11institutetext: University of Maryland, College Park, MD, 20742,
USA22institutetext: Department of Physics, Brandeis University, Waltham,
Massachusetts 02453, USA33institutetext: Department of Physics and Engineering
Physics, Tulane University, New Orleans, Louisiana, 70118, USA

# Holographic measurement in CFT thermofield doubles

Stefano Antonini 2 , Brianna Grado-White 3 , Shao-Kai Jian 2 , Brian Swingle
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We extend the results of arXiv:2209.12903 by studying local projective
measurements performed on subregions of two copies of a CFT2 in the
thermofield double state and investigating their consequences on the bulk
double-sided black hole holographic dual. We focus on CFTs defined on an
infinite line and consider measurements of both finite and semi-infinite
subregions. In the former case, the connectivity of the bulk spacetime is
preserved after the measurement. In the latter case, the measurement of two
semi-infinite intervals in one CFT or of one semi-infinite interval in each
CFT can destroy the Einstein-Rosen bridge and disconnect the bulk dual
spacetime. In particular, we find that a transition between a connected and
disconnected phase occurs depending on the relative size of the measured and
unmeasured subregions and on the specific Cardy state the measured subregions
are projected on. We identify this phase transition as an
entangled/disentangled phase transition of the dual CFT system by computing
the post-measurement holographic entanglement entropy between the two CFTs. We
also find that bulk information encoded in one CFT in the absence of
measurement can sometimes be reconstructed from the other CFT when a
measurement is performed, or can be erased by the measurement. Finally, we
show that a purely CFT calculation of the Renyi entropy using the replica
trick yields results compatible with those obtained in our bulk analysis.

##  1 Introduction

The holographic principle—and its most concrete realization, the Anti de-
Sitter/Conformal Field Theory (AdS/CFT) correspondence Maldacena:1997re ;
Witten:1998qj ; Gubser:1998bc ; Aharony:1999ti —has suggested that
gravitational spacetime is an emergent property, and in particular, that the
bulk geometry is dictated by the entanglement structure of a dual, purely
quantum system defined in one less dimension Ryu2006a ; Ryu2006b ;
Hubeny:2007xt ; Swingle:2009bg ; VanRaamsdonk:2010pw ; Maldacena:2013xja ;
Engelhardt:2014gca ; Dong:2016eik ; Harlow:2016vwg . As such, it is expected
that operations in the quantum boundary theory that modify or destroy the
entanglement structure will consequently modify or destroy any bulk geometric
structure that existed pre-operation. Recently, such phenomena have been
studied in the case of local projective measurements performed on subregions
of the boundary theory.

In particular, numasawa2016epr  initiated this work, showing that large
portions of the bulk dual spacetime could indeed be destroyed (with the
deleted region bounded by an end-of-the-world (ETW) brane) by postselecting a
boundary subregion onto a specific class of states. This line of inquiry was
furthered in Antonini:2022sfm , which gave a detailed account of this boundary
measurement in vacuum AdS3/CFT2 and in related tensor network models, and
further showed how measurement could modify the bulk geometry and holographic
dictionary via quantum teleportation. Of particular interest here,
Antonini:2022sfm  showed that when measuring two disconnected boundary
subregions, varying the measurement parameters could induce a phase transition
corresponding to whether the remaining, unmeasured boundary regions were
connected through the bulk time reflection symmetric slice or not. This bulk
phase transition corresponded to an entangled/disentangled phase transition in
the dual boundary theory.

Here, we extend the results and techniques of Antonini:2022sfm  by studying
post-selection in AdS3/CFT2 starting in the thermofield double (TFD) state of
a CFT on a line, dual to an eternal, two-sided BTZ black hole in the bulk
Maldacena:2001kr . This generalization to measurements in the TFD is of
particular interest for its potential implications for the reconstruction of
operators behind the black hole horizon (as in kourkoulou2017pure ) and for
holographic cosmologies Cooper:2018cmb ; Antonini:2019qkt ; Antonini:2021xar ,
see also Milekhin:2022bzx  for related work. A similar endeavor was undertaken
in a lower dimensional model in Antonini:2022lmg , where postselection of the
TFD state for two copies of the Sachdev-Ye-Kitaev (SYK) model dual to Jackiw-
Teitelboim (JT) gravity was considered. There, the focus was on understanding
the conditions under which the information contained in the entanglement wedge
of one side of the TFD becomes accessible by the other side after measurement.
For the $2+1$ dimensional case studied here, we also study this “bulk
teleportation” between the two sides (in the sense of Antonini:2022sfm ), and
additionally focus on how bulk connectivity is modified by measurement, aiming
to characterize when measurements destroy the Einstein-Rosen bridge stretching
between the two asymptotic regions.

We start by constructing the gravitational dual of the post-selected TFD
states. To describe a local projective measurement onto a product state of a
subregion, we start with the Euclidean path integral that prepares the
thermofield double state, and insert a slit in the time reflection symmetric
slice corresponding to the measured subregion rajabpour2015post ;
rajabpour2016entanglement . Note that this local projective measurement will
project the measured subregion onto a Cardy state cardy1989boundary ;
miyaji2014boundary . Following numasawa2016epr ; Antonini:2022sfm , we can
build the bulk duals using the AdS/BCFT proposal takayanagi2011holographic ;
fujita2011aspects . The measurement, and thus the insertion of the slit in the
path integral, will correspond to the insertion of an ETW brane in the bulk
anchored to the slit’s boundary. Note that the CFT stress-energy tensor is
divergent at the endpoints of the slit numasawa2016epr ; Antonini:2022sfm ,
leading to a singular dual metric. To avoid directly dealing with this
complication, and again following numasawa2016epr ; Antonini:2022sfm , we
perform a series of conformal transformations to map to a non-singular CFT
configuration, and thus a regular bulk.

Here, we will map our initial infinite cylinder (corresponding to the
Euclidean path integral preparing the TFD state for two copies of a CFT on a
line) with various slit configurations (corresponding to measurements of
various subregions of the two CFTs) to a finite-length cylinder. The bulk dual
of this finite cylinder can correspond to either a portion of a Euclidean BTZ
black hole (cut off by a single brane connecting the two boundaries of the
finite cylinder) or to a portion of Euclidean thermal AdS (cut off by two
branes, each one anchored to one of the two boundaries of the finite
cylinder). We then study this Hawking-Page phase transition between these two
possible geometries for measurements performed on finite and semi-infinite
intervals (in planar coordinates) on one or both CFTs in the TFD state. For
the different measurement configurations, the phase transition will have
different physical interpretations. Of particular interest are the geometrical
properties of the bulk time reflection symmetric slice. To understand this,
note that the time reflection symmetric slice in the Euclidean geometry will
also be the time reversal symmetric slice in the corresponding Lorentzian
geometry, giving the initial conditions for subsequent real time evolution.
Therefore, the presence or absence of an ETW brane on the Euclidean time
reflection symmetric slice will correspond to the presence or absence of an
ETW brane in the Lorentzian wormhole geometry. More significantly, if the bulk
Euclidean time reflection symmetric slice is connected between the two CFTs,
the two asymptotic AdS boundaries where the two CFTs live will also be
connected through an Einstein-Rosen bridge in the corresponding Lorentzian
geometry.

In section 2.1, we consider projective measurements performed on a finite
interval on one side of the TFD, followed by an additional Euclidean time
evolution performed with the full CFT Hamiltonian. By choosing a large enough
projected region, small enough euclidean time evolution, or small enough (or
negative) tension of the ETW brane (corresponding to a small or negative
boundary entropy affleck1991universal  of the specific Cardy state
cardy1989boundary  that we project onto), we can tune the system into the BTZ
phase, such that there is an ETW brane on the time reflection symmetric slice.
In the thermal AdS phase, conversely, the time reflection symmetric slice does
not contain any brane. In both phases, the two CFTs remain connected through
the time reflection symmetric slice, suggesting that they remain significantly
entangled after the measurement. We confirm this expectation by computing the
holographic entanglement entropy for subregions of one CFT via the RT formula
and noting that it remains non-zero for non-empty subregions in both phases.

A more drastic change to the bulk geometry occurs when we turn to measuring
semi-infinite intervals in either one (Section 2.2) or both CFTs (Section
2.3). In these cases, we also find two different phases and show that when the
measured region is sufficiently large or the tension of the ETW brane is
sufficiently small or negative, the time reflection symmetric slice in the
Euclidean geometry can become disconnected. This suggests that the measurement
destroys enough entanglement between the two CFTs to disconnect the Lorentzian
wormhole. This is again verified by computing the holographic entanglement
entropy between the two sides using the RT formula: in the disconnected phase,
the mutual information between the two sides vanishes.

We also find that some bulk information that would have been encoded in e.g.
the left side if no measurement were performed is accessible from the right
side in the presence of measurement. In particular, we consider insertions of
heavy operators in the Euclidean past in the path integral preparing the state
of interest. We then study, both in the absence of measurement and when
measurements of semi-infinite intervals are performed, whether the bulk
effects of such operator insertions can be reconstructed from the left or the
right CFT. If these effects can be reconstructed from the left in the absence
of measurement and from the right in the presence of measurement, we conclude
that “bulk teleportation” in the sense of Antonini:2022sfm  is taking place
between the two sides. If they can be reconstructed from the same side with
and without measurement, no bulk teleportation between the two sides is
happening. Additionally, we find that there is a third possibility, namely
that information associated with operator insertions in the Euclidean past can
be erased by the measurement, at least in the purely geometrical approximation
we work within. Our results indicate that both bulk teleportation and
information erasure take place in either the disconnected or connected phases,
with erasure occurring predominantly in the former, and teleportation
occurring predominantly in the latter.

We then confirm the phase transitions between a connected/disconnected bulk
via a boundary computation, finding the corresponding entangled/disentangled
phase transitions in the microscopic CFT description. Following the framework
of rajabpour2016entanglement , in Section 3 we calculate the Reyni entropy
between the two CFTs after the measurement is performed. By focusing on the
limits where the measured region is either very large or very small111These
limits allow us to answer the question at hand analytically thanks to some
technical simplifications arising in the corresponding formulas. and up to
subleading corrections in $N$, we again find that, upon measuring semi-
infinite intervals in either one or both CFTs, the entanglement entropy
between the two sides can be made zero for a sufficiently small unmeasured
region.

The rest of this paper is organized as follows. In Section 2, we define the
TFD state and construct the bulk duals of various boundary measurements.
Section 2.1 outlines the general strategy of constructing the bulk spacetime,
and focuses on measuring finite boundary intervals, with an additional
Euclidean time evolution performed after the measurement. We then characterize
a Hawking-Page transition corresponding to whether or not the dual Lorentzian
geometry has an ETW brane, and calculate the entanglement entropy in both
phases via the RT formula. In section 2.2, we focus on measurements of two
semi-infinite intervals on one side, and again construct the bulk duals. We
then characterize a Hawking-Page transition corresponding to whether or not
the dual Lorentzian geometry is connected, calculate the entanglement entropy
in both phases via the RT formula, and study bulk teleportation and
information erasure due to the measurement. In section 2.3 we repeat the
previous analysis for the case of semi-infinite intervals measured in both
intervals, and find analogous results. Section 3 shows the accordance between
the bulk holographic entanglement entropy calculation and a Renyi entropy
calculation in the dual CFT system for the case of semi-infinite intervals.
Finally, we end with a summary and discussion of future directions in section
4. Technical details, including the explicit construction of all conformal
transformations employed in this paper, can be found in Appendices A, B, C,
and D.

##  2 Holographic description of measurement in CFT thermofield doubles

In this section, we consider projective measurements in a thermofield double
state (TFD) of a 1+1 dimensional CFT on a line, and investigate their
consequences in the dual spacetime. Consider a CFT with Hamiltonian $H$ and a
complete set of eigenstates $|n\rangle$,

$\displaystyle H|n\rangle=E_{n}|n\rangle.$ (1)

The TFD is a purification of the thermal density matrix $e^{-\beta H}$, where
$\beta$ is the inverse temperature. It is defined as

$\displaystyle|TFD\rangle=\frac{1}{\sqrt{Z}}\sum_{n}e^{-\beta
E_{n}/2}|n\rangle_{L}|n\rangle_{R},\quad Z=\sum_{n}e^{-\beta E_{n}}$ (2)

in the doubled Hilbert space $\mathcal{H}_{L}\times\mathcal{H}_{R}$, where we
refer to the two subfactors as the left and right CFT (or side), respectively.
It is easy to check that tracing out either side yields the thermal density
matrix, e.g. $e^{-\beta H}=\text{Tr}_{R}\left(|TFD\rangle\langle TFD|\right)$,
as expected. The TFD state can be prepared via Euclidean path integral by
slicing open the path integral over an infinite cylinder along the non-compact
direction. In particular, with coordinates $x\in(-\infty,\infty)$ and
$y\in[0,\beta]$ (where $y=0,\beta$ are identified), the compact $y$ is taken
to be the Euclidean time, and $x$ the spatial coordinate along the line. After
slicing open the path integral along the $x$-axis, the open cut at $y=0$ will
define a state for the left side, and the open cut at $y=\beta/2$ will define
a state for the right side. See Fig. 1 for a depiction. The bulk geometry dual
to the TFD state is an asymptotically AdS eternal black hole, where the left
and right CFTs can be thought of as living on the left and right asymptotic
boundaries, with the spacetime between them connected by an Einstein-Rosen
bridge Maldacena:2001kr .

Figure 1: Eulidean state preparation for the TFD state at inverse temperature
$\beta$. $x\in(-\infty,\infty)$ denotes the spatial coordinate of the CFT,
while the periodic $y\in[0,\beta]$ is the Euclidean time. The TFD state is
obtained by slicing open the path integral along the Euclidean time reflection
symmetric slice $y=0,\beta/2$. The left and right CFTs are then located at the
$y=0$ and $y=\beta/2$ lines, respectively.

Here, we will consider the postselected state after a partial projective
measurement on a subregion $A$ of one or both the CFTs is performed. In
particular, the postselected state will be a Cardy state cardy2004boundary .
In a lattice-discretized version of the CFT, the measurement operator consists
of a tensor product of local projective measurements (LPM),
$M=\left(\bigotimes_{x\in
A}|\psi_{x}\rangle\langle\psi_{x}|\right)\otimes\left(\bigotimes_{x\in
A^{c}}\mathds{1}_{x}\right)$, where $|\psi_{x}\rangle$ is the state each
lattice site is projected onto, $\mathds{1}$ denotes the identity operator,
and $A^{c}$ is the unmeasured region complementary to $A$. The resulting post-
measurement state has zero spatial entanglement in region $A$, a property
typical of Cardy states miyaji2014boundary ; see also numasawa2016epr ;
Antonini:2022sfm .

On the CFT side, this measurement can be described by removing a slit in the
Euclidean path integral that prepares the thermofield double state,
corresponding to the measured subregion $A$ and having infinitesimal height in
Euclidean time. Similar CFT measurements were considered in rajabpour2015post
; rajabpour2016entanglement . Analytical formulas for the entanglement entropy
in setups involving one and two slits inserted in the Euclidean path integral
preparing a CFT ground state have been obtained and numerically verified for
simple CFTs rajabpour2015post ; rajabpour2016entanglement .

In this section, we will focus on the spacetime dual of the post-measurement
state $M|TFD\rangle$. The holographic duals of post-selected states were
explored in numasawa2016epr ; Antonini:2022sfm . The bulk dual of the measured
state can be built using the AdS/BCFT proposal takayanagi2011holographic ;
fujita2011aspects : in particular, the boundary measurement implies the
presence of an end-of-the-world (ETW) brane in the bulk. Note that, in the
absence of a proper regularization procedure, the post-measurement CFT state
is singular, with a divergent stress-energy tensor at the endpoints of the
slit. This translates into a singular bulk spacetime metric and brane
configuration. Instead of regularizing,222This could be done, for example, by
giving the slits a finite height via Euclidean time evolution of the post-
measurement state. Though it is not completely clear how to implement this,
such an approach would yield a well-defined post-measurement state, which is a
necessary pre-requisite to study post-measurement Lorentzian time evolution in
the CFT, and consequently the bulk dual. we follow numasawa2016epr ;
Antonini:2022sfm  and perform a series of conformal transformations
(necessarily containing a branch cut), which yield a regular CFT setup. The
regular bulk dual description can then be straightforwardly constructed by
means of the AdS/BCFT proposal. In particular, we will map the initial
infinite cylinder with slits to a cylinder of finite height, such that the
slits are mapped to the two boundaries of the finite cylinder, similar to
Antonini:2022sfm . Depending on the parameters of the measurement, the
resulting Euclidean bulk dual spacetime is given by either the BTZ black hole
or thermal AdS3 cut off by ETW branes anchored at the boundaries of the finite
cylinder.

