11institutetext: University of Maryland, College Park, MD, 20742,
USA22institutetext: Department of Physics, Brandeis University, Waltham,
Massachusetts 02453, USA

# Holographic measurement and quantum teleportation in the SYK thermofield
double

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

According to holography, entanglement is the building block of spacetime;
therefore, drastic changes of entanglement will lead to interesting
transitions in the dual spacetime. In this paper, we study the effect of
projective measurements on the Sachdev-Ye-Kitaev (SYK) model’s thermofield
double state, dual to an eternal black hole in Jackiw-Teitelboim (JT) gravity.
We calculate the (Renyi-2) mutual information between the two copies of the
SYK model upon projective measurement of a subset of fermions in one copy. We
propose a dual JT gravity model that can account for the change of
entanglement due to measurement, and observe an entanglement wedge phase
transition in the von Neumann entropy. The entanglement wedge for the
unmeasured side changes from the region outside the horizon to include the
entire time reversal invariant slice of the two-sided geometry as the number
of measured Majorana fermions increases. Therefore, after the transition, the
bulk information stored in the measured subsystem is not entirely lost upon
projection in one copy of the SYK model, but rather teleported to the other
copy. We further propose a decoding protocol to elucidate the teleportation
interpretation, and connect our analysis to the physics of traversable
wormholes.

##  1 Introduction

The Anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence
Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc ; Aharony:1999ti  is one of
the most promising frameworks for describing quantum gravity. Within AdS/CFT,
and more generally holography, quantum entanglement is deeply interconnected
with spacetime geometry, with the structure of boundary entanglement dictating
the structure of spacetime Ryu2006a ; Ryu2006b ; Hubeny:2007xt ;
Swingle:2009bg ; VanRaamsdonk:2010pw ; Maldacena:2013xja ; Engelhardt:2014gca
; Dong:2016eik ; Harlow:2016vwg . It is therefore reasonable to expect that
radical changes in the entanglement structure of the boundary theory, such as
those arising from a projective measurement performed on a subsystem of the
boundary, lead to radical modifications of both the dual geometry and the way
bulk information is encoded in the boundary theory. A comprehensive
description of this phenomenon in AdS3/CFT2 and related tensor network models
was recently given in Antonini:2022sfm  (see also numasawa2016epr  for earlier
work). There, it was found that performing a projective measurement on a
boundary subregion $A$ teleports the bulk information contained in $A$ into
the complementary unmeasured region $A^{c}$. In particular, while the
projective measurement destroys the entanglement structure of $A$, much of the
bulk information encoded in $A$ (i.e. much of the entanglement wedge of $A$)
is not erased. Instead, if the mesaurement outcome is known, this information
becomes accessible from $A^{c}$ after the measurement, thanks to the pre-
measurement entanglement resource between the two.111For this reason, we
should think of the projective measurement studied in Antonini:2022sfm  as
postselection. We will see in Section 4 that knowledge of the measurement
outcome is necessary to implement the teleportation protocol in our SYK setup.

In this paper, we study a related question in lower-dimensional holography. We
consider two copies (the “right” and “left” sides) of the SYK model
Sachdev:1992fk ; kitaev ; maldacena2016remarks ; maldacena2016conformal  in a
thermofield double (TFD) state maldacena2003eternal ; Maldacena:2018lmt ;
Su:2020zgc , perform a projective measurement on a subset of $M<N$ Majorana
fermions $\psi$ in the right side (where $N$ is the total number of Majorana
fermions in each copy), and ask how such a measurement affects the dual two-
sided black hole in JT gravity Teitelboim:1983ux ; Jackiw:1984je . In
particular, in analogy with the analysis of Kourkoulou and Maldacena
kourkoulou2017pure  222See also Antonini:2021xar  for a generalization to the
complex SYK model., we employ the fermion parity
$M_{k}=-i2\psi_{2k-1}\psi_{2k}$ as our measurement operator. Our results
reduce to the ones obtained in kourkoulou2017pure  in the special case $M=N$,
i.e. when all the Majorana fermions are measured in the left side 333We also
study two-sided measurement. See Appendix C..

To better understand how the bulk changes upon boundary measurement, we first
study the entanglement structure of the post-measurement state by computing
the Renyi-2 mutual information between the left side and the unmeasured
fermions in the right side. We find that it is always non-vanishing unless all
the fermions in the right side are measured. In other words, the left and
right sides always remain entangled with each other, independently of how many
fermions are measured. This can be understood as a consequence of the all-to-
all interaction of the SYK model (see Section 2.3).

We then study the holographic dual theory, JT gravity coupled to bulk matter.
The bulk matter is described by $N$ copies of a CFT, which should be
understood as the bulk dual of the $N$ Majorana fermions in the SYK model. The
effect of a measurement performed on a subset of $M$ fermions in the left
boundary 444Note that in our JT gravity analysis we assume the measurement
occurs in the left side unlike in the SYK analysis (where it is performed in
the right side). This is useful to make a direct comparison with the analysis
of Kourkoulou and Maldacena kourkoulou2017pure . Clearly, given the symmetry
of the TFD state, the two cases are equivalent.  is to create an end-of-the-
world (ETW) brane in the left asymptotic region. In particular, such a brane
is visible only to the bulk matter dual to the $M$ measured Majorana fermions,
which are now described by a boundary conformal field theory (BCFT)
cardy1989boundary . On the other hand, the brane is transparent to the
remaining bulk matter, dual to the $N-M$ unmeasured fermions. In this model,
we reproduce the results obtained in the SYK model by computing the post-
measurement mutual information between the two sides via a prescription for
the quantum extremal surface formula proposed in nezami2021quantum  (and later
discussed in chandrasekaran2022quantum ). This prescription states that the
entanglement wedge of a small number of Majorana fermions $K\ll N$ is given by
a near-boundary region with spatial size of order $\epsilon$, where $\epsilon$
is the UV cutoff of the bulk theory. A complete analysis of this new
prescription will be explored in future work newqes .

Our JT gravity analysis shows that by increasing the number $M$ of measured
Majorana fermions in the left boundary, the entanglement wedge of the right
boundary can grow to include not just the right asymptotic region, but extend
beyond the black hole horizon to include part of the left asymptotic region.
Specifically, when the majority of the left Majorana fermions are measured,
i.e. $M/N=m>m^{*}$, a phase transition occurs such that the entanglement wedge
of the right system jumps to contain the entire time reversal symmetric slice
up to an ETW brane sitting near the left boundary. The transition can be
interpreted as a quantum teleportation process Antonini:2022sfm : a projective
measurement of most of the left system teleports the bulk information it
contains into the right system. Intuitively, this is possible because in the
TFD state, a large subset $M\sim N$ of the left fermions is mostly entangled
with the right system, and very weakly entangled with the remaining $N-M$
unmeasured fermions on the left. This large entanglement resource between the
two sides allows the teleportation to take place. On the other hand, if only a
small subset $m<m^{*}$ of the left Majorana fermions are measured, the bulk
information contained in the $M$ measured fermions is teleported into the
unmeasured $N-M$ fermions in the same side. Therefore, the quantum extremal
surface is still located at the bifurcation point. See Fig. 1 for an
illustration. We also construct a simple toy model (see Appendix E) using Haar
random unitaries to show that the critical value of $M$ is determined by the
pre-measurement entanglement between the two sides. Specifically, if the
entanglement between the two sides is large, information can be teleported
more easily and the transition occurs at a smaller value of $m^{*}$.

We then proceed to make this bulk teleportation interpretation more precise.
First, we build a decoding operator able to reconstruct from the right system
information that, before the measurement, was contained in the left system.
Crucially, the decoding operator depends on the measurement outcome, as
expected. This measurement-decoding protocol can be understood as a quantum
channel acting on the pre-measurement state. Second, we study correlation
functions between operators inserted in the two sides in this setup. This
quantity is closely related to teleportation fidelity. In the large-$q$ limit
(where $q$ is the number of directly interacting fermions, see Section 2.1),
we find that at low-temperature the left-right correlation function is
enhanced to nearly its maximum value by this quantum channel. We also relate
our results to traversable wormhole setups, where the measurement-decoding
channel effectively plays the role of the two-sided deformation that makes the
wormhole traversable gao2017traversable ; maldacena2017diving ;
brown2019quantum ; nezami2021quantum ; gao2021traversable . Finally, we
provide an encoding method that reaches nearly perfect fidelity in this
teleportation protocol. By constructing such an explicit decoding protocol, we
put the teleportation and entanglement wedge transition induced by projective
measurements on a firmer ground.

(a) $m<m_{*}$

(b) $m>m_{*}$

Figure 1: Measurement-induced entanglement wedge transition. Two copies of the
SYK model, denoted by left and right sides, are in a thermofield double state,
which is dual to a two-sided eternal $AdS_{2}$ black hole. A subset of the
left Majorana fermions are measured, as indicated by blue dots. An end-of-the-
world brane is created in the left asymptotic region by the measurement. The
entanglement wedge of all of the right Majorana fermions is indicated by the
tan region. A small gray segment on the left side indicates the cutoff region
(see discussion in Section 3.2). (a) When the number of measured Majorana
fermions is less than a critical value, $M/N=m<m_{*}$, the quantum extremal
surface is located at the bifurcation surface. (b) When the number of measured
Majorana fermions exceeds a critical value, $M/N=m>m_{*}$, the entanglement
wedge includes nearly the entire time reversal invariant slice. Note that the
transition from (a) to (b) is a sharp phase transition in the bulk von Neumann
entropy calculation, while it is a crossover in the SYK Renyi-2 entropy
calculation; these are compared in Figure 9.

