aainstitutetext: Department of Physics and Engineering Physics, Tulane
University,
New Orleans, Louisiana, 70118, USAbbinstitutetext: Institute for Advanced
Study, Tsinghua University,
Beijing 100084, China

# Holographic Weak Measurement

Xinyu Sun a Shao-Kai Jian<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we study a holographic description of weak measurements in
conformal field theories (CFTs). Weak measurements can be viewed as a soft
projection that interpolates between an identity operator and a projection
operator, and can induce an effective central charge distinct from the
unmeasured CFT. We model the weak measurement by an interface brane,
separating different geometries dual to the post-measurement state and the
unmeasured CFT, respectively. In an infinite system, the weak measurement is
related to ICFT via a spacetime rotation. We find that the holographic
entanglement entropy with twist operators located on the defect is consistent
in both calculations for ICFT and weak measurements. We additionally calculate
the boundary entropy via holographic entanglement as well as partition
function. In a finite system, the weak measurement can lead to a rich phase
diagram: for marginal measurements the emergent brane separates two AdS
geometries, while for irrelevant measurements the post-measurement geometry
features an AdS spacetime and a black hole spacetime that are separated by the
brane. Although the measurement is irrelevant in the later phase, the post-
measurement geometry can realize a Python’s lunch.

##  1 Introduction

In the holographic principle, dual geometries emerge from the entanglement
structure of boundary quantum systems maldacena1999large ; gubser1998gauge ;
witten1998anti ; maldacena2003eternal ; ryu2006holographic ;
vanraamsdonk2010building . It will be interesting to explore the effect of
quantum information theoretic operations on the dual spacetime. For instance,
local projection measurements are operations that radically change the
entanglement of the wavefunction: the region being measured becomes
unentangled after the measurements. It is, in general, very difficult to
describe the effect of measurements on a many-body wavefunction because the
measurement outcome is stochastic in nature, and one would not expect a
universal description for all of them. Nevertheless, in conformal field theory
(CFT), there is a big class of local projection measurements that can be
described efficiently: when the measurement outcome is conditioned on the
conformal invariant boundary states of the CFT cardy1984conformal ;
cardy1986effect ; cardy1989boundary ; verlinde1988fusion ;
affleck1991universal . In 2d CFTs, these states are also called Cardy states
cardy2004boundary . After projecting (a subregion of) the wavefunction onto a
Cardy state, the remaining parts still respect half of the conformal symmetry
and are described by boundary conformal field theory (BCFT)
stephan2014emptiness ; rajabpour2015post ; rajabpour2016entanglement ;
najafi2016entanglement .

More concretely, the prescription to describe this class of measurements is to
cut a slit, which represents the measurement, in the complex plane that
represents the Euclidean path integral, and then map this plane with the slit,
using conformal transformations, to a BCFT living in an upper half plane.
This, according to AdS/BCFT takayanagi2011holographic ; Fujita_2011 ,
motivates the description of local projection measurements on the holography
side. As the holographic BCFT is dual to an end-of-the-world (ETW) brane
ending at the boundary, the local projection measurements are also described
by the ETW brane numasawa2016epr . When we consider the remaining state after
measurements on a contiguous interval, as shown in figure 1, it is dual to a
spacetime with an ETW brane that ends on the slit, and cuts off a spacetime
region that is homologous to the slit. We can also consider the bulk dual of
the boundary state. Because the boundary state has vanishing spatial
entanglement, it is holographically dual to a trivial spacetime
miyaji2015boundary . This is consistent with the ETW brane description of the
remaining parts, since the brane separates the dual spacetime of the remaining
state, which still holds a good amount of entanglement to support the
spacetime, and the trivial spacetime dual to the boundary state. The
informational aspect of holographic projective measurements has also been
considered antonini2022holographic ; antonini2023holographic ;
antonini2023holographicb ; goto2023entanglement .

Figure 1: Local projection onto a Cardy state in a subregion. $P$ denotes the
slit induced by the projection and $Q$ denotes the ETW brane anchoring on the
slit. The gray region is the spacetime dual to the remaining state after
measurements.

While the projection measurements onto a boundary state have a holographic
description in terms of the ETW brane, to the extent of our understanding,
much less is known about general measurements. In this paper, we consider weak
measurements, and study its holographic description. Weak measurements are
distinct from local projection measurements: the subregion being measured
still holds finite entanglement. A way to describe a weak measurement is by
introducing a parameter to describe the measurement strength. Such a parameter
ranges from zero to infinity, corresponding to no measurement and projection
measurement, respectively. We are interested in a generic measurement strength
other than zero or infinity. The weak measurement has been studied in CFTs
recently garratt2023measurements ; sun2023new ; yang2023entanglement ;
weinstein2023nonlocality . In reference sun2023new , we consider weak
measurements performed on a Luttinger liquid described by a compactified free
boson CFT with central charge $c=1$. The compactified free boson CFT can be
realized in the XXZ model, $H=-\sum_{i}(X_{i}X_{i+1}+Y_{i}Y_{i+1}+\Delta
Z_{i}Z_{i+1})$, where $X_{i},Y_{i},Z_{i}$ denote Pauli matrices and
$|\Delta|<1$. The weak measurement operator is sun2023new

$M=e^{-W\sum_{i}(-1)^{i}Z_{i}},$ (1)

where $W\in[0,\infty)$ is the parameter that captures the measurement
strength. The weak measurement is performed on all qubits, and its effect is
irrelevant, marginal, or relevant when $\Delta<0$, $\Delta=0$, or $\Delta>0$,
respectively. Compared to projection measurements, weak measurements have
richer behaviors. In particular, when it is marginal, the subsystem
entanglement entropy of the resulted state exhibits a continuous effective
central charge, $c_{\text{eff}}(W)$. More specifically, the entanglement
entropy of a subregion with length $L_{A}$ (with lattice constant $a$)

$S_{A}=\frac{c_{\text{eff}}(W)}{3}\log\frac{L_{A}}{a}.$ (2)

The continuous central charge interpolates the original value (of central
charge of the unmeasured system) and zero between the two limits of $W$, $W=0$
and $W=\infty$, that stands for no measurements and local projection
measurements respectively. Namely,

$0\leq c_{\text{eff}}(W)\leq c,\quad c_{\text{eff}}(0)=c,\quad
c_{\text{eff}}(\infty)=0.$ (3)

While in the example of the XXZ model, we have the anisotropy parameter
$\Delta$ to tune the relevance/irrelevance of the measurement strength, here,
in the holographic setup, we focus on the fixed point that is characterized by
$c_{\rm eff}$, namely, 111We will also denote the central charge of the
unmeasured CFT as $c_{1}$ in the following discussion.

$\begin{split}c_{\text{eff}}=c&\qquad\text{irrelevant}\\\
0<c_{\text{eff}}<c&\qquad\text{marginal}\\\
c_{\text{eff}}=0&\qquad\text{relevant}\end{split}$ (4)

From this example, we can deduce that weak measurements can support a state
with nontrivial spatial entanglement, and thus, a nontrivial spacetime via
AdS/CFT correspondence. In the case of local projection measurements, the ETW
brane separates a nontrivial spacetime of the remaining state and a trivial
spacetime of the boundary state. Now, different from the trivial spacetime
dual to the boundary state after projection, because the state after weak
measurements is dual to a nontrivial spacetime, the ETW brane is replaced by
an interface brane separating two nontrivial spacetimes with different central
charges. To be concrete, on the CFT side, considering a weak measurement,
given by $M=e^{-WH_{M}}$, acting on the ground state $|\Psi\rangle$, we can
use path integral to present the matrix element,
$\langle\Psi|M^{\dagger}M|\Psi\rangle=\lim_{\epsilon\rightarrow
0}\langle\Psi|M_{\epsilon}^{\dagger}M_{\epsilon}|\Psi\rangle$

$\langle\Psi|M_{\epsilon}^{\dagger}M_{\epsilon}|\Psi\rangle=\int D\phi
e^{-S},\quad S=\int
d\tau\left[L_{\text{CFT}}+f_{\epsilon}(\tau)WH_{M}\right],$ (5)

where $f_{\epsilon}(\tau)$ is a regularization function of the Dirac delta
function,

$f_{\epsilon}(\tau)=\begin{cases}\frac{1}{2\epsilon}\quad&|\tau|<\epsilon\\\
0\quad&|\tau|>\epsilon\end{cases}.$ (6)

Such a Euclidean path integral is shown in figure 2 (a), where
$|\tau|=\epsilon$ separates two regions with different (effective) central
charges. On the gravity side, we need to fill in the spacetime that is
consistent with the boundary quantum system. Because we have different regions
with different central charges at the boundary, the dual spacetime metrics
are, in general, different, and separated by interface branes, as shown in
figure 2 (b). $M_{1}$ and $M_{2}$ denote the bulk regions dual to the boundary
of the unmeasured CFT and the measurement, and they are separated by interface
branes. 222Here, $M_{1}$ is dual to the unmeasured CFT for both
$\tau>\epsilon$ and $\tau<-\epsilon$ owing to the time reflection symmetry.

Figure 2: (a) CFT with weak measurements on an infinite plane. A regularized
measurement occurs in $\tau\in(-\epsilon,\epsilon)$. The dashed line indicates
the time reflection invariant line, $\tau=0$. (b) Illustration of the bulk
duality. $M_{1}$ and $M_{2}$ denote the bulk region dual to the boundary, the
unmeasured CFT and the measurement. We also illustrate the branes separating
$M_{1}$ and $M_{2}$.

The interface branes generalize the holographic local projection measurements
to weak measurements. This spacetime is similar to AdS/ICFT, where ICFT stands
for interface CFT. The ICFT and its holographic duality have been extensively
studied kane1992transmission ; ghoshal1994boundary ; oshikawa1997boundary ;
leclair1999minimal ; peschel2005entanglement ; quella2007reflection ;
eisler2012entanglement ; gaiotto2012domain ; bachas2013fusion ;
gliozzi2015boundary ; meineri2020colliders ; karch2001open ;
bachas2002permeable ; dewolfe2002holography ; azeyanagi2007holographic ;
bachas2007fusion ; sakai2008entanglement ; gutperle2015entanglement ;
erdmenger2015bending ; brehm2015enganglement ; rozali2019information ;
bachas2020energy ; Simidzija_2020 ; chen2020quantum ; karch2021universal ;
bachas2021phases ; karch2022universal ; Anous_2022 ; bachas2022energy ;
karch2023universality ; baig2023transport ; tang2023universal ;
afrasiar2023island ; cogburn2023cft  It describes joint spatial regions of
possibly different CFTs. Nevertheless, an obvious difference is that ICFT
concerns the spatial interface, whereas weak measurements occur on an equal
time slice. In this paper, we will investigate a few key features of the
interface brane description of holographic weak measurements. Here is a brief
summary of the main contribution in this paper:

  * •

We propose that the weak measurements that support an effective central charge
$0<c_{\text{eff}}<c$ on an infinite line are described by interface branes as
shown in figure 2. The dual spacetime is given by two AdS geometries with
different radii that originate from the weak measurement and the unmeasured
CFT, respectively, see (23).

  * •

We establish the relation between the weak measurement and the spatial
interface: we show that the weak measurement of CFTs defined on an infinite
complex plane is related to ICFT via a spacetime rotation. In particular, the
logarithmic entanglement entropy with a continuous effective central charge
creates by the spatial interface is the same as that from a weak measurement
at an equal time slice. Figure 7 is a clear illustration of the spacetime
rotation between the weak measurement and the spatial interface.

  * •

We calculate the boundary entropy from holographic entanglement entropy of an
ICFT, and show that it is the same as the measurement probability by
evaluating the onshell action.

  * •

We investigate the holographic weak measurement in CFTs defined on a circle,
which leads to different phases (see figure 13): for marginal measurements,
the dual geometry is two AdS spacetimes with different central charges similar
to the case for infinite systems (this is called the no-bubble phase 333A
similar calculation with different motivations has been studied in reference
Simidzija_2020 . We use their conventions for the names of different phases
for convenience.); for irrelevant measurements, the dual geometry features an
AdS spacetime and a black hole that are separated by the brane. More
concretely, when measurements are irrelevant, the AdS spacetime dual to the
unmeasured CFT has a bubble, which is separated by an interface brane from the
black hole dual to the weak measurement region, in the time reversal invariant
slice. If the black hole horizon is (not) contained in the solution, it is the
bubble-inside (outside)-horizon phase. Interestingly, the post-measurement
geometry of the bubble-inside-horizon phase realizes a Python’s lunch
brown2019pythons .

  * •

While we primarily consider the thin brane description of the holographic weak
measurement, we extend our discussion to a thick-brane description for the
weak measurement in an infinite system (see Sec. 5). We find that a consistent
result as the thin brane construction.

The paper is organized as follows. In section 2, we consider the holographic
dual of weak measurements on a CFT defined on an infinite complex plane. The
geometry corresponds to pure AdS spacetime with different AdS radii separated
by the interface branes, as shown in figure 2. We will also show that the weak
measurement is related to interface CFT via a spacetime rotation. In section
3, we calculate the boundary entropy from holographic entanglement entropy and
from the measurement partition function. We find that they are equal. In
section 4, we consider holographic weak measurements in the CFTs defined on a
circle. Depending on the AdS radii and the brane tension, there are three
phases. We discuss the entanglement properties of different phases. In section
5, we present additional discussions of our results. We discuss the thick
brane description of weak measurements in an infinite system. Furthermore, we
point out bubble-inside-horizon phase in the case of weak measurements in
finite systems has a structure of Python’s lunch. Finally, we discuss a few
future directions. In appendix A, we consider the entanglement entropy with a
point defect, with the locations of twist operators not symmetric. Besides, we
also consider the effect of ETW brane on the geodesic. In appendix B, we
introduce some basic notations for calculating the partition function. In
appendix C, we provide a detailed derivation of geodesic solutions in general
and in the bubble-inside-horizon phase. In appendix D, we construct a tensor
network to realize Python’s lunch, whose complexity is exponential in the
difference between the maximal and outermost minimal areas.

Figure 3: Illustration of the RT surface dual to the entanglement entropy of
an interval in CFTs with different spatial interface. We plot the limit where
the left boundary of the interval is much longer than the right boundary
$\sigma_{l}\gg\sigma_{r}$. See the Appendix A1 and Appendix A2 for detailed
discussions. (a) The boundary has two CFTs with different central charges,
denoted by $c_{1}$ and $c_{2}$, respectively. They are separated by an
interface. The dual geometry is given by two AdS spacetimes with different AdS
radii separated by an interface brane colored in blue. The entanglement
entropy of an interval $(-\sigma_{l},\sigma_{r})$ is dual to the RT surface
colored in orange. (b) The boundary is a CFT with a defect. The central charge
of the CFT is denoted by $c_{1}$. The defect supports a nontrivial AdS
spacetime with a radius dual to an effective central charge $c_{2}$. These AdS
spacetimes are separated by interface branes colored in blue. The entanglement
entropy of an interval $(-\sigma_{l},\sigma_{r})$ is dual to the RT surface
colored in orange.

The calculation in Appendix A leads to some interesting results of holographic
interface CFT that we would like to summarize in the following. For the
readers who are only interested in the results of weak measurements may skip
this summary.

  * •

We consider two semi-infinite CFTs with different central charges, denoted by
$c_{1}$ and $c_{2}$. They are separated by a spatial interface. The gravity
dual of such an interface CFT is given by two AdS spacetimes with radii
determined by $c_{1}$ and $c_{2}$, respectively. The spacetimes are separated
by an interface brane, as shown in figure 3 (a). The interface is located at
$0$, and we consider the entanglement entropy of an interval
$(-\sigma_{l},\sigma_{r})$. The general result for the holographic
entanglement entropy of such a subsystem has been considered previously
Anous_2022 . We obtain a simple result at the leading order in the limit for
$\sigma_{l}\gg\sigma_{r}$ as follows 444We note that a similar result has also
been reported in reference tang2023universal .,

$S_{-\sigma_{l},\sigma_{r}}=\left(\frac{c_{1}}{6}+\frac{1}{6}\min(c_{1},c_{2})\right)\log\sigma_{l},$
(7)

where the left and right twist operators contribute to $\frac{c_{1}}{6}$ and
$\frac{1}{6}\min(c_{1},c_{2})$, respectively. It is interesting to see that
while the right twist operator is located in the region of CFT with central
charge $c_{2}$, the holographic entanglement entropy tries to “minimize” the
central charge: $\frac{1}{6}\min(c_{1},c_{2})$.

  * •

We consider a CFT of central charge $c_{1}$ with a defect. The defect supports
dual AdS spacetime with an effective central charge $c_{2}$. 555There is no
essential difference between a defect CFT and an interface CFT. The gravity
dual of such a defect CFT is given by two AdS spacetimes with radii determined
by $c_{1}$ and $c_{2}$, respectively. The spacetimes are separated by an
interface brane, as shown in figure 3 (b). Again, we consider the entanglement
entropy of an interval $(-\sigma_{l},\sigma_{r})$. We obtain a simple result
at the leading order in the limit for $\sigma_{l}\gg\sigma_{r}$ as follows,

$S_{-\sigma_{l},\sigma_{r}}=\left(\frac{c_{1}}{6}+\frac{1}{6}\min(c_{1},c_{2})\right)\log\sigma_{l},$
(8)

where the left and right twist operators contribute to $\frac{c_{1}}{6}$ and
$\frac{1}{6}\min(c_{1},c_{2})$, respectively. It is surprising to see that
while the right twist operator is located in the region of the CFT with
central charge $c_{1}$, the holographic entanglement entropy has the
information about the effective central charge created by the defect when
$c_{2}<c_{1}$. This can be intuitively understood as in the limit of
$\sigma_{l}\gg\sigma_{r}$, there is a proxy effect to the defect for the right
twist operator.

##  2 Weak measurement in an infinite system

In this section, we consider a semiclassical gravity in AdS3 describing
interface CFTs, and use Ryu-Takayanagi formula to calculate the entanglement
entropy of various subregions. We will see that the entanglement structure of
a CFT upon weak measurements is related to the interface CFTs via a spacetime
rotation.

###  2.1 Review of AdS/ICFT

The interface conformal field theory has been studied extensively both in
condensed matter kane1992transmission ; ghoshal1994boundary ;
oshikawa1997boundary ; leclair1999minimal ; peschel2005entanglement ;
quella2007reflection ; eisler2012entanglement ; gaiotto2012domain ;
bachas2013fusion ; gliozzi2015boundary ; meineri2020colliders , and in
holography karch2001open ; bachas2002permeable ; dewolfe2002holography ;
azeyanagi2007holographic ; sakai2008entanglement ; gutperle2015entanglement ;
erdmenger2015bending ; bachas2007fusion ; brehm2015enganglement ;
rozali2019information ; bachas2020energy ; Simidzija_2020 ; chen2020quantum ;
karch2021universal ; bachas2021phases ; karch2022universal ; Anous_2022 ;
bachas2022energy ; karch2023universality ; baig2023transport ;
tang2023universal ; afrasiar2023island ; cogburn2023cft . The basic structure
of interface CFTs considered in this paper is given by two CFTs, each lives in
a half plane and are separated by an interface. Here, we consider a thin brane
construction. In general, the two CFTs have different central charges. The
gravity dual of this interface CFTs consists of the corresponding AdS geometry
in bulk. There are two different AdS radii in two half spaces, accounting for
the different central charges. The bulk AdS spacetimes with different radii
are separated by a brane, connecting to the interface at the boundary. The
Euclidean action of this gravity theory reads

$\begin{split}S_{{\rm EH}}=-\frac{1}{16\pi
G_{(3)}}&\left[\int_{\mathcal{M}_{1}}{\rm d}^{3}x\
\sqrt{g_{1}}\left(R_{1}+\frac{2}{L^{2}_{1}}\right)+\int_{\mathcal{M}_{2}}{\rm
d}^{3}x\ \sqrt{g_{2}}\left(R_{2}+\frac{2}{L^{2}_{2}}\right)\right.\\\
&+\left.2\int_{\mathcal{S}}{\rm d}^{2}y\
\sqrt{h}(K_{1}-K_{2})-2T\int_{\mathcal{S}}{\rm d}^{2}y\ \sqrt{h}\right]+{\rm
corner\ and\ counter\ terms},\end{split}$ (9)

where $G_{(3)}$ is Newton constant in 3d. $\mathcal{M}_{1,2}$ denote the bulk
spacetimes dual of the two CFTs, $L_{1,2}$ are the corresponding AdS radii.
$g_{1,2}$ denote the metrics of two AdS bulks, and $R_{1,2}$ are the Ricci
scalars. $\mathcal{S}$ denotes the brane separating two AdS bulk spacetimes.
$K_{1,2}$ denote the extrinsic curvatures and $h_{ab}$ is the induced metric
on the brane. $T$ is the tension of the brane. For simplicity, we omit the
corner and counter terms.