We will use the slit prescription described above to study measurements
performed on both finite and semi-infinite intervals. For the former case, in
section 2.1 we look at projective measurements for a finite interval in one
side of the TFD state as a warm-up. While a similar measurement has been
investigated in ref. numasawa2016epr , we additionally consider a Euclidean
time evolution using the Hamiltonian of the whole CFT following the projective
measurement.333This can be regarded as another form of regularization of the
post-measurement setup yielding a regular, well-defined post-measurement state
numasawa2016epr . We will see that this imaginary time evolution leads to a
Hawking-Page phase transition: for certain measurement parameters, the bulk
time reflection symmetric slice will contain an ETW brane. For the latter
(semi-infinite interval) case, in Sections 2.2 and 2.3 we will be particularly
interested in understanding how the Lorentzian Einstein-Rosen bridge, whose
existence is associated with the entanglement between the two sides of the
thermofield double, is affected by the measurement. For measurements performed
on either one or both sides of the TFD, we again identify a Hawking-Page phase
transition and show that in this case the Einstein-Rosen bridge is destroyed
whenever we are above a phase boundary. This boundary is determined by the
size of the measured region, the temperature of the TFD state, and the brane
tension. Explicitly, this destruction is signaled by the vanishing of the
mutual information between the remaining, unmeasured regions on the two sides.

We further characterize how information associated with insertions of heavy
operators in the Euclidean past and which would be encoded in one of the two
CFTs in the absence of measurement can become accessible from the other CFT or
erased by the measurement. We call the first case “teleportation” in the sense
of the “bulk teleportation” studied in Antonini:2022sfm , which in this case
would take place between the two sides. In Section 2.2.4 we analyze these
phenomena when two semi-infinite intervals are measured on the left side and
in Section 2.3.3 when one semi-infinite interval is measured in each of the
two CFTs.

In all our measurement configurations we will assume that the two measured
intervals are projected onto the same Cardy state. This guarantees that the
associated ETW branes have the same tension $T$. In the phase where a single
connected brane is present (in the following we will call this the thermal AdS
phase), our assumption ensures that the brane is smooth and does not have any
defects. A generalization to the case where the two intervals are projected
onto different Cardy states requires a treatment of non-smooth intersections
among the branes along the lines of Miyaji:2022dna ; we leave such an analysis
to future work.

###  2.1 Finite intervals

####  2.1.1 Slit prescription

Figure 2: The Euclidean state preparation for a TFD state with a partial
projective measurement in region $A$. The red and orange lines denote the left
($y=0$) and right ($y=\beta/2$) CFTs. They are referred to as the time
reflection symmetric lines. Note that because $y=y+\beta$, the two orange
lines $y=\pm\beta/2$ are identified. (a) The removed slit region corresponding
to the measured region $A$ is indicated by the black rectangle with length is
$\Delta L$. (b) The setup involving projective measurement along a region $A$
of size $\Delta L$ followed by an amount $\Delta T/2$ of Euclidean time
evolution. Accordingly, in the reflection symmetric Euclidean path integral
there are two parallel slits separated by an imaginary time interval $\Delta
T$.

In this section, we focus on local projective measurements on a finite
interval $A$ of length $\Delta L$ in the left CFT.

As mentioned above, the states of interest will be prepared by evolving the
post-measurement state using the full CFT Hamiltonian for Euclidean time
$\Delta T/2$. This regularization procedure can be thought of as starting from
the single-slit path integral of Fig. 2 (a)—which describes the singular state
resulting from the insertion of the projection operator—and splitting this
into the two-slit Euclidean path integral depicted in Fig. 2 (b),444The
additional time evolution in the finite interval case here will allow us to
obtain a phase transition analogous to the one we will encounter in the semi-
infinite interval case. There, however, the physical consequences will be more
drastic.. This prepares the regular state

$\displaystyle|\Psi\rangle\propto e^{-H_{L}\Delta T/2}Me^{H_{L}\Delta
T/2}|TFD\rangle.$ (3)

We take the two slits to be located at

$\displaystyle\text{first slit:}\quad-\frac{\Delta L}{2}<x<\frac{\Delta
L}{2},\quad y=-\frac{\Delta T}{2},$ (4) $\displaystyle\text{second
slit:}\quad-\frac{\Delta L}{2}<x<\frac{\Delta L}{2},\quad y=\frac{\Delta
T}{2}.$ (5)

To fully define the measurement, in addition to the two parameters $\Delta L$
and $\Delta T$, it is necessary to specify the exact Cardy state we project
onto. This uniquely determines the boundary entropy of the post-measurement
state affleck1991universal ; cardy2004boundary , and in turn the tension of
the ETW brane takayanagi2011holographic ; fujita2011aspects .

(a) $x$ coordinate (b) $X$ coordinate

(c) $\zeta$ coordinate (d) $w$ coordinate

Figure 3: The four different coordinates we use to describe measurement of
finite intervals on one side of a TFD. The conformal maps between the
different coordinates are given in Appendix A.1. The red (orange) line denotes
the left CFT at $y=0$ (right CFT at $y=\beta/2$). They are time reflection
symmetric lines. The green (blue) color denotes the first (second) slit. The
black (orange) dots represent spatial negative (positive) infinity $x=-\infty$
($x=\infty$) in both left and right CFTs. (a) In the original coordinates, we
have an infinitely long cylinder with two slits, representing a measurement
described by two parameters $\Delta L$, $\Delta T$. Note $y=y+\beta$, so the
top and bottom lines are identified. (b) We map this to a 2D plane with two
radial slits. The angle of the slits are at $\pm\phi/2=\pm\pi\Delta T/\beta$,
with the first slit at negative $\phi$. The time reflection symmetric lines
are mapped to the horizontal line, where the left CFT is mapped to the
positive region, while the right is mapped to the negative region. Finally,
negative spatial infinity is mapped to origin, and positive infinity is mapped
to real infinity. (c) We next map to an annulus. The first slit maps to the
outer edge $r=1$ and the second slit to the inner edge $r=\rho$. The time
reflection symmetric lines are mapped to the circle with radius
$r=\sqrt{\rho}$, where the left CFT is mapped to the arc
$\theta\in(\phi,2\pi)$, and the right to $\theta\in(0,\phi)$. $x=-\infty$ is
mapped to $r=\sqrt{\rho},\theta=\phi$ (and $x=\infty$ is mapped to
$r=\sqrt{\rho},\theta=0$). Here we use $\zeta=re^{i\theta}$, and $\rho,\phi$
are two constants determined by the slits parameter $\Delta L,\Delta T$. (d)
Finally, we map to a cylinder with $\nu=\nu+2\pi$. The first slit is mapped to
the right edge at $\sigma=0$, and the left at $\sigma=-\pi s$). The time
reflection symmetric lines are mapped to the circle $\sigma=-\pi s/2$, where
the left CFT is mapped to the segment $\nu\in(\phi,2\pi)$ and the right to
$\nu\in(0,\phi)$. Finally, $x=-\infty$ is mapped to $\sigma=-\pi s/2,\nu=\phi$
and $x=\infty$ is mapped to $\sigma=-\pi s/2,\nu=0,2\pi$).

To study the dual spacetime, we implement a series of conformal
transformations depicted in Fig. 3. The details of the conformal
transformations are reported in Appendix A.1. The final domain is given by a
finite cylinder described by the complex coordinate $w=\sigma+i\nu$, with
$\nu\sim\nu+2\pi$ and $\sigma\in[-\pi s,0]$. The composed conformal
transformation leading from the original coordinates $(x,y)$ (Fig. 3 (a)) to
the final coordinates $(\sigma,\nu)$ (Fig. 3 (d)) is given by

$\displaystyle
x+iy=\frac{\beta}{2\pi}\log\left[e^{-i\phi}\frac{\theta_{4}\left(\frac{-i(\sigma+i\nu)-\phi}{2},e^{-\pi
s/2}\right)}{\theta_{4}\left(\frac{-i(\sigma+i\nu)}{2},e^{-\pi
s/2}\right)}\right],$ (6)

where $\phi=2\pi\Delta T/\beta$ and $s$ is a parameter related to the ratio
$\Delta L/\Delta T$, see Fig. 4 and equations (80), (81), and (85) in Appendix
A.1. Under the conformal transformation (6), the time reflection symmetric
lines are mapped to the $\sigma=-\pi s/2$ circle, with the left CFT ($y=0$)
given by the segment $\nu\in(\phi,2\pi)$ and the right CFT ($y=\beta/2$) by
$\nu\in(0,\phi)$. Negative spatial infinity $x=-\infty$ is mapped to
$\nu=\phi$, while positive spatial infinity $x=\infty$ is mapped to $\nu=0$.
Finally, the first slit is mapped to the $\sigma=0$ circle, and the second
slit to $\sigma=-\pi s$.

Figure 4: The ratio of measurement parameters $\frac{\Delta L}{\Delta T}$ as a
function of the parameter in our conformal transformation (equation 6)
$\rho=e^{-\pi s}$ as determined by equations (80) and (81) in Appendix A.1. In
this plot we choose $\phi=0.2$ as a representative example.

Figure 5: Gravity dual of the measured CFT. The color convention is the same
as in Fig. 3. The red (orange) line denotes the left CFT at the original
coordinate $y=0$ (right CFT at $y=\beta/2$). They are time reflection
symmetric lines. The green (blue) color denotes the first (second) slit. The
black (orange) dots represents spatial negative (positive) infinity in the
original coordinate $x$ in both CFTs. The light-blue (light-orange) surface
denotes the asymptotic boundary (brane). $\nu\sim\nu+2\pi$ is periodic. (a)
BTZ black hole phase. The gray circle denotes the intersection between time
reflection symmetric slice and the brane. (b) Thermal AdS phase. There are two
branes ending at $\sigma=-\pi s$ and $\sigma=0$.

####  2.1.2 Bulk spacetime and Hawking-Page transition

Having mapped our original two slit system to the finite cylinder in $w$
coordinates, we are ready to build the holographic bulk spacetime using the
AdS/BCFT duality takayanagi2011holographic ; fujita2011aspects . There are two
candidate phases, with the resulting euclidean spacetime described by either a
portion of a BTZ black hole or of thermal AdS. The bulk analysis performed in
this section is analogous to the one carried out in Antonini:2022sfm ,
although the physical interpretation of the results will be different, as the
CFT setups under consideration are different.

We consider the usual Euclidean action

$\displaystyle I=-\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}}\sqrt{g}(\mathcal{R}-2\Lambda)-\frac{1}{8\pi
G_{N}}\int_{\partial\mathcal{M}}\sqrt{h}(K-T),$ (7)

where $\Lambda=-\frac{1}{R^{2}}$ is the cosmological constant. The second term
is a boundary term on the branes, where $K$ is the trace of extrinsic
curvature. We impose Neumann boundary conditions for the brane, where $T$ is
the tension. The conventional Gibbons-Hawking-York term for the asymptotic
conformal boundary is not reported in the action.

We first consider the BTZ black hole phase555 Note that this nomenclature is
convenient but arbitrary: it depends on which coordinate ($\sigma$ or $\nu$)
we identify with the Euclidean time. Here we choose $\sigma$, implying that
when the $\sigma$ circle is contractible we call the corresponding spacetime a
BTZ black hole. When the $\nu$ circle is contractible we call the
corresponding spacetime thermal AdS. The opposite choice leads to the opposite
nomenclature. Neither choice is based on physical grounds: if we were to
analytically continue the bulk geometry to study the Lorentzian version of our
setup, neither $\sigma$ nor $\nu$ would be identified as Euclidean time. In
fact, to study the evolution of the post-measurement state in our CFT we
should instead analytically continue the original $y$ coordinate. Notice that
in this paper we adopted the opposite nomenclature with respect to
Antonini:2022sfm . with metric

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\left(\frac{dz^{2}}{h(z)}+h(z)d\sigma^{2}+d\nu^{2}\right),\quad
h(z)=1-\frac{z^{2}}{z_{H}^{2}},$ (8)

where $z=0$ corresponds to the asymptotic boundary, $z_{H}$ is the length of
the horizon (related to the temperature of the BTZ black hole by
$\beta_{H}=2\pi z_{H}$), and the periodicity of $\sigma$ is given by $2\pi
z_{H}$ (further recall the periodicity of $\nu$ is $2\pi$). See Fig. 5 (a) for
a depiction. Note that in the BCFT, $\sigma$ has a finite range
$\sigma\in[-\pi s,0]$, where the two boundaries at $\sigma=-\pi s,0$
correspond to the two slits in our original coordinates $(x,y)$.

According to the AdS/BCFT prescription, the ETW brane must anchor at the
asymptotic boundary $z=0$ at $\sigma=0$ and $\sigma=-\pi s$. The equation of
motion for the brane yields a trajectory $(\sigma(z),z,\nu)$, which for or
$T\in(0,1/R)$ gives

$\displaystyle\sigma(z)=\begin{cases}s\tan^{-1}\left[\frac{RTz}{s\sqrt{h-R^{2}T^{2}}}\right],&\quad
0<\sigma<\pi s/2\\\ \\\
s\left(\pi-\tan^{-1}\left[\frac{RTz}{s\sqrt{h-R^{2}T^{2}}}\right]\right),&\quad\pi
s/2<\sigma<\pi s.\end{cases}$ (9)

and for $T\in(-1/R,0)$666Although mathematically sound from a bulk point of
view in a bottom-up approach to holography, negative tension branes violate
standard energy conditions Barcelo:2000ta  and it is unclear whether they
correspond to physically reasonable setups in the context of AdS/BCFT. Note,
however, there are various top-down, explicit constructions of negative
tension objects within string theory (and that thus evade any potential
pathologies); see e.g. Marolf:2002np ; Burgess:2002vu  for related
discussions.,

$\displaystyle\sigma(z)=\begin{cases}s\tan^{-1}\left[\frac{RTz}{s\sqrt{h-R^{2}T^{2}}}\right],&\quad-\pi
s/2<\sigma<0\\\ \\\
s\left(-\pi-\tan^{-1}\left[\frac{RTz}{s\sqrt{h-R^{2}T^{2}}}\right]\right),&\quad-\pi
s<\sigma<-\pi s/2,\end{cases}$ (10)

where we used the relationship $z_{H}=s$, which is determined by the
requirement that the brane anchors at $\sigma=-\pi s$ and $\sigma=0$ at the
boundary. See Appendix B.1 for details, and in particular a derivation of
equations (9) and (10). Note that the periodicity of $\sigma$ at the boundary
is then given by $\sigma\sim\sigma+2\pi s$, i.e. twice the range of the
coordinate. At $z_{*}=s\sqrt{1-R^{2}T^{2}}$ and
$\sigma(z_{*})=\frac{\pi}{2}s$, $\sigma^{\prime}(z_{*})\rightarrow\infty$, so
that there is a turning point for the brane trajectory. When $RT=0$, the brane
cuts off exactly half of the bulk BTZ geometry and intersects the horizon
$z=s$, while for $RT\rightarrow\pm 1$, the brane locates near the boundary
$z=0$. For a positive tension brane, the brane will bulge outwards such that
the retained part of the geometry is larger than half of the BTZ geometry, and
it contains the horizon $z=s$. For negative tension, however, the remaining
part of the spacetime is less than half, and in particular does not contain
the horizon. More concretely, for a positive tension brane (i.e. when $RT>0$),
the bulk region contains $0<z<s$ for $\sigma\in(-\pi s,0)$ and $z(\sigma)<z<s$
for $\sigma\in(0,\pi s)$, where $z(\sigma)$ is the inverse function of (9).
For negative tension ($RT<0$), the bulk region contains $0<z<z(\sigma)$ for
$\sigma\in(-\pi s,0)$, where $z(\sigma)$ is the inverse function of (10). See
Fig. 6.