The main advantage the lower-dimensional toy-models studied here is the large
amount of control we have on both sides of the duality. For instance, we are
able to explicitly compute quantities in a boundary theory (the SYK model)
which can be regarded as a candidate UV completion of the dual bulk theory. We
can then compare the results with those obtained in the dual gravitational
system in the appropriate limits. On the other hand, thanks to the simplicity
of (1+1)-dimensional JT gravity, it is possible to analytically compute the
bulk generalized entropy in the presence of bulk matter. Both these features
would be remarkably cumbersome to obtain in higher-dimensional theories. Thus,
the present analysis allows us to explicitly realize and further extend the
constructions described in Antonini:2022sfm , gaining much additional insight
into the physical processes taking place in both the boundary and bulk theory
when projective measurements are performed. Moreover, the generalized entropy
prescription proposed here is a first step towards a more general framework
for entanglement in setups that include non-unitary operations such as
measurements.

The rest of the paper is organized as follows. In Section 2 we define the SYK
model and its TFD state and describe the measurement procedure. We then
compute the correlation functions in the post-measurement state using
Euclidean path integral techniques in the saddle point approximation
(analytically in the large-$q$ limit and numerically for any value of $q$). We
then proceed to compute the post-measurement Renyi-2 mutual information
between the two sides using a similar path integral procedure. In Section 3 we
define the bulk dual theory using JT gravity coupled to matter and study the
effects of the projective measurement on the geometry. We compute the post-
measurement mutual information between the two sides using the QES
prescription and describe the entanglement wedge phase transition and its
interpretation in terms of bulk teleportation. In Section 4 we study the
teleportation protocol and its fidelity in detail, and make the connection
between our results and traversable wormholes. In section 5 we draw our
conclusions and discuss future directions. Technical details of our analysis
are contained in Appendices A, D, F, and G. In Appendix E we give a simple toy
model of the bulk teleportation using Haar random unitaries. In Appendix B we
compare our results to the analysis of kourkoulou2017pure ;
zhang2020entanglement . In Appendix C we report SYK model results for a
measurement performed on subsets of fermions in both sides of the TFD state.

##  2 Partial measurement of a SYK thermofield double

In this section, we investigate the entanglement structure of a thermofield
double state of two copies of the SYK model (the “left” and the “right”) after
a measurement is performed on a subset of fermions in one of the two copies
(which we will conventionally identify as the “right”). A similar analysis for
a measurement performed on a subset of fermions in both sides is reported in
Appendix C. We start by defining the measurement operator and describing a
Euclidean path integral representation of the measurement procedure. We then
introduce the customary bilocal field formulation of the SYK model kitaev ;
Sachdev:2015efa ; maldacena2016remarks  and study the saddle point solution
for the post-measurement system. Finally, we compute the post-measurement
Renyi-2 mutual information between the left and the remaining (unmeasured)
fermions on the right. We find that the mutual information decreases as we
increase the fraction of measured fermions, but is always non-vanishing unless
all the right fermions are measured. The analysis of the present section
serves also as a benchmark for the results obtained in the holographic dual
description of Section 3, where we find an analogous behavior for the mutual
information in JT gravity coupled to $N$ copies of a CFT.

###  2.1 Partial measurement and path integral

The SYK model is defined by the all-to-all interacting Hamiltonian

$\displaystyle
H[\psi]=i^{q/2}\sum_{i_{1}<...<i_{q}}J_{i_{1},...,i_{q}}\psi_{i_{1}}...\psi_{i_{q}},$
(1)

where $\psi_{i},i=1,...,N$ denotes a Majorana fermion (satisfying
$\\{\psi_{i},\psi_{j}\\}=\delta_{ij}$ and $\psi_{i}^{\dagger}=\psi_{i}$), and
$J_{i_{1},...,i_{q}}$ gives the coupling between the Majorana fermions
$i_{1},...,i_{q}$. $J_{i_{1},...,i_{q}}$ is a random Gaussian variable with
average and variance given by

$\displaystyle\overline{J_{i_{1},...,i_{q}}}=0,\quad\overline{J_{i_{1},..,i_{q}}J_{i_{1}^{\prime},...,i_{q}^{\prime}}}=\prod_{k=1}^{q}\delta_{i_{k},i_{k}^{\prime}}\frac{J^{2}}{qN^{q-1}},$
(2)

where $J$ sets the overall interaction strength.

We are interested in studying the post-measurement properties of the TFD state
of two copies of the SYK model. The unnormalized TFD state, which is defined
in a doubled Hilbert space, is given by

$\displaystyle|TFD\rangle=e^{-\frac{\beta}{4}(H_{L}+H_{R})}|\infty\rangle$ (3)

where $\beta$ is a parameter representing the inverse temperature of the
thermal system obtained by tracing out one of the two sides,
$H_{L}=H[\psi_{L}]$, $H_{R}=H[i\psi_{R}]$ (with $\psi_{L}$ and $\psi_{R}$
Majorana fermion operators on the left and right Hilbert space respectively),
and $\ket{\infty}$ is the thermofield double state at infinite temperature
defined by $(\psi_{L,i}+i\psi_{R,i})|\infty\rangle=0$ Maldacena:2018lmt .

Given the thermofield double state (3), we now want to perform a projective
measurement on $M$ of the $N$ Majorana fermions in the right system. Inspired
by the local projective measurement considered in $AdS_{3}/BCFT_{2}$
numasawa2016epr ; Antonini:2022sfm  that retains bulk geometry, the
measurement operator of interest here will be the fermion parity operator
$M_{R,k}$ formed by the $2k-1$ and $2k$ right Majorana fermions in analogy
with the analysis of Kourkoulou and Maldacena kourkoulou2017pure :

$\displaystyle M_{R,k}=-i2\psi_{R,2k-1}\psi_{R,2k}$ (4)

where the factor of 2 accounts for the chosen normalization of the Majorana
fermions and we assume for simplicity that $M$ is an even integer.
Intuitively, if we divide Majorana fermions on the right side in pairs
$(\psi_{R,2k-1},\psi_{R,2k})_{k=1,...,N/2}$ and identify each pair with a
complex fermion $\tilde{\psi}_{R,k}$, the fermion parity (4) corresponds to
the Pauli-$Z$ operator $\sigma_{R,k}^{z}$ for the $k$-th complex fermion on
the right side. The raising and lowering operators for the complex fermions
are then

$\sigma_{R,k}^{+}=\sqrt{2}\left(\psi_{R,2k-1}+i\psi_{R,2k}\right),\quad\sigma_{R,k}^{-}=\sqrt{2}\left(\psi_{R,2k-1}-i\psi_{R,2k}\right).$
(5)

To perform a projective measurement on the first $M$ fermions on the right
side we must measure the fermi parity (4) for $k=1,...,M/2$. The measurement
operators for different $k$ commute. The eigenvalues of each operator are
$r_{k}=\pm 1$ 555Corresponding to a spin-up ($r_{k}=+1$) or spin-down
($r_{k}=-1$) state for the complex fermion.. Therefore, the measurement record
is given by a string of binary numbers $\textbf{r}=(r_{1},...,r_{M/2})$. In
our analysis below we will study how properties of the post-measurement state
change when varying the number of measurement operators applied on the right
side—given by $M/2$. More precisely, since we are interested in studying the
SYK model in the large-$N$ limit (where it admits a semiclassical bulk dual
spacetime), we define $m\equiv M/N$, take the large-$N$ limit keeping $m$
fixed, and then vary the value of $m$.

Suppose we measure the fermion parity (4) for $k=1,...,M/2$ in the TFD state
(3) and get a measurement outcome $\textbf{r}=(r_{1},...,r_{M/2})$. The post-
measurement state $|R_{\textbf{r}}(m)\rangle$ for the $M$ measured right
Majorana fermions is an eigenstate of all the $M/2$ fermion parity operators
666We take the post-measurement state for the $M$ measured Majorana fermions
to be normalized, i.e. $\langle
R_{\textbf{r}}(m)|R_{\textbf{r}}(m)\rangle=1$.:

$\displaystyle
M_{R,k}|R_{\textbf{r}}(m)\rangle=r_{k}|R_{\textbf{r}}(m)\rangle,\quad
k=1,...,M/2.$ (6)

Using the raising and lowering operators introduced in equation (5), equation
(6) can be rewritten in the form

$\displaystyle\psi_{R,2k-1}|R_{\textbf{r}}(m)\rangle=-ir_{k}\psi_{R,2k}|R_{\textbf{r}}(m)\rangle,\quad
k=1,...,M/2,$ (7)

which will be useful for identifying the correct boundary conditions in the
path integral formalism. Therefore, the full post-measurement (unnormalized)
state of the two-sided system can be obtained by inserting the projection
operator $|R_{\textbf{r}}(m)\rangle\langle R_{\textbf{r}}(m)|$:

$\displaystyle|\Psi_{\textbf{r}}(m)\rangle=\left(\mathds{1}^{(1-m)}\otimes|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right)|TFD\rangle$ (8)

where we use $\mathds{1}^{(1-m)}$ to denote the identity operator acting on
the unmeasured Majorana fermions on the right side 777Here we are omitting the
identity operator which acts on all the Majorana fermions of the left side..
The Born probability of observing this state is
$p_{\textbf{r}}(m)=\frac{\langle\Psi_{\textbf{r}}(m)|\Psi_{\textbf{r}}(m)\rangle}{\langle
TFD|TFD\rangle}$. Note that the simultaneous eigenstates of all the fermion
parity operators (4) for $k=1,...,M/2$ form a complete basis of the measured
subspace, i.e.