The central charge of boundary CFT is related to the AdS radius via

$c_{1,2}=\frac{3L_{1,2}}{2G_{(3)}}.$ (10)

We work in the semiclassical limit, $c_{1,2}\gg 1$. There are two junction
conditions for induced metric $h_{ab}$ and extrinsic curvatures $K_{1,2}$,

$\displaystyle h_{ab}=\frac{\partial x_{1}^{\mu}}{\partial
y^{a}}\frac{\partial x_{1}^{\nu}}{\partial y^{b}}g_{\mu\nu}^{1}=\frac{\partial
x_{2}^{\mu}}{\partial y^{a}}\frac{\partial x_{2}^{\nu}}{\partial
y^{b}}g_{\mu\nu}^{2},$ (11a) $\displaystyle\Delta K_{ab}-h_{ab}\Delta
K=-Th_{ab},$ (11b)

where $\Delta K_{ab}=K_{1,ab}-K_{2,ab}$. After tracing (11b) we have $\Delta
K=2T$, and then $\Delta K_{ab}=Th_{ab}$.

Before we solve the junction conditions, we introduce a few coordinate systems
that will be used in this paper. A Euclidean ${\rm AdS_{3}}$ can be considered
as a hyperboloid

$-(X^{0})^{2}+(X^{3})^{2}+\sum_{i=1}^{2}(X^{i})^{2}=-L^{2},$ (12)

in four dimensional Minkowski spacetime with metric $G_{\mu\nu}={\rm
Diag}(-1,1,1,1)$. The first set of coordinates, which is useful for solving
the junction conditions, is given by

$\begin{split}X^{0}=&\frac{L^{2}+\tau^{2}+y^{2}}{2y}\cosh{\frac{\rho}{L}},\qquad
X^{1}=L\sinh{\frac{\rho}{L}},\\\
X^{2}=&\frac{L^{2}-\tau^{2}-y^{2}}{2y}\cosh{\frac{\rho}{L}},\qquad
X^{3}=\frac{L\tau}{y}\cosh{\frac{\rho}{L}},\end{split}$ (13a) ${\rm
d}s^{2}={\rm
d}\rho^{2}+L^{2}\cosh^{2}{\left(\frac{\rho}{L}\right)}\left(\frac{{\rm
d}\tau^{2}+{\rm d}y^{2}}{y^{2}}\right),$ (13b)

where $-\infty<\rho<\infty$ and $y>0$. Under a coordinate transformation,

$z=\frac{y}{\cosh{\left(\frac{\rho}{L}\right)}},\qquad
x=y\tanh{\left(\frac{\rho}{L}\right)},$ (14)

we get the second set of coordinates $(\tau,x,z)$ with the familiar metric

${\rm d}s^{2}=L^{2}\frac{{\rm d}\tau^{2}+{\rm d}x^{2}+{\rm d}z^{2}}{z^{2}}.$
(15)

To simplify the coordinate transformation, we introduce an angular variable
$\sin{\chi}=\tanh{\left(\frac{\rho}{L}\right)}$, such that $z=y\cos{\chi},\
x=y\sin{\chi}$. The asymptotic boundary is $z=0$ or equivalently
$\chi=\pm\frac{\pi}{2}$.

To solve the junction condition (11a), we use (13a) and consider a brane
located at $\rho_{i}=\rho_{i}^{*}$, $i=1,2$, or equivalently,
$\sin\psi_{i}=\tanh\left(\frac{\rho_{i}^{\ast}}{L_{i}}\right)$. The convention
for $\chi_{i}$ is that $\chi_{i}\rightarrow-\pi/2$ corresponds to the
asymptotic boundary of the region $M_{i}$. An example of the branes is shown
in figure 4. Then the junction condition requires

$L_{1}^{2}\cosh^{2}{\left(\frac{\rho_{1}^{*}}{L_{1}}\right)}\left(\frac{{\rm
d}\tau_{1}^{2}+{\rm
d}y_{1}^{2}}{y_{1}^{2}}\right)=L_{2}^{2}\cosh^{2}{\left(\frac{\rho_{2}^{*}}{L_{2}}\right)}\left(\frac{{\rm
d}\tau_{2}^{2}+{\rm d}y_{2}^{2}}{y_{2}^{2}}\right),$ (16a)
$\frac{1}{L_{1}}\tanh{\frac{\rho_{1}^{*}}{L_{1}}}+\frac{1}{L_{2}}\tanh{\frac{\rho_{2}^{*}}{L_{2}}}=T,$
(16b)

which the first equation leads to $y_{1}=y_{2},\ \tau_{1}=\tau_{2}$ and
$L_{1}\cosh{\left(\frac{\rho_{1}^{*}}{L_{1}}\right)}=L_{2}\cosh{\left(\frac{\rho_{2}^{*}}{L_{2}}\right)}$.
Combining with the second equation, we have

$\tanh{\frac{\rho_{1}^{*}}{L_{1}}}=\frac{L_{1}}{2T}\left(T^{2}+\frac{1}{L_{1}^{2}}-\frac{1}{L_{2}^{2}}\right),\quad\tanh{\frac{\rho_{2}^{*}}{L_{2}}}=\frac{L_{2}}{2T}\left(T^{2}+\frac{1}{L_{2}^{2}}-\frac{1}{L_{1}^{2}}\right).$
(17)

Here $T\in(T_{\rm min},T_{\rm max})$ where $T_{\rm
max,min}=\left|\frac{1}{L_{1}}\pm\frac{1}{L_{2}}\right|$. In terms of the
angular variable, we have

$\sin{\psi_{1,2}}=\tanh{\left(\frac{\rho^{\ast}_{1,2}}{L_{1,2}}\right)}.$ (18)

Therefore, the brane is determined by the AdS radii and the tension.

Figure 4: (a) An interface brane between two AdS spacetimes. The two branes
are identified. (b) RT surface that connects $[-\sigma_{2},\sigma_{1}]$.

In 3d gravity, the entanglement entropy of an interval
$[\sigma_{1},\sigma_{2}]$ in the boundary theory is given by the geodesic
length $d(\sigma_{1},\sigma_{2})$ via the RT formula,

$S_{\sigma_{1},\sigma_{2}}=\frac{d(\sigma_{1},\sigma_{2})}{4G_{(3)}}.$ (19)

In general, we have to solve the geodesic equation subjected to the connection
conditions to get the RT surface. In the simple metric (15), the relevant
geodesic is given by an arc with the arc center located at the boundary (or
the extension line of the boundary which will be clear later), and the
connection condition is that the tangents of two arcs at the junction are the
same. With these simple rules, we are able to determine the RT surface and
calculate the geodesic length.

In the following, we will investigate a special case with three AdS bulk
regions separated by two interfaces. The full theory has a reflection
symmetry, as indicated in figure 5.

###  2.2 Weak measurement and effective central charge

Consider a weak measurement on the ground state $|\Psi\rangle$ defined in an
infinite line, we can use path integral to present (unnormalized) measurement
amplitude, $\langle\Psi|M^{\dagger}M|\Psi\rangle=\lim_{\epsilon\rightarrow
0}\langle\Psi|M_{\epsilon}^{\dagger}M_{\epsilon}|\Psi\rangle$

$\langle\Psi|M_{\epsilon}^{\dagger}M_{\epsilon}|\Psi\rangle=\int D\phi
e^{-S},\quad S=\int
d\tau\left[L_{\text{CFT}}+f_{\epsilon}(\tau)WH_{M}\right],$ (20)

where $f_{\epsilon}(\tau)$ is a regularization function of the Dirac delta
function,

$f_{\epsilon}(\tau)=\begin{cases}\frac{1}{2\epsilon}\quad&|\tau|<\epsilon\\\
0\quad&|\tau|>\epsilon\end{cases}.$ (21)

Such a Euclidean path integral is shown in figure 2 (a). We assume that the
state after weak measurement, $M|\Psi\rangle$, has an effective central charge
$c_{\text{eff}}$. Because there are different regions with different central
charges at the boundary, the dual spacetime metrics are separated by interface
branes, as in figure 2 (b). $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ denote the
bulk regions dual to the boundary of the unmeasured CFT and the measurement,
and they are separated by interface branes.

Figure 5: The gravity dual of weak measurements. $\tau=\pm\epsilon$ denote the
regularized weak measurement. Note that the geometry is symmetric w.r.t.
$\tau=0$. Two branes connected by the dashed line are identified as the same
one.

The holographic dual of the weak measurements is shown in figure 5. $AdS_{1}$
and $AdS_{2}$ are dual to the original CFT and the weak measurement. The
action is the same as (9), except that the brane ends at $\tau=\pm\epsilon$,
$z=0$. The metric is

${\rm d}s^{2}=L_{i}^{2}\frac{{\rm d}\tau^{2}+{\rm d}x^{2}+{\rm
d}z^{2}}{z^{2}},$ (22)

with the AdS radii given respectively by

$c_{1}=\frac{3L_{1}}{2G_{\text{(3)}}},\quad
c_{\text{eff}}=\frac{3L_{2}}{2G_{\text{(3)}}}.$ (23)

Because $c_{\text{eff}}\leq c_{1}$ for weak measurements, we have $L_{1}\geq
L_{2}$.

The two AdS spacetimes are joint via an interface brane, which is determined
by the junction condition. Because the metric is symmetric under the exchange
of $x$ and $\tau$, we can directly use the solution from AdS/ICFT in the
appendix via an exchange of $x$ and $\tau$. Because the weak measurements
respect translational symmetry in the spatial direction, the interface brane
can be described by $(z,\tau_{i}(z),x)$. Here $\tau_{1,2}$ denotes the
interface bane in $AdS_{1,2}$. The solutions of the interface branes ending at
$\tau=\epsilon$ 666The solution of the interface branes ending at
$\tau=-\epsilon$ can be obtained by time reflection transformation,
$\tau\rightarrow-\tau$. are given by,

$\begin{split}&\tau_{1}(z)=-\tan\psi_{1}z+\epsilon,\quad\sin\psi_{1}=\frac{L_{1}}{2T}\left(T^{2}+\frac{1}{L_{1}^{2}}-\frac{1}{L_{2}^{2}}\right),\\\
&\tau_{2}(z)=\tan\psi_{2}z+\epsilon,\quad\sin\psi_{2}=\frac{L_{2}}{2T}\left(T^{2}+\frac{1}{L_{2}^{2}}-\frac{1}{L_{1}^{2}}\right).\end{split}$
(24)

For the solution to exist, it is easy to see that the tension should satisfy
$T\in(T_{\rm min},T_{\rm max})$, where $T_{\rm
max}=\frac{1}{L_{1}}+\frac{1}{L_{2}}$, $T_{\rm
min}=|\frac{1}{L_{2}}-\frac{1}{L_{1}}|$. The minimal tension is non-negative,
and $c_{\text{eff}}\leq c$ implies $\psi_{2}\geq 0$ with (24). The two branes
in $AdS_{2}$ will not have a crossing even at the limit $\epsilon\rightarrow
0$ as seen from figure 5. 777Here we start from the CFT side with an
assumption $c_{\rm eff}<c$, so we have $\psi_{2}>\psi_{1}$ and two brane will
not cross. On the other hand, if we start from the gravity side and require
the two branes not to cross, then we can actually allow $L_{2}>L_{1}$, which
means $c_{\rm eff}>c$. It is because, for $L_{2}>L_{1}$, there are the two
branes which cross at some $T$ and don’t cross at other $T$. Therefore, non-
crossing condition from the gravity side is not enough to require $c_{\rm
eff}<c$ in the CFT side. In the following, we will find stricter conditions
from the gravity side for the marginal (relevant) phase, which turns out to
imply $c_{\rm eff}<c$ in the CFT side.

In the following, we consider entanglement entropy of the post-measurement
state $|\Psi_{M}\rangle=M|\Psi\rangle$. To calculate holographic entanglement
entropy of a subregion $\\{x:x\in(\sigma_{1},\sigma_{2})\\}$ of
$|\Psi_{M}\rangle$, we will use the RT formula,

$S_{\sigma_{1},\sigma_{2}}=\frac{d(\sigma_{1},\sigma_{2})}{4G_{(3)}}.$ (25)

In 3d, the RT surface is nothing but geodesics. $d(\sigma_{1},\sigma_{2})$
denotes the length of a geodesic ending at $\sigma_{1}$ and $\sigma_{2}$.

Generally, for a geodesic with two end points $(\tau,x,z)$ and
$(\tau^{\prime},x^{\prime},z^{\prime})$ in one uniform AdS spacetime, its
length reads

$d=L\cosh^{-1}{\left(\frac{(\tau-\tau^{\prime})^{2}+(x-x^{\prime})^{2}+z^{2}+z^{\prime
2}}{2zz^{\prime}}\right)}.$ (26)

To this end, we consider two end points at the time reversal invariant plane,
two end points are located on $(x=0,z=\varepsilon,\tau=0)$ and
$(x^{\prime}=l,z=\varepsilon,\tau^{\prime}=0)$ 888$\varepsilon$ is the UV
cutoff in the $z$ direction near the boundary. Do not confuse with $\epsilon$
that is introduced to regularize the measurements.. Because of the time
reflection symmetry, the geodesic will be located at $\tau=0$ slice, which is
contained in $AdS_{2}$. A schematic illustration is shown in figure 7 (a). The
orange curve denotes the geodesic at the time reversal invariant slice.
Therefore, the geodesic length is

$d=L_{2}\cosh^{-1}{\left(\frac{l^{2}}{2\varepsilon^{2}}\right)}\approx
L_{2}\log{\left(\frac{l^{2}}{\varepsilon^{2}}\right)},$ (27)

which leads to

$S_{0,l}=\frac{c_{\text{eff}}}{3}\log{\frac{l}{\varepsilon}}.$ (28)

The prefactor of $\log$-term is $\frac{c_{\text{eff}}}{3}$, and it is because
both end points are located in the region $M_{2}$.

From this result, we can find that without measurement, the entanglement
entropy is $S=\frac{c}{3}\log{\frac{l}{\varepsilon}}$; while in the presence
of weak measurements, the prefactor will depend on the AdS radius of
$AdS_{2}$, which can be continuously tuned. This is consistent with the
marginal weak measurements in sun2023new . If we take the limit
$c_{\text{eff}}\rightarrow 0$, there is no $\log$-term, which means we also
retain the relevant phase with an area law. This is consistent with local
projection measurements, which cannot support nontrivial spatial entanglement.
On the gravity side, this corresponds to $L_{2}=0$, and the interface brane
reduces to the ETW brane. We can recover the AdS/BCFT description of local
projection measurements.

###  2.3 Spacetime rotation

In this subsection, we show that a spacetime rotation can relate the spatial
ICFT and the weak measurement. Consider an ICFT on a disk with a defect at
$x=0$. We are interested in the entanglement entropy of $\\{x:x\in(0,l)\\}$,
where $l>0$ is the disk radius. To calculate this entanglement entropy, a
twist operator is inserted at the defect, and it creates a branch cut
$\\{x:x\in(0,l)\\}$. This is shown in the top panel of figure 6. We denote the
partition function with this branch cut by $Z_{n}$. The entanglement entropy
reads

$S=\lim_{n\rightarrow 1}S_{n},\quad
S_{n}=\frac{1}{1-n}\log\frac{Z_{n}}{(Z_{1})^{n}}$ (29)

where $Z_{1}$ is the original partition function without twist operator, and
$S_{n}$ is the Renyi entanglement entropy. To evaluate $Z_{n}$, we can make a
coordinate transformation $z=\log w$, as shown in the top panel of figure 6.
The defect line is mapped to $\text{Im}(z)=(k-\frac{1}{2})\pi$,
$k=0,1,...,2n-1$, and in the $z$ plane, it has the periodic boundary condition
$z=z+i2n\pi$. After this map, $Z_{n}$ is a partition function on a cylinder
with defect lines $\text{Im}(z)=(k-\frac{1}{2})\pi$, $k=0,1,...,2n-1$.

Figure 6: Illustration of spacetime rotation between ICFT and weak
measurements. The blue region denotes an interface with regularization. The
wavy line denotes the branch cut created by the twist operator at the defect.
Top penal: ICFT defined in the $w$ plane. The defect is located at
$(x=0,-\infty<\tau<\infty)$. Bottom penal: Weak measurements in the $w$ plane.
The defect is located at $(-\infty<x<\infty,\tau=0)$. $z=\log w$ is the
coordinate transformation to map the Riemann sheet with defects to a cylinder.
The defect lines are mapped onto $(k-1/2)\pi$ and $k\pi$ for ICFT and for weak
measurements, respectively.

It has been shown that the entanglement entropy across the defect is
characterized by a logarithm with a continuous coefficient depending on the
strength of the defect, namely,

$S=\frac{c_{\rm eff}}{6}\log\frac{l}{\varepsilon},$ (30)

where ${c_{\rm eff}}$ denotes the effective central charge. $\varepsilon$
stands for the lattice constant, serving as a UV cutoff. We have assumed the
other boundary of the system (as it is defined in a disk) has a normal
boundary condition such that it does not contribute to the entanglement
entropy.

Consider a spacetime rotation of the defect from $(x=0,-\infty<\tau<\infty)$
to $(-\infty<\tau<\infty,\tau=0)$ 999On a disk, this should be from
$(x=0,-l<\tau<l)$ to $(-l<x<l,\tau=0)$. This corresponds to the weak
measurement given by (20). The entanglement entropy of the subregion
$\\{x:x\in(0,l)\\}$ is again calculated by inserting a twist operator at
$x=0$, as shown in the bottom panel of figure 6. We can use the same
coordinate transformation $z=\log w$ to evaluate $\tilde{Z}_{n}$ in this case.
Note that we use $\tilde{Z}_{n}$ to denote the partition function with branch
cut in the measurement case. In the bottom panel of figure 6, the defect line
is mapped to $\text{Im}(z)=k\pi$, $k=0,1,...,2n-1$, with $z=z+i2n\pi$. After
this map, $\tilde{Z}_{n}$ is a partition function on a cylinder with defect
lines $\text{Im}(z)=k\pi$, $k=0,1,...,2n-1$. However, since the $\text{Im}(z)$
is periodic, we can shift the coordinate by $z\rightarrow z-i\pi/2$.
$\tilde{Z}_{n}$ is equivalent to $Z_{n}$ in the spatial ICFT.

