis of ${\cal H}_{\alpha
aII^{\prime}_{10}}\cong{\cal H}_{i}$ again ignoring any negligibly small
corrections. Note also that $\tilde{U}$ depends on a specific $I_{20}$, which
indicates the state dependence in the above construction of $\tilde{U}$.

Figure 8:  We draw the ${\tilde{U}}^{\dagger}$ operation in terms of
$U^{\dagger}$. One requires postselection of the $2n$ $I^{\prime}_{20}$ qubits
to the $|I_{20}\rangle$ state. These legs are denoted by dotted lines. In
addition, the number of the $i$-qubits is denoted by $m_{s}\,(=\log_{2}|i|)$.

The Petz map of our interest can reconstruct a code space operation $W$, which
may be understood as a black hole interior operator in the Hayden-Preskill
protocol [29],

$|a\rangle\longrightarrow W|a\rangle=\sum_{b}|b\rangle W_{ba}\ ,$ (6.12)

from the state $|\psi\rangle_{RL}$ by some operation within the Hilbert space
${\cal H}_{i}$ alone. The map is realized by [24, 25]

$M_{P}=\sigma^{-\frac{1}{2}}\,{\rm
tr}_{\alpha,I_{L}}VWV^{\dagger}\,\sigma^{-\frac{1}{2}}\ ,$ (6.13)

with

$\sigma={\rm tr}_{\alpha,I_{L}}VV^{\dagger}\ .$ (6.14)

One then finds that, for a specific $I_{10}$ inherited from $V$,

$\sigma=\frac{1}{|\alpha||I|}\sum_{\alpha,a,I}|e_{\alpha
aII_{10}}\rangle\langle e_{\alpha aII_{10}}|\ .$ (6.15)

It is clear that the reconstruction is made by the operation within the space
of ${\cal H}_{i}$. This map may be rewritten as

$M_{P}=\tilde{U}W{\tilde{U}}^{\dagger}\,.$ (6.16)

As was already shown, ${\tilde{U}}^{\dagger}$ may be expressed in terms of
matrix $U^{\dagger}$ with some postselection, which is illustrated in Figure
8. It is also clear that the ${\tilde{U}}^{\dagger}$ action involves a
Python’s lunch configuration of $2n$ extra qubits. In addition, its
realization by $U^{\dagger}$ requires the postselection to the
$|I_{20}\rangle$ state as illustrated by dotted legs in the Figure 8. Hence we
conclude that the operator complexity of the Petz map becomes

$C_{\rm rec}\sim 2^{n}=e^{\Delta S_{P}}\ ,$ (6.17)

which was first shown in [50].

Finally we would like to discuss observational difficulties in detecting the
Python’s lunch degrees of freedom (see [51] for some related discussions). In
our setup, a computationally bounded observer performs relatively simple
operations that are limited to the Hilbert space ${\cal H}_{i}$ introduced
above. Of course, such probe of the Python’s lunch degrees of freedom will not
be feasible. For instance, the bulk causality, in the classical regime,
forbids the boundary observer to detect these degrees of freedom since they
lie behind the horizon. Hence any of these probes ought to be based on certain
quantum effects such as Hawking radiation. Below we shall show that their
observational signature is suppressed exponentially in $-1/G$ and, hence,
nonperturbative.

As before, we use the isometry map $V$ in (6.5) where we take an empty code
space, i.e. $\\{a\\}=\emptyset$. Then the state in (6.6) may be rewritten as

$|\psi\rangle_{RL}=\frac{1}{\sqrt{|\alpha||I|}}\sum_{\alpha,I}|\alpha\rangle|P(\alpha,I)\rangle|I\rangle_{L}\
,$ (6.18)

where the $(\alpha,I)$ state reads666 Strictly speaking, the case of
$\\{a\\}=\emptyset$ was not included in the consideration of [50].

$|P(\alpha,I)\rangle=\sum_{i}|i\rangle{\tilde{U}}^{i}_{\alpha
II_{10}}=\sum_{i}|i\rangle{{U}}^{\alpha i}_{II_{10}I_{20}}\sqrt{|\alpha|}\ ,$
(6.19)

with the orthonormality

$\langle
P(\alpha,I)|P(\alpha^{\prime},I^{\prime})\rangle=\delta_{\alpha\alpha^{\prime}}\delta_{II^{\prime}}\
.$ (6.20)

One may view the $(\alpha,I)$ state as the maximally-entangled counterpart of
the Python’s lunch state $|\alpha\rangle|I\rangle_{L}$.

We would like to probe the state $|P(\alpha,I)\rangle$ by a certain
interaction amplitude that induces a transition from $|i_{0}\rangle$ to
$|j_{0}\rangle$ with an amplitude $A^{j_{0}}_{i_{0}}$
($i_{0},j_{0}\in\\{i\\}$). This amplitude is assumed to be ${{\cal O}(1)}$. We
may introduce $K_{0}$ such amplitudes
$A^{j_{(k)0}}_{i_{(k)0}}\,(k=1,2,\cdots,K_{0})$ to confirm the
$|P(\alpha,I)\rangle$ state but let us drop the $k$ index below for the
simplicity of presentation.