$\displaystyle\sum_{\textbf{r}}|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|=\mathds{1}^{(m)},$ (9)

which leads to $\sum_{\textbf{r}}p_{\textbf{r}}(m)=1$, as expected.

Now that we have fully characterized the measurement operator and the post-
measurement state, we can compute the amplitude
$\overline{\langle\Psi_{\textbf{r}}(m)|\Psi_{\textbf{r}}(m)\rangle}$ using a
Euclidean path integral representation. Here, as above, the overline indicates
averaging over the Gaussian couplings (2). The amplitude can be written as

$\displaystyle\begin{aligned}
\overline{\langle\Psi_{\textbf{r}}(m)|\Psi_{\textbf{r}}(m)\rangle}&=\overline{\langle
TFD|\left(|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right)|TFD\rangle}\\\
&=\overline{\text{Tr}_{R}\left[e^{-\beta
H_{R}}|R_{\textbf{r}}(m)\rangle\langle R_{\textbf{r}}(m)|\right]}=\int D\psi
e^{-I},\end{aligned}$ (10)

where in the second equality we used the properties of the TFD state (3),
$\text{Tr}_{R}$ denotes the trace over the right system, and the action
appearing in the Euclidean path integral is given by 888Note that this action
is valid only in the large-$N$ limit, in which replica non-diagonal terms
arising when averaging over the gaussian couplings are suppressed by
$N^{1-q}$.

$\displaystyle-I=-\int
d\tau\sum_{j}\frac{1}{2}\psi_{j}(\tau)\partial_{\tau}\psi_{j}(\tau)+\int
d\tau_{1}d\tau_{2}\frac{J^{2}}{2qN^{q-1}}\left(\sum_{j}\psi_{j}(\tau_{1})\psi_{j}(\tau_{2})\right)^{q}.$
(11)

The insertion of the projection operator $|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|$ in the trace corresponds to the following boundary
conditions in the path integral:

$\displaystyle\begin{aligned}
&\psi_{2k-1}(0)=-ir_{k}\psi_{2k}(0),\quad\psi_{2k-1}(\beta)=ir_{k}\psi_{2k}(\beta),\quad
k=1,...,M/2,\\\ &\psi_{M+j}(0)=-\psi_{M+j}(\beta),\quad
j=1,...,N-M\end{aligned}$ (12)

where we inserted the projection operator at Euclidean time $\tau=0$
(corresponding to the right system). The boundary conditions for the measured
Majorana fermions—i.e. $\psi_{k}$ with $1\leq k\leq M$—reported in the first
line of equation (12) can be obtained from equation (7) and its complex
conjugate 999The boundary conditions can be easily derived by noticing that
the Euclidean path integral prepares the ket state for the right system at
$\tau=0$ and the bra state for the right system at $\tau=\beta$.. On the other
hand, the unmeasured Majorana fermions $\psi_{k}$ with $M+1\leq k\leq N$
satisfy the conventional anti-periodic boundary conditions for fermionic
fields, reported in the second line of equation (12). The boundary conditions
on the Euclidean time circle are illustrated in Fig. 2.

If $m=0$—which means no fermions are measured— the path integral clearly
computes the usual thermal partition function with inverse temperature
$\beta$. On the other hand, if $m=1$—which means that the whole right system
is measured—the path integral can be interpreted as preparing a Kourkoulou-
Maldacena state kourkoulou2017pure  at $\tau=\beta/2$ (see Appendix B).

Figure 2: Different boundary conditions for Majorana fields in the imaginary
time contour. (a) The measurement operator $|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|$ is inserted on the contour at $\tau=0$, which is indicated
by two small triangles, for $\psi_{i},i=1,...,M$. (b) The conventional contour
for $\psi_{j+M},j=1,...,N-M$.

It is now useful to introduce bilocal collective fields, in terms of which the
equations of motion derived in the saddle point approximation of the Euclidean
path integral are substantially simplified kitaev ; Sachdev:2015efa ;
maldacena2016remarks . The form of the boundary conditions (12) suggests that
we should introduce different types of bilocal fields. First, we introduce the
“diagonal” bilocal fields

$\displaystyle\begin{aligned}
&\tilde{G}_{11}(\tau_{1},\tau_{2})=\frac{2}{M}\sum_{k=1}^{M/2}\psi_{2k-1}(\tau_{1})\psi_{2k-1}(\tau_{2}),\qquad\tilde{G}_{22}(\tau_{1},\tau_{2})=\frac{2}{M}\sum_{k=1}^{M/2}\psi_{2k}(\tau_{1})\psi_{2k}(\tau_{2}),\\\
&\tilde{G}_{33}(\tau_{1},\tau_{2})=\frac{1}{N-M}\sum_{j=1}^{N-M}\psi_{M+j}(\tau_{1})\psi_{M+j}(\tau_{2}).\end{aligned}$
(13)

For later convenience, we also introduce the following “off-diagonal” bilocal
fields (similar to the ones introduced in kourkoulou2017pure ):

$\displaystyle\tilde{G}_{12}(\tau_{1},\tau_{2})=\frac{2}{M}\sum_{k=1}^{M/2}r_{k}\psi_{2k-1}(\tau_{1})\psi_{2k}(\tau_{2}),\quad\tilde{G}_{21}(\tau_{1},\tau_{2})=\frac{2}{M}\sum_{k=1}^{M/2}r_{k}\psi_{2k}(\tau_{1})\psi_{2k-1}(\tau_{2}).$
(14)

In terms of these bilocal fields (the arguments of which are in the range
$\tau_{1,2}\in(0,\beta)$), the large-$N$ action becomes

$\displaystyle-I=\int
d\tau_{1}d\tau_{2}\Big{[}-\frac{1}{2}\Big{(}\sum_{k=1}^{M/2}\psi_{2k-1}(\tau_{1})(\partial-\tilde{\Sigma}_{11})\psi_{2k-1}(\tau_{2})+\sum_{k=1}^{M/2}\psi_{2k}(\tau_{1})(\partial-\tilde{\Sigma}_{22})\psi_{2k}(\tau_{2})$
$\displaystyle+\sum_{j=M+1}^{N}\psi_{j}(\tau_{1})(\partial-\tilde{\Sigma}_{33})\psi_{j}(\tau_{2})\Big{)}-\frac{1}{2}\left(\frac{M}{2}\tilde{\Sigma}_{11}\tilde{G}_{11}+\frac{M}{2}\tilde{\Sigma}_{22}\tilde{G}_{22}+(N-M)\tilde{\Sigma}_{33}\tilde{G}_{33}\right)$
$\displaystyle+\frac{J^{2}}{2qN^{q-1}}\left(\frac{M}{2}\tilde{G}_{11}(\tau_{1},\tau_{2})+\frac{M}{2}\tilde{G}_{22}(\tau_{1},\tau_{2})+(N-M)\tilde{G}_{33}(\tau_{1},\tau_{2})\right)^{q}\Big{]},$
(15)

where $\partial=\delta(\tau_{1}-\tau_{2})\partial_{\tau_{2}}$, and we have
introduced the Lagrange multiplier fields
$\tilde{\Sigma}_{11}(\tau_{1},\tau_{2})$,
$\tilde{\Sigma}_{22}(\tau_{1},\tau_{2})$ and
$\tilde{\Sigma}_{33}(\tau_{1},\tau_{2})$ (which play the role of the self-
energy). It is easy to check that integrating out the Lagrange multiplier
fields enforces the definitions (13) of the bilocal fields.