Therefore, under the spacetime rotation, we can deduce that after weak
measurements, the entanglement entropy is the same as that in the case of ICFT
when the twist operator is located at the defect,

$\tilde{S}=\frac{c_{\rm eff}}{6}\log\frac{l}{\varepsilon}.$ (31)

Indeed, in reference sun2023new , the effective central charge from weak
measurement reads

$c_{\rm eff}=-\frac{6}{\pi^{2}}\left([(1+s)\log(1+s)+(1-s)\log(1-s)]\log
s+(1+s){\rm Li}_{2}(-s)+(1-s){\rm Li}_{2}(s)\right),$ (32)

where $s$ is a function of measurement parameter $s=1/\cosh(2W)$ in (1). The
same effective central charge 101010Actually, $c_{\rm eff}/2$ appears in the
entanglement entropy across a defect of Ising CFT because reference sun2023new
investigated a Luttinger liquid, that can decouple into two Ising CFT at the
noninteracting limit. also appears in the entanglement entropy across a defect
of Ising CFT.

On the gravity side, because the metric is symmetric (22) under spacetime
rotation, the solution of brane can also be obtained via a spacetime rotation.
To explore the entanglement structure in both cases, we consider the
entanglement entropy of a subregion with length $l$. For the ICFT, we consider
that one of the twist operators is located at the defect. The RT surface in
the ICFT case and the weak measurement case is illustrated in figure 7. It is
given by (28) for weak measurements. For ICFT, we expect the entanglement
entropy to be

$S_{0,l}=\frac{c+c_{\rm eff}}{6}\log\frac{l}{\varepsilon},$ (33)

because the two ends of the RT surface are located in $M_{1}$ and $M_{2}$
regions, respectively. We will calculate the entanglement entropy for the ICFT
case in the next subsection, and find that this is indeed the case.

Figure 7: (a) The spacetime due to weak measurements. $M_{1,2}$ denote the
bulk dual to the unmeasured CFT and the weak measurements, respectively.
$\epsilon$ denotes the regularization of the weak measurement. The blue
surface indicates the interface brane. (b) The spacetime due to the spatial
ICFT. Upon a spacetime rotation in $(x,\tau)$. The orange curve denotes the RT
surface of a subregion.

###  2.4 Entanglement entropy in holographic ICFTs

Upon a spacetime rotation, the interface brane dual to the weak measurement
becomes a spatial interface brane of ICFT. In the spatial ICFT, we arrive at
branes $(z,\tau,x_{i}(z))$ with

$\begin{split}&x_{1}(z)=-\tan\psi_{1}z+\epsilon,\quad\sin\psi_{1}=\frac{L_{1}}{2T}\left(T^{2}+\frac{1}{L_{1}^{2}}-\frac{1}{L_{2}^{2}}\right),\\\
&x_{2}(z)=\tan\psi_{2}z+\epsilon,\quad\sin\psi_{2}=\frac{L_{2}}{2T}\left(T^{2}+\frac{1}{L_{2}^{2}}-\frac{1}{L_{1}^{2}}\right),\end{split}$
(34)

as shown in figure 8 (a).

Figure 8: (a) Interface branes of the spatial ICFT. It is related to the weak
measurements via a spacetime rotation. (b) RT surface ending at $[0,l]$

We consider the entanglement entropy of a subregion $\\{x:x\in[0,l]\\}$. The
RT surface that ends on these two points is illustrated in figure 8. Because
we take limit $\sigma_{2}\rightarrow 0$, we may refer to figure 9. To solve
the geodesic, we just need to compute $|{\rm OA}|$. For $\triangle{\rm OAP}$,
we have $\alpha=\frac{\pi}{2}-\psi_{2}$ and $\beta=\frac{\pi}{2}+\psi_{1}$.
Then the law of sines gives

$\frac{\sigma_{1}-R}{\sin{\alpha}}=\frac{R}{\sin{\beta}}=\frac{x}{\sin{(\pi-\alpha-\beta)}}$
(35)

where $|{\rm OA}|=x$. The solutions are
$R=\frac{\cos{\psi_{1}}}{\cos{\psi_{1}}+\cos{\psi_{2}}}\sigma_{1}$ and
$x=\frac{\sin{(\psi_{2}-\psi_{1})}}{\cos{\psi_{1}}}R$. Now we can calculate
the length of geodesic. For ${\rm OA}$ we have ${\rm O}=(0,\varepsilon)$ and
${\rm A}=(x\sin{\psi_{2}},x\cos{\psi_{2}})$, and

$\begin{split}d_{\rm
OA}=&L_{2}\cosh^{-1}{\frac{\varepsilon^{2}+(x\cos{\psi_{2}})^{2}+(x\sin{\psi_{2}})^{2}}{2\varepsilon
x\cos{\psi_{2}}}}\approx
L_{2}\cosh^{-1}{\left(\frac{x}{2\varepsilon\cos{\psi_{2}}}\right)}\\\
\approx&L_{2}\log{\left(\frac{\sigma_{1}}{\varepsilon}\frac{\sin{(\psi_{2}-\psi_{1})}}{\cos{\psi_{2}}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}.\end{split}$
(36)

For ${\rm AB}$, with origin ${\rm P}$ and
$\gamma=\frac{\pi}{2}+\psi_{1}-\psi_{2}$ we have ${\rm
A}=(-R\sin{\gamma},R\cos{\gamma})$ and ${\rm B}=(R,\varepsilon)$, so

$\begin{split}d_{\rm
AB}=&L_{1}\cosh^{-1}{\left(\frac{\varepsilon^{2}+(R\cos{\gamma})^{2}+(R+R\sin{\gamma})^{2}}{2\varepsilon
R\cos{\gamma}}\right)}\approx
L_{1}\cosh^{-1}{\left(\frac{R(2+2\sin{\gamma})}{2\varepsilon\cos{\gamma}}\right)}\\\
\approx&L_{1}\log{\left(\frac{\sigma_{1}}{\varepsilon}\frac{(2+2\cos{(\psi_{2}-\psi_{1})})\cos{\psi_{1}}}{\sin{(\psi_{2}-\psi_{1})}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}=L_{1}\log{\left(\frac{\sigma_{1}}{\varepsilon}\frac{2\cos{\psi_{1}}}{\tan{(\frac{\psi_{2}-\psi_{1}}{2})}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}.\end{split}$
(37)

And total length of geodesic with $\sigma_{1}=l$ is

$d(0,l)=d_{\rm OA}+d_{\rm AB}=(L_{1}+L_{2})\log{\frac{l}{\varepsilon}}+d_{\rm
sub},$ (38)

where $d_{\rm sub}$ is given by

$\begin{split}d_{\rm
bdy}=&L_{2}\log{\left(\frac{\sin{(\psi_{2}-\psi_{1})}}{\cos{\psi_{2}}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}+L_{1}\log{\left(\frac{2\cos{\psi_{1}}}{\tan{(\frac{\psi_{2}-\psi_{1}}{2})}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}.\end{split}$
(39)

Figure 9:  AdS dual of CFT with interface defect for infinite system and its
geodesic.

Therefore, with (25) we have

$S_{0,\sigma_{2}}=\frac{c_{1}+c_{\rm
eff}}{6}\log{\frac{l}{\varepsilon}}+S_{\rm sub},$ (40)

where $S_{\rm sub}=\frac{d_{\rm sub}}{4G_{(3)}}$. The subleading entropy $S$
is UV dependent because the leading term is a logarithm with the UV
dependence, and there is no way to subtract the leading term. 111111We will
see in the symmetric case discussed later, the subleading term is independent
of the UV cutoff because there is a well-defined procedure to subtract the
leading term. Here the prefactor of $\log$-term is $\frac{c_{1}+c_{\rm
eff}}{6}$, which is because the location of each end points are in region $1$
and $2$.

Now we discuss the condition of existence of nontrivial geodesic solution.
Because we need $\triangle{\rm OAP}$, it means there must be a crossing point
for lines ${\rm AP}$ and ${\rm OB}$. Therefore, $\alpha+\beta<\pi$, which is
equivalent to $\psi_{1}<\psi_{2}$. With (24), we have

$\begin{split}\sin{\psi_{2}}-\sin{\psi_{1}}=\frac{L_{2}}{2T}\left(T^{2}+\frac{1}{L_{2}^{2}}-\frac{1}{L_{1}^{2}}\right)-\frac{L_{1}}{2T}\left(T^{2}+\frac{1}{L_{1}^{2}}-\frac{1}{L_{2}^{2}}\right)=\frac{(\frac{1}{L_{1}}+\frac{1}{L_{2}})^{2}-T^{2}}{2T}(L_{1}-L_{2}).\end{split}$
(41)

Because $T<T_{\rm max}=\frac{1}{L_{1}}+\frac{1}{L_{2}}$, $\psi_{1}<\psi_{2}$
corresponds to $L_{1}>L_{2}$, which means $c_{1}>c_{\rm eff}$. Therefore, the
existence of a nontrivial geodesic corresponds to marginal (relevant) phase,
which is consistent with the CFT results.

This motives us to derive $c_{\rm eff}<c_{1}$ for marginal (relevant)
measurement in the CFT side from the gravity perspective. In the two regions
as shown in figure 4, if one region is dual to the measurement or defect after
the spacetime rotation, 121212In general, we have three regions with a
reflection symmetry, where two are dual to the unmeasured CFT and one is dual
to the measurement. In the discussion here, only two regions are relevant, so
we refer to figure 4 for simplicity. the boundary of that region has the
length scale in the order of regularization parameter $\epsilon$. The limit of
$\sigma\rightarrow 0$ must be well-defined. For a geodesic ending at a general
$\sigma_{l}$ and $\sigma_{r}$ in figure 4, it will cross the interface brane.
For the case of measurements, we expect that the geodesic continues to be
well-defined in the limit $\epsilon\rightarrow 0$. We find that this is only
true when $\psi_{1}<\psi_{2}$ because if $\psi_{1}>\psi_{2}$, the condition
that $\alpha+\beta<\pi$ is not satisfied. Therefore, the existence of a limit
$\epsilon\rightarrow 0$ with a well-defined geodesic that crosses the
interface brane implies $L_{2}<L_{1}$.

##  3 Boundary entropy of holographic measurement in an infinite system

As mentioned above, the entanglement of a spatial ICFT is related to the weak
measurement via spacetime rotation. In particular, the same effective central
charge appears in both cases. It is given by the AdS radius that is dual to
the weak measurement regions, $c_{\text{eff}}=\frac{3L_{2}}{2G_{\text{(3)}}}$.
To determine the interface brane solution, we also need the tension $T$ of the
brane. How is the tension related to the weak measurement? In this section, we
will show that the tension $T$ is related to the boundary entropy of weak
measurements. To this end, we first calculate the entanglement entropy of a
region symmetric w.r.t. the defect in the spatial ICFT. The boundary entropy,
$S_{\rm bdy}$, appears as the excess of entanglement entropy induced by the
defect. Then we calculate the amplitude in the weak measurement case,

$\begin{split}Z&=\frac{\langle\Psi|M^{\dagger}M|\Psi\rangle}{\langle\Psi|\Psi\rangle}\\\
&=e^{-(I_{M}-I_{0})}\end{split}$ (42)

where in the second line, we use the saddle point approximation, and $I_{M}$
and $I_{0}$ denote the on-shall action with and without weak measurements,
respectively. We find that the same boundary entropy 131313We call it a
boundary entropy because the ICFT can be mapped to a BCFT by folding trick.
is given by the partition function $S_{\rm bdy}=\log Z$.

Figure 10:  AdS dual of CFT with interface defect for infinite system with
symmetry and its geodesic.

###  3.1 Entanglement entropy of a symmetric region in the spatial ICFT

We consider a symmetric configuration in the spatial ICFT, as shown in figure
10. Assuming $\sigma_{l}=\sigma_{r}=\sigma$, we can get analytical result for
entanglement entropy and corresponding boundary entropy in the following. By
symmetry, the center of the circle must be on the defect line, and its radius
is $\sigma$. Therefore, for ${\rm AC}$ with ${\rm A}=(-\sigma,\varepsilon),\
{\rm C}=(\sigma\sin{\psi_{1}},\sigma\cos{\psi_{1}})$, the length of the
geodesic connecting them is

$d_{\rm
AC}=L_{1}\cosh^{-1}{\left(\frac{(-\sigma-\sigma\sin{\psi_{1}})^{2}+\varepsilon^{2}+(\sigma\cos{\psi_{1}})^{2}}{2\varepsilon\sigma\cos{\psi_{1}}}\right)}\approx
L_{1}\log{\left(\frac{2\sigma}{\varepsilon}\frac{1+\sin{\psi_{1}}}{\cos{\psi_{1}}}\right)}.$
(43a) For ${\rm CD}$ with ${\rm C}=(-\sigma\sin{\psi_{2}},\
\sigma\cos{\psi_{2}})$ and ${\rm D}=(\sigma\sin{\psi_{2}},\
\sigma\cos{\psi_{2}})$, the length of the geodesic connecting them is $d_{\rm
CD}=L_{2}\cosh^{-1}{\left(\frac{(-2\sigma\sin{\psi_{2}})^{2}+2(\sigma\cos{\psi_{2}})^{2}}{2(\sigma\cos{\psi_{2}})^{2}}\right)}=L_{2}\cosh^{-1}{\left(\frac{1+\sin^{2}{\psi_{2}}}{\cos^{2}{\psi_{2}}}\right)}.$
(43b)

By symmetry, we also have $d_{\rm AC}=d_{\rm DB}$, so the total entanglement
entropy for the symmetric case in figure 10 is

$\begin{split}S_{-\sigma,\sigma}=\frac{d_{\rm AC}+d_{\rm CD}+d_{\rm
DB}}{4G_{(3)}}=\frac{c_{1}}{3}\log{\left(\frac{2\sigma}{\varepsilon}\right)}+S_{\rm
bdy},\end{split}$ (44)

where the boundary entropy $S_{\rm bdy}$ is

$S_{\rm
bdy}=\frac{c_{1}}{3}\log{\left(\frac{1+\sin{\psi_{1}}}{\cos{\psi_{1}}}\right)}+\frac{c_{\text{eff}}}{6}\cosh^{-1}{\left(\frac{1+\sin^{2}{\psi_{2}}}{\cos^{2}{\psi_{2}}}\right)}.$
(45)

Actually, the two terms in the boundary entropy have the same forms after some
manipulation. For the first term, we have

$\frac{1+\sin{\psi_{1}}}{\cos{\psi_{1}}}=\frac{1-\cos{\left(\psi_{1}+\frac{\pi}{2}\right)}}{\sin{\left(\psi_{1}+\frac{\pi}{2}\right)}}=\frac{2\sin^{2}{\frac{\psi_{1}+\frac{\pi}{2}}{2}}}{2\sin{\frac{\psi_{1}+\frac{\pi}{2}}{2}}\cos{\frac{\psi_{1}+\frac{\pi}{2}}{2}}}=\tan{\left(\frac{\psi_{1}}{2}+\frac{\pi}{4}\right)}.$
(46)

To simplify the second term, (i) we first prove that

$\cosh^{-1}{\left(\frac{1+\sin^{2}{\psi}}{\cos^{2}{\psi}}\right)}=2\tanh^{-1}{\left(\sin{\psi}\right)}.$
(47a) Define ${\rm l.h.s.}=2p$, then $\frac{2}{\cos^{2}{\psi}}-1=\cosh{2p}$,
so
$\cos^{2}{\psi}=\left[\frac{1}{2}\left(\cosh{2p}+1\right)\right]^{-1}=\frac{1}{\cosh^{2}{p}}$.
Therefore,
$\sin^{2}{\psi}=1-\cos^{2}{\psi}=1-\frac{1}{\cosh^{2}{p}}=\tanh^{2}{p}$.
Finally, we get $p=\tanh^{-1}{\left(\sin{\psi}\right)}$, and finish the proof.
(ii) Now we prove
$\tanh^{-1}{\left(\sin{\psi}\right)}=\log{\left(\tan{\left(\frac{\psi}{2}+\frac{\pi}{4}\right)}\right)}.$
(47b) Similar to the proof above, we define ${\rm r.h.s.}=p$, then
$e^{p}=\tan{\left(\frac{\psi}{2}+\frac{\pi}{4}\right)}$. Therefore,
$\sin{\psi}=-\cos{\left(\psi+\frac{\pi}{2}\right)}=\frac{1-2\cos^{2}{\frac{\psi+\frac{\pi}{2}}{2}}}{\sin^{2}{\frac{\psi+\frac{\pi}{2}}{2}}+\cos^{2}{\frac{\psi+\frac{\pi}{2}}{2}}}=\frac{\tan^{2}{\frac{\psi+\frac{\pi}{2}}{2}}-1}{\tan^{2}{\frac{\psi+\frac{\pi}{2}}{2}}+1}=\frac{e^{2p}-1}{e^{2p}+1}=\tanh{p}$.
It means $p=\tanh^{-1}{\left(\sin{\psi}\right)}$ and we finish the proof.

With (47a) and (47b), the second term in (45) can be expressed as

$\cosh^{-1}{\left(\frac{1+\sin^{2}{\psi_{2}}}{\cos^{2}{\psi_{2}}}\right)}=2\tanh^{-1}{\left(\sin{\psi_{2}}\right)}=2\log{\left(\tan{\left(\frac{\psi_{2}}{2}+\frac{\pi}{4}\right)}\right)}.$
(48)

Therefore, the boundary entropy (45) can be simplified as

$S_{\rm
bdy}=\frac{c_{1}}{3}\log{\left(\tan{\left(\frac{\psi_{1}}{2}+\frac{\pi}{4}\right)}\right)}+\frac{c_{\text{eff}}}{3}\log{\left(\tan{\left(\frac{\psi_{2}}{2}+\frac{\pi}{4}\right)}\right)}.$
(49)

Here are some remarks about the results: (i) One may wonder if the boundary
entropy is UV dependent from (44) by rescaling the UV cutoff $a$. However, we
can properly regularize this dependence by defining the boundary entropy as
the excess of entanglement entropy due to the defect

$S_{\rm bdy}=S_{-\sigma,\sigma}-S^{(0)}_{-\sigma,\sigma},$ (50)

where the second term $S^{(0)}_{-\sigma,\sigma}$ denotes the entanglement
entropy of the same region but without a defect. (ii) While (49) seemingly has
two independent terms from two regions, this is not the case. Recall that the
solutions of the branes $\psi_{1,2}$ in (34) depend on both radii $L_{1,2}$
and the tension $T$. It is clear that the parameters in the holographic
description are uniquely determined by the central charge of the unmeasured
CFT $c_{1}$, the effective central charge after measurement $c_{\text{eff}}$,
and the boundary entropy $S_{\text{bdy}}$. (iii) In the symmetric case
considered here, the prefactor of the logarithm in (44) is $\frac{c_{1}}{3}$.
This can be intuitively understood: the two twist operators are located in
region 1, and when we take the long-wave length limit, they approach deep in
the bulk of the system symmetrically. However, when the interval is not
symmetric and the limit is not taken symmetrically, the prefactor of the
logarithm can also be related to $c_{\rm eff}$ as we will show in appendix A.