Figure 6: Brane trajectories projected on the $(\sigma,z)$ plane. Note that
the brane trajectories do not depend on $\nu$. (a) Brane trajectories from (9)
for $RT=0.9,0.7,0.2$. (b) Brane trajectories from (10) for
$RT=-0.9,-0.7,-0.2$. (c) Schematic trajectories of branes with different
tensions. The bottom semi-circle $\sigma\in(-\pi s,0)$ is the asymptotic
boundary.

Next, we consider the thermal AdS777Again, we emphasize that this nomenclature
is arbitrary and opposite with respect to the one used in Antonini:2022sfm ,
see footnote 5. phase, see Fig. 5 (b). In this case, the roles of $\sigma$ and
$\nu$ switch, such that the metric becomes

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\left(\frac{dz^{2}}{f(z)}+f(z)d\nu^{2}+d\sigma^{2}\right),\quad
f(z)=1-z^{2}.$ (11)

where the periodicity of $\nu\sim\nu+2\pi$ fixes the maximum value of $z$ to
be $z=1$. Again, the brane must anchor at the boundary $z=0$ at $\sigma=0$ and
$\sigma=-\pi s$. In the thermal AdS phase, there are two disconnected ETW
branes whose trajectories are now given by

$\displaystyle\sigma(z)=\begin{cases}\sinh^{-1}\left(\frac{RTz}{\sqrt{1-R^{2}T^{2}}}\right),&\quad\sigma>-\pi
s/2,\\\ \\\ -\pi
s-\sinh^{-1}\left(\frac{RTz}{\sqrt{1-R^{2}T^{2}}}\right),&\quad\quad\sigma<-\pi
s/2,\end{cases}$ (12)

where the first line corresponds to the brane anchored at $\sigma=0$, and the
second corresponds to the brane anchored at $\sigma=-\pi s$. See Appendix B.2
for a derivation. The retained bulk geometry is given by $\sigma\in\left(-\pi
s-\sinh^{-1}\left(\frac{RTz}{\sqrt{1-R^{2}T^{2}}},\right),\sinh^{-1}\left(\frac{RTz}{\sqrt{1-R^{2}T^{2}}}\right)\right)$,
$z\in(0,1]$, and $\nu\in[0,2\pi]$—see Fig. 5 (b). We plot several brane
trajectories in Fig. 7 for different tensions, both positive and negative. As
above in the BTZ phase, here in the thermal AdS phase the remaining bulk
region is larger for greater tension. Moreover, if the tension is sufficiently
negative, the two branes anchored at the boundary at $\sigma=-\pi s$ and
$\sigma=0$ will intersect. To avoid the complication of this possibility, we
restrict the tension to be $T\in(-T_{*},1/R)$, where the minimal tension
$T_{*}$ is implicitly defined through

$\displaystyle\sinh\frac{\pi s}{2}=\frac{RT_{*}}{\sqrt{1-R^{2}T^{2}_{*}}}.$
(13)

Figure 7: Brane trajectories projected on the $(\sigma,z)$ plane with fixed
$\nu$ and $\nu+\pi$. Note that the brane trajectory does not depend on $\nu$.
The trajectories are given by (12) for $RT=\pm 0.9,\pm 0.7,\pm 0.2$.  Figure
8: The phase diagram in the $s$-$RT$ plane.

In order to find whether the BTZ or thermal AdS phase is the dominant one in
the Euclidean gravitational path integral, we can evaluate the Euclidean
action (7) on-shell to find the phase with the least action fujita2011aspects
. The details of the action evaluation are left to Appendix C, though we
summarize the results here. The phase boundary between the BTZ black hole and
thermal AdS is given by a curve in the $s$-$RT$ plane given by

$\displaystyle
RT=\tanh\left[\frac{\pi}{2}\left(\frac{1}{s_{c}}-s_{c}\right)\right].$ (14)

where $s_{c}=s_{c}(T)$ denotes the critical value of $s$ for any given brane
tension $T$. As shown in Fig. 8, for a fixed value of the tension $T$
(corresponding to the specific choice of Cardy state we are projecting on),
the BTZ black hole phase is dominant for small values of $s$, and thus to
measurement parameters satisfying $\Delta L\gg\Delta T$ (see Fig. 4 and
Appendix A.1). On the other hand, the thermal AdS phase is dominant for large
values of $s$ (corresponding to the measurement parameters satisfying $\Delta
L\ll\Delta T$). Alternatively, if we first fix a value of $s$ (i.e. for given
values of the measurement parameters $\Delta L$ and $\Delta T$), the BTZ black
hole phase is dominant for small values of the tension $T$, while the thermal
AdS phase is dominant for large values of $T$. It is thus possible to tune
across the phase boundary by appropriately choosing either the measurement
parameters $\Delta L$ and $\Delta T$ or the specific Cardy state we postselect
on.

The two different phases described will correspond to two qualitatively
different dual spacetimes. To better understand the properties of the
Lorentzian spacetimes of interest888Which are those obtained by analytic
continuation after identifying the original $y$ coordinate to be the Euclidean
time. dual to these two phases, we focus on the time reflection symmetric
slice, which is invariant under analytic continuation and is therefore a
spatial slice of the Lorentzian geometry as well. In our final coordinate
system, this slice is given by $\sigma=-\pi s/2$ on the boundary and by the
$\sigma=\pm\pi s/2$ in the bulk (see Figs. 3 (d) and 5). This corresponds to
the slice $y=0,\pm\beta/2$ in the original boundary coordinates, i.e. the
slice where the measured TFD state for the two CFTs is prepared by the
boundary Euclidean path integral.

In the BTZ black hole phase, the time reflection symmetric slice spans from
the asymptotic boundary (at $\sigma=-\pi s/2,z=0,\nu\in(0,2\pi)$) to the brane
(at $\sigma=\pm\pi s/2,z=z(\pi/2),\nu\in(0,2\pi)$), where the $+$ sign
corresponds to positive tension branes, and $z(\sigma)$ is given by inverting
the first line of 9. See fig. 5 (a). Therefore, in the BTZ phase, the bulk
time reflection symmetric slice is cut off by the ETW brane.

On the other hand, in the thermal AdS phase, the time reflection symmetric
slice is bounded only by the asymptotic boundary $\sigma=-\pi
s/2,z=0,\nu\in(0,2\pi)$ and it does not intersect the brane. See Fig. 5 (b).

In both cases, note that the bulk time reflection symmetric slice is connected
between the left and right CFTs. This signals the fact that if the measured
region is finite-sized, the Einstein-Rosen bridge (in the original $x$
coordinates) remains intact after measurement. Note that this will not be true
for the infinite intervals case considered in the next subsections. Before
moving on to this case, however, we verify bulk connectivity by explicitly
computing the holographic entanglement entropy of boundary subregions on the
time reflection symmetric slice, making use of the bulk spacetimes we have
just constructed.

####  2.1.3 Holographic entanglement entropy in the post-measurement geometry

With the dual spacetime in hand, we can now turn to computing the holographic
entanglement entropy of intervals $[x_{1},x_{2}]$ at $y=0$ on the boundary. We
will restrict our attention to the leading order approximation given by the
Ryu-Takayanagi (RT) formula Ryu2006a ; Ryu2006b . Recall that in the presence
of ETW branes, the RT surface is allowed to end anywhere along the branes
while respecting the homology constraint (i.e. homology is computed relative
to the brane).

We first consider the BTZ black hole phase, focusing on the $\Delta L\gg\Delta
T$ limit for simplicity (which implies $s\ll 1$). In this case, the two
parameters $\phi,s$ of the conformal map are determined by the measurement
parameters $\Delta L$, $\Delta T$, and $\beta$ via

$\displaystyle\phi=\frac{2\pi}{\beta}\Delta T,\quad s=\frac{2\Delta T}{\Delta
L}\left(1-\frac{\Delta T}{\beta}\right).$ (15)

See Appendix A.1 for a derivation. For the time reflection symmetric lines we
focus on here, the coordinate transformation between the original CFT and the
final BTZ black hole coordinates (6) reduces to

$\displaystyle x_{1}=\frac{\beta}{2\pi}\log[\pm
X(e^{-\frac{\pi}{2}s+i\nu_{1}})],\quad x_{2}=\frac{\beta}{2\pi}\log[\pm
X(e^{-\frac{\pi}{2}s+i\nu_{2}})],$ (16)

where the $+$ sign corresponds to the left system at $y=0$ (and the $-$ sign
corresponds to the right system at $y=\beta/2$), and $X$ is given by

$\displaystyle
X(e^{-\frac{\pi}{2}s+i\nu})=\frac{\theta_{1}(\frac{\nu-\phi}{2},e^{-\frac{\pi}{2}s})}{\theta_{1}(\frac{\nu}{2},e^{-\frac{\pi}{2}s})}$
(17)

(see Appendix A.1 for details and the definition of the theta functions).

We also need to map the cutoff from the original CFT to the BTZ/thermal AdS
coordinates. According to the general coordinate transformation constructed in
roberts2012time , we have

$\displaystyle\epsilon=a\left|\frac{\partial X}{\partial\nu}\right|^{-1},$
(18)

where $a$ is the UV cutoff in the CFT.

Now, let us compute the holographic entanglement entropy of the interval
$[x_{1},x_{2}]$ in the right system (we therefore pick the $-$ sign in (16))
using the RT formula. In the BTZ black hole coordinates, these two points are
given by

$\displaystyle(\sigma_{1},\nu_{1},z_{1})=(-\frac{\pi}{2}s,\nu_{1},\epsilon(\nu_{1})),$
(19)
$\displaystyle(\sigma_{2},\nu_{2},z_{2})=(-\frac{\pi}{2}s,\nu_{2},\epsilon(\nu_{2})).$
(20)

There are two candidate RT surfaces, one connected and one disconnected that
reaches from the $y=\beta/2$ line to the time reflection symmetric slice on
the brane. In the connected phase, the length of the RT surface is given by
(see Appendix D)

$\displaystyle
D_{12}=R\cosh^{-1}\left\\{\frac{z_{H}^{2}}{z_{1}z_{2}}\left[\cosh\frac{\nu_{1}-\nu_{2}}{z_{H}}-\sqrt{\left(1-\frac{z_{1}^{2}}{z_{H}^{2}}\right)\left(1-\frac{z_{2}^{2}}{z_{H}^{2}}\right)}\cos\frac{\sigma_{1}-\sigma_{2}}{z_{H}}\right]\right\\},$
(21)

where we have used the fact that $z_{H}=s$. In the disconnected phase, the two
segments of the RT surface end on the brane, with endpoints fixed by symmetry
to be

$\displaystyle\begin{aligned}
(\sigma_{3},\nu_{3},z_{3})&=(\frac{\pi}{2}s,\nu_{1},z_{H}\sqrt{1-(RT)^{2}}),\\\
(\sigma_{4},\nu_{4},z_{4})&=(\frac{\pi}{2}s,\nu_{2},z_{H}\sqrt{1-(RT)^{2}})\end{aligned}$
(22)

for the segments anchored at $(\sigma_{1},\nu_{1},z_{1})$ and
$(\sigma_{2},\nu_{2},z_{2})$ respectively. The length of the geodesic segments
are again given by equation (21), such that the holographic entanglement
entropy is given by

$\displaystyle S_{12}^{r}=\frac{1}{4G_{N}}\min\\{D_{12},D_{13}+D_{24}\\}.$
(23)

The entanglement entropy for a specific choice of $\Delta L\gg\Delta T$ and
$\beta$ is depicted in Fig. 9. We plot the entanglement entropy of the
interval $[-\frac{\Delta x}{2},\frac{\Delta x}{2}]$ in the right system as a
function of its size $\Delta x$. As expected, the connected phase dominates
for small $\Delta x$, while the disconnected phase dominates for large $\Delta
x$, i.e. when the interval is large enough that having two disconnected
segments ending on the brane reduces the length of the RT surface.

Figure 9: Entanglement entropy of interval $[-\frac{\Delta x}{2},\frac{\Delta
x}{2}]$ in the right system in the BTZ phase. The tension is taken to be zero
and we set $R=1$.

The thermal AdS phase corresponds (for fixed tension $T$) to large values of
$s$. We can then study the limit $\frac{\Delta
L}{\beta}\ll\sin(\frac{\pi}{\beta}\Delta T)$ (see Appendix A.1 and Fig. 4),
for which

$\displaystyle e^{-\pi s}=\left(\frac{\pi\Delta
L}{4\beta\sin\frac{\phi}{2}}\right)^{2},\quad\phi=\frac{2\pi}{\beta}\Delta T.$
(24)

The conformal maps (16)-(17) for the time symmetric slice and the relationship
(18) between cutoffs in the two coordinate systems still hold in this limit.
In the thermal AdS phase there is only one, connected candidate class of RT
surfaces because, as we have pointed out at the end of Section 2.1.2, the
brane does not intersect the time reflection symmetric slice. In the thermal
AdS spacetime, the geodesics between the two endpoints
$(\sigma_{1},\nu_{1},z_{1})=(-\frac{\pi}{2}s,\nu_{1},\epsilon(\nu_{1}))$ and
$(\sigma_{2},\nu_{2},z_{1})=(-\frac{\pi}{2}s,\nu_{2},\epsilon(\nu_{2}))$ can
again be computed via equation 21, except with $\nu$ and $\sigma$ switched,
and with $z_{h}$ set to one. The entropy is then given by

$S^{r}_{12}=\frac{1}{4G_{N}}R\cosh^{-1}\left\\{\frac{1}{z_{1}z_{2}}\left[1-\sqrt{(1-z_{1}^{2})(1-z_{2}^{2})}\cos[\nu_{1}-\nu_{2}]\right]\right\\}$
(25)

The entanglement entropy for an interval $\left[-\frac{\Delta
x}{2},\frac{\Delta x}{2}\right]$ in the left system as a function of $\Delta
x$ for a specific example is depicted in Fig. 10.

We remark that in both phases the entanglement entropy grows as $\Delta x$
increases. In particular, this implies that the entanglement entropy of the
whole right (and therefore left) CFT after measurement is large. In other
words, the two CFTs are still highly entangled after measurement. This result
was to be expected, given that, as we have discussed, the bulk time reflection
symmetric slice is connected between the two CFTs in both phases.

Figure 10: Entanglement entropy of interval $[-\frac{\Delta x}{2},\frac{\Delta
x}{2}]$ in the right system int the thermal AdS phase. The tension is taken to
be zero and we set $R=1$.

###  2.2 Infinite intervals: one-sided measurement

####  2.2.1 Slit prescription

As mentioned at the beginning of this section, the Lorentzian spacetime dual
to the TFD state is a double-sided black hole, with two asymptotically AdS
boundaries connected via an Einstein-Rosen bridge. Each CFT copy can be
thought of as living on one of the asymptotic AdS boundaries. From the
analysis of the previous subsection, it is clear that when only finite
intervals are measured, there is always a connected portion of Lorentzian
spacetime separating the two CFTs, as evidenced by the fact that the
entanglement entropy between the two sides remains non-zero. Further, recall
that both CTFs are mapped to different segments of the $\sigma=-\pi s/2$
circle in BTZ/thermal AdS coordinates, and that the time reflection symmetric
slice (whose boundary contains the two CFTs) is connected both in the BTZ and
in the thermal AdS phase. This slice corresponds to the time reversal
invariant slice in the Lorentzian geometry where the initial conditions for
the real time evolution are given. Its connectivity implies the connectivity
of the Lorentzian wormhole, which is therefore preserved after the
measurement. In other words, the Einstein-Rosen bridge remains intact after
the measurement, and so the two CFTs are necessarily still highly entangled.

In this subsection, we will see that the measurement of infinitely long
intervals in one of the two CFTs can instead “destroy” the Einstein-Rosen
bridge and disconnect the two sides. This change in the bulk geometry reflects
the fact that the two CFTs are being disentangled by the measurement. In
particular, we will consider the measurement of two semi-infinite intervals in
the left CFT (as depicted in Fig. 11). We find that in the BTZ black hole
phase, the time reflection symmetric slice is disconnected. This implies that
the connectivity of the corresponding Lorentzian wormhole is destroyed by the
measurement. Therefore, the analogous Hawking-Page transition described in the
previous subsections will now correspond to a connected-disconnected phase
transition in the Lorentzian geometry associated with the measured TFD state,
and to an entangled/disentangled phase transition in the microscopic boundary
theory.