Before deriving the equations of motion, it is useful to define a new Majorana
field zhang2020entanglement

$\displaystyle\chi_{k}(s)=\begin{cases}\psi_{2k-1}(s),\quad&0<s<\beta,\\\
ir_{k}\psi_{2k}(2\beta-s),\quad&\beta<s<2\beta.\end{cases}$ (16)

In the rest of the present subsection and in the next subsection we denote by
$s$ coordinates in the range $(0,2\beta)$ and by $\tau$ coordinates in the
range $(0,\beta)$. The advantage of introducing this new field is that the
boundary conditions (12) reduce to the conventional anti-periodic boundary
conditions for $\chi_{k}(s)$, i.e. $\chi_{k}(0)=-\chi_{k}(2\beta)$. In terms
of the new field $\chi_{k}(s)$ the first line of equation (2.1) becomes

$\displaystyle-\frac{1}{2}\int_{0}^{2\beta}ds_{1}\int_{0}^{2\beta}ds_{2}\chi_{k}(s_{1})\left(\delta(s_{1}-s_{2})\partial_{s_{2}}-\hat{\tilde{\Sigma}}(s_{1},s_{2})\right)\chi_{k}(s_{2}),$
(17)

where we used $r_{k}^{2}=1$ and introduced a compact notation for the self-
energy $\hat{\tilde{\Sigma}}(s_{1},s_{2})$ (with $0<s_{1,2}<2\beta$)

$\displaystyle\hat{\tilde{\Sigma}}(s_{1},s_{2})=\left[\begin{array}[]{cccc}\tilde{\Sigma}_{11}(s_{1},s_{2})&0\\\
0&-\tilde{\Sigma}_{22}(2\beta-s_{1},2\beta-s_{2})\end{array}\right].$ (20)

In this matrix notation, the range of the $s_{i}$ coordinates for the
$\hat{\tilde{\Sigma}}_{a_{1}a_{2}}(s_{1},s_{2})$ element is
$(a_{i}-1)\beta<s_{i}<a_{i}\beta$ with $a_{i}=1,2$, for $i=1,2$, respectively.
More explicitly,
$\hat{\tilde{\Sigma}}_{11}(s_{1},s_{2})=\tilde{\Sigma}_{11}(s_{1},s_{2})$ with
$0<s_{1},s_{2}<\beta$ and
$\hat{\tilde{\Sigma}}_{22}(s_{1},s_{2})=-\tilde{\Sigma}_{22}(2\beta-
s_{1},2\beta-s_{2})$ with $\beta<s_{1},s_{2}<2\beta$. The ranges
$0<s_{1}<\beta$, $\beta<s_{2}<2\beta$ and $\beta<s_{1}<2\beta$,
$0<s_{2}<\beta$ correspond to the (vanishing) off-diagonal elements
$\hat{\tilde{\Sigma}}_{12}(s_{1},s_{2})$ and
$\hat{\tilde{\Sigma}}_{21}(s_{1},s_{2})$, respectively.

Now both $\psi_{j},j=M+1,...,N$ and $\chi_{k},k=1,...,M/2$ satisfy the
conventional anti-periodic boundary conditions typical of fermionic fields. We
can then integrate out the Majorana fermions to get

$\displaystyle-\frac{I}{N}$ $\displaystyle=$
$\displaystyle\frac{m}{2}\log\text{Pf}\left(\partial-\hat{\tilde{\Sigma}}\right)+(1-m)\log\text{Pf}(\partial-\tilde{\Sigma}_{33})$
(22) $\displaystyle-\int
d\tau_{1}d\tau_{2}\frac{1}{2}\left(\frac{m}{2}\tilde{\Sigma}_{11}\tilde{G}_{11}+\frac{m}{2}\tilde{\Sigma}_{22}\tilde{G}_{22}+(1-m)\tilde{\Sigma}_{33}\tilde{G}_{33}\right)$
$\displaystyle+\int
d\tau_{1}d\tau_{2}\frac{J^{2}}{2q}\left(\frac{m}{2}\tilde{G}_{11}+\frac{m}{2}\tilde{G}_{22}+(1-m)\tilde{G}_{33}\right)^{q}$

where Pf indicates the Pfaffian. Owing to the large $N$ structure of the
action, we can derive the equations of motion in the saddle point
approximation, which are given by the Schwinger-Dyson equations (here and in
the following we drop the $\tilde{}$ to indicate on-shell solutions)

$\displaystyle\hat{G}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\left(\partial-\hat{\Sigma}\right)^{-1}(s_{1},s_{2}),\quad
G_{33}(\tau_{1},\tau_{2})=(\partial-\Sigma_{33})^{-1}(\tau_{1},\tau_{2}),$
(23) $\displaystyle\Sigma_{ii}(\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle
J^{2}\left(\frac{m}{2}G_{11}(\tau_{1},\tau_{2})+\frac{m}{2}G_{22}(\tau_{1},\tau_{2})+(1-m)G_{33}(\tau_{1},\tau_{2})\right)^{q-1},\quad
i=1,2,3.$

where we have introduced a compact notation for $\hat{G}$ analogous to the one
introduced in equation (20) for the self-energy:101010Note that, although
$\hat{\Sigma}$ is diagonal, $(\partial-\hat{\Sigma})^{-1}$ is not, and
therefore $\hat{G}$ contains off-diagonal components.

$\displaystyle\hat{G}(s_{1},s_{2})=\left[\begin{array}[]{cccc}G_{11}(s_{1},s_{2})&iG_{12}(s_{1},2\beta-
s_{2})\\\ iG_{21}(2\beta-s_{1},s_{2})&-G_{22}(2\beta-s_{1},2\beta-
s_{2})\end{array}\right].$ (26)

Note that for the off-diagonal components, $\hat{G}_{a_{1}a_{2}}(s_{1},s_{2})$
with $a_{1}\neq a_{2}$, the matrix notation introduced after equation (20)
implies that the ranges of the $s_{1}$ and $s_{2}$ coordinates are different
(i.e. $0<s_{1}<\beta$, $\beta<s_{2}<2\beta$ for $\hat{G}_{12}$ and
$\beta<s_{1}<2\beta$, $0<s_{2}<\beta$ for $\hat{G}_{21}$). It is not hard to
see that
$\hat{G}(s_{1},s_{2})=\frac{2}{M}\sum_{k=1}^{M/2}\chi_{k}(s_{1})\chi_{k}(s_{2})$.

$\hat{G}$ and $G_{33}$ are on-shell Green’s functions. In the next subsection,
we will compute $\hat{G}$ and $G_{33}$ by solving the Schwinger-Dyson
equations (23) analytically in the large-$q$ limit and numerically for any
value of $q$.

###  2.2 Saddle point solution

The action (11) is $O(N)$-symmetric. In particular, it is invariant under the
action of a subgroup of $O(N)$, the “Flip group” kourkoulou2017pure , the
elements of which flip the sign of any number of even-indexed Majorana
fermions. In other words, the action of an element of the Flip group is given
by $\psi_{2k-1}\rightarrow\psi_{2k-1},\psi_{2k}\rightarrow-\psi_{2k}$ for one
or more given values of $k$ with $1\leq k\leq N/2$. In terms of the complex
fermions $\tilde{\psi}_{k}$ associated with pairs of Majorana fermions, an
element of the Flip group flips the sign of one or more spins. Note that both
the diagonal and the-off diagonal bilocal fields (13) and (14) are invariant
under the Flip group as well as the on-shell Green’s functions.

The invariance under the Flip group clearly implies that all our results are
independent of the measurement outcome $\textbf{r}=(r_{1},...,r_{M/2})$: the
Flip group acting on any number of the first $M$ Majorana fermions can map a
given measurement outcome to any other measurement outcome. One consequence of
this invariance is that the on-shell solutions $\hat{G}(s_{1},s_{2})$ and
$G_{33}(\tau_{1},\tau_{2})$ in our setup involving measurement are related to
the thermal Green’s function $G_{th}(\tau_{1},\tau_{2})$. In fact, the on-
shell Green’s function $G_{11}(\tau_{1},\tau_{2})$ can be written as

$\displaystyle G_{11}(\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle\frac{2}{M}\sum_{k}\frac{1}{Z}\text{Tr}\left[e^{-\beta
H[\psi]}\mathcal{T}\left(\psi_{2k-1}(\tau_{1})\psi_{2k-1}(\tau_{2})\right)|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right],$ (27)

where $Z=\text{Tr}\left[e^{-\beta H[\psi]}|R_{\textbf{s}}(m)\rangle\langle
R_{\textbf{s}}(m)|\right]$ is the normalization factor,
$\psi(\tau)=e^{H[\psi]\tau}\psi e^{-H[\psi]\tau}$, and $\mathcal{T}$ denotes
imaginary time ordering. Using the invariance under measurement outcome, we
can then write

$\displaystyle G_{11}(\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle\frac{2}{M}\sum_{k}\frac{1}{Z}\left(\frac{1}{2^{M/2}}\sum_{\textbf{r}}\text{Tr}\left[e^{-\beta
H[\psi]}\mathcal{T}\left(\psi_{2k-1}(\tau_{1})\psi_{2k-1}(\tau_{2})\right)|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right]\right)$ (28) $\displaystyle=$
$\displaystyle\frac{2}{M}\sum_{k}\frac{\text{Tr}\left[e^{-\beta
H[\psi]}\mathcal{T}\left(\psi_{2k-1}(\tau_{1})\psi_{2k-1}(\tau_{2})\right)\right]}{\text{Tr}\left[e^{-\beta
H[\psi]}\right]}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i}\frac{\text{Tr}\left[e^{-\beta
H[\psi]}\mathcal{T}\left(\psi_{i}(\tau_{1})\psi_{i}(\tau_{2})\right)\right]}{\text{Tr}\left[e^{-\beta
H[\psi]}\right]}=G_{th}(\tau_{1},\tau_{2})$

where to go from the first to the second line we have used the completeness
relation for the measured subspace
$\sum_{\textbf{r}}|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|=\mathds{1}^{(m)}$ and
$Z=\frac{1}{2^{M/2}}\sum_{\textbf{r}}\text{Tr}\left[e^{-\beta
H[\psi]}|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right]=\frac{1}{2^{M/2}}\text{Tr}\left[e^{-\beta
H[\psi]}\right]$. The equality leading from the second to the third line is
true in virtue of the $O(N)$ symmetry of the model. An analogous relation can
be obtained for $G_{22}(\tau_{1},\tau_{2})$ and $G_{33}(\tau_{1},\tau_{2})$.
Therefore, $G_{ii},i=1,2,3$ are all given by the usual imaginary time
correlation.