###  3.2 Boundary entropy from weak measurements partition function

In a BCFT, the boundary entropy (49) can be related to g-theorem with the
definition of boundary entropy $S_{\rm bdy}=\log{g}$ where
$g=\left<0|B\right>$, and $\left|B\right>$ is the boundary state of the BCFT.
We consider a similar quantity in this subsection: the partition function with
the measurement, as is given by (42). In reference Fujita_2011 , the authors
have given the boundary entropy of one AdS region with a single ETW brane, for
completeness, we review the calculation in appendix B. For our interested case
shown in figure 10, we apply a similar procedure and obtain the boundary
entropy in the following. As mentioned in appendix B, we first apply a special
conformal transformation, which will lead to the geometry in figure 11. There
are three bulk regions and two interface branes. Although it seems that there
is overlap between two connected regions, we should calculate the action
separately and consider that the different regions are connected by some magic
glue.

Figure 11:  Spacial conformal transformation that maps the brane into part of
a sphere. (a) Illustration of the map in the xz-plane. (b) Illustration of the
map in the xt-plane.

To this end, we define the corresponding action

$\begin{split}&I=-\frac{1}{16\pi G_{(3)}}\left[\int_{\mathcal{M}_{1}}{\rm
d}^{3}x\
\sqrt{g_{1}}\left(R_{1}+\frac{2}{L^{2}_{1}}\right)+\int_{\mathcal{M}_{2}}{\rm
d}^{3}x\
\sqrt{g_{2}}\left(R_{2}+\frac{2}{L^{2}_{2}}\right)+\int_{\mathcal{M}_{3}}{\rm
d}^{3}x\ \sqrt{g_{3}}\left(R_{3}+\frac{2}{L^{2}_{3}}\right)\right.\\\
&+\left.2\int_{\mathcal{S}_{12}}{\rm d}^{2}y\
\sqrt{h_{12}}(K_{1}-K_{2})-2T_{12}\int_{\mathcal{S}_{12}}{\rm d}^{2}y\
\sqrt{h_{12}}+2\int_{\mathcal{S}_{23}}{\rm d}^{2}y\
\sqrt{h_{23}}(K_{2}-K_{3})-2T_{23}\int_{\mathcal{S}_{23}}{\rm d}^{2}y\
\sqrt{h_{23}}\right].\end{split}$ (51)

By the symmetry, we have $L_{1}=L_{3}$ and $T_{12}=T_{23}=T$. The connection
conditions (11b) lead to $K_{1}-K_{2}=K_{2}-K_{3}=2T$. With a similar method
in appendix B, we have the onshell action

$\begin{split}I=&-\frac{1}{16\pi G_{(3)}}\left[\int_{\mathcal{M}_{1}}{\rm
d}^{3}x\
\sqrt{g_{1}}\left(R_{1}+\frac{2}{L^{2}_{1}}\right)+\int_{\mathcal{M}_{2}}{\rm
d}^{3}x\
\sqrt{g_{2}}\left(R_{2}+\frac{2}{L^{2}_{2}}\right)+\int_{\mathcal{M}_{3}}{\rm
d}^{3}x\ \sqrt{g_{3}}\left(R_{3}+\frac{2}{L^{2}_{3}}\right)\right.\\\
&+\left.\int_{\mathcal{S}_{12}}{\rm d}^{2}y\
\sqrt{h_{12}}(K_{1}-K_{2})+\int_{\mathcal{S}_{23}}{\rm d}^{2}y\
\sqrt{h_{23}}(K_{2}-K_{3})\right]\\\ =&-\frac{1}{16\pi
G_{(3)}}\sum_{\alpha=1,2,3}\left[\int_{\mathcal{M}_{\alpha}}{\rm d}^{3}x\
\sqrt{g_{\alpha}}\left(R_{\alpha}+\frac{2}{L^{2}_{\alpha}}\right)+\sum_{\beta={\rm
L,R}}\left(\int_{\mathcal{S}_{\alpha,\beta}}{\rm d}^{2}y\
\sqrt{h_{\alpha,\beta}}K_{\alpha,\beta}\right)\right]\\\
=&\sum_{\alpha=1,2,3}I_{\alpha},\end{split}$ (52)

where we define $\beta$ as the label of boundary of the region $\alpha$, so
$K_{\alpha,\beta}$ corresponds to left and right brane of the region $\alpha$,
and the corresponding signs are absorbed by redefining the directions for
$K_{\alpha,\beta}$ point outside from their bulk regions. Here for $AdS_{1}$
we only have one boundary brane and one boundary term in action, while for
region $AdS_{2}$ we have two boundary terms. However, from the discussion
before, for the largest region after SCT (here it is right $AdS_{1}$), we must
apply UV-cutoff, which can be an ETW brane on the right of right $AdS_{1}$ in
figure 11. The ETW brane is perpendicular to the boundary CFT and has no
contribution to the boundary entropy. Besides, we can also add another ETW′
brane on the left on left $AdS_{1}$ without contribution. Then each region can
be calculated with the same method in appendix B with (199), and the result of
each part is

$\begin{split}I_{\alpha}=&\frac{L}{4G_{N}}\left[\left(\frac{r_{D}^{2}}{2\varepsilon^{2}}+\log{\frac{\varepsilon}{r_{D}}}-\frac{\rho^{*}}{L}-\frac{1}{2}+\frac{r_{D}}{\varepsilon}\sinh{\frac{\rho^{*}}{L}}\right)-\left(\frac{{r_{D}^{\prime}}^{2}}{2\varepsilon^{2}}+\log{\frac{\varepsilon}{r_{D}^{\prime}}}-\frac{\rho^{*}{}^{\prime}}{L}-\frac{1}{2}+\frac{r_{D}^{\prime}}{\varepsilon}\sinh{\frac{\rho^{*}{}^{\prime}}{L}}\right)\right],\end{split}$
(53)

where $r_{D},\rho^{*}$ correspond to one brane and
$r_{D}^{\prime},\rho^{*}{}^{\prime}$ correspond to another brane. 141414For
example, for $AdS_{2}$ we can consider its region after special conformal
transformation is bounded by two branes. We can first calculate the
contribution without the left brane, which is the first term. Then the region
bounded by the left brane which we should subtract corresponds to the second
term. So actually the direction of ${\rho^{*}}^{\prime}$ in the second term is
opposite to the direction of the brane of $AdS_{2}$. Therefore, the total
boundary entropy is

$\begin{split}S_{\rm
bdy}=&\frac{\rho_{1}^{*}-(-\rho_{0}^{*})}{4G_{(3)}}+\frac{\rho_{2}^{*}-(-\rho_{2}^{*})}{4G_{(3)}}+\frac{\rho_{0}^{*}-(-\rho_{1}^{*})}{4G_{(3)}}=\frac{2\rho_{1}^{*}+2\rho_{2}^{*}}{4G_{(3)}}\\\
=&\frac{c_{1}}{3}\log{\left(\tan{(\frac{\psi_{1}}{2}+\frac{\pi}{4})}\right)}+\frac{c_{2}}{3}\log{\left(\tan{(\frac{\psi_{2}}{2}+\frac{\pi}{4})}\right)},\end{split}$
(54)

where $\rho_{0}^{*}=0$. As mentioned before, here we change the sign of second
$\rho^{*}$ in the numerator because we use the proper definition in reference
Anous_2022 . Besides, although here we add ETW brane which is perpendicular to
CFT surface artificially, we can remove it as discussed in appendix B. Without
ETW brane, we can consider it as the case with ETW brane with location
$x\rightarrow\infty$. Under corresponding conformal transformation, we can map
the ETW brane to a semi-sphere at the infinity, and it covers the whole space
with $r_{D}\rightarrow\infty$. As shown in (54), the final boundary entropy
doesn’t rely on $r_{D}$, so we can safely apply the limit above and get the
same result. And this result is also consistent with (49) without ETW brane.

##  4 Holographic weak measurements in a finite system: phase transition and
entanglement entropy

In the previous section about weak measurements in an infinite system, the
holographic description is given by interface branes. The effective central
charge $c_{\text{eff}}\leq c_{1}$ distinguishes the different cases of the
weak measurements:

$\begin{split}c_{\text{eff}}=c_{1}&\qquad\text{irrelevant}\\\
0<c_{\text{eff}}<c_{1}&\qquad\text{marginal}\\\
c_{\text{eff}}=0&\qquad\text{relevant}\end{split}$ (55)

The interface brane solution exists for the irrelevant and marginal cases. In
the relevant case, the interface brane becomes an ETW brane.

In this section, we consider weak measurements in a finite system. We will see
that in the finite system, the transition between the irrelevant and marginal
weak measurements is dual to a transition of interface branes with different
topologies. After discussing the phase diagram, we calculate the subregion
entanglement entropy in different phases, and identify the fate of weak
measurements.

###  4.1 Phase diagram

The Euclidean path integral of a CFT, defined on a circle $x=x+2\pi$, with
weak measurements is shown by the cylinder in figure 12. $|\Psi\rangle$
denotes the ground state of the CFT and $M|\Psi\rangle$ denotes the state
after the weak measurements. The measurements are performed in the full
system, and preserve translational symmetry. Again, we denote the central
charge of the unmeasured CFT as $c_{1}$ and the effective central charge of
the measured state $M|\Psi\rangle$ as $c_{\rm eff}\leq c_{1}$.

Figure 12: Euclidean path integral of a CFT, defined in a circle, with weak
measurements regularized at $\tau=\pm\epsilon$. $|\Psi\rangle$
($\langle\Psi|$) indicates the ket (bra) of the ground state prepared from
Euclidean path integral.

The gravity dual is to “fill in” the bulk of the cylinder with the boundary
condition given by the Euclidean path integral defined on the surface of the
cylinder. In reference Simidzija_2020 , the authors have discussed the phase
diagram of the dual gravity solution, though they considered a different
problem. In the following, we summarize the results in reference
Simidzija_2020 .

To this end, we start from a similar action in (9). This time,
$\mathcal{M}_{1,2}$ denote regions dual to the original unmeasured CFT on a
circle and the post-measurement state, respectively, and we use a different
convention for the metric:

${\rm d}s^{2}=f_{i}(r){\rm d}t^{2}+\frac{{\rm d}r^{2}}{f_{i}(r)}+r^{2}{\rm
d}x^{2},$ (56)

where $f_{i}(r)=(1-\mu_{i})+\lambda_{i}r^{2}$ with $\lambda_{i}=L_{i}^{-2}$.
151515Here we can consider that the dimension of $r$ is $[r]=[L_{i}]$, which
means $[f(r)]=[x]=[1]$ and $[s]=[t]=[r]=[L_{i}]$. In this metric, $t$ denotes
the imaginary time, and $x=x+2\pi$ denotes the spatial coordinate. $r$ is the
radius, with $r\rightarrow\infty$ corresponding to the boundary. $\mu_{i}$
determines whether the metric is thermal AdS ($\mu_{i}<1$) or BTZ black hole
($\mu_{i}>1$). Again, we have the relation

$c_{1}=\frac{3L_{1}}{2G_{\text{(3)}}},\quad
c_{\text{eff}}=\frac{3L_{2}}{2G_{\text{(3)}}}.$ (57)

The parameter $\mu_{i}$ and the brane solution are determined by the radius
$L_{i}$ and the tension $T$. Notice that there is a regularization parameter
$\epsilon$, at which the interface brane is located. To determine the phase
diagram of weak measurements, we take $\epsilon\rightarrow 0$.

Similar to the previous case, the solution exists when the tension falls in
the following range:

$\frac{1}{L_{2}}-\frac{1}{L_{1}}\leq T\leq\frac{1}{L_{2}}+\frac{1}{L_{1}}.$
(58)

Note that we focus on $L_{2}\leq L_{1}$. In this range of parameters, there
are three different phases, which are called “no-bubble”, “bubble-inside-
horizon” and ‘bubble-outside-horizon” in reference Simidzija_2020 . We show
the phase diagram in figure 13. In all three phases, for the region dual to
the unmeasured CFT, the metric is given by thermal AdS space with $\mu_{1}=0$.
For simplicity, we define $\mu_{2}=\mu$ without loss of generality. Depending
on whether $\mu<1$ or $\mu>1$, the spacetime dual to the weak measurement
region is given by AdS or BTZ black hole, respectively.

The no-bubble phase is given by $\mu=0$. The dual gravity theories of the
unmeasured CFT and the weak measurement are given by AdS spacetimes,
$AdS_{1,2}$, with different radii, similar to the weak measurement in the
infinite system. The two AdS spacetimes are separated by interface branes, as
is illustrated in figure 13 (b). In this phase, the time reversal invariant
slice $\tau=0$ is included in $AdS_{2}$ that is dual to the weak measurement
region. Hence, it is referred to as the no-bubble phase.

On the other hand, in the bubble-inside-horizon and bubble-outside-horizon
phases, $\mu>1$. The dual gravity theory of the weak measurement region is
given by a BTZ black hole. The interface brane separates the AdS spacetime and
the black hole spacetime, as is illustrated in figure 13 (c). In this phase,
the time reversal invariant slice will cut through both the AdS spacetime and
the black hole spacetime. The AdS spacetime emerges as a bubble, surrounded by
the interface brane, in the black hole spacetime. Furthermore, depending on
whether the horizon of the black hole is included or not, the bubble solution
splits into two cases. When the horizon is not included in the solution, we
have the bubble-outside-horizon phase. Namely, the interface brane (the
boundary of the bubble) is located outside the black hole horizon, while the
interior of the bubble is the AdS spacetime, so the solution does not include
the horizon. When the horizon is included in the solution, we have the bubble-
inside-horizon phase.

Figure 13:  (a) The phase diagram of the weak measurement in a finite system.
The vertical cross-section of the cylinder in the no-bubble phase (b) and in
the bubble phase (c). The light (dark) region is dual to the unmeasured CFT
(the weak measurement). (b) The dual geometry consists of two AdS spacetimes
with different radii. (c) The dual geometry of the unmeasured CFT (the weak
measurement) is given by the AdS (black hole) spacetime.

The phase boundary between the no-bubble phase and the bubble phase is given
by $f(\hat{R},\hat{T})=0$, where

$\begin{split}f(\hat{R},\hat{T})=&2\pi\Theta\left(\frac{\hat{T}}{\hat{R}}-1\right)-\frac{1}{\sqrt{\hat{R}\hat{T}}}\Pi\left(0,\frac{1}{2}\sqrt{\frac{1-(\hat{R}-\hat{T})^{2}}{\hat{R}\hat{T}}}\right)\\\
&+\frac{1}{\sqrt{\hat{R}\hat{T}}}\frac{\hat{R}+\hat{T}}{\hat{R}-\hat{T}}\Pi\left(1-\frac{1}{(\hat{R}-\hat{T})^{2}},\frac{1}{2}\sqrt{\frac{1-(\hat{R}-\hat{T})^{2}}{\hat{R}\hat{T}}}\right),\end{split}$
(59)

where $\hat{T}=TL_{2}$, $\hat{R}=L_{2}/L_{1}$, and the function $\Pi(\nu,z)$
is defined by

$\Pi(\nu,z)=\int_{0}^{1}\frac{{\rm d}t}{(1-\nu
t^{2})\sqrt{(1-t^{2})(1-z^{2}t^{2})}}.$ (60)

The phase boundary between the bubble-outside-horizon phase and the bubble-
inside-horizon phase is given by $TL_{1}=1$. Notice that these phase
boundaries are obtained in the limit $\epsilon\rightarrow 0$.

We further review a few useful quantities from reference Simidzija_2020  that
will be used later in the discussion of bubble phases. The black hole metric
dual to the weak measurement has a horizon at

$r_{H}=\sqrt{\frac{\mu-1}{\lambda_{2}}}.$ (61)

The interface brane, on the other hand, is located at

$r_{0}=\frac{\mu}{\sqrt{2\sqrt{\mu^{2}T^{2}\lambda_{1}+\mu
T^{2}(\lambda_{2}-\lambda_{1}-T^{2})+T^{4}}+\mu(\lambda_{2}-\lambda_{1}-T^{2})+2T^{2}}}.$
(62)

The transition point is when these two are degenerate. In the black hole
solution, there is a relation between the measurement regularization
$\epsilon$ and $\mu$,

$2\epsilon=\frac{1}{\sqrt{\mu}}f(\hat{R},\hat{T}).$ (63)

where $f(\hat{R},\hat{T})$ is given in (59). Apparently, the existence of the
measurement solution requires $\epsilon>0$. The onshell action difference
between the black hole phase and the AdS phase dual to the weak measurement is
given by
$S^{BH}-S^{AdS}=-\frac{\mu\epsilon}{\sqrt{\lambda_{2}}}+O(\mu^{-1/2})$, which
means that as long as the black hole solution exists, it dominates over the
AdS solution. This also gives the phase boundary between the no-bubble phase
and the bubble phase discussed before. This also indicates that in the bubble
phase, $f(\hat{R},\hat{T})>0$, if we take $\epsilon\ll 1$, it means
$\mu\propto\epsilon^{-2}\gg 1$.

###  4.2 Brief summary of geodesic equations and geodesic length

Later we will discuss the length of geodesic in different phases, which may
lead to different entanglement behaviors. Here we briefly summarize geodesic
equations and their length for different metrics. The detail of the derivation
is given in appendix C.1.

Owing to the time reversal symmetry, we consider the geodesics at the time
reversal invariant slice $\tau=0$. They can be expressed as $(\tau=0,x,r(x))$.
With metric (56), the geodesics for $\mu>1$ and $\mu<1$ read

$r^{-2}+(\frac{1}{L^{2}(\mu-1)}-c_{1})\sinh^{2}{[\sqrt{\mu-1}(x+c_{2})]}=c_{1}\qquad\mu>1,$
(64a)
$r^{-2}+(\frac{1}{L^{2}(1-\mu)}+c_{1})\sin^{2}{[\sqrt{1-\mu}(x+c_{2})]}=c_{1}\qquad\mu<1,$
(64b)

where $c_{1},c_{2}$ are two parameters to be determined. The geodesics above
are usual geodesics for black hole and AdS metrics. While for the black hole
metric, $\mu>1$, we have another geodesic solution,

$r^{-2}+(\frac{1}{L^{2}(\mu-1)}-c_{1})\sinh^{2}{[\sqrt{\mu-1}(x+c_{2})]}=\frac{1}{L^{2}(\mu-1)}.$
(65)

This is an unusual geodesic because it is always longer than (64a) in a pure
black hole spacetime. The comparison of the length of the two geodesics is
given in appendix C.1.

For the length of geodesics, we start from

$d=\int{\rm d}l\ \sqrt{g}=\int{\rm
d}s=\int\sqrt{\frac{r^{\prime}(x)}{f}+r^{2}}\ {\rm d}x,$ (66)

where $f$ is the function defined in metric (56), and $r^{\prime}(x)$ is the
derivative. Then the integral can be evaluated to give

$d=L\tanh^{-1}{\left[\frac{\tanh{(\sqrt{\mu-1}(x+c_{2}))}}{L\sqrt{c_{1}(\mu-1)}}\right]}\qquad\mu>1,$
(67a)
$d=L\tanh^{-1}{\left[\frac{\tan{(\sqrt{1-\mu}(x+c_{2}))}}{L\sqrt{c_{1}(1-\mu)}}\right]}\qquad\mu<1.$
(67b)

Actually, in (64) and (67) we notice that the solutions of two cases are
related: for example, starting from $\mu>1$ in (67a), it can be continued to
$\mu<1$, then $\sqrt{\mu-1}={\rm i}\sqrt{1-\mu}$ leads to (67b).