We again start with a thermofield double state with temperature $\beta$,
prepared by a Euclidean path integral on a 2D infinite cylinder with
coordinates $(x,y)$, and $y\sim y+\beta$. We perform a local projective
measurement on two semi-infinite segments of the left CFT given by

$\displaystyle\text{first slit:}\quad x<-\frac{\Delta L}{2},\quad y=0,$ (26)
$\displaystyle\text{second slit:}\quad\frac{\Delta L}{2}<x,\quad y=0,$ (27)

where $\Delta L>0$ now denotes the length of the unmeasured interval. This
parameter, in addition to the tension of the ETW brane (corresponding to the
boundary entropy of the specific Cardy state we postselect on), fully
specifies the measurement.

As in the finite interval case, in order to obtain a non-singular
configuration to analyze, we implement a series of conformal transformations
to map the two slits on the infinite Euclidean cylinder in $(x,y)$ coordinates
to the boundaries of a finite cylinder in $(\sigma,\nu)$ coordinates. Note
that here (unlike in the previous section) we do not implement any additional
Euclidean time evolution, simply because we do not need to in order to obtain
the phase transition of interest here.

The details of the conformal transformations are reported in Appendix A.2.
This procedure allows us to build the bulk dual spacetime using the AdS/BCFT
proposal in complete analogy with the previous subsection. However, note that
here the two time reflection symmetric slices in the original infinite
cylinder, corresponding to the locations of the two CFTs, get mapped to two
disconnected segments of the final, finite cylinder.

We make use of three conformal transformations that map between four different
sets of coordinates as indicated in Fig. 11. The composed conformal
transformation from the original $(x,y)$ coordinates (Fig. 11 (a)) to the
final $w$ coordinate (Fig. 11 (d)) is given by

$\displaystyle
x+iy=\frac{\beta}{2\pi}\log\left[\frac{\theta_{4}^{2}\left(-\frac{i(\sigma+i\nu)}{2},e^{-\pi
s}\right)}{\theta_{1}^{2}\left(-\frac{i(\sigma+i\nu)}{2},e^{-\pi
s}\right)}\right].$ (28)

The parameter $s$ is related to the parameter $\Delta L$ through equation (99)
(see Fig. 12). Recall that here, as above, in the original $(x,y)$ coordinates
the left CFT corresponds to the $y=0$ line, and the right CFT to $y=\beta/2$.
In the final $(\sigma,\nu)$ coordinates, the unmeasured part of the left CFT
is mapped to the $\nu=\pi$ segment and the right CFT is mapped to the
$\nu=0,2\pi$ segment. The two slits are again mapped to the circles
$\sigma=-\pi s$ (first slit) and $\sigma=0$ (second slit), whereas negative
(positive) infinity in $x$ is mapped to $\sigma=-\pi s$, $\nu=0,2\pi$
($\sigma=0$, $\nu=0,2\pi$).

(a) $x$ coordinate (b) $X$ coordinate

(c) $\zeta$ coordinate (d) $w$ coordinate

Figure 11: The four different sets of coordinates used to describe the
measurement of two semi-infinite intervals in the left CFT. The conformal maps
between the different coordinates are given in Appendix A.2. The red (orange)
line denotes the unmeasured part of the left side CFT at $y=0,x\in(-\Delta
L/2,\Delta L/2)$ (the right side CFT at $y=\beta/2$). They are time reflection
symmetric lines. The blue (green) color denotes the first (second) slit. The
black (orange) dots represent negative (positive) infinity in $x$ in both left
and right CFTs. (a) The initial, infinitely long cylinder with two infinite
slits in the left CFT. $\Delta L$ is the length of the unmeasured interval.
Also note $y\sim y+\beta$, so the top and bottom lines should be identified.
The black (orange) dots represents spatial negative (positive) infinity
$x=-\infty$ ($x=\infty$) in both left and right CFTs. (b) We first map to a 2D
plane with two radial slits on the real line. The time reflection symmetric
lines are mapped to the real line, where the left (right) CFT is mapped to a
positive (negative) region. negative (positive) infinity in $x$ is mapped to
origin (infinity). (c) We next map to the annulus. The second slit is mapped
to the outer edge $r=1$ and the first slit to the inner edge $r=\rho$. The
unmeasured part of the left CFT is mapped to the segment $\zeta\in(-\rho,-1)$,
and the right CFT is mapped to $\zeta\in(\rho,1)$. negative (positive)
infinity in $x$ is mapped to $r=\rho,\theta=0$ ($r=1,\theta=0$). Here we use
$\zeta=re^{i\theta}$, and $\rho$ is a constant determined by the size $\Delta
L$ of the unmeasured part of the left CFT. See equation (99). (d) Finally, we
map to the finite cylinder with $\nu\sim\nu+2\pi$. The first (second) slit is
mapped to the left (right) edge at $\sigma=-\pi$ ($\sigma=0$). The unmeasured
part of the left CFT is mapped to the segment $\nu=\pi$, and the right CFT is
mapped to $\nu=0,2\pi$. negative (positive) infinity in $x$ is mapped to
$\sigma=-\pi s,\nu=0,2\pi$ ($\sigma=0,\nu=0,2\pi$), with $\rho=e^{-\pi s}$.

As we will see, the fact that the right CFT and the unmeasured part of the
left CFT are mapped to two disconnected segments (instead of to two segments
on the same circle, as in the finite interval case of Section 2.1) implies
that it is possible for the measurement of infinite intervals to destroy the
Einstein-Rosen bridge and disconnect the Lorentzian geometry between the two
CFTs. If a given measurement of infinite intervals sufficiently disentangles
the two CFTs, then the remaining systems must be conformally mapped to two
segments in the final, finite cylinder that are not on the boundaries of a
connected bulk time reflection symmetric slice. We will see how, in the BTZ
black hole phase, this is indeed the case.

Figure 12: The relationship between the measurement parameter $\Delta L/\beta$
and the conformal transformation parameter $s$ for the case where infinite
intervals are measured on one-side. Note that the parameter $s$ is a monotonic
function of $\Delta L$.

####  2.2.2 Bulk spacetime and Hawking-Page transition

We have now mapped the infinite cylinder with two semi-infinite slits to the
same finite cylinder we had in the finite interval case of the previous
subsections. Therefore, the Euclidean bulk dual spacetime is the same as in
Sec. 2.1.2. In particular, we have two phases determined by the same phase
boundary (14), see Fig. 8. Note that here, $s$ is related to $\Delta L$ by
equation (99) in Appendix A.2. When the tension is zero, the transition is at
$s=1$, corresponding to $\frac{2\pi}{\beta}\Delta L=\log 2$.

While the discussion of the phase structure is analogous to the finite
interval measurement case, the physical interpretation of the phase transition
is different. In fact, in the semi-infinite interval case of this subsection
we used different coordinate transformations that mapped the unmeasured part
of the left CFT and the right CFT to two disconnected segments, see Fig. 11.
As a result, the “measurement-induced” Hawking-Page transition between the BTZ
black hole phase and the thermal AdS phase corresponds now to a connected-
disconnected phase transition of the Einstein-Rosen bridge in the associated
Lorentzian spacetime, and to an entangled/disentangled phase transition
between the two CFTs. To see why this is the case, consider the Euclidean
spacetime associated with the two phases, depicted in Fig. 13, and focus on
the bulk time reflection symmetric slice (whose boundary contains the right
CFT and the unmeasured part of the left CFT). In the $w$ coordinate (see Fig.
11 (d)), this is given by the $\nu=0,\pi$ bulk slice. In the BTZ black hole
phase, the time reflection symmetric slice clearly has two disconnected
components: both span from the asymptotic boundary $z=0,\sigma\in(-\pi s,0)$
to the brane $z=z(\sigma),\sigma\in(0,\pi s)$, but one is located at $\nu=0$
while the other is located at $\nu=\pi$. Here $z(\sigma)$ is the inverse
function of
$\sigma(z)=s\tan^{-1}\left[\frac{RTz}{s\sqrt{h-R^{2}T^{2}}}\right]$. Now
recall that the time reflection symmetric slice in the Euclidean geometry
corresponds to the time reversal symmetric slice in the associated Lorentzian
geometry, on which initial data for the real time evolution is defined. Since
the time reflection symmetric slice is disconnected in the BTZ black hole
phase, the associated Lorentzian geometry is also disconnected. When $s$ is
sufficiently small for a fixed $RT$ (so that the size $\Delta L$ of the
unmeasured region in the left CFT is sufficiently small) or when the brane
tension $T$ is sufficiently small or negative for a fixed value of $\Delta L$,
the measurement “destroys” the Einstein-Rosen bridge and disconnects the
Lorentzian geometry between the two CFTs, as expected. This corresponds to
having zero entanglement between the unmeasured part of the left CFT and the
right CFT after the measurement, as we will see shortly.

On the other hand, in the thermal AdS phase the time reflection symmetric
slice is bounded by the asymptotic boundary—at $\nu=0,z=0,\sigma\in(-\pi s,0)$
and $\nu=\pi,z=0,\sigma\in(-\pi s,0)$— and the branes—at
$\nu=0\cup\pi,\sigma=\sigma(z),z\in(0,1)$, where
$\sigma(z)=\sinh^{-1}\left(\frac{RTz}{\sqrt{1-R^{2}T^{2}}}\right)$. In this
phase, the time reflection symmetric bulk slice has only one connected
component. Following the reasoning above, this implies that the Lorentzian
geometry is also connected, meaning that the measurement does not destroy the
Einstein-Rosen bridge.

As expected, for fixed $RT$, the thermal AdS phase is dominant for large $s$
(i.e. when the size $\Delta L$ of the unmeasured region in the left CFT is
large, see Fig. 12). Note that the connected thermal phase is dominant also
for large negative values of the tension for any fixed value of $s$. Having a
connected post-measurement Lorentzian spacetime corresponds to having a large
amount of entanglement between the unmeasured part of the left CFT and the
right CFT after the measurement, as we will now describe.

Figure 13: Gravity dual of the infinite interval measurement setup (with two
infinite intervals measured in the left CFT). The color convention is the same
as in Fig. 11. The red line denotes the unmeasured part of the left CFT at
$y=0$, and the orange line denotes the right CFT at $y=\beta/2$. These are
time reflection symmetric lines. The blue (green) color denotes the first
(second) slit. The black (orange) dots represents negative (positive) infinity
in $x$ in both left and right CFTs. The light-blue (light-orange) surface
denotes the asymptotic boundary (the ETW brane). The gray color denotes the
intersection between the time reflection symmetric slice and the brane.
$\nu\sim\nu+2\pi$ is periodic. (a) BTZ black hole phase involving a single
connected brane. (b) Thermal AdS phase involving two disconnected branes
anchored at $\sigma=-\pi s$ and $\sigma=0$.

####  2.2.3 Holographic entanglement entropy in the post-measurement geometry

In order to characterize the phase transition in terms of the post-measurement
entanglement between the right CFT and the unmeasured part of the left CFT, we
can look at the post-measurement mutual information $I^{lr}=S(l)+S(r)-S(lr)$,
where $S(l)$ is the entanglement entropy of the unmeasured part of the left
CFT, $S(r)$ is the entanglement entropy of the right CFT, and $S(lr)=0$ is the
entanglement entropy of the union of the two, which is vanishing since the
full system is in a pure state. The entanglement entropies $S(l)$ and $S(r)$
can be computed applying the Ryu-Takayanagi formula to the segments
$\nu=0,z=0,\sigma\in(-\pi s,0)$ and $\nu=\pi,z=0,\sigma\in(-\pi s,0)$,
respectively. In the BTZ phase, the dominant RT surface is clearly the empty
surface for both left and right CFTs, and therefore we get
$S^{l}=S^{r}=I^{lr}=0$. In the thermal AdS phase, the dominant RT surface sits
at $z=1$, starting at the left brane and ending at the right brane. Its area
is then given by

$\displaystyle S^{l}=S^{r}=\frac{R}{4G_{N}}\int
d\sigma=\frac{R}{4G_{N}}\left[\pi
s+2\sinh^{-1}\left(\frac{RT}{\sqrt{1-(RT)^{2}}}\right)\right].$ (29)

Thus, the mutual information between the right CFT and the unmeasured part of
the left CFT is

$\displaystyle I^{lr}=\begin{cases}0,\quad&\text{BTZ phase: }s<s_{c}(T)\\\ \\\
\frac{R}{2G_{N}}\left[\pi
s+2\sinh^{-1}\left(\frac{RT}{\sqrt{1-(RT)^{2}}}\right)\right],\quad&\text{thermal
AdS phase: }s>s_{c}(T).\end{cases}$ (30)

where $s_{c}(T)$ is given in (14). This result (which is analogous to the one
obtained in Antonini:2022sfm , although its physical interpretation is
different) clearly shows how the Hawking-Page transition, which corresponds to
a Lorentzian bulk connected/disconnected phase transition, also corresponds to
an entangling/disentangling phase transition in the boundary theory. This can
be readily understood: it tells us that, for any given Cardy state we project
on, if we measure a region of the CFT which is too large we disentangle the
right CFT from the remainder of the left CFT.

####  2.2.4 Heavy operator insertions, teleportation and information erasure

Now that we have identified our Hawking-Page phase transition as a
connected/disconnected phase transition for the Lorentzian bulk (or,
equivalently, an entangling/disentangling phase transition for the dual
boundary theory), we can ask what portions of the bulk are encoded in each of
the CFTs after the postselection is performed. In particular, we will be
interested in whether bulk information encoded in one CFT in the absence of
measurement becomes accessible from the other CFT in the presence of a
measurement. In Antonini:2022sfm  a similar question could be sharply
answered, thanks to the specific choice of setup, in terms of the connectivity
of the post-measurement geometry. The results of Antonini:2022sfm  revealed
that portions of the bulk that in the absence of measurement would have been
encoded in some region $A$ of the boundary CFT become accessible from the
complementary region $\bar{A}$ when region $A$ is measured, provided that the
measurement outcome (i.e. the specific Cardy state we are postselecting on) is
known.999We work here under the assumption that the measurement outcome is
known to all boundary observers. This feature was dubbed “bulk teleportation”
Antonini:2022sfm  and showed (using tensor network models) to be in fact due
to quantum teleportation of the bulk information from region $A$ to region
$\bar{A}$.

In the setup considered here, there is no analogously direct way to understand
if bulk teleportation between the two sides is taking place. While studying
the post-measurement bulk connectivity allowed us to determine whether the two
CFTs remained significantly entangled after the measurement, it does not tell
us if and how much of the pre-measurement left wedge is accessible from the
right CFT in the post-measurement geometry. Fortunately, there is another way
we can answer this question, namely, by studying the insertions of heavy
operators in the Euclidean path integral preparing the state.

#### Operator insertions in the absence of measurement

Consider a CFT operator $\mathcal{O}$ dual to a bulk scalar field $\phi$ with
mass $m$ in the range $1\ll mR\ll R/\ell_{P}$, where $\ell_{P}$ is the Planck
length. The large mass condition $mR\gg 1$—corresponding to a large conformal
dimension $\Delta_{\mathcal{O}}$ for the dual operator—guarantees that
spacelike correlators of the operator $\mathcal{O}$ can be computed in the
geodesic approximation Faulkner:2018faa . The condition $mR\ll R/\ell_{P}$
ensures that the backreaction of the scalar field on the background geometry
can be neglected. Now consider an insertion of the operator $\mathcal{O}$ at
some value of the Euclidean time $y=-y^{*}$ for $y*\in(0,\beta/2)$ and at
$x=x^{*}$ with $x^{*}\in(-\infty,\infty)$ in the boundary Euclidean path
integral preparing the TFD state. Because a well-defined real Lorentzian
theory can be obtained by analytic continuation only if the associated
Euclidean path integral is time reflection symmetric,101010Equivalently, a
well-defined real Lorentzian geometry can be obtained by analytic continuation
only if the associated Euclidean gravitational path integral, and in
particular the associated Euclidean saddle, is time reflection symmetric. we
must consider a second insertion of the operator $\mathcal{O}$ at $y=y^{*}$,
$x=x^{*}$.