The Schwinger-Dyson equations (23) can then be rewritten in a simpler form
111111We omit here and in the following analysis the equations for $G_{33}$
and $\Sigma_{33}$, which can be solved in complete analogy with the $11$
component.

$\displaystyle\hat{G}=(\partial-\hat{\Sigma})^{-1},\quad\hat{\Sigma}(s_{1},s_{2})=\frac{\mathcal{J}^{2}}{q}P(s_{1},s_{2})[2\hat{G}(s_{1},s_{2})]^{q-1},$
(29)

where we defined $\mathcal{J}^{2}=qJ/2^{q-1}$ for later convenience and

$\displaystyle P(s_{1},s_{2})=\theta(\beta-s_{1})\theta(\beta-
s_{2})+\theta(s_{1}-\beta)\theta(s_{2}-\beta).$ (30)

The boundary conditions now read

$\displaystyle\hat{G}(s,s)=\frac{1}{2},\quad\hat{G}(s_{1},s_{2})=-\hat{G}(s_{2},s_{1}),\quad\hat{G}(s_{1}+2\beta,s_{2})=-\hat{G}(s_{1},s_{2}).$
(31)

The first and second equations are a consequence of the commutation relations
and the definition of the time ordering operator, whereas the third condition
implements the anti-periodic boundary conditions for the fermionic field
$\chi(s)$.

Figure 3: The domain of the bilocal Green’s function $\hat{G}(s_{1},s_{2})$.
$P(s_{1},s_{2})$ is nonzero in the shaded regions I and IV. According to our
matrix notation, $\hat{G}_{11}(s_{1},s_{2})=G_{11}(s_{1},s_{2})$ denotes the
Green’s function in region I, $\hat{G}_{21}(s_{1},s_{2})=iG_{21}(2\beta-
s_{1},s_{2})$ denotes the Green’s function in region II,
$\hat{G}_{12}(s_{1},s_{2})=iG_{12}(s_{1},2\beta-s_{2})$ denotes the Green’s
function in region III, and $\hat{G}_{22}(s_{1},s_{2})=-G_{22}(2\beta-
s_{1},2\beta-s_{1})$ denotes the Green’s function in region IV.

The Schwinger-Dyson equations can be solved analytically in the large-$q$
limit maldacena2016remarks . Using the ansatz
$\hat{G}(s_{1},s_{2})=\hat{G}_{0}(s_{1},s_{2})\left(1+\frac{1}{q}g(s_{1},s_{2})+...\right)$,
where we suppressed terms of higher order in $q$ and introduced the bare
propagator

$\displaystyle\hat{G}_{0}(s_{1},s_{2})=\frac{1}{2}\left[\begin{array}[]{cccc}\text{sgn}(s_{1}-s_{2})&-1\\\
1&\text{sgn}(s_{1}-s_{2})\end{array}\right],$ (34)

the Schwinger-Dyson equations (29) at leading order in $q$ give an equation
for $g(s_{1},s_{2})$ maldacena2016remarks :

$\displaystyle\partial_{s_{1}}\partial_{s_{2}}[\hat{G}_{0}(s_{1},s_{2})g(s_{1},s_{2})]=\mathcal{J}^{2}P(s_{1},s_{2})[2\hat{G}_{0}(s_{1},s_{2})]^{q-1}e^{g(s_{1},s_{2})}.$
(35)

Note that the measurement introduces a factor $P(s_{1},s_{2})$ on the right-
hand side which is not present in the conventional Liouville equation
maldacena2016remarks . Given the expression (30) for $P(s_{1},s_{2})$, we can
divide the domain into four regions as shown in Fig. 3. In region I and IV,
the Liouville equation (35) takes the form 121212Here, we used the fact that
$q$ is an even number in order for the Hamiltonian to be a bosonic operator.

$\displaystyle\partial_{s_{1}}\partial_{s_{2}}g(s_{1},s_{2})=2\mathcal{J}^{2}e^{g(s_{1},s_{2})}.$
(36)

The relationship (28) between the diagonal Green’s functions in our setup and
the thermal Green’s function implies that we can assume the diagonal elements
of $\hat{G}(s_{1},s_{2})$ (i.e. the Green’s function in regions I and IV) are
time translation invariant. Therefore in region I and IV,
$g(s_{1},s_{2})=g(s_{12})$ where $s_{12}\equiv s_{1}-s_{2}$. The solution of
the Liouville equation is then

$g_{\text{I}}(s_{12})=g_{\text{IV}}(s_{12})=2\log\left[\frac{\sin\gamma}{\sin(\alpha|s_{12}|+\gamma)}\right]$
(37)

where $\alpha=\mathcal{J}\sin\gamma$ and $\gamma$ is a constant to be
determined by imposing the anti-periodic boundary condition. At leading order
in $q$ we then obtain the following result for the diagonal elements of
$\hat{G}(s_{1},s_{2})$:

$\displaystyle\hat{G}_{11}(s_{1},s_{2})=\hat{G}_{22}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{12})\left(\frac{\sin\gamma}{\sin(\alpha|s_{12}|+\gamma)}\right)^{2/q}.$
(38)

In regions II and III, the Liouville equation (35) takes the form
$\partial_{s_{1}}\partial_{s_{2}}g(s_{1},s_{2})=0$, which is solved by the
ansatz $g(s_{1},s_{2})=f_{1}(s_{1})+f_{2}(s_{2})$. The form of $f_{1}(s_{1})$
and $f_{2}(s_{2})$ can be obtained by matching the solutions for
$\hat{G}(s_{1},s_{2})$ at the boundaries between regions I,IV and regions
II,III. This yields the following off-diagonal elements of
$\hat{G}(s_{1},s_{2})$:

$\displaystyle\begin{aligned}
&\hat{G}_{21}(s_{1},s_{2})=\frac{1}{2}\left(\frac{\sin\gamma}{\sin(\alpha(s_{1}-\beta)+\gamma)}\frac{\sin\gamma}{\sin(\alpha(\beta-
s_{2})+\gamma)}\right)^{2/q},\\\
&\hat{G}_{12}(s_{1},s_{2})=-\frac{1}{2}\left(\frac{\sin\gamma}{\sin(\alpha(s_{2}-\beta)+\gamma)}\frac{\sin\gamma}{\sin(\alpha(\beta-
s_{1})+\gamma)}\right)^{2/q}.\end{aligned}$ (39)

Finally, imposing the anti-periodic boundary condition
$\hat{G}(s_{1}+2\beta,s_{2})=-\hat{G}(s_{1},s_{2})$, the constant $\gamma$ is
implicitly determined by

$\displaystyle\gamma+\frac{\mathcal{J}\beta\sin\gamma}{2}=\frac{\pi}{2}.$ (40)

Note that our results for the off-diagonal Green’s function (39) are in
agreement with the $m=1$ results of kourkoulou2017pure .

(a) $G_{33}$ (b) $G_{11,22}$

(c) $G_{21}$

Figure 4: A comparison between numerical (represented by blue dots) and
large-$q$ analytical (represented by gray curves) solutions for the diagonal
components of the Green’s function. The parameters are $\beta\mathcal{J}=20$,
$q=4$. The imaginary time contour with length $\beta$ is discretized into
$100$ intervals. (a) The unmeasured Majorana fermions’ Green’s function
$G_{33}(\tau_{1},\tau_{2})$. (b) The measured Majorana fermions’ Green’s
functions $G_{11}(\tau_{1},\tau_{2})$, $G_{22}(\tau_{1},\tau_{2})$. Note that
$G_{11}(\tau_{1},\tau_{2})=G_{22}(\tau_{1},\tau_{2})=G_{33}(\tau_{1},\tau_{2})$
as expected by the flip symmetry argument in equation (28). (c) The measured
Majorana fermions’ Green’s function $G_{21}(\tau_{1},\tau_{2})$.

Alternatively, the Schwinger-Dyson equation (29) can be solved numerically by
iteration for any value of $q$. Given that our results are independent of the
measurement outcome, we take for simplicity $r_{k}=1$ for all $k=1,...,M/2$. A
comparison of the numerical results with the large $q$ solution described
above is shown in Fig. 4, where the dots are numerical results and the gray
curves are the analytical large-$q$ solutions derived above. Note that, as
expected, the numerical solutions for the diagonal Green’s functions of the
measured Majorana fermions $G_{11}(\tau_{1},\tau_{2})$,
$G_{22}(\tau_{1},\tau_{2})$ and the ummeasured Majorana fermions
$G_{33}(\tau_{1},\tau_{2})$ are identical.

###  2.3 Mutual information

In order to set the stage for the holographic calculation carried out in
Section 3, we now compute the post-measurement Renyi-2 mutual information
between the left side and the unmeasured Majorana fermions on the right side.
The SYK mutual information computed in the present subsection will serve as a
benchmark to test our results for the holographic mutual information, which
will give us information about the effect of the measurement on the wormhole
dual to the SYK thermofield double state. The choice of computing the Renyi-2
mutual information in the SYK setup rather than the Von Neumann mutual
information is due to the fact that it can be easily computed in the saddle
point approximation using a Euclidean path integral for two replicas of our
system with appropriate boundary conditions. The Renyi-2 entropy is also known
to provide a good qualitative approximation of the von Neumann entropy for the
SYK model (see the discussion and comparison between Renyi-2 entropy and von
Neumann entropy from exact diagonalization liu2018quantum  in Ref.
zhang2020subsystem ). This is confirmed by our comparison between the SYK
Renyi-2 mutual information and the holographic von Neumann entropy computed in
the bulk dual setup in Section 3.