Let’s discuss the asymptotic behavior of the geodesic length. Assuming two end
points located at $(r=\frac{1}{\varepsilon},x=\pm\frac{1}{2}x_{0})$, in the
AdS spacetime for $\mu<1$, we can obtain $c_{1,2}$, and thus the total length
of the geodesic,

$\begin{split}\Delta
d=&2L\tanh^{-1}{\left[\frac{\tan{(\sqrt{1-\mu}\frac{x_{0}}{2})}}{L\sqrt{c_{1}(1-\mu)}}\right]}\approx
2L\log{\left(\frac{2\sin{(\sqrt{1-\mu}\frac{x_{0}}{2})}}{L\sqrt{1-\mu}\varepsilon}\right)}.\end{split}$
(68)

Therefore, for $x_{0}\ll\frac{1}{\sqrt{1-\mu}}$, we obtain a logarithm $\Delta
d=2L\log{\frac{x_{0}/L}{\varepsilon}}$, as expected.

On the other hand, in the black hole spacetime for $\mu>1$, the total length
of the geodesic is

$\begin{split}\Delta
d=&2L\tanh^{-1}{\left[\frac{\tanh{(\sqrt{\mu-1}\frac{x_{0}}{2})}}{L\sqrt{c_{1}(\mu-1)}}\right]}\approx
2L\log{\left(\frac{2\sinh{(\sqrt{\mu-1}\frac{x_{0}}{2})}}{L\sqrt{\mu-1}\varepsilon}\right)},\end{split}$
(69)

Therefore, for $x_{0}\ll\frac{1}{\sqrt{\mu-1}}$, we obtain $\Delta
d=2L\log{\frac{x_{0}/L}{\varepsilon}}$. But, for
$x_{0}\gg\frac{1}{\sqrt{\mu-1}}$, we obtain $\Delta
d=2L\log{(\frac{1}{\varepsilon L\sqrt{\mu-1}})}+L\sqrt{\mu-1}x_{0}$, which
corresponds to volume-law entanglement.

For the unusual geodesic (65), it has a fixed point
$(x=-c_{2},r^{-1}=\sqrt{\frac{1}{L^{2}(\mu-1)}})$ located at the black hole
horizon. The length of the unusual geodesic (65) is

$d=L\tanh^{-1}{\left[\sqrt{2-c_{1}L^{2}(\mu-1)}{\tanh{(\sqrt{\mu-1}(x+c_{2}))}}\right]}.$
(70)

Again if we assume two end points
$(r_{0}=\varepsilon^{-1},\pm\frac{x_{0}}{2})$ on the geodesic, then $c_{2}=0$,
and

$c_{1}=\frac{r^{-2}+\frac{1}{L^{2}(\mu-1)}\left(\sinh^{2}{(\sqrt{\mu-1}\frac{x_{0}}{2})}-1\right)}{\sinh^{2}{(\sqrt{\mu-1}\frac{x_{0}}{2})}},$
(71)

which leads to the total length

$\begin{split}\Delta
d=&2L\tanh^{-1}{\left[\left(1-\frac{L^{2}(\mu-1)\varepsilon^{2}}{\cosh^{2}{[\sqrt{\mu-1}\frac{x_{0}}{2}]}}\right)^{\frac{1}{2}}\right]}\approx
2L\log{\left(\frac{2\cosh{(\sqrt{\mu-1}\frac{x_{0}}{2})}}{L\sqrt{\mu-1}\varepsilon}\right)}.\end{split}$
(72)

However, different from (69), for $x_{0}\ll\frac{1}{\sqrt{\mu-1}}$, it does
not lead to a logarithmic function. While for
$x_{0}\gg\frac{1}{\sqrt{\mu-1}}$, we have $\Delta
d=2L\log{(\frac{1}{\varepsilon L\sqrt{\mu-1}})}+L\sqrt{\mu-1}x_{0}$, which is
the same as the usual geodesic.

###  4.3 No-bubble phase: marginal weak measurements

In the no-bubble phase, the time reversal invariant slice is included in the
$AdS_{2}$ spacetime with radius $L_{2}$. Due to the time reversal symmetry,
the RT surface of a subregion is located within the time reversal invariant
slice. Therefore, the entanglement entropy is given accordingly by

$S=\frac{c_{\text{eff}}}{3}\log\frac{l}{\varepsilon},$ (73)

where $l$ is the length of the subregion, and we have taken the limit
$\varepsilon/l\ll 1$ and $l/(2\pi)\ll 1$. Notice that
$c_{\text{eff}}=\frac{3L_{2}}{2G_{\text{(3)}}}$. This corresponds to the
marginal phase with a continuous effective central charge.

If we take the limit $L_{2}\rightarrow 0$, then the entanglement will be an
area law. This corresponds to the relevant case. In this case, the interface
brane becomes the ETW brane. The region dual to the weak measurement is a
trivial spacetime, and it is similar to the gravity dual of a boundary state
resulted from projection measurements.

###  4.4 Geodesic in the bubble-outside-horizon phase

We first consider the bubble-outside-horizon phase. The geometry of the
bubble-outside-horizon phase is shown in figure 14. We expect the entanglement
entropy to satisfy a $\log$-law, because the horizon is not included in the
black hole solution and the RT surface can cross the interface brane to enter
the AdS spacetime.

Figure 14:  The light (dark) shaded region denotes the AdS (black hole)
spacetime. (a) The time reversal invariant slice in the bubble-outside-horizon
phase. The blue circle indicates the interface brane at $r_{0}$ and the dashed
circle indicates the horizon $r_{H}$. (b) The vertical cross-section of the
cylinder indicates the AdS spacetime, and the circle indicates the black hole
spacetime.

We can focus on the entanglement entropy of a subregion in the AdS space, as
shown in figure 15. Naively, we would expect that the geodesic in region II
has the form (64a), and the geodesic in region I has the form (64b). Let’s try
to solve the geodesic that crosses the brane with the connection conditions:
the geodesic and its derivative are continuous on the crossing point.

Figure 15:  (a) Illustration of the geodesic in the bubble-outside-horizon
phase. It consists of two parts, one in the AdS spacetime (region I), and the
other in the black hole spacetime (region II). The AdS geodesic solution
(green) is given in (64), while the black hole geodesic solution (orange) is
the unusual one (65). (b) Zooming the portion of geodesics in (a). The
geodesic solution is symmetric in the coordinate $x$.

We assume that the subregion in the boundary of time reversal invariant slice
is located at $(-\delta,\delta)$. For the limit we considered, the RT surface
will cross the interface brane. We assume the crossing point to be
$(r,x)=(r_{0},-x_{0})$, then for the geodesic in the region I, we have

$r_{0}^{-2}+(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}=c_{1},$ (74a) and for the
geodesic in the region II, we have
$r_{0}^{-2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}\right)\sinh^{2}{\sqrt{\mu-1}(x_{0}-b_{2})}=b_{1}.$
(74b) Because the geodesic in region II has the end point
$(r,x)=(\varepsilon^{-1},-\delta)$, we also have
$\varepsilon^{2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}\right)\sinh^{2}{\sqrt{\mu-1}(\delta-
b_{2})}=b_{1}.$ (74c) Besides, the connection condition requires
$\begin{split}&\left.\frac{{\rm d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
I}}=-\frac{(L_{1}^{-2}+c_{1})\sin{x_{0}}\cos{x_{0}}}{\sqrt{c_{1}-(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}}}\\\
=&\left.\frac{{\rm d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
II}}=-\sqrt{\mu-1}\frac{(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh{[\sqrt{\mu-1}(x_{0}-b_{2})]}\cosh{[\sqrt{\mu-1}(x_{0}-b_{2})]}}{\sqrt{b_{1}-(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh^{2}{[\sqrt{\mu-1}(x_{0}-b_{2})]}}}.\end{split}$
(74d)

These four equations are enough to determine the four unknown parameters
$x_{0},c_{1},b_{1}$ and $b_{2}$. However, it is difficult to obtain a general
analytical solution. In the following, we focus on the limit

$\varepsilon\ll r_{H}^{-1}\ll x_{0}\ll 1,$ (75)

that can give analytical results. This limit is justified as follows:

  * •

$\varepsilon\ll r_{H}^{-1}$ is due to the requirement that the UV-cutoff
should be the minimal length scale.

  * •

$r_{H}\propto\sqrt{\mu}$ in the large $\mu$ limit. It means
$r_{H}^{-1}\propto\epsilon$, the regularization parameter of the weak
measurement. 161616Note that we use $\epsilon$ to denote the regularization of
the weak measurement, and $\varepsilon$ to denote the UV cutoff near the
boundary. This justifies the limit $r_{H}^{-1}\ll x_{0}$.

  * •

$x_{0}\ll 1$ is because for original CFT we set it on a circle with radius
$1$, and we want our subsystem is much smaller than the total system
171717While $x_{0}$ is not the length of the subsystem, we will see that at
this limit, $\delta\sim x_{0}$. Therefore, the limit (75) is equivalent to
$\varepsilon\ll r_{H}^{-1}\ll\delta\ll 1$. .

Besides, we assume $L_{i}$ for $i=1,2$ are order-one numbers $\mathcal{O}(1)$.

In such a limit, we first estimate the order of $r_{0}$. From (61) and (62),
up to $\mathcal{O}(\mu^{-1})$ we have

$\left(\frac{r_{H}}{r_{0}}\right)^{2}\approx\lambda_{2}^{-1}(-T+\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}})(T-\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}}),$
(76)

which leads to $r_{H}/r_{0}\rightarrow\eta$ with

$\eta=\sqrt{1-\left(\frac{T-\sqrt{\lambda_{1}}}{\sqrt{\lambda_{2}}}\right)^{2}}.$
(77)

In the bubble phase, $\eta<1$ and is an order-one number. There is a subtlety:
we take the limit $\mu\rightarrow\infty$ to determine the order of $r_{0}$,
but when we solve equations (74), we do not directly take this limit but leave
it at the end of the calculation. However, it is unimportant because the
results won’t change in the order we consider in the discussion below.

Let’s try to solve the equations (74). From (74b) and (74c), the denominators
of the two sides in (74d) are both $r_{0}^{-1}$. So we only need to look at
the numerator. From (74a) we know
$c_{1}=[r_{0}^{-2}+L_{1}^{-2}\sin^{2}{x_{0}}]/\cos^{2}{x_{0}}\sim\mathcal{O}(x_{0}^{2})$,
then the numerator on the left-hand side of (74d) is in the order of
$\mathcal{O}(x_{0})$. However, with the help of (74b), the numerator on the
right-hand side of (74d) can be simplified as

$\sqrt{(\mu-1)(b_{1}-r_{0}^{-2})\left[\frac{1}{L_{2}^{2}(\mu-1)}-r_{0}^{-2}\right]}<\sqrt{(\mu-1)(r_{H}^{-2}-r_{0}^{-2})^{2}}=L_{2}^{-1}(1-\eta^{2})r_{H}^{-1}\sim\mathcal{O}(r_{H}^{-1})$
(78)

where we have used $b_{1}<\frac{1}{L_{2}^{2}(\mu-1)}=r_{H}^{-2}$. Therefore,
in (74d) we have $\mathcal{O}(x_{0})$ for the numerator of the left-hand side
and $\mathcal{O}(r_{H}^{-1})$ for the numerator of the right-hand side. With
the assumption $r_{H}^{-1}\ll x_{0}$ in (75), there is no solution for (74).
The naive geodesic we use does not have a solution at the limit. The key is
the derivative of geodesic in region II is too small to satisfy the connection
condition. This problem can be solved by considering a different geodesic in
the region II.

We consider unusual geodesic in the region II. The four new equations are
listed as follows

$r_{0}^{-2}+(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}=c_{1},$ (79a)
$r_{0}^{-2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}\right)\sinh^{2}{[\sqrt{\mu-1}(x_{0}-b_{2})]}=\frac{1}{L_{2}^{2}(\mu-1)},$
(79b)
$\varepsilon^{2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}\right)\sinh^{2}{[\sqrt{\mu-1}(\delta-
b_{2})]}=\frac{1}{L_{2}^{2}(\mu-1)},$ (79c) $\begin{split}&\left.\frac{{\rm
d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
I}}=-\frac{(L_{1}^{-2}+c_{1})\sin{x_{0}}\cos{x_{0}}}{\sqrt{c_{1}-(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}}}\\\
=&\left.\frac{{\rm d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
II}}=-\sqrt{\mu-1}\frac{(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh{[\sqrt{\mu-1}(x_{0}-b_{2})]}\cosh{[\sqrt{\mu-1}(x_{0}-b_{2})]}}{\sqrt{\frac{1}{L_{2}^{2}(\mu-1)}-(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh^{2}{[\sqrt{\mu-1}(x_{0}-b_{2})]}}}.\end{split}$
(79d)

In the following we will again consider the limit (75) that gives an
analytical solution.

We first check if (79) has a solution that satisfies the connection condition
(79d). Again, the denominators of the two sides in (79d) are both $r_{0}$
owing to (79a) and (79b). The numerator for the left-hand side is the same as
before, and is again in the same order of $\mathcal{O}(x_{0})$. But for the
right-hand side, with (79b) we have

$\begin{split}&\sqrt{(\mu-1)\left[\frac{1}{L_{2}^{2}(\mu-1)}-r_{0}^{-2}\right]\left[\frac{2}{L_{2}^{2}(\mu-1)}-r_{0}^{-2}-b_{1}\right]}\\\
&=L_{2}^{-1}\sqrt{(1-\eta^{2})(2r_{H}^{-2}-r_{0}^{-2}-b_{1})}.\end{split}$
(80)

So we require $b_{1}\sim-x_{0}^{2}$ to satisfy the connection condition.

In the limit (75), we expand variables in the order of $x_{0}$ and $r_{H}$.
From (79a), (79b), and (79d), we can express $c_{1}$, $b_{1}$, and $b_{2}$ as
a function of $x_{0}$, i.e.,

$\begin{split}c_{1}=&L_{1}^{-2}x_{0}^{2},\\\
b_{1}=&-L_{2}^{2}(1-\eta^{2})^{-1}L_{1}^{-4}x_{0}^{2},\\\
b_{2}=&x_{0}-L_{1}^{2}(1-\eta^{2})\frac{r_{H}^{-2}}{{x_{0}}}.\end{split}$ (81)

We then substitute (81) into (79c) to get an equation that only involves
$x_{0}$. (79c) can be brought to $\sinh^{2}{[\sqrt{\mu-1}(\delta-
b_{2})]}=\frac{\frac{1}{L_{2}^{2}(\mu-1)}-\varepsilon^{2}}{\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}}$.
We can use (81) to simplify both the left-hand side and right-hand side to get

$\begin{split}&L_{1}^{2}\sqrt{1-\eta^{2}}r_{H}^{-2}x_{0}^{-1}\left[1+\frac{1}{2}L_{2}^{-2}L_{1}^{4}(1-\eta^{2})^{2}\frac{r_{H}^{-2}}{x_{0}^{2}}\right]\\\
&=\delta-
x_{0}+L_{1}^{2}(1-\eta^{2})\frac{r_{H}^{-2}}{{x_{0}}}\left(1+\frac{1}{2}L_{2}^{-2}L_{1}^{4}(1-\eta^{2})^{2}\frac{r_{H}^{-2}}{x_{0}^{2}}\right).\end{split}$
(82)

Therefore, at the leading order, we obtain a quadratic equation for $x_{0}$

$L_{1}^{2}\sqrt{1-\eta^{2}}r_{H}^{-2}x_{0}^{-1}=\delta-
x_{0}+L_{1}^{2}(1-\eta^{2})\frac{r_{H}^{-2}}{{x_{0}}}.$ (83)

There are two solutions. Because $x_{0}\approx\delta$, the relevant solution
reads

$\begin{split}x_{0}=&\frac{1}{2}[\delta+\delta(1-4L_{1}^{2}\sqrt{1-\eta^{2}}(1-\sqrt{1-\eta^{2}})r_{H}^{-2}\delta^{-2})^{\frac{1}{2}}]\\\
\approx&\delta-
L_{1}^{2}\sqrt{1-\eta^{2}}(1-\sqrt{1-\eta^{2}})r_{H}^{-2}\delta^{-1},\end{split}$
(84)

where the second term has the order
$\mathcal{O}(\frac{r_{H}^{-2}}{x_{0}^{2}}\cdot x_{0})\ll\mathcal{O}(x_{0})$.
Having obtained $x_{0}$, we then substitute it back into (81) to get the other
parameters.

To verify our analytical geodesic solution, we also numerically solve (79).
With parameters
$L_{1}=1,L_{2}=1.1,\mu=1000,\varepsilon=0.0001,T=0.6,\delta=0.1$, and the
solutions are

$x_{0}=0.0979312,b_{1}=-0.0593183,b_{2}=0.0962979,c_{1}=0.0103255.$ (85)

From our analytical solution, we can get the following results

$x_{0}=0.0979612,b_{1}=-0.0598777,b_{2}=0.0963236,c_{1}=0.0095964.$ (86)

Thus, our analytical results are closed to numerical results in the limit
(75).

With the solution above, we can calculate the geodesic length. The geodesic
can be split into two halves, each of which ranges from the boundary point
$x=\pm\delta$ to the symmetric point at $x=0$. We focus on the length of one
half, and the total length is simply $\Delta d=2(\Delta d_{1}+\Delta d_{2})$,
where $\Delta d_{1,2}$ denotes the length in region I and II, respectively.
For the geodesic in region II, from (70) we have

$\begin{split}\Delta d_{2}&=(\Delta d_{2})_{1}-(\Delta d_{2})_{2},\\\ (\Delta
d_{2})_{1}&=L_{2}\tanh^{-1}{\left[\sqrt{2-b_{1}L_{2}^{2}(\mu-1)}{\tanh{(\sqrt{\mu-1}(\delta-
b_{2}))}}\right]},\\\ (\Delta
d_{2})_{2}&=L_{2}\tanh^{-1}{\left[\sqrt{2-b_{1}L_{2}^{2}(\mu-1)}{\tanh{(\sqrt{\mu-1}(x_{0}-b_{2}))}}\right]}.\end{split}$
(87)

With the solution, we can simplify these two distances separately. For
$(\Delta d_{2})_{1}$, we arrive at

$\begin{split}(\Delta
d_{2})_{1}\approx&L_{2}\left\\{\log{\left[\frac{2}{L_{2}\sqrt{\mu-1}\varepsilon}\right]}+\frac{1}{2}(\mu-1)(\delta-
b_{2})^{2}\right\\},\end{split}$ (88)

where $\delta-b_{2}\approx L_{1}^{2}\sqrt{1-\eta^{2}}r_{H}^{-2}\delta^{-1}$.
And for $(\Delta d_{2})_{2}$, we arrive at

$\begin{split}(\Delta
d_{2})_{2}\approx&L_{2}\left[\tanh^{-1}{(\sqrt{1-\eta^{2}})}+\frac{1}{2}(1-\eta^{2})^{\frac{3}{2}}L_{1}^{4}L_{2}^{-2}r_{H}^{-2}\delta^{-2}\right].\end{split}$
(89)

Therefore, the geodesic length in region II reads

$\begin{split}\Delta
d_{2}=&L_{2}\left\\{\log{\left(\frac{2r_{H}^{-1}}{\varepsilon}\right)}+\frac{1}{2}L_{2}^{-2}L_{1}^{4}(1-\eta^{2})(1-\sqrt{1-\eta^{2}})r_{H}^{-2}\delta^{-2}-\tanh^{-1}{(\sqrt{1-\eta^{2}})}\right\\}.\end{split}$
(90)

Besides, for the geodesic in region I, it is

$\begin{split}\Delta
d_{1}=&L_{1}\tanh^{-1}{\left[\frac{\tan{x_{0}}}{L_{1}\sqrt{c_{1}}}\right]}=L_{1}\tanh^{-1}{\left[\left(\frac{\sin^{2}{x_{0}}}{L_{1}^{2}r_{0}^{-2}+\sin^{2}{x_{0}}}\right)^{\frac{1}{2}}\right]},\end{split}$
(91)

With (84), we have

$\begin{split}\frac{\sin^{2}{x_{0}}}{L_{1}^{2}r_{0}^{-2}+\sin^{2}{x_{0}}}\approx&1-L_{1}^{2}\eta^{2}r_{H}^{-2}\delta^{-2}.\end{split}$
(92)

Therefore, the length of geodesic in region I is

$\begin{split}\Delta
d_{1}=L_{1}\tanh^{-1}{\left(1-\frac{1}{2}L_{1}^{2}\eta^{2}r_{H}^{-2}\delta^{-2}\right)}\approx
L_{1}\cosh^{-1}{\left(\frac{L_{1}^{-1}\eta^{-1}\delta}{r_{H}^{-1}}\right)}\approx
L_{1}\log{\left(\frac{2L_{1}^{-1}\eta^{-1}\delta}{r_{H}^{-1}}\right)}.\end{split}$
(93)

Finally, we have the total length of geodesic

$\begin{split}\Delta
d=&2L_{2}\left\\{\log{\left(\frac{2r_{H}^{-1}}{\varepsilon}\right)}-\tanh^{-1}{(\sqrt{1-\eta^{2}})}\right\\}+2L_{1}\log{\left(\frac{2L_{1}^{-1}\eta^{-1}\delta}{r_{H}^{-1}}\right)},\end{split}$
(94)

where we have taken the limit $\mu\rightarrow\infty$ and
$r_{H}^{-1}\rightarrow 0$. Since the subsystem size is $2\delta$, the
corresponding entanglement entropy satisfies a $\log$-law. Namely, the
entanglement entropy of the subsystem is in turn given by
$S=\frac{c_{1}}{3}\log(2\delta)$, where we only keep the dependence of
$\delta$. The prefactor of the logarithm is the same as the central charge of
the unmeasured CFT, which means the weak measurement is irrelevant.