As we have discussed, the Euclidean path integral without any operator
insertions prepares the TFD state on the $y=0,\pm\beta/2$ slice. The same path
integral modified by the operator insertions at $y=\pm y^{*}$, $x=x^{*}$
prepares on the same slice a TFD state perturbed by the operator insertion. We
can then ask whether the effect of the operator insertion is detectable from
the left (at $y=0$) or the right (at $y=\pm\beta/2$) CFT. In the geodesic
approximation, this question is straightforward to answer — we can consider
the time reflection symmetric bulk geodesic in the $y-z$ plane connecting
$y=-y^{*}$ and $y=y^{*}$, and determine whether it crosses the time reflection
symmetric slice at $y=0$ or $y=\pm\beta/2$.111111As noted above, these
geodesics can be simply computed using the background Euclidean BTZ metric
because we are working within a range of scalar masses in which backreaction
of the scalar field on the geometry is negligible. In fact, in the Lorentzian
spacetime obtained by analytic continuation, the $y=0$ slice corresponds to
the portion of the time reversal symmetric slice in the left wedge and encoded
in the left CFT, whereas the $y=\pm\beta/2$ slice corresponds to the portion
in the right wedge and encoded in the right CFT. If, for example, the geodesic
intersects the symmetric slice at $y=0$, the effect of the operator insertion
is to create a particle in the left wedge, whose properties can be
reconstructed from the left CFT. For $y^{*}\in(0,\beta/4)$ the geodesics cross
the time reflection symmetric slice at $y=0$ and for
$y^{*}\in(\beta/4,\beta/2)$ they cross the reflection symmetric slice at
$y=\pm\beta/2$ (see Figure 14).

Figure 14: The time reflection symmetric geodesics in the $x=x^{*}$ slice of
the BTZ black hole. The insertion points are at $y=\pm y^{*}$ on the boundary
and the associated bulk Euclidean geodesics are depicted in blue. The red
dashed line indicates the $y=0,\pm\beta/2$ time reflection symmetric slice.
For $y^{*}\in(0,\beta/4)$ the geodesics cross the time reflection symmetric
slice at $y=0$ and for $y^{*}\in(\beta/4,\beta/2)$ they cross the reflection
symmetric slice at $y=\pm\beta/2$.

#### Bulk teleportation and information erasure in the presence of measurement

In order to understand how the information associated with these operator
insertions is affected by the measurement, we can carry out the following
procedure:

  1. 1.

Consider operator insertions analogous to those described above at given
points $(x^{*},-y^{*})$ and $(x^{*},y^{*})$ in our path integral with slits in
the original $(x,y)$ coordinates depicted in Figure 11 (a).

  2. 2.

Map the two points to the points $P_{1}=(\sigma^{*},\nu^{*})$,
$P_{2}=(\sigma^{*},2\pi-\nu^{*})$ in BTZ/thermal AdS coordinates using the map
(28), with $\sigma^{*}\in(-\pi s,0)$ and $\nu^{*}\in(0,\pi)$.121212Recall that
in BTZ/thermal AdS coordinates the unmeasured part of the left CFT is mapped
to $\nu=\pi$, the right CFT to $\nu=0,2\pi$ and the time reflection symmetric
slice is therefore mapped to the $\nu=0,\pi$ slice, with $\nu=0$ and
$\nu=2\pi$ identified.

  3. 3.

In the bulk dual spacetime—which can be either thermal AdS or the BTZ black
hole cut off by ETW branes—compute the shortest, reflection symmetric (around
the $\nu=0,\pi$ slice) geodesic anchored at the points $P_{1}$ and $P_{2}$.
For each choice of points $P_{1}$, $P_{2}$ there are two candidate geodesics:
a constant-$\sigma$ connected geodesic which crosses the reflection symmetric
slice, and a constant-$\nu$ geodesic with two disconnected segments ending on
the ETW brane (see Figure 15). Since both geodesic are anchored at the points
$P_{1}$, $P_{2}$ and they are both reflection symmetric, the correct geodesic
to consider in our analysis is the shorter of the two.

  4. 4.

If the constant-$\sigma$ connected geodesic is the shortest one, determine
whether it crosses the reflection symmetric slice at $\nu=0$ or at
$\nu=0,2\pi$. Similar to what we observed in the absence of measurement, if
$\nu^{*}\in(0,\pi/2)$ the geodesic crosses the reflection symmetric slice at
$\nu=0,2\pi$, whereas if $\nu^{*}\in(\pi/2,\pi)$ it crosses the reflection
symmetric slice at $\nu=\pi$.

If the constant-$\sigma$ connected geodesic is the shortest one, there are two
possibilities. First, notice that the RT surface calculation carried out in
the previous subsection implies that the $\nu=0,2\pi$ portion of the
reflection symmetric slice is part of the entanglement wedge of the right CFT,
whereas the $\nu=\pi$ portion of the same slice is part of the entanglement
wedge of the left CFT. Now consider an operator insertion with
$y^{*}\in(0,\beta/4)$, whose associated geodesic in the absence of measurement
crosses the time reflection symmetric slice at $y=0$ and whose effect is
therefore detectable from the left CFT. If, in the presence of measurement,
the point is mapped to an insertion point with $\nu^{*}\in(\pi/2,\pi)$, the
constant-$\sigma$ geodesic crosses the reflection symmetric slice at
$\nu=\pi$, implying that the effect of the operator insertion is still
detectable from the left CFT. However, if the point is mapped to an insertion
point with $\nu^{*}\in(0,\pi/2)$, then the constant-$\sigma$ geodesic crosses
the reflection symmetric slice at $\nu=0,2\pi$, implying that the effect of
the operator insertion is detectable from the right CFT after the measurement
is performed. In the latter case, bulk teleportation in the sense of
Antonini:2022sfm  is taking place between the two sides: part of the pre-
measurement entanglement wedge of the left CFT is encoded in the right CFT
after the measurement is performed. A completely analogous reasoning can be
carried out for insertion points with $y^{*}\in(\beta/4,\beta/2)$.

On the other hand, if the shortest geodesic associated with an insertion with
$y^{*}\in(0,\beta/4)$ is the constant-$\nu$ disconnected one, the geodesic
ends on the brane and it does not intersect the portion of the time reflection
symmetric slice accessible from either CFT. Therefore, we can conclude that
the operator insertion does not affect the post-measurement Lorentzian
geometry (at least in the purely geometrical approximation we are focusing
on): the measurement “erases” the information associated with the operator
insertion. This suggests that part of the pre-measurement entanglement wedge
of the left side is destroyed by the measurement, as we could have suspected.

Figure 15: Time reflection symmetric geodesics in the BTZ (left) and thermal
AdS (right) phases. The symmetric insertion points on the AdS boundary are
indicated by black dots. The constant-$\sigma$ connected geodesic is
represented by a brown line connecting the two insertion points and passing
through the $\nu=0,\pi$ reflection symmetric slice. The constant-$\nu$
disconnected geodesic is represented by two purple lines anchored at the
insertion points and ending on the ETW brane, with the intersection points
between the geodesic and the brane represented by gray dots. Note that the
disconnected geodesic does not intersect the reflection symmetric slice.

In order to complete our analysis, we compute the length of the connected and
disconnected geodesics in the BTZ and thermal AdS spacetimes cut off by ETW
branes. Let us first restrict our analysis to $\nu^{*}\in(\pi/2,\pi)$ and
$\sigma^{*}\in(-\pi s/2,0)$. In the BTZ phase we again use equation (21),
obtaining

$L_{conn}^{btz}=R\cosh^{-1}\left\\{\frac{s^{2}}{\epsilon^{2}}\left[\cosh\left(\frac{2(\pi-\nu^{*})}{s}\right)-\left(1-\frac{\epsilon^{2}}{s^{2}}\right)\right]\right\\}$
(31)

for the connected geodesic and

$L_{disc}^{btz}=2\min_{z_{b}}\left(R\cosh^{-1}\left\\{\frac{s^{2}}{\epsilon
z_{b}}\left[1-\sqrt{\left(1-\frac{\epsilon^{2}}{s^{2}}\right)\left(1-\frac{z_{b}^{2}}{s^{2}}\right)}\cos\left(\frac{\sigma_{b}^{btz}(z_{b})-\sigma^{*}}{s}\right)\right]\right\\}\right)$
(32)

for the disconnected geodesic, where $\epsilon$ is the bulk IR cutoff, we have
identified $z_{H}=s$, and $\sigma_{B}^{btz}(z_{b})$ is the brane position
given by the first lines of equations (9)-(10) for positive and negative
tension branes, respectively. Note that to obtain the length of the
disconnected geodesic we minimized over the location of the endpoint of the
geodesic on the brane—given by $(z_{b},\sigma_{b}^{btz}(z_{b}),\nu^{*})$ for
the component of the geodesic at $\nu=\nu^{*}$—and we multiplied by 2 to
account for the two disconnected components of the geodesic at $\nu=\nu^{*}$
and $\nu=2\pi-\nu^{*}$. By symmetry, the geodesic lengths for
$\nu^{*}\in(0,\pi/2)$ and $\sigma^{*}\in(-\pi s,-\pi s/2)$ are given by the
same formulae after replacing $\pi-\nu^{*}\to\nu^{*}$ and
$\sigma^{*}\to-\sigma^{*}-\pi s$, with $\sigma_{B}^{btz}(z_{b})$ still given
by the first lines of equations (9)-(10).

In the thermal AdS phase, we can again use equation (21), exchanging
$\sigma\leftrightarrow\nu$ and setting $z_{H}=1$. For $\nu^{*}\in(\pi/2,\pi)$
and $\sigma^{*}\in(-\pi s/2,0)$ we get

$L_{conn}^{th}=R\cosh^{-1}\left\\{\frac{1}{\epsilon^{2}}\left[1-\left(1-\epsilon^{2}\right)\cos(2\left(\pi-\nu^{*}\right))\right]\right\\}$
(33)

for the connected geodesic and

$L_{disc}^{th}=2\min_{z_{b}}\left(R\cosh^{-1}\left\\{\frac{1}{\epsilon
z_{b}}\left[\cosh\left(\sigma_{b}^{th}(z_{b})-\sigma^{*}\right)-\sqrt{\left(1-\epsilon^{2}\right)\left(1-z_{b}^{2}\right)}\right]\right\\}\right)$
(34)

for the disconnected geodesic, where $\sigma_{b}^{th}(z_{b})$ is the brane
position now given by the first line of equation (12). As in the BTZ phase,
the geodesic lengths for $\nu^{*}\in(0,\pi/2)$ and $\sigma^{*}\in(-\pi s,-\pi
s/2)$ are given by the same formulae after replacing $\pi-\nu^{*}\to\nu^{*}$
and $\sigma^{*}\to-\sigma^{*}-\pi s$, with $\sigma_{B}^{th}(z_{b})$ still
given by the first lines of equation (12).

The results of our analysis for the measurement of two semi-infinite intervals
in the left CFT are reported in Figure 16. We represent the boundary domain in
BTZ/thermal AdS coordinates $(\sigma,\nu)$ and shade with different colors the
regions in which operator insertions lead to teleportation, erasure, or
neither. We find that bulk teleportation from the left to the right side
occurs more extensively in the (connected) thermal phase and erasure is more
likely in the (disconnected) BTZ phase. Within either of the two phases,
teleportation is favored for large positive values of the tension, while
erasure is favored for small (or negative) values of the tension. In
particular, we focused here on the better-understood $T\geq 0$ case. However,
applying our analysis to the $T<0$ case we find that even information
associated with operator insertions with $y^{*}\in(\beta/4,\beta/2)$—which for
$T\geq 0$ is accessible from the right CFT both in the absence and in the
presence of measurement—can be erased.131313This seems to suggest that by
projecting part of the left CFT on a Cardy state with negative boundary
entropy we are able to affect the entanglement wedge of the right CFT. This
puzzling feature, which arose in a similar fashion in Antonini:2022sfm , could
be evidence of the non-physical nature of solutions involving negative tension
branes.

Note that this result is qualitatively different from the one obtained in
lower dimensional models kourkoulou2017pure ; Antonini:2022lmg , where
teleportation between the two sides occurs when most of the microscopic system
living on one side is measured. We would also like to remark that, although
the present analysis does not allow us to detect bulk teleportation from the
measured part to the unmeasured part of the left CFT, the results of
Antonini:2022sfm  suggest that this also generically takes place.

Figure 16: Bulk teleportation and erasure in the one-sided infinite
measurement setup. For heavy operator insertions in the white, blue, and green
regions the connected geodesic is the shorter one, whereas for insertions in
the orange regions the disconnected geodesic is shorter. The effect of heavy
operator insertions in the blue (white) region can be reconstructed from the
left (right) CFT independently of whether a measurement is performed or not.
The effect of insertions in the green region can be reconstructed from the
left CFT in the absence of measurement and from the right CFT in the presence
of measurement, signaling bulk teleportation between the two sides. The effect
of insertions in the orange regions cannot be reconstructed from either CFT in
the presence of measurement: this information is being erased by the
measurement. Note that teleportation occurs more extensively in the
(connected) thermal phase and for large positive tensions, whereas erasure is
more likely in the (disconnected) BTZ phase and for small (or negative)
tensions. In all plots we set $R=1$, $\beta=2$ and choose the value of the IR
cutoff to be $\epsilon=10^{-4}$. (a) BTZ phase with parameters $s=0.1$,
$T=0.0001$. (b) BTZ phase with parameters $s=0.1$, $T=0.9$. (c) Thermal AdS
phase with parameters $s=2$, $T=0.0001$. (d) Thermal AdS phase with parameters
$s=2$, $T=0.9$.

Finally, we report for completeness in Figure 17 a schematic illustration of
the $y=0,\pm\beta/2$ slice of the post-measurement geometry in the BTZ and
thermal AdS phases. In the thermal AdS phase we also qualitatively depict the
RT surface for the whole right CFT or, equivalently, for the unmeasured part
of the left CFT (in the BTZ phase the RT surface is simply the empty set). The
portions of the pre-measurement entanglement wedge of the left CFT
“teleported” into the right CFT by the measurement in the two phases are also
shown. We remark that this should be regarded only as a useful and intuitive
cartoon of what the $y=0,\pm\beta/2$ slice would look like in a setup where
the measurement is physically regularized; we remind that the measurement
setup studied in the present paper is not regularized and the $y=0,\pm\beta/2$
bulk time reflection symmetric slice is actually singular.

Figure 17: A schematic illustration of the $y=0,\pm\beta/2$ bulk time
reflection symmetric slice for the one-sided infinite measurement setup with
positive tension. The red (orange) line indicates the unmeasured part of the
left CFT (the right CFT) and the two slits are represented in green and blue.
The region shaded in red is bounded and cut off by the ETW brane. The blue
region is the post-measurement entanglement wedge of the left CFT, whereas the
union of the the yellow and the green regions is the post-measurement
entanglement wedge of the right CFT. In particular, the yellow region is part
of the entanglement wedge of the right CFT even without measurement, whereas
the green region is “teleported” from the left to the right CFT by the
measurement. (a) BTZ phase. The time-reflection symmetric slice is
disconnected, implying that the RT surface for the right CFT (or,
equivalently, the unmeasured part of the left CFT) is the empty set and the
two CFTs are disentangled by the measurement. Only a small portion of the pre-
measurement entanglement wedge of the left CFT is teleported into the right
CFT. (b) Thermal AdS phase. The time reflection symmetric slice is connected
and the RT surface, which has a finite length given by equation (29), is
depicted as a dark green line. A large portion of the pre-measurement
entanglement wedge of the left CFT is teleported into the right CFT.

###  2.3 Infinite intervals: two-sided measurement

####  2.3.1 Slit prescription

In the previous subsection we saw that the measurement of two semi-infinite
intervals on one side of the TFD was able to disconnect the Einstein-Rosen
bridge. In this subsection, we investigate an analogous question in the case
where one semi-infinite interval is measured in each one of the two CFTs.

As usual, we start with a TFD state with temperature $\beta$ living in a 2D
cylinder $(x,y)$ with $y\sim y+\beta$. The measurement is now described by two
infinite slits located on the time reflection symmetric slice $y=0,\beta/2$
(see Fig. 18 (a))

$\displaystyle\text{first slit:}\quad x>\frac{\Delta L}{2},\quad
y=\frac{\beta}{2},$ (35) $\displaystyle\text{second slit:}\quad
x<-\frac{\Delta L}{2},\quad y=0.$ (36)

$\Delta L$ now controls the size of the region which is unmeasured in both
CFTs. If $\Delta L>0$, there is an unmeasured region corresponding to the same
range of $x$ coordinate in both CFTs, while if $\Delta L<0$ there is no such
“overlapping” region.