The transition amplitude from $|P(\alpha,I)\rangle$ to $|j_{0}\rangle$ then
becomes

$A=\langle i_{0}|P(\alpha,I)\rangle A^{j_{0}}_{i_{0}}\ ,$ (6.21)

whose magnitude squared may be estimated as

$p=|A|^{2}\sim|\langle i_{0}|P(\alpha,I)\rangle|^{2}\sim{\cal O}(C^{-2})\ .$
(6.22)

To show this, we note that $|P(\alpha,I)\rangle=\tilde{U}|\alpha
II_{10}\rangle$. As was just shown, the complexity of the unitary $\tilde{U}$
(or equivalently ${\tilde{U}}^{\dagger}$) is given by $C_{rec}\sim e^{\Delta
S_{P}}$. This implies that the $\tilde{U}|\alpha II_{10}\rangle$ state is
spreading nearly uniformly over some $C_{rec}$-number of its basis states.
Therefore we conclude that

$|\langle i_{0}|P(\alpha,I)\rangle|\sim{\cal O}(C_{rec}^{-1/2})\sim{\cal
O}(C^{-1})\ .$ (6.23)

Thus the amplitude is indeed exponentially suppressed in $-1/G$ and the
$(\alpha,I)$ state cannot be detected through any perturbative interactions on
the boundary. Namely the Python’s lunch degrees lie within the almost
completely dark sector of the system. This is also consistent with the claim
[52] that any ghost operators acting on the Python’s lunch Hilbert space
${\cal H}_{\alpha}$ nearly commute with any operation on ${\cal H}_{i}$
applied by any computationally bounded observer. As in [52], our argument for
the above exponential suppression involves a state dependence since the
$P(\alpha,I)$ construction depends on $I_{10}$ and $I_{20}$.

Finally we would like to comment on the degrees of freedom lying outside the
outermost appetizer but still inside the horizon. The restricted complexity
$\tilde{C}_{V}$ associated with these degrees of freedom may be estimated as
the volume of the region divided by ${G\ell}$ following for instance the
volume conjecture [26] of holographic complexity. Repeating a similar
reasoning as in the above, the observational probability through some boundary
${\cal O}(1)$ interactions may be estimated as

${\tilde{p}}\sim{\cal O}({\tilde{C}}^{-1}_{V})\sim{\cal O}(G)\,.$ (6.24)

Hence these degrees of freedom can be probed perturbatively by appropriate
boundary experiments. This is consistent with the fact that, in evaporating
black holes, information on such degrees of freedom may be transferred to
their Hawking radiation after their Page time, which is a semiclassical effect
of ${\cal O}(G)$. This is also consistent with the observation [46] that the
reconstruction of such degrees of freedom may be achieved with relatively
simple boundary operations.

##  7 Conclusions

In this work, we have presented the deformation of the two-dimensional black
hole by a massless scalar field which realizes Python’s lunch geometries.
First, we have studied the deformation of JT gravity by the massless scalar
field. With general source and state deformations, a full general solution of
the dilaton has been derived. And then, from the asymptotic behavior of the
dilaton solution, we have identified the locations of the deformed horizons
and the future/past infinities. Here, we have shown that the difference of the
future infinity of one side and the past infinity of the other side is less
than $\pi$ in the global time coordinate, and from the NEC we have also
demonstrated that any extremal points of the dilaton are located behind of the
horizon. In addition, we have calculated the extremal points of the deformed
dilaton solution to find the QESmin, Python’s lunches and appetizers. We have
shown that the position and the number of Python’s lunches and appetizers can
vary drastically by tuning the deformation parameters. Finally, we have
discussed the restricted complexity of the general Python’s lunches and the
reconstruction of a code space operator via a Petz map. We have explained why
it is difficult to probe the Python’s lunch degrees of freedom and have shown
that the observational probability of Python’s lunch degrees of freedom is
exponentially suppressed.

The undeformed eternal black hole can be holographically described by the TFD
state

$|{\rm TFD}\rangle\,=\,{1\over\sqrt{Z}}\sum_{n}e^{-{\beta E_{n}\over
2}}|E_{n}\rangle_{L}|E_{n}\rangle_{R}\,,$

where $Z$ is a normalization factor. The deformation by a massless scalar
field produces an excitation of the black hole on its left and right sides
asymmetrically, which results in the holographic dual state to the Python’s
lunch geometry of which the left and right sides are asymmetric in general. To
the leading order in source and vev deformations, the corresponding
deformations of the TFD state can be worked out; See [43] for the detailed
construction of such states in a three dimensional AdS gravity with a scalar
field. It would be interesting to analyze the entanglement structure of the
deformed state and to understand how the Python degrees of freedom are
incorporated in the state. We leave this general construction of the deformed
state dual to Python’s lunches for future works.

It is well-known that the JT gravity for the nearly-AdS2 can be reduced to a
Schwarzian theory on the boundary. In this reduction, the scalar field in the
AdS bulk will be coupled to the Schwarzian mode on the boundary. Under the
deformation by the massless scalar field, the Schwarzian action will be also
deformed in a way that the left and right Schwarzian modes are now asymmetric.
In this deformed Schwarzian theory, many questions are left unanswered. How
does the Python’s lunch appear in this deformed Schwarzian theory? How can we
evaluate the complexity and estimate the difficulty in detecting the Python’s
lunch degrees of freedom? We also leave those questions for future studies.

### Acknowledgement

We would like to thank Andreas Gustavsson for careful reading of the
manuscript. D.B. was supported in part by NRF 2020R1A2B5B01001473 and by Basic
Science Research Program through NRF funded by the Ministry of Education
(2018R1A6A1A06024977). C.K. was supported in part by NRF 2019R1F1A1059220.
S.-H.Y. was supported in part by NRF 2018R1D1A1A09082212 and NRF
2021R1A2C1003644 and supported by Basic Science Research Program through the
NRF funded by the Ministry of Education 2020R1A6A1A0304787. JY was supported
by NRF (2019R1F1A1045971, 2022R1A2C1003182) and by an appointment to the JRG
Program at the APCTP through the Science and Technology Promotion Fund and
Lottery Fund of the Korean Government. This is also supported by the Korean
Local Governments - Gyeongsangbuk-do Province and Pohang City.

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