As we discussed in the previous subsection, the unnormalized post-measurement
state of the two-sided system is $|\Psi_{\textbf{r}}(m)\rangle$ given by (8),
and the corresponding unnormalized density matrix is
$\Psi_{\textbf{r}}(m)\equiv|\Psi_{\textbf{r}}(m)\rangle\langle\Psi_{\textbf{r}}(m)|$.
Because $|\Psi_{\textbf{r}}(m)\rangle$ is a pure state, we have
$S_{L}(\Psi_{\textbf{r}}(m))=S_{R}(\Psi_{\textbf{r}}(m))$, and
$S_{LR}(\Psi_{\textbf{r}}(m))=0$. Here $S_{L}$ ($S_{R}$) denotes the
entanglement entropy of the left (right) side, and $S_{LR}$ denotes the
entanglement entropy of the full two-sided system. Therefore, the mutual
information is twice the entanglement entropy of the left system

$\displaystyle
I_{LR}(\Psi_{\textbf{r}}(m))=S_{L}(\Psi_{\textbf{r}}(m))+S_{R}(\Psi_{\textbf{r}}(m))-S_{LR}(\Psi_{\textbf{r}}(m))=2S_{L}(\Psi_{\textbf{r}}(m)).$
(41)

We focus our attention on the Renyi-2 entanglement entropy—denoted by
$S^{(2)}(\Psi_{\textbf{r}}(m))$. This can be obtained via numerical evaluation
of a path integral for two replicas of our system, with appropriate boundary
conditions which we will derive below. The Renyi-2 entropy of the left system
is defined by

$\displaystyle
e^{-S_{L}^{(2)}(\Psi(m))}=\frac{\overline{\text{Tr}_{L}\left[(\text{Tr}_{R}[\Psi(m)])^{2}\right]}}{\overline{\left(\text{Tr}[\Psi(m)]\right)^{2}}},$
(42)

where we used the fact that disorder-averaging the ratio is equivalent to
disorder-averaging the numerator and denominator separately up to
$1/N$-suppressed corrections. We have also omitted the subscript r indicating
the measurement outcome due to the invariance under measurement outcome
discussed above. In particular, in the following calculation we set
$r_{k}=1,k=1,...,M/2$ for simplicity.

(a) $i=1,...,M$ (b) $i=M+1,...,N$

Figure 5: Imaginary time contour $\tau\in(0,2\beta)$ for the Euclidean path
integral computing the numerator of equation (42). The points $\tau=0$,
$\beta\pm\delta$, $\beta/2\pm\delta$, $3\beta/2\pm\delta$ and $2\beta$ where
boundary conditions must be imposed are labeled in the figure. The twist
operator inserted in the left system is indicated by the dashed line. (a) The
effect of the measurement operator on $\psi_{i},i=1,...,M$ is indicated by the
triangles, corresponding to the second and third lines of the boundary
conditions (45). At the insertion point of the twist operator, the boundary
conditions are given by the first line of (45). (b) There is no measurement
operator for $\psi_{i},i=M+1,...,N$. The boundary conditions are then simply
set by the first line of (45).

Equation (42) can be expressed in terms of Euclidean path integrals computing
the numerator and denominator separately. Because the Renyi-2 entropy involves
the square of the density matrix, we extend the imaginary time contour to
$\tau\in(0,2\beta)$ with $\tau\in(0,\beta)$ corresponding to the first replica
and $\tau\in(\beta,2\beta)$ corresponding to the second replica. With a change
of notation from the previous subsections, we consider the left system to be
at $\tau=0,\beta,2\beta$ (for the two replicas) and the right system where the
measurement is performed to be at $\tau=\beta/2,3\beta/2$. When computing the
numerator of (42), a twist operator must be inserted in the left system,
setting anti-periodic boundary conditions at $\tau=0,2\beta$ for the Majorana
fermions with period $2\beta$; there is no twist operator for the denominator.
Additionally, as in the previous subsections, the measurement operator
inserted in the right side sets different boundary conditions for measured and
unmeasured Majorana fermions. See Fig. 5 for a representation of the boundary
conditions in the path integral computing the numerator of (42).

The path integral for the numerator reads

$\displaystyle\overline{\text{Tr}_{L}\left[(\text{Tr}_{R}[\Psi(m)])^{2}\right]}=\int
D\psi e^{-I},$ (43)

where the action is given by

$\displaystyle-I=-\int_{0}^{2\beta}d\tau\sum_{j}\frac{1}{2}\psi_{j}(\tau)\partial_{\tau}\psi_{j}(\tau)+\int_{0}^{2\beta}d\tau_{1}\int_{0}^{2\beta}d\tau_{2}\frac{J^{2}}{2qN^{q-1}}\left(\sum_{j}\psi_{j}(\tau_{1})\psi_{j}(\tau_{2})\right)^{q},$

and the boundary conditions are

$\displaystyle\begin{aligned}
&\psi_{i}(0)=-\psi_{i}(2\beta),\quad\psi_{i}(\beta_{-})=\psi_{i}(\beta_{+}),\quad
i=1,...,N,\\\
&\psi_{2k-1}\left(\frac{\beta_{-}}{2}\right)=i\psi_{2k}\left(\frac{\beta_{-}}{2}\right),\quad\psi_{2k-1}\left(\frac{\beta_{+}}{2}\right)=-i\psi_{2k}\left(\frac{\beta_{+}}{2}\right),\quad
k=1,...,M/2,\\\
&\psi_{2k-1}\left(\frac{3\beta_{-}}{2}\right)=i\psi_{2k}\left(\frac{3\beta_{-}}{2}\right),\quad\psi_{2k-1}\left(\frac{3\beta_{+}}{2}\right)=-i\psi_{2k}\left(\frac{3\beta_{+}}{2}\right),\quad
k=1,...,M/2.\end{aligned}$ (45)

Here we used $\beta_{\pm}$ as a shorthand notation to indicate that the
quantity in parenthesis is shifted by an infinitesimal positive number
$\delta$ (for instance, $3\beta_{\pm}/2=3\beta/2\pm\delta$). The first line of
equation (45) is a consequence of the presence of the twist operator in the
left system; the second and third lines are a consequence of the insertion of
the measurement operator (4) in the right side for $k=1,...,M/2$ (see Fig. 5
for an illustration).

Similarly, the denominator of equation (42) can also be represented by a path
integral with the same action (2.3). The only difference is in the boundary
conditions. In fact, the absence of a twist operator implies that the boundary
conditions for the denominator read

$\displaystyle\begin{aligned}
&\psi_{i}(0)=-\psi_{i}(\beta_{-}),\quad\psi_{i}(\beta_{+})=-\psi_{i}(2\beta),\quad
i=1,...,N,\\\
&\psi_{2k-1}\left(\frac{\beta_{-}}{2}\right)=i\psi_{2k}\left(\frac{\beta_{-}}{2}\right),\quad\psi_{2k-1}\left(\frac{\beta_{+}}{2}\right)=-i\psi_{2k}\left(\frac{\beta_{+}}{2}\right),\quad
k=1,...,M/2,\\\
&\psi_{2k-1}\left(\frac{3\beta_{-}}{2}\right)=i\psi_{2k}\left(\frac{3\beta_{-}}{2}\right),\quad\psi_{2k-1}\left(\frac{3\beta_{+}}{2}\right)=-i\psi_{2k}\left(\frac{3\beta_{+}}{2}\right),\quad
k=1,...,M/2.\end{aligned}$ (46)

The next step is to rewrite the large-$N$ action in terms of bilocal fields
analogous to the ones defined in equations (13) and (14), but with arguments
in the range $\tau_{1,2}\in(0,2\beta)$. We can then derive and numerically
solve the equations of motion in the saddle point approximation for the
numerator and denominator of equation (42), and compute the corresponding on-
shell actions $I_{num}(m)$ and $I_{den}(m)$. The details of this procedure are
reported in Appendix A. Finally, the Renyi-2 mutual information in the saddle
point approximation is immediately given by
$I_{LR}^{(2)}(\Psi(m))=2S_{L}^{(2)}(\Psi(m))=2(I_{num}(m)-I_{den}(m))$.

The result of the numerical evaluation of the Renyi-2 mutual information
$\mathcal{I}_{LR}^{(2)}(\Psi(m))=I_{LR}^{(2)}(\Psi(m))/N$ is shown in Fig. 6.
Note that the mutual information is finite for any value of $0<m<1$, i.e. for
any size of the subset $M$ of measured Majorana fermions. Only when all
Majorana fermions are measured ($m=1$, corresponding to the analysis of
kourkoulou2017pure , see Appendix B) does the mutual information vanish. In
other words, the left and right systems always remain entangled, no matter how
large the measured subsystem in the right is. In the next section, we will
find an analogous result by computing the holographic entanglement entropy in
the bulk dual theory (given by JT gravity coupled to $N$ copies of a CFT).
This will allow us to give in Section 4 a holographic interpretation of our
results in terms of bulk teleportation Antonini:2022sfm : the effect of the
partial measurement performed on the right system is to teleport bulk
information previously contained in the entanglement wedge of the right system
into the left system. We also observe a phase transition in the behavior of
the mutual information at large $\beta$ from nearly flat to linearly decaying.
In the bulk dual picture, this corresponds to an entanglement wedge phase
transition triggered by the measurement, as we will describe in Section 3.