As above, we implement a series of conformal transformations to map our domain
to the finite cylinder so that we can construct the bulk dual. The three
conformal transformations, which are described in detail in Appendix A.3, are
depicted in Fig. 18. The composed conformal transformation from the original
$(x,y)$ coordinates (Fig. 18 (a)) to the final $w$ coordinate (Fig. 18 (d)) is
given by

$\displaystyle
x+iy=\frac{\beta}{2\pi}\log\left[-\frac{\theta_{3}^{2}\left(-\frac{i(\sigma+i\nu)}{2},e^{-\pi
s}\right)}{\theta_{1}^{2}\left(-\frac{i(\sigma+i\nu)}{2},e^{-\pi
s}\right)}\right].$ (37)

The parameter $s$ is related to the parameter $\Delta L$, see Fig. 19 and
equation (108) in Appendix A.3. Recall that in the original $(x,y)$
coordinates, the left CFT corresponds to the $y=0$ line, and the right CFT to
$y=\beta/2$. In the final $(\sigma,\nu)$ coordinates, the unmeasured part of
the left CFT is mapped to the $\nu=0,2\pi$ segment and the unmeasured part of
the right CFT is mapped to the $\nu=\pi$ segment. The two slits are again
mapped to the circles $\sigma=0$ (first slit) and $\sigma=-\pi s$ (second
slit), while negative (positive) infinity in $x$ is mapped to $\sigma=-\pi s$,
$\nu=0,2\pi$ ($\sigma=0$, $\nu=\pi$).

(a) $x$ coordinate (b) $X$ coordinate

(c) $\zeta$ coordinate (d) $w$ coordinate

Figure 18: The four different sets of coordinates used to describe the
measurement of a semi-infinite interval in both of the two CFTs. The conformal
maps between the different coordinates are given in Appendix A.3. The red
(orange) line denotes the unmeasured part of the left CFT at $y=0$ (the right
CFT at $y=\beta/2$). They are time reflection symmetric lines. The green
(blue) color denotes the first (second) slit. The black (orange) dots
represent negative (positive) infinity in $x$ in both left and right CFTs. (a)
We start with an infinitely long cylinder with one infinite slit in each of
the two CFTs. The first (second) slit is located at $y=\beta/2$ ($y=0$), i.e.
in the right (left) CFT. $\Delta L$ denotes the region which is unmeasured in
both CFTs. Recall that $y\sim y+\beta$, so that $y=-\beta/2=\beta/2$. The
black (orange) dots represents negative (positive) infinity in $x$ for both
left and right CFTs. (b) We map this to the 2D plane with two radial slits on
the real axis. The first (second) slit is mapped to a semi-infinite (finite)
segment. The unmeasured parts of the two CFTs are also mapped to the real
axis, where the left (right) CFT is mapped to a semi-infinite (finite)
segment. Negative (positive) infinity is mapped to the origin (infinity). (c)
We next map to the annulus. The first slit is the outer edge $r=1$ and the
second slit is the inner edge $r=\rho$. The unmeasured part of the left
(right) CFT is mapped to a segment $r\in(\rho,1)$ ($r\in(-1,-\rho)$). Negative
(positive) infinity in $x$ is mapped to $r=\rho,\theta=\pi$ ($r=1,\theta=0$).
Here we use $\zeta=re^{i\theta}$, and $\rho$ is a constant determined by the
measurement parameter $\Delta L$. (d) Finally, we map to a finite cylinder
with $\nu\sim\nu+2\pi$. The fist (second) slit is mapped to the right (left)
edge at $\sigma=0$ ($\sigma=-\pi s$). The unmeasured part of the left (right)
CFT is mapped to a segment $\nu=0,2\pi$ ($\nu=\pi$). Negative (positive)
infinity in $x$ is mapped to $\sigma=-\pi s,\nu=0,2\pi$ ($\sigma=0,\nu=\pi$).

Note that the final domain in $w$ coordinates is almost identical to the one
obtained in Section 2.2 for the measurement of two semi-infinite intervals on
only one of the two CFTs (see Fig. 11). The only differences are where
positive infinity in $x$ is mapped, by the relationship between $\Delta L$ and
$s$, and by the fact that the location of the unmeasured parts of the left and
right CFTs are switched. Therefore, all the discussion (see Sections 2.2.1 and
2.2.2) about the phase structure and its implications for the connectivity of
the Lorentzian geometry and the entanglement structure of the dual microscopic
theory remains valid and unmodified in the two-sided measurement setup of the
present subsection.

Figure 19: The relationship between the measurement parameter $\Delta L/\beta$
and the conformal transformation parameter $s$ for the case where semi-
infinite intervals are measured on each side of the TFD. The parameter $s$ is
a monotonically increasing function of $\Delta L$. In particular, when $\Delta
L=0$, $s=1$.

####  2.3.2 Bulk spacetime and Hawking-Page transition

Given the similarities between the final domains, the spacetime dual to the
final finite cylinder (in $w$ coordinates) will also be completely analogous
to the one studied in Subsection 2.2.2, see Fig. 13. In particular, the phase
boundary between the BTZ black hole phase and the thermal AdS phase is still
determined by the value of the parameter $s$ and the brane tension $T$. The
BTZ black hole phase again corresponds to a “disconnected” phase, in which the
Einstein-Rosen bridge connecting the two CFTs in the Lorentzian spacetime dual
to the TFD state is destroyed by the measurement. On the other hand, the
thermal AdS phase is a “connected” phase, in which the Einstein-Rosen bridge
is preserved after the measurement. The two phases correspond to a
disentangled and an entangled phase in the dual CFTs, and the mutual
information between the unmeasured regions in the two CFTs takes the same form
(30) as in the one-sided measurement case.

The only differences reside in the switched locations of the left and right
CFTs (which is irrelevant for our discussion), in the relationship between $s$
and $\Delta L$—which is now given by equation (108)—and in the meaning of the
parameter $\Delta L$. As we have discussed, this quantifies the size of the
region in $x$ coordinate which is unmeasured in both CFTs. Intuitively, if
$\Delta L=0$ we are measuring exactly half of the system ($x>0$ for the right
CFT and $x<0$ for the left CFT), if $\Delta L>0$ we are measuring less than
half of the system, and vice versa for $\Delta L<0$. As we have seen, $\Delta
L=0$ corresponds to $s=1$, which is the critical point for vanishing tension
$T=0$. This leads to an intuitive result: in the $T=0$ case, i.e. if we
postselect on a Cardy state with zero boundary entropy, measurement of more
than half of the system will disentangle the two CFTs, destroy the Einstein-
Rosen bridge, and disconnect the dual Lorentzian spacetime.

####  2.3.3 Heavy operator insertions, teleportation and information erasure

The analysis of heavy operator insertions in the two-sided measurement setup
is completely analogous to the one carried out for the one-sided measurement
in Section 2.2.4. The only difference is that the map between the original
$(x,y)$ coordinates and the BTZ/thermal AdS coordinates $(\sigma,\nu)$ is now
given by equation (37). Recall that with this map the left CFT is mapped to
the $\nu=0,2\pi$ slice, whereas the right CFT is mapped to the $\nu=\pi$
slice. As a result, for given insertion points $P_{1}=(\sigma^{*},\nu^{*})$,
$P_{2}=(\sigma^{*},2\pi-\nu^{*})$ for which the constant-$\sigma$ connected
geodesic is dominant, if the geodesic crosses the reflection symmetric slice
at $\nu=0,2\pi$ (i.e. if $\nu^{*}\in(0,\pi/2)$) the effect of the operator
insertion can be detected from the left CFT; if the geodesic crosses the
reflection symmetric slice at $\nu=\pi$ instead (i.e. if
$\nu^{*}\in(\pi/2,\pi)$), the effect of the operator insertion can be detected
from the right CFT. The lengths of the connected and disconnected geodesics in
the two phases are still given by equations (31), (32), (33), and (34).

The results of our analysis for the two-sided measurement case are reported in
Figure 20. We again represent the boundary domain in the BTZ/thermal AdS
coordinates $(\sigma,\nu)$ and shade with different colors the regions in
which operator insertions lead to teleportation, erasure, or neither. Similar
to the one-sided measurement analysis, we find that teleportation between the
two sides occurs more extensively in the (connected) thermal phase and erasure
is more likely in the (disconnected) BTZ phase, with teleportation favored for
large positive values of the tension and erasure favored for small (or
negative) values of the tension. We again focused our attention on the better-
understood $T\geq 0$ case. Note that in the double-sided measurement case,
part of the pre-measurement entanglement wedge of the left CFT is “teleported”
into the right CFT by the measurement and, at the same time, part of the pre-
measurement entanglement wedge of the right CFT is “teleported” into the left
CFT.

Figure 20: For heavy operator insertions in the white, blue, green, and gray
regions the connected geodesic is the shorter one, whereas for insertions in
the orange regions the disconnected geodesic is shorter. The effect of heavy
operator insertions in the blue (white) region can be reconstructed from the
left (right) CFT independently of whether a measurement is performed or not.
The effect of insertions in the green (gray) region can be reconstructed from
the left (right) CFT in the absence of measurement and from the right (left)
CFT in the presence of measurement, signaling symmetric bulk teleportation
between the two sides. The effect of insertions in the orange regions cannot
be reconstructed from either CFT in the presence of measurement: this
information is being erased by the measurement. Teleportation occurs more
extensively in the (connected) thermal phase and erasure is more likely in the
(disconnected) BTZ phase, with teleportation favored for large positive values
of the tension and erasure favored for small (or negative) values of the
tension. In all plots we set $R=1$, $\beta=2$ and choose the value of the IR
cutoff to be $\epsilon=10^{-4}$. (a) BTZ phase with parameters $s=0.5$,
$T=0.0001$. (b) BTZ phase with parameters $s=0.5$, $T=0.9$. (c) Thermal AdS
phase with parameters $s=2$, $T=0.0001$. (d) Thermal AdS phase with parameters
$s=2$, $T=0.9$.

In Figure 21 we depict a schematic illustration of the $y=0,\pm\beta/2$ slice
of the post-measurement geometry in the two phases for the two-sided
measurement setup, showing the bulk regions teleported from the left to the
right and vice versa. In the thermal phase we also depict the RT surface for
the unmeasured part of the left CFT (or, equivalently, the unmeasured part of
the right CFT). In the BTZ phase the RT surface is again the empty set. As
above, this illustration is just a cartoon of the $y=0,\pm\beta/2$ slice we
would obtain in a physically regularized setup.

Figure 21: Schematic illustration of the $y=0,\pm\beta/2$ bulk time reflection
symmetric slice for the two-sided infinite measurement setup with positive
tension. The red (orange) line indicates the unmeasured part of the left
(right) CFT and the two slits are represented in green and blue. The region
shaded in red is bounded and cut off by the ETW brane. The union of the blue
and the gray region is the post-measurement entanglement wedge of the left
CFT, whereas the union of the yellow and green regions is the post-measurement
entanglement wedge of the right CFT. In particular, the blue (yellow) region
is part of the entanglement wedge of the left (right) CFT even without
measurement, whereas the gray (green) region is “teleported” from the right
(left) to the left (right) CFT by the measurement. (a) BTZ phase. The time-
reflection symmetric slice is disconnected, implying that the RT surface for
the unmeasured part of the right CFT (or, equivalently, the unmeasured part of
the left CFT) is the empty set and the two CFTs are disentangled by the
measurement. Only a small portion of the pre-measurement entanglement wedge of
the left (right) CFT is teleported into the right (left) CFT. (b) Thermal AdS
phase. The time reflection symmetric slice is connected and the RT surface,
which has a finite length given by equation (29), is depicted as a dark green
line. A large portion of the pre-measurement entanglement wedge of the left
(right) CFT is teleported into the right (left) CFT.

##  3 Projective measurement in CFT thermofield doubles

In this section, we study the measurements considered in Section 2 purely from
the boundary CFT point of view. In particular, we will focus on the cases
considered in Sections 2.2 and 2.3 where infinite intervals are measured
either in one or in both of the CFTs, and calculate the entanglement entropy
between the two sides after the measurement is performed. This will allow us
to characterize the measurement-induced Hawking-Page transition seen above in
terms of an entangling phase transition in the microscopic system, and to
reproduce (up to corrections due to $1/N$-suppressed effects) the results for
the mutual information obtained by computing the holographic entanglement
entropy in the bulk dual geometry. A general framework to carry out the
calculations of interest has been developed in Ref. rajabpour2016entanglement
. We will first outline the main steps of the calculation, which are common to
the two setups, and then discuss the two cases in detail in Sections 3.1 and
3.2, respectively.

We start from the $X$ coordinates (see Fig. 11 for the one-sided measurement
and Fig. 18 for the two-sided measurement). Because we are interested in
computing the von Neumann entropy between the two sides, we can insert a twist
operator on either side, i.e. at either $y=0$ or $y=\beta/2$ (since the full
state is pure, the two choices lead to the same result). Here we will insert a
twist operator in the left CFT ($y=0$) for measurements on one side, and in
the right CFT ($y=\beta/2$) for the measurements on both sides. Recall that
the Renyi entropy is given by

$\displaystyle S_{\alpha}=\frac{1}{1-\alpha}\left(\log Z_{\alpha}-\alpha\log
Z_{1}\right),$ (38)

where $Z_{\alpha}$ is the partition function of the CFT on a Riemann surface
given by $\alpha$ copies of the original domain, and with a branch cut at
$y=0$ or $y=\beta/2$ (depending on which side we are computing the Renyi
entropy of) that connects the $i$-th and the $(i\pm 1)$-th Riemann sheet
calabrese2009entanglement . This is equivalent to computing an $\alpha$-point
function of appropriate twist fields in a single copy. The role of the twist
operator is precisely to account for the correct boundary condition at the
branch cut. The von Neumann entropy can then be obtained by analytically
continuing $\alpha$ and taking the $\alpha\rightarrow 1$ limit.

In order to use the results of Ref. rajabpour2016entanglement  directly, we
consider a conformal transformation from the $X$ coordinate to a plane with
two symmetric slits at $(-\frac{1}{k},-1)\cup(1,\frac{1}{k})$, which we denote
by $\tilde{X}$. The form of the conformal transformation in the two cases is
given in Appendices A.2 and A.3, respectively. For both, the branch cut is
mapped to $(-1,1)$, see Fig. 22 (a) for an illustration.

To evaluate the Renyi entropy (38), we follow rajabpour2016entanglement  and
consider the free energy

$\displaystyle F_{\alpha}=-\log Z_{\alpha}$ (39)

first, which can be computed as follows by considering a small shift $\delta
l$ of one of the slits. Let us denote by $l$ the distance between the two
slits in $X$ and horizontally shift the second slit (the one mapped to
$(1,1/k)$ in $\tilde{X}$) by an amount $\delta l$. The change in the free
energy of the system on the $\alpha$-sheeted Riemann surface due to the shift
can then be written as rajabpour2016entanglement

$\displaystyle\delta F_{\alpha}(X)=-\frac{\delta l}{2\pi i}\oint dX\langle
T(X)\rangle+c.c.$ (40)

where $\langle T(X)\rangle$ denotes the vacuum expectation value of the stress
tensor in $X$, and the integral’s contour encloses the second slit. The next
step is to map our domain in $\tilde{X}$ to an annulus in $\zeta$ coordinates
using the conformal transformation

$\displaystyle\zeta(\tilde{X})=e^{-\frac{\pi s}{2}}\exp\left[\pi
s\frac{\text{sn}^{-1}(\tilde{X},k^{2})}{2K(k^{2})}\right]$ (41)

where sn is the elliptic sine function, $K$ is the complete elliptic integral
of the first kind, and we have defined

$\displaystyle s=\frac{2K(k^{2})}{K(1-k^{2})}.$ (42)

The slit at $(1,1/k)$ is mapped to the outer edge $|\zeta|=1$, the slit at
$(-1/k,-1)$ is mapped to the inner edge $|\zeta|=e^{-\pi s}$, and the branch
cut at $\tilde{X}\in(-1,1)$ is mapped to $\zeta\in(e^{-\pi s},1)$, see Fig.
22. We can now perform an additional conformal transformation
$\zeta_{\alpha}=\zeta^{1/\alpha}$ to “unwind” and obtain an annulus without
branch cuts.