We numerically compute the geodesic length for different sizes of the
subsystem, ranging from $0.025$ to $0.5$. In figure 16 we plot the numerical
results and analytical results of (94), it shows that they are consistent with
each other.

Figure 16:  Geodesic length as a function of the subsystem size in the bubble-
outside-horizon phase. The parameters are chosen to be
$L_{1}=1,L_{2}=1.1,\mu=1000,\varepsilon=0.0001,T=0.6$.

###  4.5 Geodesic in the bubble-inside-horizon phase

Now we consider the entanglement entropy of a subregion in the bubble-inside-
horizon phase. Again, we assume that the subregion in the boundary of time
reversal invariant slice is located at $(-\delta,\delta)$. In this phase, the
bubble is outside the horizon, in other words, the black hole horizon exists
in the bulk solution, as shown in the left panel of figure 21. Naively, we
would expect that the geodesic starting from the boundary is located outside
the horizon, and is contained in the back hole dual to the region II. If this
was true, then the entanglement entropy would satisfy a volume law. However,
in the following, we will show that there exists a new kind of geodesic that
can cross the horizon. The new geodesic has a shorter length and leads to a
$\log$-law, and, therefore, is preferred.

The new geodesic has two sections located in the region I, II, respectively.
In region II, the usual geodesic in black hole geometry will never reach the
horizon, so the unusual geodesic (65) that has a fix point
$(x,r)=(-c_{2},r_{H})$ is needed to cross the horizon and connect to the
section in the region II. Besides, this unusual geodesic (65) is symmetric at
$x=-c_{2}$, so the derivative vanishes at the horizon $\left.\frac{{\rm
d}r^{-1}}{{\rm d}x}\right|_{x=-c_{2}}=0$. The possible geodesic is shown in
figure 17. We emphasize that this new geodesic (including two sections in
region I and II) is shorter because of the existence of region I: it has a
section in region I that shortens the length.

Figure 17:  Illustration of the RT surface in the bubble-inside-horizon phase.
The blue line indicates the interface brane, and the dashed line indicates the
horizon. The RT surface has three parts, denoted by $l_{1,2,3}$. The region II
has two parts $l_{1,2}$ that connect at the horizon. They are given by the
unusual geodesic solution. $\pm\beta$ is the symmetric point of the solution
of the parts $l_{1,2}$. $(-x_{0},r_{0}^{-1})$ is the connecting point between
the geodesics of the part $l_{3}$ and $l_{2}$.

In figure 17, the geodesic has three parts, denoted as $l_{1,2,3}$. For parts
$l_{1}$ and $l_{2}$, the corresponding geodesics are given by (65), with the
same $c_{2}$ but different $c_{1}$. The part $l_{3}$ is the usual geodesic in
the AdS metric.181818Actually, there is another possibility that between part
$l_{1}$ and $l_{2}$ there can be an additional geodesic along the horizon, but
in the appendix C.2 we show that this possibility always leads to a longer
geodesic. So we only consider three parts $l_{1,2,3,}$. Assuming that two end
points are $(r^{-1},x)=(\varepsilon,\pm\delta)$, then for any $\beta$ (which
is defined as the symmetric point of the geodesic of part $l_{1,2}$ as shown
in the figure) we have geodesic solutions as shown in figure 17. In the
following, we will first solve the general geodesic equations for any given
$\beta$ and then determine the $\beta$ such that the total length of the
geodesic is minimal. To obtain an analytical result, we again consider the
same limit,

$\varepsilon\ll r_{H}^{-1}\ll\delta\ll 1.$ (95)

For part $l_{1}$, we have the following equation for any given $\beta$

$\varepsilon^{2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}^{\prime}\right)\sinh^{2}{[\sqrt{\mu-1}(\delta-\beta)]}=\frac{1}{L_{2}^{2}(\mu-1)}.$
(96)

For the parts $l_{2}$ and $l_{3}$, we have the following equations, where the
last one is from the connection condition

$r_{0}^{-2}+(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}=c_{1},$ (97a)
$r_{0}^{-2}+\left(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1}\right)\sinh^{2}{[\sqrt{\mu-1}(x_{0}-\beta)]}=\frac{1}{L_{2}^{2}(\mu-1)},$
(97b) $\begin{split}&\left.\frac{{\rm d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
I}}=-\frac{(L_{1}^{-2}+c_{1})\sin{x_{0}}\cos{x_{0}}}{\sqrt{c_{1}-(L_{1}^{-2}+c_{1})\sin^{2}{x_{0}}}}\\\
=&-\left.\frac{{\rm d}r^{-1}}{{\rm d}x}\right|_{r=r_{0},{\rm
II}}=\sqrt{\mu-1}\frac{(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh{[\sqrt{\mu-1}(x_{0}-\beta)]}\cosh{[\sqrt{\mu-1}(x_{0}-\beta)]}}{\sqrt{\frac{1}{L_{2}^{2}(\mu-1)}-(\frac{1}{L_{2}^{2}(\mu-1)}-b_{1})\sinh^{2}{[\sqrt{\mu-1}(x_{0}-b_{2})]}}}.\end{split}$
(97c)

Notice that the minus sign in the last equation is because the definition of
the geodesic in region II is opposite in the $r$ direction. It is clear if we
compare the part $l_{2}$ and the part $l_{1}$ in figure 17.

From (97a) and (97b) we can get $c_{1}$ and $b_{1}$ as a function of $x_{0}$

$c_{1}=\frac{r_{0}^{-2}+L_{1}^{-2}\sin^{2}{x_{0}}}{\cos^{2}{x_{0}}},$ (98)
$b_{1}=\frac{1}{L_{2}^{2}(\mu-1)}-\frac{\frac{1}{L_{2}^{2}(\mu-1)}-r_{0}^{-2}}{\sinh^{2}{[\sqrt{\mu-1}(x_{0}-\beta)]}}.$
(99)

Substituting these two expressions into (97c), we arrive at

$\tan{x_{0}}\tanh{[\sqrt{\mu-1}(\beta-
x_{0})]}=\frac{\sqrt{\mu-1}(\frac{1}{L_{2}^{2}(\mu-1)}-r_{0}^{-2})}{(L_{1}^{-2}+r_{0}^{-2})}.$
(100)

While we leave $\beta$ an undetermined parameter to get an expression for
geodesic length and determine $\beta$ by demanding the geodesic length to be
minimal, we can first estimate the order of $\beta$. We argue that $\beta$
must have the same order of $\delta$ and $|\delta-\beta|\ll\delta$. The
intuitive argument is that if $|\delta-\beta|\sim\mathcal{O}(\delta)$, the
geodesic length will lead to a volume law: when $\beta=\eta^{\prime}\delta$
with $\eta^{\prime}<1$ and $\eta^{\prime}\sim\mathcal{O}(1)$, the length of
the part $l_{1}$ will be proportional to $\mathcal{O}(\delta)$. A precise
estimate is given in the appendix C.2. But we can construct a geodesic with a
$\log$-law for $|\delta-\beta|\ll\delta$.

The right-hand side of (100) is of the order $\mathcal{O}(r_{H}^{-1})$. Then
the left-hand side implies that the solution is $x_{0}\sim r_{H}^{-1}$ or
$x_{0}\sim\beta$. If $x_{0}\sim r_{H}^{-1}$, with (97b) and (72), the length
of the part $l_{2}$ is

$\begin{split}L_{2}\tanh^{-1}{\left[\left(1-\frac{L_{2}^{2}(\mu-1)r_{0}^{-2}}{\cosh^{2}{[\sqrt{\mu-1}(\beta-
x_{0})]}}\right)^{\frac{1}{2}}\right]}\approx r_{H}(\beta-
x_{0})-L_{2}\log{\eta}.\end{split}$ (101)

Because $\beta-x_{0}\approx\delta$, the geodesic length leads to volume-law
entanglement. Therefore, we should consider the solution $x_{0}\sim\beta$,
which will later be shown to have $\log$-law entanglement. In this case, (100)
can be expanded as

$L_{2}^{-1}r_{H}(\beta-x_{0})x_{0}\approx
L_{2}^{-1}L_{1}^{2}(1-\eta^{2})r_{H}^{-1}.$ (102)

which leads to the solution

$x_{0}=\beta-(1-\eta^{2})L_{1}^{2}r_{H}^{-2}\beta^{-1}.$ (103)

With this solution, the other parameters, (98), (99), and thus the geodesic
length can be expressed as a function of $\beta$.

We can verify our analytical result by numerically calculation. For instance,
using parameters
$L_{1}=1,L_{2}=1.1,\mu=10000,\varepsilon=0.00001,T=1.6,\beta=0.1$, the
numerical result of (100) is

$x_{0}=0.0996397,$ (104)

and the analytical solution, (103) gives

$x_{0}=0.09964.$ (105)

Therefore, our analytical result is close to numerical calculation in the
limit (95).

We can calculate the geodesic length with the solution above. For the part
$l_{1}$, with (96) and (72), we have

$\begin{split}\Delta
d_{l_{1}}\approx{\left(\frac{\cosh{(\sqrt{\mu-1}(\delta-\beta))}}{L_{2}\sqrt{\mu-1}\varepsilon}\right)}&L_{2}\log{\left(\frac{r_{H}^{-1}}{\varepsilon}\right)}+L_{2}\log{\left(2\cosh{(\sqrt{\mu-1}(\delta-\beta))}\right)}.\end{split}$
(106)

If $|\delta-\beta|\lesssim O(r_{H}^{-1})$, the second term in (106) will at
most contribute to $\mathcal{O}(1)$. For the part $l_{2}$ with (97b) and (72),
we have

$\begin{split}\Delta
d_{l_{2}}\approx&L_{2}\tanh^{-1}{\left[\left(1-\frac{\eta^{2}}{\cosh^{2}{[\sqrt{\mu-1}(\beta-
x_{0})]}}\right)^{\frac{1}{2}}\right]}\\\
\approx&L_{2}\left\\{\tanh^{-1}{(\sqrt{1-\eta^{2}})}+\frac{(\mu-1)(\beta-
x_{0})^{2}}{2\sqrt{1-\eta^{2}}}\right\\},\end{split}$ (107)

where we have used $\sqrt{\mu-1}(\beta-x_{0})\approx
L_{1}^{2}L_{2}^{-1}(1-\eta^{2})r_{H}^{-1}\beta^{-1}\sim\mathcal{O}(\frac{r_{H}^{-1}}{\delta})\ll
1$. For the part $l_{3}$ with (97a) and (68), we have

$\begin{split}\Delta
d_{l_{3}}=&L_{1}\tanh^{-1}{\sqrt{\frac{\sin^{2}{x_{0}}}{L_{1}^{2}r_{0}^{-2}+\sin^{2}{x_{0}}}}}\approx
L_{1}\log{\left(\frac{2L_{1}^{-1}\sin{x_{0}}}{r_{0}^{-1}}\right)},\end{split}$
(108)

where we have used $x_{0}\approx\beta\sim\mathcal{O}(\delta)$ and
$r_{0}^{-1}\sim\mathcal{O}(r_{H}^{-1})\ll\mathcal{O}(x_{0})$.

Using the approximation above and (103), we have

$\begin{split}\Delta d_{l_{1}}+\Delta d_{l_{2}}+\Delta
d_{l_{3}}\approx&L_{2}\log{\left(2\cosh{(\sqrt{\mu-1}(\delta-\beta))}\right)}+\frac{1}{2}L_{2}^{-1}(1-\eta^{2})^{\frac{3}{2}}L_{1}^{4}r_{H}^{-2}\beta^{-2}\\\
&+L_{1}\log{\left(\beta-(1-\eta^{2})L_{1}^{2}r_{H}^{-2}\beta^{-1}\right)}+{\rm
const},\end{split}$ (109)

where ${\rm const}$ denotes the terms independent of $\beta$. We minimize the
geodesic length (109) w.r.t. $\beta$. Denoting the right-hand side of (109) as
$f(\beta)$, its derivative is

$\begin{split}f^{\prime}(\beta)=&-r_{H}\tanh{(L_{2}^{-1}r_{H}(\delta-\beta))}-L_{2}^{-1}(1-\eta^{2})^{\frac{3}{2}}L_{1}^{4}r_{H}^{-2}\beta^{-3}+L_{1}\frac{1+(1-\eta^{2})L_{1}^{2}r_{H}^{-2}\beta^{-2}}{\beta-(1-\eta^{2})L_{1}^{2}r_{H}^{-2}\beta^{-1}}\\\
\approx&\frac{L_{2}}{\beta}\left\\{\frac{L_{1}}{L_{2}}-L_{2}^{-1}\beta
r_{H}\tanh{(L_{2}^{-1}r_{H}(\delta-\beta))}\right\\},\end{split}$ (110)

where in the last equation we only keep the order $\mathcal{O}(\beta^{-1})$
and ignore higher orders, like $O(\beta^{-1}\frac{r_{H}^{-2}}{\beta^{2}})\ll
O(\beta^{-1})$. It turns out that the only consistent solution of
$f^{\prime}(\beta)=0$ is given by

$\beta\approx\delta-L_{1}L_{2}r_{H}^{-2}\delta^{-1},$ (111)

corresponding to the minimal point of the geodesic length.

Figure 18:  (a) The geodesic in the region II in the bubble-outside-horizon
phase. (b) The geodesic in the region II in the bubble-inside-horizon phase.

Finally, substituting (111) into (109), the geodesic length reads

$\begin{split}\Delta d=&2(\Delta d_{l_{1}}+\Delta d_{l_{2}}+\Delta
d_{l_{3}})\\\
\approx&2L_{2}\left[\log{\left(\frac{2r_{H}^{-1}}{\varepsilon}\right)}+\tanh^{-1}{(\sqrt{1-\eta^{2}})}\right]+2L_{1}\log{\left(\frac{2L_{1}^{-1}\eta^{-1}\delta}{r_{H}^{-1}}\right)},\end{split}$
(112)

where in the derivation we have ignored higher orders in
$\mathcal{O}(\frac{\delta^{2}}{r_{H}^{-2}})$. Since the subsystem size is
$2\delta$, the corresponding entanglement entropy satisfies a $\log$-law. The
prefactor of the logarithmic entanglement entropy will be the same as the
central charge of the unmeasured CFT, which means the weak measurement is
irrelevant.

Here is one additional remark. Similar to (94), there is an order-one term in
(112), $2L_{2}\tanh^{-1}{(\sqrt{1-\eta^{2}})}$, but in a different sign. It is
because for (94) we have a part of the geodesic in figure 18 (a), while for
(112) we have a part of the geodesic in figure 18 (b). A half of the symmetric
geodesic has no constant term, so for figure 18 (a) we have
$0-2L_{2}\tanh^{-1}{(\sqrt{1-\eta^{2}})}$, while for figure 18 (b) we have
$0+2L_{2}\tanh^{-1}{(\sqrt{1-\eta^{2}})}$.

Finally, we numerically compute the length of the geodesic for different sizes
of the subsystem, ranging from $0.025$ to $0.5$. In figure 19, we plot the
numerical results and analytical results of (112), and it shows that they are
consistent with each other.

Figure 19:  Geodesic length as a function of the subsystem size in the bubble-
outside-horizon phase. The parameters are chosen to be
$L_{1}=1,L_{2}=1.1,\mu=10000,\varepsilon=0.00001,T=1.6$. We do not use the
variation to determine $\beta$ from $\delta$, instead, we use the
approximation (111).

##  5 Discussion

###  5.1 Thick brane description

In Sec. 2, we have considered weak measurements in an infinite system using a
bottom-up thin-brane description. Here, we discuss the thick-brane description
of the weak measurement and the relation to spatial interface CFT. In the
following, we set $L=1$ for simplicity.

To this end, we consider the foliation of a general 3d Euclidean metric

${\rm d}s^{2}={\rm d}\rho^{2}+e^{2A(\rho)}\frac{{\rm d}x^{2}+{\rm
d}y^{2}}{y^{2}},$ (113)

where $A(\rho)$ is a warpfactor that controls the size of each AdS2 slice.
Figure 20 (a) illustrates the foliation. The Euclidean Poincare coordinate is
related to the foliation by $z(y,\rho)$ and $\tau(y,\rho)$. As before, the
weak measurement occurs at $\tau=0$. The post-measurement state has an
effective central charge denoted by $c_{\rm eff}$. For $\tau\neq 0$, the
asymptotic region $z\rightarrow 0$ is given by the unmeasured CFT with central
charge $c_{1}$. In the foliation coordinate, the measurement region at the
boundary is reached by $\lim_{y\rightarrow 0}\tau(y,\rho)=0$, and the
unmeasured CFT region at the boundary is reached by $\rho\rightarrow\pm\infty$
as shown in figure 20 (a).

It is helpful to look at the pure AdS3 metric: when

$e^{A(\rho)}=\cosh\rho,$ (114)

the metric is reduced to an AdS3 metric. One can check this via a coordinate
transformation similar to (14),

$z=\frac{y}{\cosh\rho},\qquad\tau=y\tanh{\rho}.$ (115)

The only difference is that $x$ and $\tau$ are exchanged in (14). From this
coordinate transformation, because $y\geq 0$, if the metric has time reversal
symmetry, we can deduce that the metric is invariant under
$\rho\rightarrow-\rho$. Of course, this is a trivial example because there is
no measurement in the pure AdS3 metric. Nevertheless, we expect this property
is general in the case of weak measurement. Namely, the imaginary time
satisfies $\tau(y,-\rho)=-\tau(y,\rho)$. We make the following assumptions for
the warpfactor:

  1. 1.