Figure 6: Renyi-2 mutual information $\mathcal{I}_{LR}^{(2)}(\Psi(m))$ between
the left system and the unmeasured Majorana fermions on the right side.
$m=M/N$ is the ratio of measured Majorana fermions on the right side. The
mutual information is always non-vanishing for $m<1$, and vanishes only when
the whole right side is measured ($m=1$).

##  3 Partial measurement in JT gravity

In this section, we explore the gravity dual of the SYK measurements
considered above. In particular, our dual system will be Jackiw-Teitelboim
(JT) gravity coupled to bulk matter, which we take to be described by $N$
copies of a CFT (capturing the $N$ Majorana fermions in the SYK model). In
this and the following sections, we switch the side that is being measured;
namely, we will measure the left side of the TFD. We can then calculate the
mutual information in the post-measurement state between the right side and
remaining fermions on the left side, this time using quantum extremal surface
(QES) formula Engelhardt:2014gca . In particular, the bulk description (which
relies on the location of the QES) will make manifest the entanglement wedge
of the right side and thus, in turn, demonstrate how information about the
center of the bulk will shift around, becoming encoded in different boundary
regions as a result of measurement. Specifically, when the number of measured
fermions remains small ($m\lesssim 0.26$), a significant amount of the left
asymptotic region will be encoded in the remaining (unmeasured) fermions of
the left system. However, as the number of measured fermions becomes large
($m\gtrsim 0.26$), the quantum extremal surface will undergo a phase
transition, and information about nearly all of the left side of the bulk will
become accessible from the right side. As in Antonini:2022sfm , we refer to
this as teleportation of the bulk geometry, a notion which we will make more
manifest in Section 4 below.

Here, we start with a more in depth description of our bulk model in Section
3.1, before moving to the holographic description of partial measurement and a
computation of the mutual information in Section 3.2. Specifically, we propose
for the QES prescription that the measured matter can be modeled by a boundary
conformal field theory, as boundary measurements create an end-of-the-world
brane in the bulk that serves as a boundary for the bulk conformal matter.
Along with a comparison with the SYK results above, we find strong agreement
between the QES prescription and the SYK calculation.

###  3.1 JT gravity with matter CFT

The action for JT gravity coupled to matter CFTs is given by

$\displaystyle\begin{aligned} I=&I_{JT}(g,\phi)+I_{M}(g,\chi)\\\
I_{JT}(g,\phi)=&-\frac{\phi_{0}}{16\pi G_{N}}\Big{[}\int
d^{2}x\sqrt{g}R+2\int_{\partial\mathcal{M}}K\Big{]}\\\ &-\frac{1}{16\pi
G_{N}}\Big{[}\int
d^{2}x\sqrt{g}\phi(R+2)+2\int_{\partial\mathcal{M}}\phi|_{\text{bdy}}K\Big{]},\end{aligned}$
(47)

where $\Phi=\phi_{0}+\phi$ is a dilaton field, with $\phi$ a small fluctuation
around a uniform background $\phi_{0}$. If we view this theory as coming from
the dimensional reduction of a higher dimensional theory, $\phi_{0}$ would
denote the area of the black hole horizon at extremality (with $\phi$ giving
the area deviations for near extremal black holes), and is therefore related
to ground state entropy of an extremal black hole. The first line of the JT
action is a purely topological term, so that the dynamical part of the action
comes solely from the second line of equation (47). We take the matter action
$I_{M}(g,\chi)$ to be given by $N$ copies of a CFT with central charge $c$
(which we will take to be $c=1/2$ when comparing with the SYK results of the
previous section, as each Majorana fermion can be thought of as half of a
Dirac fermion).

The resulting equations of motion read

$\displaystyle\begin{aligned} R&=-2,\\\ T_{\mu\nu}&=\frac{1}{8\pi
G_{N}}(\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\nabla^{2}\phi+g_{\mu\nu}\phi),\end{aligned}$ (48)

where $T_{\mu\nu}$ is the stress tensor associated with the matter fields of
$I_{M}$. The metric of the (Euclidean) eternal black hole is given by (see
Appendix D for a derivation)

$\displaystyle
ds^{2}=\frac{4\pi^{2}}{\beta^{2}}\frac{d\sigma^{2}+d\tau^{2}}{\sinh^{2}\frac{2\pi}{\beta}\sigma},$
(49)

where $\tau$ is the imaginary time. In these coordinates, which cover one
exterior portion of the two-sided black hole, the horizon is located at
$\sigma\rightarrow\infty$, and the asymptotic boundary is located at
$\sigma=\epsilon$. At the boundary, the induced metric satisfies the condition
$g=\frac{1}{\epsilon^{2}}$, and the dilaton field satisfies
$\phi(\epsilon)=\frac{\phi_{r}}{\epsilon}$. This leads to the dilaton profile,

$\displaystyle\phi(\sigma)=\phi_{r}\frac{2\pi}{\beta}\frac{1}{\tanh\frac{2\pi}{\beta}\sigma}.$
(50)

Figure 7:  The doubled spacetime used to compute the bulk matter entropy. The
boundary at the center of the figure is transparent. The left-most boundary
and the right-most boundary are identified. $\sigma_{1}$ and $-\sigma_{2}$
denote twist operators inserted in the first and second copies of the black
hole.

We will compute the entropy of a subset $R$ of the boundary system via the
quantum extremal surface formula

$\displaystyle
S(R)=\min\\{\underset{\sigma}{\text{ext}}S_{\text{gen}}(\sigma)\\},$ (51)

i.e. the smallest, extremal value of $S_{\text{gen}}$, the generalized
entropy, which is given by the sum of the area of $\sigma$ (divided by
$4G_{N}$) plus the bulk matter entropy in a region bound by $\sigma$. More
explicitly, we compute the entropy by
$S_{\text{gen}}=\frac{\Phi(\sigma)}{4G_{N}}+S_{\text{CFT}}(\sigma)$, evaluated
at an extremal value of $\sigma$, which we call the quantum extremal surface.
In order to do this, we need to first calculate the bulk entanglement entropy
of some region extending out to the boundary. We can do this by employing a
“doubling trick,” as in nezami2021quantum . The matter in the spacetime will
obey reflecting boundary condition at the asypmtotic boundary
$\sigma=\epsilon$. We can related the bulk entropy in this space to the
entropy on a cylinder – specifically, a doubled space constructed by gluing
together two copies of a black hole at their boundaries, this time with
transparent boundary conditions (with a left moving mode in one copy joined to
a right moving mode in the second copy). See Fig. 7 for a depiction.

In this doubled space, we can calculate the entropy of a bulk subregion (at
$t=0$) by inserting twist operators at points $(\sigma_{1},0)$ in the first
copy and $(-\sigma_{2},0)$ in the second copy. The entanglement entropy of the
matter is then given by (see Appendix D)

$\displaystyle
S_{\text{double}}([-\sigma_{2},\sigma_{1}])=\frac{c}{6}\log\left[\frac{4\sinh^{2}\frac{\pi}{\beta}(\sigma_{1}+\sigma_{2})}{\sinh\frac{2\pi}{\beta}\sigma_{1}\sinh\frac{2\pi}{\beta}\sigma_{2}}\right],$
(52)

where the subscript indicates the doubled theory. Therefore, in the original
theory, a single twist operator at $(\sigma,0)$ gives an entanglement entropy

$\displaystyle
S_{\text{CFT}}(\sigma)=\frac{1}{2}S_{\text{double}}([-\sigma,\sigma])=\frac{c}{6}\log
2.$ (53)

Note that from (52), the entanglement entropy does not depend on the value of
$\sigma$.

If we consider the entanglement entropy of the full right system, the quantum
extremal surface will be at the bifurcation surface as determined by symmetry.
The generalized entropy is then given by

$\displaystyle\begin{aligned}
S_{\text{gen}}&=\frac{\phi_{0}}{4G_{N}}+\frac{\phi_{r}}{4G_{N}}\frac{2\pi}{\beta}+\frac{Nc}{6}\log
2,\\\
&=\frac{\tilde{\phi}_{0}}{4G_{N}}+\frac{\phi_{r}}{4G_{N}}\frac{2\pi}{\beta},\end{aligned}$
(54)

where in the first line, we have included $N$ copies of the matter CFT, and in
the second line for simplicity, we have defined
$\frac{\tilde{\phi}_{0}}{4G_{N}}=\frac{\phi_{0}}{4G_{N}}+\frac{Nc}{6}\log 2$.
To make contact with the SYK model, we note that the first term corresponds to
the ground state entropy, and the second term is a correction linear in
temperature.