(a) $\tilde{X}$ coordinate (b) $\zeta$ coordinate

Figure 22:  The green (blue) color denotes the first (second) slit. The dashed
line denotes a branch cut $(-1,1)$ induced by the twist operator. (a) 2D plane
with two symmetric slits $(-1/k,-1)\cup(1,1/k)$, where $k<1$ is a real number.
(b) Annulus. The first slit is the outer edge $|\zeta|=1$, the second slit is
the inner edge $|\zeta|=e^{-\pi s}$, and the branch cut is indicated by the
dashed line.

The purpose of this coordinate transformation is to simplify the evaluation of
the free energy, of which we report here the final result (we refer the
readers to Ref. rajabpour2016entanglement  for a detailed derivation). The
free energy consists of two parts:

$\displaystyle F_{\alpha}=F^{\text{annu}}_{\alpha}+F^{\text{geom}}_{\alpha},$
(43)

where $F^{\text{annu}}_{\alpha}$ is the free energy of the annulus and
$F^{\text{geom}}_{\alpha}$ denotes the geometric contribution originating from
the coordinate transformation between the plane in $X$ coordinates and the
annulus. The free energy in the annulus geometry is known and can be written
in two equivalent forms cardy2004boundary ; rajabpour2016entanglement :

$\displaystyle-F^{\text{annu}}_{\alpha}=\log\left(q_{\alpha}^{-\frac{c}{24}}(1+\sum_{j}n_{j}q^{\Delta_{j}}_{\alpha})\right)-c\frac{\pi
s}{12\alpha}$ (44)
$\displaystyle-F^{\text{annu}}_{\alpha}=\log\left(\tilde{q}_{\alpha}^{-\frac{c}{24}}(b_{0}^{2}+\sum_{j}b_{j}^{2}\tilde{q}^{\Delta_{j}}_{\alpha})\right)-c\frac{\pi
s}{12\alpha},$ (45)

where $c$ is the central charge, $n_{j}$ is the number of degenerate states
with scaling dimension $\Delta_{j}$, $b_{j}$ are real numbers, and
$q_{\alpha}=e^{-\frac{2\pi\alpha}{s}}$, $\tilde{q}_{\alpha}=e^{-\frac{2\pi
s}{\alpha}}$. The first (second) expression is appropriate for the open
(closed) string channel and is useful to compute the annulus partition
function in the $s\ll 1$ ($s\gg 1$) limit. On the other hand, the geometric
contribution is implicitly given by

$\displaystyle\frac{\delta F_{\alpha}^{\text{geom}}}{\delta
l}=\frac{ic}{12\pi}\oint dX\\{\zeta_{\alpha}(X),X\\},$ (46)

where again the integral’s contour encloses the second slit, and
$\\{f(x),x\\}=\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}-\frac{3}{2}(\frac{f^{\prime\prime}(x)}{f^{\prime}(x)})^{2}$
is the Schwarzian derivative. Note that $\zeta_{\alpha}(X)$ denotes the
composed conformal transformation leading from the $X$ coordinates to the
$\zeta_{\alpha}$ coordinates. In terms of the free energy, the Renyi entropy
is finally given by

$\displaystyle
S_{\alpha}=S_{\alpha}^{\text{annu}}+S_{\alpha}^{\text{geom}}=\frac{1}{\alpha-1}\left[F_{\alpha}^{\text{annu}}+F_{\alpha}^{\text{geom}}-\alpha(F_{1}^{\text{annu}}+F_{1}^{\text{geom}}\right)].$
(47)

With these results in hand, we are now ready to analyze the two cases in more
detail and compute their respective entanglement entropies.

###  3.1 Infinite intervals: one-sided measurement

The position of the two slits in $X$ coordinates in the one-sided measurement
case is given in equation (94) and (95). The conformal transformation from $X$
to $\tilde{X}$ is a simple Mobius transformation

$\displaystyle\tilde{X}(X)=\frac{2a}{k}\frac{X}{aX+1}-\frac{1}{k}$ (48)

where we have defined

$\displaystyle a=\frac{1}{\sqrt{h(h+l)}},\quad
k=\frac{l+2h-2\sqrt{h(l+h)}}{l},$ (49)

for $h$ the length of the first slit and $l$ the distance between the two
slits in $X$, and noting that the second slit has infinite length. Using
equations (94) and (95), the explicit expressions for $h$ and $l$ are

$\displaystyle h=e^{-\frac{\pi\Delta L}{\beta}},\quad l=e^{\frac{\pi\Delta
L}{\beta}}-e^{-\frac{\pi\Delta L}{\beta}}.$ (50)

The geometric contribution can now be obtained in implicit form by evaluating
equation (46). Noting that there are poles at $\tilde{X}=\pm 1,\pm 1/k$ and
that the contour integral picks up two poles at $\tilde{X}=1,1/k$, we arrive
at rajabpour2016entanglement

$\displaystyle\frac{\delta F_{\alpha}^{\text{geom}}}{\delta
l}=\frac{c}{6}\frac{a(1-k)\left[-2\pi^{2}+(1+6k+k^{2})\alpha^{2}K^{2}(1-k^{2})\right]}{16k(1+k)\alpha
K^{2}(1-k^{2})}.$ (51)

This equation cannot be solved analytically for general values of $h$ and $l$,
so we will focus on the $l\gg h$ ($\Delta L\gg\beta$, or $s\gg 1$) and $l\ll
h$ ($\Delta L\ll\beta$, or $s\ll 1$) limits, which correspond to being safely
in the thermal AdS and BTZ black hole phases, respectively.141414This can be
easily seen by inspection of Figs. 8 and 12: the thermal AdS (BTZ) phase is
dominant for large (small) values of $s$, and $s$ is a monotonically
increasing function of the ratio $\Delta L/\beta$.

####  3.1.1 $\Delta L\gg\beta$

In the limit $l\gg h$, we can expand equation (51) at leading order in $h/l$
and then integrate over $l$. The result at leading order reads

$\displaystyle
F_{\alpha}^{\text{geom}}=\frac{c(\alpha^{2}-1)}{{12}\alpha}\log\frac{l}{h}.$
(52)

Since in this limit we have $s\gg 1$, the last term in the annulus free energy
(45) is cancelled by the $-c\log(\tilde{q}_{\alpha})/24$ term, and the leading
order annulus free energy is simply given by $F_{\alpha}^{\text{annu}}=-2\log
b_{0}$. Summing such annulus contribution to the geometric free energy (52),
the Renyi entropy (47) therefore gives

$\displaystyle\begin{split}S_{\alpha}&=\frac{(1+\alpha)c}{12\alpha}\log\frac{l}{h}+2\log
b_{0}\\\ &\approx\frac{(1+\alpha)c}{12\alpha}\frac{2\pi\Delta L}{\beta}+2\log
b_{0}\end{split}$ (53)

where $\log b_{0}$ is the boundary entropy affleck1991universal ;
rajabpour2016entanglement  and in the last equality we used equation (50) and
kept only leading order terms in $\Delta L\gg\beta$.

####  3.1.2 $\Delta L\ll\beta$

In the limit $l\ll h$, we can compute the geometric contribution to the free
energy by expanding equation (51) at leading order in $l/h$ and integrating
over $l$. This leads to

$\displaystyle
S_{\alpha}^{\text{geom}}=\frac{1}{\alpha-1}\left(F_{\alpha}^{\text{geom}}-\alpha
F_{1}^{\text{geom}}\right)$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha)\pi^{2}c}{12\alpha\log\frac{16h}{l}}\approx\frac{(1+\alpha)\pi^{2}c}{12\alpha\log\frac{8\beta}{\pi\Delta
L}}$ (54)

where in the last equality we used equation (50) and kept only leading order
terms in $\Delta L\ll\beta$. Next, we consider the annulus free energy. In the
limit $\Delta L\ll\beta$, according to (42), we have

$\displaystyle s=\frac{\pi}{\log\frac{8\beta}{\pi\Delta L}},$ (55)

which implies $q_{\alpha}\ll 1$. Using equation (55), the contribution to the
Renyi entropy that comes from the last term of the annulus free energy (44) is

$\displaystyle-\frac{(1+\alpha)c\pi
s}{12\alpha}=-\frac{(1+\alpha)\pi^{2}c}{12\alpha\log\frac{8\beta}{\pi\Delta
L}}.$ (56)

This term exactly cancels the geometric contribution (54). The
$c\log(q_{\alpha})/24$ term also gives a vanishing contribution to the Renyi
entropy. We are therefore left with only one term in (44) which we can write
as

$\displaystyle\log\left(1+\sum_{j}n_{j}q^{\Delta_{j}}_{\alpha}\right)\approx
n_{1}e^{-\Delta_{1}\frac{2\pi\alpha}{s}}=n_{1}\left(\frac{8\beta}{\pi\Delta
L}\right)^{-2\alpha\Delta_{1}},$ (57)

where in the first step we assumed that $\Delta_{1}$ is the smallest scaling
dimension, and in the second step we used the relation (55) for $s$. This
gives the Renyi entropy,

$\displaystyle
S_{\alpha}=\begin{cases}\frac{\alpha}{\alpha-1}n_{1}\left(\frac{8\beta}{\pi\Delta
L}\right)^{-2\Delta_{1}}\quad&\alpha>1\\\
2\Delta_{1}n_{1}\left(\frac{8\beta}{\pi\Delta
L}\right)^{-2\Delta_{1}}\log\frac{8\beta}{\pi\Delta L}\quad&\alpha=1\\\
\frac{1}{1-\alpha}n_{1}\left(\frac{8\beta}{\pi\Delta
L}\right)^{-2\alpha\Delta_{1}}\quad&\alpha<1\end{cases}$ (58)

To compare these results with those from the bulk dual construction in Section
2.2, we focus on the von Neumann entropy (e.g. the $\alpha=1$ case). From the
CFT calculations above we obtained the von Neumann entropy

$\displaystyle S^{lr}=\begin{cases}\frac{c}{6}\frac{2\pi\Delta L}{\beta}+2\log
b_{0}\quad&\Delta L\gg\beta\\\ 2\Delta_{1}n_{1}\left(\frac{\pi\Delta
L}{8\beta}\right)^{2\Delta_{1}}\log\frac{8\beta}{\pi\Delta L}\quad&\Delta
L\ll\beta\end{cases},$ (59)

where the $lr$ superscript is a reminder that this is the entanglement entropy
between left and right sides after measurement.

We can compare this with the expression (30) for the mutual information
obtained computing the holographic entanglement entropy in the bulk dual
setup. In the $\Delta L\gg\beta$ limit we have $s\approx\frac{2\Delta
L}{\beta}$. By further identifying $c=\frac{3R}{2G_{N}}$ and the boundary
entropy with the contribution dependent on the brane tension $T$
takayanagi2011holographic ; fujita2011aspects , we recover the result obtained
for the thermal AdS phase of the gravity calculation of section 2.2151515We
remind that $I^{lr}=2S^{lr}$ because the full system is in a pure state. (see
the second line of equation (30)).

In the BTZ phase, the classical bulk calculation gives vanishing mutual
information between the two sides. This is what we should expect from a pure
classical gravity calculation in the bulk: the $\Delta L\ll\beta$ result found
here comes purely from the operator content in the CFT calculation and is
independent of the central charge. It is therefore a subleading-in-$N$
contribution which we can expect to be captured by the holographic
entanglement entropy only if we include bulk quantum fields (it would in fact
be given by the bulk fields’ entanglement entropy).

###  3.2 Infinite intervals: two-sided measurement

The slits in the two-sided measurement case are given in equations (103) and
(104). The map from the $X$ to the $\tilde{X}$ coordinates is achieved via a
Mobius transformation:

$\displaystyle\tilde{X}=\frac{-X/k+a}{X+a},$ (60)

with

$\displaystyle a=-h-\sqrt{h(l+h)},\quad k=\frac{l+2h-2\sqrt{h(l+h)}}{l},$ (61)

where $h$ is the length of the second slit and $l$ is the distance between the
two slits in $X$ coordinates:

$\displaystyle h=e^{-\frac{\pi\Delta L}{\beta}},\hskip
56.9055ptl=e^{\frac{\pi\Delta L}{\beta}}.$ (62)

The geometric contribution can be obtained by evaluating (46). Noting that
there are poles at $\tilde{X}=\pm 1,\pm 1/k$ and that the contour integral
picks up two poles at $\tilde{X}=1,1/k$, we arrive at

$\displaystyle\frac{\delta F_{\alpha}^{\text{geom}}}{\delta
l}=\frac{c\alpha}{6}\left(\frac{(1-k)\pi^{2}}{4ak(1+k)^{2}\alpha^{2}K^{2}(1-k^{2})}+\frac{(k-1)(1+6k+k^{2})}{8ak(1+k)^{2}}\right).$
(63)

Once again, equation (63) does not have an analytic solution for general $l$
and $h$. We will then discuss again two limits, namely $\Delta L/\beta\gg 1$
($l\gg h$, or $s\gg 1$) and $\Delta L/\beta\ll-1$ ($l\ll h$, or $s\ll 1$).

####  3.2.1 $\Delta L/\beta\gg 1$

In the limit $l\gg h$, we get the geometric contribution to the free energy by
expanding equation (63) as a function of $h/l$ and then integrating over $l$.
Then combining the contribution from the annulus free energy, we arrive at

$\displaystyle S_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha)c}{12\alpha}\log\frac{l}{h}+2\log b_{0},$ (64)
$\displaystyle\approx$
$\displaystyle\frac{(1+\alpha)c}{12\alpha}\frac{2\pi\Delta L}{\beta}+2\log
b_{0}.$ (65)

This is similar to the one-sided measurement result, where $\log b_{0}$ is the
boundary entropy originated from the annulus contribution affleck1991universal
; rajabpour2016entanglement . In the second line we used equation (62) and
took the limit $\Delta L/\beta\gg 1$.

####  3.2.2 $\Delta L/\beta\ll-1$

In the limit $l\ll h$, we can expand equation (63) as a function of $l/h$ and
integrate over $l$ to get the geometric free energy. This leads to

$\displaystyle
S_{\alpha}^{\text{geom}}=\frac{1}{\alpha-1}\left(F_{\alpha}^{\text{geom}}-\alpha
F_{1}^{\text{geom}}\right)$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha)\pi^{2}c}{12\alpha\log\frac{16h}{l}}\approx-\frac{(1+\alpha)\pi
c}{24\alpha}\frac{\beta}{\Delta L},$ (66)

where in the second equality we used equation (62) and $\Delta L/\beta\ll-1$.
Next, we consider the annulus free energy. In this limit, from equation (42)
we have

$\displaystyle s=-\frac{\beta}{2\Delta L},$ (67)

which also indicates $q_{\alpha}\ll 1$. Using this relation, the contribution
to the Renyi entropy that comes from the last term in (44) is

$\displaystyle-\frac{(1+\alpha)c\pi s}{12\alpha}=\frac{(1+\alpha)\pi
c}{24\alpha}\frac{\beta}{\Delta L}.$ (68)

This term exactly cancels the geometric contribution (66).

What is left is then again the first term in (44), which we can write as (like
in the one-sided case, the contribution from the $c\log(q_{\alpha})/24$ term
is vanishing)

$\displaystyle\log\left(1+\sum_{j}n_{j}q^{\Delta_{j}}_{\alpha}\right)\approx
n_{1}e^{-\Delta_{1}\frac{2\pi\alpha}{s}}=n_{1}e^{2\alpha\Delta_{1}\frac{2\pi\Delta
L}{\beta}},$ (69)

where in the second equality we used the relation (67). This gives the Renyi
entropy

$\displaystyle
S_{\alpha}=\begin{cases}\frac{\alpha}{\alpha-1}n_{1}e^{2\Delta_{1}\frac{2\pi\Delta
L}{\beta}}\quad&\alpha>1\\\ -2\Delta_{1}n_{1}\frac{2\pi\Delta
L}{\beta}e^{2\Delta_{1}\frac{2\pi\Delta L}{\beta}}\quad&\alpha=1\\\
\frac{1}{1-\alpha}n_{1}e^{2\alpha\Delta_{1}\frac{2\pi\Delta
L}{\beta}}\quad&\alpha<1\end{cases}$ (70)

Note that in this limit $\Delta L<0$. As we have discussed, this means that
there is no unmeasured region corresponding to the same range of $x$
coordinate in both CFTs (intuitively, we have measured “more than half" of
both sides, see Fig. 18 for an illustration). Notice that in the two-sided
measurement case of this subsection the entanglement entropy between the two
sides decays exponentially in $\Delta L/\beta$ as opposed to the power-law
behavior (58) found in the one-sided measurement case of the previous
subsection.