$A(\rho)$ is an even function of $\rho$, i.e., $A(\rho)=A(-\rho)$. This is
because of the time reversal invariance.

  2. 2.

For $\rho\rightarrow\pm\infty$, we have $e^{A(\rho)}\sim\cosh\rho$, i.e., we
recover AdS3 away from the measurement in the unmeasured CFT region.

  3. 3.

The minimal value of the warpfactor is located at $\rho=\pm\rho^{\ast}$ with
$A^{*}=A(\rho^{\ast})$. We assume $\rho^{\ast}\geq 0$ without loss of
generality. At this minimal point, the first-order derivative vanishes
$A^{\prime}(\rho^{\ast})=0$. Notice that $\rho^{\ast}=0$ is a special case of
this.

Figure 20: (a) An illustration of the foliation coordinate (113). Weak
measurements occur at $\tau=0$. For $\tau\neq 0$, the asymptotic boundary is
given by the unmeasured CFT $\rho\rightarrow\pm\infty$. (b) The RT surface
colored in orange connecting the two ends of the interval $x\in(0,l)$ is
located in the $\rho=\pm\rho^{\ast}$ slice colored in blue for the weak
measurement case. (c) The RT surface colored in orange connects two endpoints
of $x\in(0,l)$ for the spatial interface CFT at the $\tau=0$ slice colored in
blue. Note that the coordinate $x$ and $\tau$ are exchanged.

We consider entanglement entropy of a subsystem $x\in[0,l]$. The corresponding
RT surface is parametrized by $(y(x),\rho(x))$. Then the surface area of the
RT surface can be rewritten as an action $\mathcal{A}=\int{\rm d}x\mathcal{L}$
with the Lagrangian,

$\mathcal{L}=\sqrt{e^{2A(\rho)}\frac{\dot{y}^{2}+1}{y^{2}}+\dot{\rho}^{2}},\quad\dot{y}=\frac{{\rm
d}y}{{\rm d}x},\quad\dot{\rho}=\frac{{\rm d}\rho}{{\rm d}x}.$ (116)

The equation of motion is

$\begin{split}&e^{2A(\rho)}A^{\prime}(\rho)(1+\dot{y}^{2})^{2}-y[(1+\dot{y}^{2})(\dot{\rho}\dot{y}+y(-2A^{\prime}(\rho)\dot{\rho}^{2}+\ddot{\rho}))-y\dot{\rho}\dot{y}\ddot{y}]=0,\\\
&e^{2A(\rho)}[(1+\dot{y}^{2})(1+yA^{\prime}(\rho)\dot{\rho}\dot{y})+y\ddot{y}]+y^{2}\dot{\rho}[2yA^{\prime}(\rho)\dot{\rho}^{2}\dot{y}-y\dot{y}\ddot{\rho}+\dot{\rho}(1-\dot{y}^{2}+y\ddot{y})]=0.\end{split}$
(117)

A solution to the first equation is given by $\rho(x)=\rho^{\ast}$ because
$A^{\prime}(\rho^{\ast})=0$ and $\dot{\rho}=0$. It means that the RT surface
is located at the slice with the minimal warpfunction
$A^{\ast}=A(\rho^{\ast})$. Therefore, the equation for the geodesic is

$1+\dot{y}^{2}+y\ddot{y}=0,$ (118)

which gives a well-known solution

$y^{2}+(x-c_{1})^{2}=c_{2}^{2},$ (119)

where $c_{1,2}$ are integration constants. Therefore, the geodesic is also an
arc in $\rho=\rho^{*}$ slice. In this slice, the metric is $AdS_{2}$ with
radius $e^{A^{*}}$. With the results before, the area of the RT surface (119)
at the leading order of $l$ is

$\mathcal{A}=2e^{A^{\ast}}\log l,$ (120)

which gives the effective central charge $c_{{\rm
eff}}=\frac{3e^{A^{\ast}}}{2G_{(3)}}$, and $S_{l}=\frac{c_{\rm
eff}}{3}\log{l}$. Notice that for $\rho^{\ast}\neq 0$, the RT surface is not
located at the time reversal invariant slice. There is also a degenerate RT
surface at $\rho=-\rho^{\ast}$. This is illustrated in figure 20 (b). There is
also a solution at the time reversal invariant slice because $A^{\prime}(0)=0$
for the ever function $A(\rho)$. This solution has the area given by
$2e^{A(0)}\log l>2e^{A^{\ast}}\log l$ because $A(0)>A^{\ast}$.

Now, let’s consider a spatial interface CFT via a spacetime rotation
$\tau\leftrightarrow x$. The metric is

${\rm d}s^{2}={\rm d}\rho^{2}+e^{2A(\rho)}\frac{{\rm d}\tau^{2}+{\rm
d}y^{2}}{y^{2}}.$ (121)

The spatial coordinate is given by $x=x(y,\rho)$, and it satisfies
$x(y,-\rho)=-x(y,\rho)$. The asymptotic boundary $z\rightarrow 0$ for $x\neq
0$ is given by two CFTs. These two boundaries are given by
$\rho\rightarrow\pm\infty$, respectively. The interface is located at $x=0$.
An illustration of this coordinate is shown in figure 20.

We consider the entanglement entropy of an interval $x\in(0,l)$. In this case,
the RT surface is parametrized by $y(\rho)$ at a fixed time slice $\tau=0$. In
the following, we use a similar approach in reference karch2021universal . The
area functional is $\mathcal{A}=\int{\rm d}\rho\mathcal{L}$ with the
Lagrangian

$\mathcal{L}=\sqrt{1+e^{2A}\frac{\dot{y}^{2}}{y^{2}}},\quad\dot{y}=\frac{{\rm
d}x}{{\rm d}\rho}.$ (122)

The Lagrangian leads to the geodesic equation

$\frac{\dot{y}}{y}=\frac{c_{s}e^{-A}}{\sqrt{e^{2A}-c_{s}^{2}}},$ (123)

where $0\leq c_{s}\leq e^{A^{\ast}}$ is an integration constant. In general,
$\rho$ runs from $-\infty$ to $\infty$, and $c_{s}$ determines the end points
of the interval $l_{L}$ and $l_{R}$. A spatial case is $c_{s}=e^{A^{\ast}}$,
the right-hand-side of (123) diverges at $\rho=\rho^{\ast}$. The only possible
solution is $y=0$ at $\rho^{\ast}$. Therefore, the end point of the interval
is right on the interface $l_{L}=0$, and for simplicity we denote $l_{R}=l$.
In this spatial case, the parameter runs in the range
$\rho\in(\rho_{0},\rho_{+})$ such that $\rho_{0}\rightarrow\rho^{*}$,
$\rho_{+}\rightarrow\infty$. 191919We do not directly take the limit because
we need to regularize the area. An illustration of the RT surface is shown in
figure 20 (c). With a solution of $A(\rho)$, the RT surface satisfies

$\frac{\dot{y}}{y}=\frac{e^{A^{\ast}-A(\rho)}}{\sqrt{e^{2A(\rho)}-e^{2A^{\ast}}}}$
(124)

and its area reads

$\mathcal{A}=\int_{\rho_{0}}^{\rho_{+}}{\rm
d}\rho\mathcal{L},\quad\mathcal{L}=\frac{1}{\sqrt{1-e^{2(A^{\ast}-A(\rho))}}}.$
(125)

We are interested in the leading order of the area as a function of the
interval length. To proceed, we can consider the variation of the area with
respect to $l$:

$\frac{\delta\mathcal{A}}{\delta
l}=\mathcal{L}|_{\rho=\rho_{+}}\frac{\delta\rho_{+}}{\delta
l}-\mathcal{L}|_{\rho=\rho_{0}}\frac{\delta\rho_{0}}{\delta l}.$ (126)

At $\rho_{+}\rightarrow\infty$, $e^{A}\approx\cosh\rho$, the metric asymptotes
to an AdS3. We impose a conventional regularization $z=\varepsilon$ at $x=l$.
Then according to the coordinate transformation (14),
$\varepsilon\approx\frac{y}{\cosh\rho_{+}}$, $l=y\tanh\rho_{+}$, and this
leads to

$\rho_{+}=\log\frac{2l}{\varepsilon},\quad\frac{\delta\rho_{+}}{\delta
l}=\frac{1}{l}.$ (127)

At a finite $\rho_{0}\rightarrow\rho^{\ast}$, the time component of the metric
is ${\rm d}s^{2}=e^{2A^{*}}\frac{{\rm d}\tau^{2}}{y^{2}}$. We again impose the
conventional cutoff in an AdS2 Poincare coordinate, i.e., ${\rm
d}s^{2}=\frac{{\rm d}\tau^{2}}{\tilde{z}^{2}}$ with $\tilde{z}=\varepsilon$
karch2021universal . This will lead to $y(\rho_{0})\approx
e^{A^{\ast}}\varepsilon$. Then we have $y(\rho_{0})\approx
e^{A^{*}}\varepsilon$ and $y(\rho_{+})\approx l$ at the two ends. Integrating
over the differential equation (124),

$\log\frac{l}{e^{A^{\ast}}\varepsilon}=\int_{\rho_{0}}^{\rho_{+}}{\rm
d}\rho\frac{e^{A^{\ast}-A(\rho)}}{\sqrt{e^{2A(\rho)}-e^{2A^{\ast}}}}.$ (128)

Taking the derivative with respect to $l$, we obtain

$\frac{1}{l}=-\frac{e^{A^{\ast}-A(\rho_{0})}}{\sqrt{e^{2A(\rho_{0})}-e^{2A^{\ast}}}}\frac{\delta\rho_{0}}{\delta
l}.$ (129)

Consequently, the variation of $\rho_{0}$ is

$\frac{\delta\rho_{0}}{\delta
l}=-\frac{1}{l}\frac{\sqrt{e^{2A(\rho_{0})}-e^{2A^{\ast}}}}{e^{A^{\ast}-A(\rho_{0})}}\approx-\frac{1}{l}\sqrt{e^{2A(\rho_{0})}-e^{2A^{\ast}}}+\mathcal{O}(\rho_{0}-\rho^{\ast}).$
(130)

Finally, for the Lagrangian at $\rho_{+}\rightarrow\infty$, because
$e^{A}=\cosh\rho_{+}$, $\mathcal{L}|_{\rho=\rho_{+}}=1$. For Lagrangian at
$\rho_{0}$, we have

$\mathcal{L}_{\rho=\rho_{0}}=\frac{e^{A^{\ast}}}{\sqrt{e^{2A(\rho_{0})}-e^{2A^{\ast}}}}+\mathcal{O}(\rho_{0}-\rho^{\ast}).$
(131)

Now put everything together, the variation of the area with respect to $l$
reads

$\frac{\delta\mathcal{A}}{\delta l}=(1+e^{A^{\ast}})\frac{1}{l}.$ (132)

Then by integration, we get the leading contribution

$\mathcal{A}=(1+e^{A^{\ast}})\log l,$ (133)

which gives the entanglement entropy $S_{l}=\frac{c_{1}+c_{\rm
eff}}{6}\log{l}$ with $c_{{\rm eff}}=\frac{3e^{A^{\ast}}}{2G_{(3)}}$ and
$c_{1}=\frac{3}{2G_{(3)}}$ denoting the effective central charge induced by
the defect and the central charge of the original CFT (corresponding to the
unmeasured CFT in our case), respectively.

A few remarks follow:

  * •

Comparing with the result of weak measurements in (120), we can see the weak
measurement induced effective central charge is the same as the effective
central charge from the interface. Clearly, the two endpoints of the interval
in the measurement case are both located at $\tau=0$, leading to a factor
$2\times e^{A^{\ast}}$, and in the case of spatial interface CFT, one of the
end points is located at $x=0$ while the other is located at $x>0$, leading to
a factor of $1+e^{A^{\ast}}$.

  * •

While our calculation is valid for a general $\rho^{\ast}$, we expect that the
weak measurement will lead to a post-measurement metric with $\rho^{\ast}=0$
and $A^{\prime}(\rho)>0$ for $\rho>0$. The intuition is that weak measurements
gradually decrease the entanglement of the state. Then, according to reference
karch2023universality , there is a c-theorem stating that $c_{\rm eff}\leq
c_{1}$. It means that the effective central charge induced by weak
measurements is not greater than that of the unmeasured CFT. This is
consistent with the calculation in the CFT side in reference sun2023new .

###  5.2 Python’s lunch

In Sec. 4.5, we consider the bubble-inside-horizon phase and its geodesics.
Although it has the same behavior of entanglement entropy as the bubble-
outside-horizon phase, its geometry structure is different with the bubble-
outside-horizon phase, and will lead to much larger complexity.

Figure 21:  (a) The reversal invariance slice for the bubble-inside-horizon
phase. The blue circle indicates the interface brane and the dashed circle
indicates the horizon. (b) The metric component in the $x$ direction as a
function of a global coordinate $\tilde{r}$.

The time reversal invariant slice of the bubble-inside-horizon phase is shown
in figure 21 (a). We define $\tilde{r}$ as the global coordinate that
corresponds to the distance to center and increases monotonically. Then the
relation between the local coordinate and the global coordinate is

$r=\begin{cases}\tilde{r}\quad&\tilde{r}\leq r_{0},\\\
2r_{0}-\tilde{r}\quad&r_{0}<\tilde{r}\leq 2r_{0}-r_{H},\\\
\tilde{r}-2r_{0}+2r_{H}\quad&2r_{0}-r_{H}<\tilde{r}.\end{cases}$ (134)

Here $r$ is the local AdS coordinate in region I, and the local black hole
coordinate in region II. Notice that $r$ is also the metric component in the
$x$ direction, i.e., $\sqrt{g_{xx}}=r$. The function $r(\tilde{r})$ is
monotonic in the bubble-outside-horizon phase, but is not monotonic in the
bubble-inside-horizon phase. As shown in figure 21, there are two minimal
points at the center $r=0$ of the AdS spacetime and at the horizon $r=r_{H}$
of the black hole spacetime, respectively, and a maximal point at the
interface brane $r=r_{0}$. This realizes a Python’s Lunch geometry. The
details of the definition are given in appendix D.

The simple example of Python’s lunch geometry is shown in figure 34, with a
“minmaxmin” structure. For the complexity of this tensor network, beside the
contribution from total number of tensors, the post-selection will play an
important role and lead to exponentially large complexity. An intuitive
understanding is that, for the tensor network from maximum to minimum, we need
to apply post-selection to decrease the number of legs to realize the final
state. In this process, we must select one state from the whole Hilbert space,
with exponentially small probability. Crudely, to have one successful outcome,
we must repeat this experiment for exponential times, which leads to
exponentially large complexity.

Similarly, for the bubble-inside-horizon phase, the region from maximum to
minimum between $r=r_{0}$ and $r=r_{H}$ corresponds to the post-selection part
and leads to exponentially big complexity. While for the bubble-outside-
horizon phase, $r(\tilde{r})$ is a monotonic function so that no post-
selection is needed. Therefore, we conclude that, two irrelevant phases with
and without a horizon will have completely different behaviors in terms of
their complexity. Especially, the bubble-inside-horizon phase has the geometry
of Python’s lunch.

To summarize, in this paper, we consider the holographic dual of weak
measurements in conformal field theory. Because of the logarithmic scaling
entanglement entropy (characterized by a distinct effective central charge)
supported by post-measurement states from marginal measurements, the
holographic dual involves interface branes separating spacetimes dual to the
post-measurement state and the unmeasured CFT, respectively, generalizing the
holographic dual of the conformal boundary state. We also establish the
correspondence between the weak measurement and the spatial interface. In a
finite system, while the irrelevant measurements will not change the
entanglement scaling for the post-measurement state, it may create a Python’s
lunch.

We conclude this paper by mentioning a few open questions we would like to
explore in the future: (1) From an information perspective, the weak
measurements result in several interesting scenarios for the post-measurement
holographic dual. For the irrelevant case, the weak measurement can create a
Python’s lunch, greatly increasing the complexity for bulk reconstructions.
How is the reconstruction map related to the measurement operators? For the
marginal case, while the AdS spacetime dual to the unmeasured CFT is replaced
by a new different AdS metric, is the bulk information erased by the
measurements? (2) While we briefly discuss the thick-brane description, where
the bulk solution is continuous, of the weak measurements and find that it is
consistent with the thin-brane description, it is worthwhile to explore in
more detail about the universal features of weak measurements using the
general thick brane construction. Moreover, a top-down construction of weak
measurements is also of great interest because we can get a handle from both
sides of the theory. A classic example is the so-called Janus solution
bak2003a ; bak2007three . While such a solution is not directly related to
weak measurements via a spacetime rotation, it would be interesting to explore
other deformations or scenarios that can be related to measurements.

###### Acknowledgements.

We thank Zhuo-Yu Xian for fruitful discussions. We are grateful to Sanjit
Shashi for helpful communication regarding the thick brane construction. XS
acknowledges the support from Tsinghua Visiting Doctoral Students Foundation.
XS is also supported by the Lavin-Bernick Grant during the visit to Tulane
university. The work of SKJ is supported by a start-up fund at Tulane
university.

##  Appendix A Geodesic with spatial point defect

In this section, we will use the method in reference Anous_2022  to derive the
bipartite entanglement entropy for finite or infinite systems. With AdS/CFT
duality, we just need to calculate the length of geodesic in different AdS
geometry.

We first introduce a few coordinates which may be useful later. With (12), we
define

$\begin{split}X^{0}=&L\frac{r_{b}}{r_{H}}\cosh{\frac{r_{H}}{L}\theta},\qquad
X^{1}=L\sqrt{\frac{r_{b}^{2}}{r_{H}^{2}}-1}\cos{\frac{r_{H}\tau_{b}}{L^{2}}},\\\
X^{2}=&L\frac{r_{b}}{r_{H}}\sinh{\frac{r_{H}}{L}\theta},\qquad
X^{3}=L\sqrt{\frac{r_{b}^{2}}{r_{H}^{2}}-1}\sin{\frac{r_{H}\tau_{b}}{L^{2}}},\end{split}$
(135)

with corresponding metric

${\rm d}s^{2}=\left(\frac{r_{b}^{2}-r_{H}^{2}}{L^{2}}\right){\rm
d}\tau_{b}^{2}+\left(\frac{r_{b}^{2}-r_{H}^{2}}{L^{2}}\right)^{-1}{\rm
d}r_{b}^{2}+r_{b}^{2}{\rm d}\theta^{2}.$ (136)

The parametrization of (135) requires $\tau_{b}\sim\tau_{b}+\frac{2\pi
L^{2}}{r_{H}}$, which means there is a periodic boundary condition on the
$\tau_{b}$ direction.

Now we want to solve the junction condition (11a) and (11b) with coordinate
(13a). Consider a brane located on $\rho_{i}=\rho_{i}^{*}$ where $i=1,2$. The
first condition gives (16a), which leads to $y_{1}=y_{2},\ \tau_{1}=\tau_{2}$
and
$L_{1}\cosh{\left(\frac{\rho_{1}^{*}}{L_{1}}\right)}=L_{2}\cosh{\left(\frac{\rho_{2}^{*}}{L_{2}}\right)}$.
The second condition can be derived below. The extrinsic curvature can be
expressed as $K_{ab}=\frac{1}{2}n_{i}g^{ij}\partial_{j}g_{ab}$ where $g_{ij}$
is metric in the original space and $g_{ab}$ is metric on the hypersurface. We
define $\vec{n}=(n_{\rho}=1,0,0)$ with metric (13b), then
$K_{ab}=\frac{1}{2}g^{\rho\rho}\partial_{\rho}g_{ab}=\frac{1}{2}\partial_{\rho}g_{ab}=\frac{1}{2}\partial_{\rho}(\frac{L^{2}}{y^{2}}\cosh^{2}{\frac{\rho}{L}})=\frac{1}{L}\tanh{\frac{\rho}{L}}g_{ab}$.
Because for two regions 1 and 2 the directions of $\vec{\rho}$ are opposite
and here $g_{ab}\equiv h_{ab}$, then (11b) leads to (16b). Solving these two
conditions, we have (24). Finally, we can apply RT formula (25) to calculate
the entanglement entropy with the length of different geodesics, and we will
consider different cases in the following.

In reference Anous_2022 , the authors give a general geodesic solution for any
$\sigma_{1,2}$ with two regions ${1,2}$ in AdS space. Assuming the interface
brane is located at $\chi_{1,2}=\psi_{1,2}$ and two end points are located on
$x_{1}=-\sigma_{1}$ and $x_{2}=\sigma_{2}$, the length of the geodesic is

$d(\sigma_{1},\sigma_{2})=L_{1}\log{\left[\frac{2r}{\varepsilon_{1}}\tan{\left(\frac{\varphi}{2}\right)}\right]}+L_{2}\log{\left[\frac{2R}{\varepsilon_{2}}\tan{\left(\frac{\theta}{2}\right)}\right]},$
(137)

where

$\begin{split}r=\frac{1}{2}\csc{\left(\frac{\varphi}{2}\right)}\sec{\left(\frac{\psi_{1}+\psi_{2}}{2}\right)}\left[\sigma_{2}\cos{\left(\frac{\theta}{2}\right)}-\sigma_{1}\cos{\left(\frac{\theta}{2}+\varphi\right)}\right],\\\
R=\frac{1}{2}\csc{\left(\frac{\theta}{2}\right)}\sec{\left(\frac{\psi_{1}+\psi_{2}}{2}\right)}\left[\sigma_{1}\cos{\left(\frac{\varphi}{2}\right)}-\sigma_{2}\cos{\left(\frac{\varphi}{2}+\theta\right)}\right],\end{split}$
(138)

$\varepsilon_{1,2}$ are UV-cutoff, $\varphi=\pi+\psi_{1}+\psi_{2}-\theta$ and
$\theta$ is expressed as

$\begin{split}&\cos{\theta}=\frac{\cos{\left(\frac{\psi_{1}-\psi_{2}}{2}\right)}}{\sigma_{1}^{2}+\sigma_{2}^{2}+2\sigma_{1}\sigma_{2}\cos{(\psi_{1}+\psi_{2})}}\left\\{-\sigma_{2}^{2}\cos{\left(\frac{\psi_{1}-\psi_{2}}{2}\right)}+\sigma_{1}^{2}\cos{\left(\frac{\psi_{1}+3\psi_{2}}{2}\right)}\right.\\\
&+2\sigma_{1}\sigma_{2}\sin{\psi_{2}}\sin{\left(\frac{\psi_{1}+\psi_{2}}{2}\right)}-\left[\sigma_{1}\sin{\left(\frac{\psi_{1}+3\psi_{2}}{2}\right)}-\sigma_{2}\sin{\left(\frac{\psi_{1}-\psi_{2}}{2}\right)}\right]\times\\\
&\left.\sqrt{\left[\frac{(\sigma_{1}+\sigma_{2})^{2}-(\sigma_{1}-\sigma_{2})^{2}\cos(\psi_{1}-\psi_{2})+4\sigma_{1}\sigma_{2}\cos{(\psi_{1}+\psi_{2})}}{2\cos^{2}{\frac{\psi_{1}-\psi_{2}}{2}}}\right]}\right\\}.\end{split}$
(139)

Generally, the geodesic equation of equal time slice is

$\left(x-\frac{\sigma_{1}+\sigma_{2}}{2}\right)^{2}+z^{2}=\left(\frac{\sigma_{1}-\sigma_{2}}{2}\right)^{2},$
(140)

and the length of the geodesic is (26), which corresponds to two end points
$(\tau,x,z)$ and $(\tau^{\prime},x^{\prime},z^{\prime})$ in (140).

###  A.1 Defect/CFT geodesic in an infinite system

####  A.1.1 General solution of defect/CFT geodesic in an infinite system

The first case is a subsystem with two end points on the defect and the
boundary, respectively, as shown in figure 9. Here, we label the AdS dual of
two half-plane CFT as region 1, and the dual of the defect as region 2. To
show the geodesic explicitly, we extend the width of the defect, which is
approximately zero in CFT. Now we want to calculate the entanglement entropy
of the subsystem using the RT formula. The orange curve constituted by two
arcs is a smooth geodesic which connects two end points.

With (137), we can take the limit $\sigma_{2}\rightarrow 0$ and
$\sigma_{1}=l$. Then (139) can be simplified as
$\cos{\theta}=-\cos{(\psi_{2}-\psi_{1})}$ for $\psi_{2}>\psi_{1}$, otherwise
$\cos{\theta}=-1$. Actually, $\cos{\theta}=-1$ with $\psi_{2}<\psi_{1}$ is not
relevant because for $\theta=\pi$, we have $r=0$ in (138), which means the
whole geodesic is in region $1$. With (24), we have

$\begin{split}\sin{\psi_{2}}-\sin{\psi_{1}}=\frac{L_{2}}{2T}\left(T^{2}+\frac{1}{L_{2}^{2}}-\frac{1}{L_{1}^{2}}\right)-\frac{L_{1}}{2T}\left(T^{2}+\frac{1}{L_{1}^{2}}-\frac{1}{L_{2}^{2}}\right)=\frac{(\frac{1}{L_{1}}+\frac{1}{L_{2}})^{2}-T^{2}}{2T}(L_{1}-L_{2}).\end{split}$
(141)

Because $T<T_{\rm max}=\frac{1}{L_{1}}+\frac{1}{L_{2}}$, $\psi_{2}<\psi_{1}$
corresponds to $L_{1}<L_{2}$, which means $c_{1}<c_{2}$. Therefore, the
induced effective central charge of entanglement entropy by defect cannot be
larger than the original CFT without defect, which is consistent with the
results we get in the main text.

Now let’s focus on the nontrivial case with $\psi_{2}>\psi_{1}$ and
$L_{2}<L_{1}$. With $\theta=\pi-\psi_{2}+\psi_{1},\ \varphi=2\psi_{2}$,
(compared with reference we exchange label 1 and 2), we have

$R=\frac{\cos{\psi_{1}\sigma_{1}}}{\cos{\psi_{1}}+\cos{\psi_{2}}},\quad
r=\frac{\sin{(\psi_{2}-\psi_{1})}\sigma_{1}}{2\sin{\psi_{2}}(\cos{\psi_{1}}+\cos{\psi_{2}})}.$
(142)

So the geodesic length with the same UV-cutoff for two regions is

$d_{0,\sigma_{1}}=(L_{1}+L_{2})\log{\frac{\sigma_{1}}{\varepsilon}}+d_{\rm
sub},$ (143)

where $d_{\rm sub}$ corresponding to the sub-leading term of entanglement
entropy induced by the brane in AdS space.

$\begin{split}d_{\rm
sub}=&L_{2}\log{\left(\frac{\sin{(\psi_{2}-\psi_{1})}}{\cos{\psi_{2}}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}+L_{1}\log{\left(\frac{2\cos{\psi_{1}}}{\tan{\frac{\psi_{2}-\psi_{1}}{2}}(\cos{\psi_{1}}+\cos{\psi_{2}})}\right)}\\\
=&L_{2}\log{\left(\frac{\sin{\frac{\psi_{2}-\psi_{1}}{2}}}{\cos{\psi_{2}}\cos{\frac{\psi_{1}+\psi_{2}}{2}}}\right)}+L_{1}\log{\left(\frac{\cos{\psi_{1}}}{\sin{\frac{\psi_{2}-\psi_{1}}{2}}\cos{\frac{\psi_{1}+\psi_{2}}{2}}}\right)}.\end{split}$
(144)

We can find that the results of (143) and (144) are consistent with the
results of (36) and (37).

Therefore, with (25) we have

$S_{0,\sigma_{1}}=\frac{c_{1}+c_{2}}{6}\log{\frac{l}{\varepsilon}}+S_{\rm
sub},$ (145)

where $S_{\rm sub}=\frac{d_{\rm sub}}{4G_{(3)}}$. Here, the prefactor of the
$\log$-term is $\frac{c_{1}+c_{2}}{6}$ because the locations of each end
points are in region $1$ and $2$, respectively.

####  A.1.2 Large size limit for the original region

Figure 22:  AdS dual of CFT with interface defect for infinite system and the
defect/CFT geodesic.

In the following, we consider the limit case $\sigma_{1}\rightarrow\infty$
with fixed finite $\sigma_{2}$ with figure 22. We denote the end points of the
geodesic to be Q in region 1 and P in region 2, respectively. In the limit
$\sigma_{1}\rightarrow\infty$, the figure does not show Q explicitly. To have
an intuition about the limit, we can calculate the limit of (139).

$\begin{split}&\cos{\theta}=\left\\{\begin{aligned}
\cos{(\psi_{1}+\psi_{2})}+\frac{4\sigma_{2}\cos{\psi_{1}}(\cos{\psi_{2}}-\cos{\psi_{1}})\cos^{2}{\frac{\psi_{1}+\psi_{2}}{2}}}{-1+\cos{(\psi_{1}-\psi_{2})}}\sigma_{1}^{-1},\quad&\psi_{1}>\psi_{2},\\\
\cos{(2\psi_{2})}+4\sigma_{2}\cos^{2}{\psi_{2}}\cos{\frac{\psi_{1}+\psi_{2}}{2}}\csc{\frac{\psi_{1}-\psi_{2}}{2}}\sin{\psi_{2}}\sigma_{1}^{-1},\quad&\psi_{1}<\psi_{2}.\end{aligned}\right.\end{split}$
(146)

Therefore, the relation between $\psi_{1}$ and $\psi_{2}$ plays an important
role for the limit case. In the following, we will discuss this property in
detail.

We first consider $\psi_{2}<\psi_{1}$ case shown in figure 22 (a). The key is
that, for $\sigma_{1}\rightarrow\infty$ we have ${\rm
SA}\mathrel{/\mskip-2.5mu/}{\rm CFT_{left}}$, which can be proven that it only
exists for $\psi_{2}<\psi_{1}$ as follows. For $\triangle{\rm ASO}$, because
$\sigma_{2}>0$, we have $\angle{\rm SOA}=\frac{\pi}{2}-\psi_{2}>\angle{\rm
SPA}=\frac{\pi-(\psi_{1}+\psi_{2})}{2}$, which requires $\psi_{2}<\psi_{1}$.
Then we can the geodesic. With the law of sines, because $\angle{\rm
ASO}=\pi-(\psi_{1}+\psi_{2})-(\frac{\pi}{2}-\psi_{2})=\frac{\pi}{2}-\psi_{1}$,
we have

$\frac{R-\sigma_{2}}{\sin{\left(\frac{\pi}{2}-\psi_{1}\right)}}=\frac{R}{\sin{\left(\frac{\pi}{2}-\psi_{2}\right)}}.$
(147)

The solution is
$R=\frac{\cos{\psi_{2}}\sigma_{2}}{\cos{\psi_{2}}-\cos{\psi_{1}}}$. It
satisfies $R>\sigma_{2}$ because only one of $\psi_{1}$ and $\psi_{2}$ can be
negative, which means $\cos{\psi_{2}}-\cos{\psi_{1}}>0$ with
$\sin{\psi_{1}}+\sin{\psi_{2}}>0$. Then we can get $|{\rm SO}|$ in
$\triangle{\rm SAO}$. With the law of sines, we have $|{\rm
SO}|=\frac{R\sin{(\psi_{1}+\psi_{2})}}{\cos{\psi_{2}}}$, which is a constant
for fixed $\sigma_{2}$. Therefore, for the limit case
$\sigma_{1}\rightarrow\infty$, the geodesic length in region 2 is a constant
and contributes a sub-leading term.

Now we can calculate the entanglement entropy for $\psi_{2}<\psi_{1}$. For the
left part of the geodesic ${\rm QS}$ with end points ${\rm Q}$ and ${\rm S}$
we have ${\rm Q}=(-\sigma_{1},\varepsilon)$ and ${\rm S}=(|{\rm
OS}|\sin{\psi_{1}},|{\rm OS}|\cos{\psi_{1}})$, which gives

$d_{\rm QS}=L_{1}\cosh^{-1}{\frac{(|{\rm
OS}|\sin{\psi_{1}}+\sigma_{1})^{2}+(|{\rm
OS}|\cos{\psi_{1}})^{2}+\varepsilon^{2}}{2|{\rm
OS}|\cos{\psi_{1}}\varepsilon}}.$ (148a) Besides, in the large $\sigma_{1}$
limit we keep the leading term $\sigma_{1}^{2}$ in the numerator, which gives
$d_{\rm QS}\approx
L_{1}\log{\frac{\sigma_{1}^{2}}{\varepsilon[\sigma_{2}\sin{(\psi_{1}+\psi_{2})}\cos{\psi_{1}}/(\cos{\psi_{2}}-\cos{\psi_{2}})]}}=L_{1}\log{\frac{\sigma_{1}^{2}/\sigma_{2}}{\varepsilon}}+d_{\rm
QS,sub},$ (148b) where $d_{\rm
QS,sub}=L_{1}\log{\frac{\cos{\psi_{2}}-\cos{\psi_{2}}}{\sin{(\psi_{1}+\psi_{2})}\cos{\psi_{1}}}}$.
For the right part of the geodesic ${\rm SP}$ we have ${\rm S}=(-|{\rm
OS}|\sin{\psi_{2}},|{\rm OS}|\cos{\psi_{2}})$ and ${\rm
P}=(\sigma_{2},\varepsilon)$, which gives $d_{\rm
SP}=L_{2}\cosh^{-1}{\frac{(|{\rm OS}|\sin{\psi_{2}}+\sigma_{2})^{2}+(|{\rm
OS}|\cos{\psi_{2}})^{2}+\varepsilon^{2}}{2|{\rm
OS}|\cos{\psi_{2}}\varepsilon}}\approx
L_{2}\log{\frac{\sigma_{2}}{\varepsilon}}+d_{\rm SP,sub}.$ (148c)

where $d_{\rm
SP,sub}=L_{2}\log{\frac{\sin^{2}{(\psi_{1}+\psi_{2})}+(\cos{\psi_{2}}-\cos{\psi_{1}})^{2}+2\sin{\psi_{2}}\sin{(\psi_{1}+\psi_{2})}(\cos{\psi_{2}}-\cos{\psi_{1}})}{\cos{\psi_{2}}\sin{(\psi_{1}+\psi_{2})}(\cos{\psi_{2}}-\cos{\psi_{1}})}}$.
Therefore, the total length of the geodesic reads

$d=L_{1}\log{\frac{\sigma_{1}^{2}/\sigma_{2}}{\varepsilon}}+L_{2}\log{\frac{\sigma_{2}}{\varepsilon}}+d_{\rm
sub},$ (149)

where $d_{\rm sub}=d_{\rm SP,sub}+d_{\rm QS,sub}$. Therefore, for
$\sigma_{1}\gg 1$ with a fixed $\sigma_{2}$, the prefactor of
$\log{\sigma_{1}}$ is $2L_{1}$, which leads to the prefactor $\frac{c_{1}}{3}$
for the entanglement entropy.

Now we consider $\psi_{2}>\psi_{1}$ case shown in figure 22 (b). For
$\sigma_{1}\rightarrow\infty$ we expect $|{\rm OS}|\rightarrow\infty$, which
means $|{\rm AO}|\approx|{\rm AP}|$. Then $\triangle{\rm SAO}$ is an isosceles
triangle with $\angle{\rm ASP}=\angle{\rm APS}=\frac{\pi}{2}-\psi_{2}$. To
solve the geodesic, let’s look at $\triangle{\rm SBO}$. With the law of sines,
we have

$\frac{\sigma_{1}-r}{\sin{(\frac{\pi}{2}-\psi_{2})}}=\frac{r}{\sin{(\frac{\pi}{2}+\psi_{1}})}=\frac{|{\rm
SO}|}{\sin{(\psi_{2}-\psi_{1})}}.$ (150)

The solutions are
$r=\frac{\cos{\psi_{1}}\sigma_{1}}{\cos{\psi_{2}}+\cos{\psi_{1}}}\rightarrow\infty$
and $|\rm
SO|=\frac{\sin{(\psi_{2}-\psi_{1})}\sigma_{1}}{\cos{\psi_{2}}+\cos{\psi_{1}}}$.

With the results above, we just need to plug new $|{\rm OS}|$ into (148a) and
(148c). Finally, the total geodesic length is

$d=L_{1}\log{\frac{\sigma_{1}}{\varepsilon}}+L_{2}\log{\frac{\sigma_{1}}{\varepsilon}}+d_{\rm
sub},$ (151)

where

$d_{\rm
sub}=L_{1}\log{\frac{A^{2}+1+2A\sin{\psi_{1}}}{A\cos{\psi_{1}}}}+L_{2}\log{\frac{A}{\cos{\psi_{2}}}}$
(152)

and $A=\frac{\sin{(\psi_{2}-\psi_{1})}}{\cos{\psi_{2}}+\cos{\psi_{1}}}$.
Therefore, in this case, the prefactor of $\log{\sigma_{1}}$ is $L_{1}+L_{2}$,
which leads to the prefactor $\frac{c_{1}+c_{2}}{6}$ for the entanglement
entropy.

Here is an additional remark about the results above. For two cases
$L_{1}<L_{2}$ and $L_{1}>L_{2}$, the prefactor of $\log{\sigma_{1}}$ are
different at the large $\sigma_{1}$ limit. The prefactor of $\log$-term in
entanglement entropy is always the smaller one: for $c_{1}<c_{2}$ we have
$\frac{c_{1}}{6}+\frac{c_{1}}{6}$, but for $c_{1}>c_{2}$ we have
$\frac{c_{1}}{6}+\frac{c_{2}}{6}$. In summary, the leading term of the
entanglement entropy at $\sigma_{1}\rightarrow\infty$ reads

$S_{-\sigma_{1},\sigma_{2}}=\left(\frac{c_{1}}{6}+\frac{1}{6}\min(c_{1},c_{2})\right)\log\sigma_{1}.$
(153)

This property also exists in more complicated cases, which will be considered
later.

###  A.2 CFT/CFT geodesic in an infinite system

####  A.2.1 General case for CFT/CFT geodesic in an infinite system

For a 1d quantum system with a defect located at $x=0$, we can choose a
subsystem with two end points located on different sides. In the following, we
will construct the geodesic with several arcs and use (26) to calculate its
length. The diagram is shown in figure 23.

Figure 23:  AdS dual of CFT with interface defect for infinite system and the
geodesic. (a) General CFT/CFT geodesic for $\psi_{2}>\psi_{1}$. (b) Geodesic
for limit case $\sigma_{l}=0$ with $\psi_{2}<\psi_{1}$.

Here we denote the location of two end points
$(x=-\sigma_{l},z=\varepsilon,\tau=0)$ and
$(x^{\prime}=\sigma_{r},z^{\prime}=\varepsilon,\tau^{\prime}=0)$, and the