To compare these results with the gravity calculation, we can again focus on
the von Neumann entropy. From the CFT calculations we obtained the von Neumann
entropy

$\displaystyle S^{lr}=\begin{cases}\frac{c}{6}\frac{2\pi\Delta L}{\beta}+2\log
b_{0}\quad&\Delta L/\beta\gg 1\\\ -2\Delta_{1}n_{1}\frac{2\pi\Delta
L}{\beta}e^{2\Delta_{1}\frac{2\pi\Delta L}{\beta}}\quad&\Delta
L/\beta\ll-1\end{cases}$ (71)

Noticing that $s\approx\frac{2\Delta L}{\beta}$ in the $\Delta L/\beta\gg 1$
limit and using the relation $c=\frac{3R}{2G_{N}}$, we obtain the same result
obtained from the bulk holographic entanglement entropy calculation in the
thermal AdS phase, see equation (30). Similar to the one-sided case, the bulk
calculation gives vanishing mutual information between the two sides in the
BTZ phase. As we have already explained, this is expected because we neglected
the contribution of bulk quantum fields to the holographic entanglement
entropy.

##  4 Discussion

In this paper we considered local projective measurements performed on the TFD
state of two copies of a 2D CFT on a line and studied the effects of the
measurement on the entanglement structure of the microscopic system as well as
on the bulk dual spacetime. We found that measuring finite intervals in one of
the CFTs is not enough to completely disentangle the two CFTs, and the
corresponding dual Lorentzian spacetime remains connected. On the other hand,
measuring semi-infinite intervals in one or both the CFTs triggers an
entangled/disentangled phase transition between the two CFTs, corresponding to
a connected/disconnected phase transition in the bulk dual geometry. The
disconnected phase is dominant when the measured regions are large enough and
the boundary entropy of the Cardy state we postselect on (corresponding to the
brane tension in the bulk description) is sufficiently small or negative.
Therefore, we conclude that measurement of infinite subregions of the CFTs can
destroy the Einstein-Rosen bridge connecting the two AdS asymptotic boundaries
in the double-sided BTZ black hole geometry dual to the TFD state. We also
quantified these statements by computing the post-measurement holographic
mutual information between the two CFTs, and verified via a microscopic
calculation performed in the dual CFT system, which yielded compatible
results. Finally, we showed how inserting heavy operators in the boundary
Euclidean path integrals preparing the microscopic state in the absence and
presence of measurement allows us to detect whether, when a measurement is
performed, the information associated with such operator insertions is
“teleported” (in the sense of the “bulk teleportation” studied in
Antonini:2022sfm ) from one CFT to the other, erased, or neither. The results
of the present paper extend those obtained in Antonini:2022sfm  and represent
a higher-dimensional analogue of those presented in Antonini:2022lmg .

Several open questions still remain, which could orient the direction of
future research on this topic. First, it would be interesting to understand
whether postselection procedures such as those investigated here facilitate
the reconstruction of regions of bulk spacetime behind a black hole horizon.
In fact, as we have seen, when one or two semi-infinite intervals are measured
in, say, the left CFT, information that without measurement would be
accessible from the left CFT becomes accessible from the right CFT. In other
words, part of the pre-measurement entanglement wedge of the left CFT becomes
part of the post-measurement entanglement wedge of the right CFT. Naively, one
would expect the reconstruction of physics in such a “teleported” region to be
extremely complex because it involves accessing information that, at least in
the pre-measurement geometry, sits behind a horizon. However, provided that
the observer on the right has knowledge of the measurement outcome (i.e. of
the specific Cardy state we postselected on), it could be possible to
reconstruct physics in this region in a much simpler way by evolving the right
CFT in real time with a modified Hamiltonian conditioned on the measurement
outcome, similar to the work of Kourkoulou and Maldacena kourkoulou2017pure .
This suggests that (partial) knowledge of the bulk state behind a horizon
could largely simplify reconstruction of behind-the-horizon physics in various
setups, possibly also in Python’s lunch geometries. However, the
implementation of this idea in our setup requires the definition of a non-
singular Lorentzian geometry associated with the post-measurement state of
interest, which in turn requires a well-defined regularization of the
singularities arising in our setup and discussed in Sections 1 and 2. A second
interesting direction would be to study tensor network models able to
reproduce the results we obtained. This could shed light on the consequences
of the measurement procedure for the complexity of bulk reconstruction, and in
particular of the reconstruction of behind-the-horizon physics.

It would also be interesting to investigate whether the effects of heavy
operator insertions which in our purely geometric approximation are “erased”
by the measurement (in the sense we have explained) could still be accessible
when bulk fields are included in our analysis. For example, a possible way to
access the heavy operator insertion is to insert bulk operators in the left or
right CFT on the time reversal invariant slice, which can be done, for
instance, using HKLL reconstruction hamilton2006holographic , and calculate
the matrix element between these two insertions: the heavy operator insertion
in the Euclidean past and the bulk operator insertion on the time reversal
invariant slice. It is natural to expect that the matrix element increases to
an order one number in a proper normalization, when the geodesic from the
heavy operator insertion is inside the entanglement wedge of either CFT.
Nevertheless, the matrix element can be small but nonvanishing even if the
disconnected geodesic from the heavy operator insertion ends on the brane. It
would be interesting to investigate the behavior of the matrix element and its
implication on the accessibility of the information of the heavy operator
insertion.

Finally, a general and fundamental question is how generic the prescriptions
used here and in numasawa2016epr ; Antonini:2022sfm  are. In particular, the
presence of ETW branes in the bulk description of the post-measurement state
seems to be strictly related to the choice of postselecting on Cardy states.
The dual description of different classes of measurement, however, are yet
unknown. At the current stage, it is unclear whether the constructions used
here could be generalized to describe more complicated setups by including
additional elements (for instance branes that are not pure-tension, or
additional modifications of the bulk spacetime besides the insertion of
branes, such as the inclusion of shock waves or additional matter), or if a
completely different framework is needed. Equally intriguing is the
possibility that holographic tools similar to those employed here could be
applied to describe the physics of decoherence Zou:2023rmw .

###### Acknowledgements.

We would like to thank Raphael Bousso, Alexey Milekhin, Vincent Su for helpful
discussions. We acknowledge support from the Simons Foundation via It From
Qubit (S-K.J.), from the U.S. Department of Energy grant DE-SC0009986 (S-K.J.
and B.G.S.), from the AFOSR under FA9550-19-1-0360 (B.G-W.), and from the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research, Accelerated Research for Quantum Computing program “FAR-
QC” (S.A.). Part of S.-K. J.’s work is supported by a startup fund at Tulane
University.

##  Appendix A Details of the conformal transformations

In this appendix we will delve into greater detail about the conformal
transformations used in the main sections of the paper above. To better
understand the effect of each conformal transformation, we will describe how
the slits, the time reflection symmetric lines, and $x=\pm\infty$ are mapped
through the different coordinates in the various measurement configurations
considered above.

###  A.1 Finite intervals

We start with the conformal transformations used to describe measurement of
finite intervals considered in Section 2.1 and illustrated in Fig. 3.

Firstly, the infinitely long cylinder in $(x,y)$ coordinates (Fig. 3 (a)) is
mapped to a 2D plane in the $X$ coordinate system161616When, like in this
case, we indicate a coordinate system by a single symbol, it is understood
that we are considering complex coordinates, e.g. $X$ coordinate stands for
the $(X,\bar{X})$ coordinate system. (Fig. 3 (b)) by

$\displaystyle
X=e^{\frac{2\pi}{\beta}(x+iy)},\quad\bar{X}=e^{\frac{2\pi}{\beta}(x-iy)}.$
(72)

Under this conformal map, the two parallel slits are mapped to two radial
slits with angles $\pm\phi/2$, where we define

$\displaystyle\phi=2\pi\Delta T/\beta.$ (73)

Also, notice that both of the slits span from $|X|=e^{-\pi\Delta L/\beta}$ to
$|X|=e^{\pi\Delta L/\beta}$. The time reflection symmetric lines are both
mapped to the horizontal line $X=\bar{X}$, with negative (positive) infinity
in $x$ coordinates mapped to the origin (infinity) in $X$ coordinates.

Secondly, the 2D plane in $X$ coordinates with two radial slits (Fig. 3 (b))
is mapped to an annulus in $\zeta$ coordinates (Fig. 3 (c)), in which the two
radial slits are mapped to the two circular edges of the annulus. The
conformal transformation mapping the annulus in $\zeta$ with external radius
$|\zeta|=1$ and internal radius $|\zeta|=\rho<1$ to two radial slits in $X$ is
given by crowdy2006conformal :

$\displaystyle
X(\zeta)=C(\alpha,\beta)\frac{(\zeta-\alpha)(\zeta-\bar{\alpha}^{-1})}{(\zeta-\beta)(\zeta-\bar{\beta}^{-1})}\prod_{k=1}^{\infty}\frac{(1-\rho^{2k}\frac{\zeta}{\alpha})(1-\rho^{2k}\frac{\alpha}{\zeta})(1-\rho^{2k}\bar{\alpha}\zeta)(1-\rho^{2k}\frac{1}{\bar{\alpha}\zeta})}{(1-\rho^{2k}\frac{\zeta}{\beta})(1-\rho^{2k}\frac{\beta}{\zeta})(1-\rho^{2k}\bar{\beta}\zeta)(1-\rho^{2k}\frac{1}{\bar{\beta}\zeta})}$
(74)

where $\zeta=\alpha$ is mapped to $X=0$, and $\zeta=\beta$ to $|X|=\infty$.
$C(\alpha,\beta)$ is a normalization prefactor. By choosing
$\alpha=\sqrt{\rho}e^{i\phi}$, $\beta=\sqrt{\rho}$, and $C=e^{-i3\phi/2}$, the
two circles are mapped to two finite length radial slits, and equation (74)
can be simplified to

$\displaystyle
X(\zeta)=e^{-i\frac{\phi}{2}}\frac{\theta_{4}(\frac{-i\log\zeta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{-i\log\zeta}{2},\sqrt{\rho})},$
(75)

where $\theta_{i}(\varphi,q)$ is the Jacobi theta function. In our
conventions, the theta functions are defined as

$\displaystyle\theta_{3}(\varphi,q)=\theta_{4}(\varphi+\frac{\pi}{2},q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}e^{i2n\varphi}$
$\displaystyle=\prod_{m=1}^{\infty}(1-q^{2m})\left(1+q^{2m-1}e^{i2\varphi}\right)\left(1+q^{2m-1}e^{-i2\varphi}\right),$
(76)
$\displaystyle\theta_{2}(\varphi,q)=\theta_{1}(\varphi+\frac{\pi}{2},q)=\sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^{2}}e^{i(2n+1)\varphi}$
$\displaystyle=2q^{1/4}\cos\varphi\prod_{m=1}^{\infty}(1-q^{2m})\left(1+q^{2m}e^{i2\varphi}\right)\left(1+q^{2m}e^{-i2\varphi}\right).$
(77)

The conformal transformation (75) is determined by two parameters, $\rho$ and
$\phi$. $\phi$ is given by (73); in order to see how $\rho$ is related to the
slit parameters, we note that the first (second) slit in $\zeta$ is given by
$|\zeta|=1$ ($|\zeta|=\rho$). Setting $\zeta=re^{i\theta}$, the two slits are

$\displaystyle\text{first slit:
}X(e^{i\theta})=e^{-i\frac{\phi}{2}}\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})},\quad\theta\in(0,2\pi),$
(78) $\displaystyle\text{second slit: }X(\rho
e^{i\theta})=e^{i\frac{\phi}{2}}\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})},\quad\theta\in(0,2\pi).$
(79)

Since $\theta_{4}$ is a real function when its arguments are real, as $\theta$
moves from $\theta=0$ to $\theta=2\pi$, $X(e^{i\theta})$ ($X(\rho
e^{-i\theta})$) is a radial slit with angle $-\phi/2$ ($\phi/2$). According to
(72), the slits span from $|X|=e^{-\pi\Delta L/\beta}$ to $|X|=e^{\pi\Delta
L/\beta}$. Thus, when $\theta$ moves from $\theta=0$ to $\theta=2\pi$, the
function
$\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})}$
will move between $e^{-\pi\Delta L/\beta}$ and $e^{\pi\Delta L/\beta}$, i.e.,

$\displaystyle\frac{\theta_{4}(\frac{\theta_{\text{max}}-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta_{\text{max}}}{2},\sqrt{\rho})}\equiv\max_{\theta\in(0,2\pi)}\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})}=e^{\pi\Delta
L/\beta},$ (80)
$\displaystyle\frac{\theta_{4}(\frac{\theta_{\text{min}}-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta_{\text{min}}}{2},\sqrt{\rho})}\equiv\min_{\theta\in(0,2\pi)}\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})}=e^{-\pi\Delta
L/\beta},$ (81)

where the first equality should be regarded as the definition of
$\theta_{\text{max}}$ and $\theta_{\text{min}}$, and the second equality gives
a relation between $\rho$ and the slit parameters $\Delta L$ and $\Delta T$.
Unfortunately, there is no simple expression for this relation, but as an
illustration, we plot $\frac{\Delta L}{\Delta T}$ as a function of $\rho$ in
Fig. 4 for $\phi=0.2$. Note that when $\Delta T\ll\Delta L$, $\rho\rightarrow
1$. In particular, when $\Delta T\rightarrow 0$, the slit will reduce to the
case in Fig. 2 (a), and the dual spacetime is described by an ETW brane
anchored at the slit boundaries. In the following, we give a simple relation
near $\rho=1$.

The asymptotic form of $\theta_{4}(\varphi,q)$ near $q=1$ is
banerjee2016lambert

$\displaystyle\theta_{4}(\varphi,q)\approx\sqrt{\frac{\pi}{\log
q^{-1}}}e^{\frac{((\varphi)_{\pi}-\pi/2)^{2}}{\log
q}},\quad(\varphi)_{\pi}=\varphi\text{ mod }\pi,$ (82)

from which, we arrive at

$\displaystyle\frac{\theta_{4}(\frac{\theta-\phi}{2},\sqrt{\rho})}{\theta_{4}(\frac{\theta}{2},\sqrt{\rho})}\approx\exp\left[\frac{((\theta-\phi)_{2\pi}-\pi)^{2}-((\theta)_{2\pi}-\pi)^{2}}{2\log\rho}\right].$
(83)

In this case, it is not hard to see that $\theta_{\text{min}}=\phi$ and
$\theta_{\text{max}}=0$, which according to (80) leads to
$\frac{2\pi}{\beta}\Delta L=-\frac{(2\pi-\phi)\phi}{\log\rho}$. Thus we have
the following relation near $\rho=1$,

$\displaystyle\rho=\exp\left[-2\pi\frac{\Delta T}{\Delta
L}\left(1-\frac{\Delta
T}{\beta}\right)\right],\quad\phi=\frac{2\pi}{\beta}\Delta T.$ (84)

Thus, $\log\rho\geq-\frac{\pi}{2}\frac{\beta}{\Delta L}$.171717The exponent on
the right hand side of equation (84) is minimized for $\Delta T=\beta/2$, and
this approximation is therefore valid in the limit $\frac{\Delta L}{\beta}\gg
1$. We can then introduce another parameter $s$ defined as

$\displaystyle\log\rho=-\pi s,$ (85)

which near $\rho=1$ takes the form

$\displaystyle s=\frac{2\Delta T}{\Delta L}\left(1-\frac{\Delta
T}{\beta}\right),$ (86)

and so is roughly the ratio between the imaginary time evolution and the
length of the measured interval. As we have seen in Section 2.1.2, a phase
transition in the brane configuration in the bulk—corresponding to an
entanglement phase transition in the dual CFT—can be characterized in terms of
a critical value $s_{c}(T)$ of the parameter $s$ we just introduced. Note that