###  3.2 Measurement-induced entanglement wedge transition

With an expression for the bulk generalized entropy in hand, we can now
investigate the effect of performing a measurement on one side, and in
particular, calculate the mutual information between the two sides. We know
that measuring the full left side will create an end-of-the-world brane in the
left asymptotic region kourkoulou2017pure  (and, of course, measuring nothing
will leave the bulk unchanged). We thus expect that a natural bulk dual to
partial measurement is the creation of an end-of-the-world brane in the bulk,
but one only visible to the bulk fields dual to the measured fermions. As
such, we model the bulk matter for the measured fields by a BCFT living in the
bulk, with their boundary at the end-of-the-world brane. Thus, if we measure
$M$ Majorana fermions on the left side, we will have $N-M$ copies of a normal
matter CFT corresponding to the unmeasured fermions, and $M$ copies of a BCFT
corresponding to the measured fermions, as shown in Fig. 8.

Figure 8: A boundary CFT in the $AdS_{2}$ background. The right boundary is
the conventional $AdS$ boundary. Boundary conditions associated with the
projective measurement of part of the left Majorana fermions are imposed at
the left boundary. An ETW brane visible to $M$ bulk fermions is depicted in
blue, where $M$ is the number of fermions measured in the left side.

To calculate the bulk entropy arising from the measured fermions, we insert a
twist operator for the BCFT at $(\sigma,0)$, which leads to the following
entanglement entropy

$\displaystyle
S_{\text{BCFT}}(\sigma)=\frac{c}{6}\log\left[\frac{2\sinh\frac{\pi}{\beta}(2\sigma)}{\sinh\frac{2\pi}{\beta}\sigma}\right]+\log
g=\frac{c}{6}\log 2+\log g,$ (55)

where $\log g$ is the boundary entropy of the BCFT cardy1989boundary ;
affleck1991universal .

For $M\ll N$, we do not expect the position of the quantum extremal surface to
change much. We can verify this by calculating the entanglement entropy of the
right side,

$\displaystyle\begin{aligned}
S_{\text{gen}}(M)&=\frac{\phi_{0}}{4G_{N}}+\frac{\phi_{r}}{4G_{N}}\frac{2\pi}{\beta}\frac{1}{\tanh\frac{2\pi}{\beta}\sigma}+(N-M)S_{\text{CFT}}(\sigma)+MS_{\text{BCFT}}(\sigma),\\\
&=\frac{\tilde{\phi}_{0}}{4G_{N}}+\frac{\phi_{r}}{4G_{N}}\frac{2\pi}{\beta}+M\log
g.\end{aligned}$ (56)

In the second line, because $S_{\text{CFT}}(\sigma)$ and
$S_{\text{BCFT}}(\sigma)$ do not depend on the end point, we find
$\sigma=\infty$ as the location for the quantum extremal surface, as expected.

Because $M\ll N$, there will still be a large mutual information between the
right side and the unmeasured Majorana on the left after projection. Further,
because the full state is pure, the entanglement entropy of the unmeasured
Majorana on the left side will be equal to that of the right. As a
consequence, the entanglement wedge of the left side will also extend up to
the bifurcation surface. As above, we include twist operators for all copies
of matter fields at the QES, including $K=N-M$ copies of a CFT, and $M$ copies
of a BCFT, yielding the same entropy as in (56).

We can also consider the opposite limit, when we measure a large number of
Majorana fermions, i.e., $M\sim N$ (such that the remaining number of
unmeasured fermions is small $K\ll N$). To understand this limit, we invoke
the bulk picture suggested in nezami2021quantum : if we have access to the
remaining $K\ll N$ unmeasured boundary Majorana fermions of the left system,
we expect to be able to reconstruct their dual bulk fields in a small region
near the boundary. In particular, the state of the fermions remains constant
on time scales $\sim 1/J$. HKLL bulk reconstruction hamilton2006holographic
then implies we can reconstruct the dual fields in a small wedge near the
boundary (the bulk domain of dependence of this boundary timelike region).
This wedge will have size on the order of the UV cutoff $\epsilon\sim 1/J$. As
a result, these fermions will have a bulk entropy associated to this UV bulk
region. Since the entanglement wedge is now at the UV cutoff, we take the QES
to be empty (and hence take the area contribution from the dilaton to be
zero).

To compute the bulk entropy associated with the unmeasured fermions in the UV
cutoff region, we note that the reduced density matrix of the unmeasured
Majoranas can be found by projecting the thermal density matrix onto a fixed
state. In the low temperature limit, the thermal density matrix reduces the
effective available space to a low energy subsector. Therefore, the effect of
measurement is constrained to that reduced low energy subspace as well.
Because the SYK model is chaotic, we expect the entanglement entropy of the
unmeasured fermions when $K\ll N$ will be dominated by subsystem entanglement
entropy evaluated in the ground state, which is given by huang2019eigenstate

$\displaystyle\frac{S_{\text{UV}}}{N}=\lambda\left(\frac{\log
2}{2}-\frac{1}{16}\arcsin\lambda^{3/2}\right),\quad\lambda=K/N.$ (57)

where the second term originates from the nontrivial spectral density near the
spectrum edge in the SYK model garcia2017analytical .

With these two limits fairly well understood, in the intermediate regime
($K\sim M$), we take the entropy to be given by
$S_{\text{QES}}=\min\\{S_{\text{gen}},S_{\text{UV}}\\}$. While this bulk
protocol may potentially seem ad hoc, we can justify the prescription by
comparing it to the SYK calculations done in Section 2. As demonstrated in
Fig. 9, we find a good match between the two approaches. To make this
comparison, we take

$\displaystyle\frac{\tilde{\phi}_{0}}{4G_{N}}=S_{0},\quad\frac{\phi_{r}}{4G_{N}}=\frac{NC}{2\pi\mathcal{J}}.$
(58)

and $\log g=-\frac{\log 2}{12}$ 131313This value of the boundary entropy is
determined by matching the SYK data. However, it is interesting to observe
that if we imagine the bulk matter is dual to $N$ decoupled EPR states, then
after one partner in each EPR state is measured, the entanglement entropy for
the other partner simply vanishes. This means that the boundary entropy should
cancel the bulk contribution to the entropy for the BCFT, which would predict
$\log g=-\frac{c}{6}\log 2$ from (55). In the $c=1/2$ case of our interest,
this gives exactly the value of the boundary entropy obtained by matching the
SYK result.. Notice that since the full state is pure, the mutual information
is given by $I_{LR}=2S_{\text{QES}}$.

For $S_{0}=0.2324N$, $C=0.28$, $c=0.5$, and $\beta\mathcal{J}=50$, the phase
transition in the quantum extremal surface occurs at $\lambda_{*}\approx
0.72$, see Fig. 9. Therefore, when $M>m_{*}N\approx 0.26N$ (with
$m_{*}=1-\lambda_{*}$) Majorana fermions are measured in the left system, the
information about the bulk that would have been encoded in those fermions
becomes available to the right system, similar to Antonini:2022sfm . In other
words, we find that performing a projective measurement on about a third of
the left system is sufficient to allow us to teleport the information about
the left side of the bulk into the right boundary, except for a small wedge at
the cutoff of the left boundary. On the other hand, when a small number of
fermions are measured, i.e. $M<m_{*}N\approx 0.26N$, the bulk information
contained in the $M$ measured fermions is teleported into the unmeasured
$K=N-M$ fermions in the same side. Therefore, the quantum extremal surface
still sits at the bifurcation surface and the information about the left side
of the bulk can still be accessed from the unmeasured fermions in the left
side. An illustration of the entanglement wedge transition is shown in Fig.
10.

We will make efforts to better understand this transfer of bulk information by
exploring quantum teleportation and traversable wormholes in the next section.

Figure 9: Comparison of the Von Neumann mutual information computed in this
section using the QES prescription and the Renyi-2 mutual information compued
in Section 2.3 for the SYK model. We choose the parameters $S_{0}=0.2324N$,
$C=0.28$, $c=0.5$, and $\beta\mathcal{J}=50$. The two results are in good
agreement. In the bulk calculation, the phase transition is sharp, while in
the SYK calculation it is smoothed out.

Figure 10: Measurement-induced entanglement wedge transition. The tan color
denotes the entanglement wedge of the right side. (a) When a small subset of
Majorana fermions is measured in the left side, i.e. $m<m_{*}$, the
entanglement wedge of the right side is the right Rindler patch. (b) When a
large subset of Majorana fermions is measured in the left side, i.e.
$m>m_{*}$, the entanglement wedge of the right side includes part of the
interior and of the left Rindler patch.

##  4 Quantum teleportation

As noted in the previous section, when a sufficient number of boundary
fermions are measured ($m>m^{*}$), the entanglement wedge of the right
boundary jumps to contain nearly the entire bulk (see Fig. 10 (b)).
Consequently, almost all of the bulk (including most of the left wedge) can be
reconstructed solely from the right system. To make this concrete, we can
imagine releasing a particle on the left side at time $t=-t_{0}$, then
measuring the left side at time $t=0$. The measurement will create an end-of-
the world brane in the bulk. However, this will not effect the particle, which
will continue to fall into the black hole, and in particular will not escape
out to the right boundary, as illustrated in Fig. 11 (a). To extract the
information from the right side, we must construct a decoding operator using
information from the measurement. In the following section, we will see that
by applying such a decoding operator to the right side, we will be able to
extract the particle at $t=t_{0}$ in the right side. This is simply a quantum
teleportation protocol, which can be used to explain the quantum information
origin of the entanglement wedge transition explored above. Unsurprisingly,
this boundary teleportation protocol can be related to a traversable wormhole
in the bulk 141414Note that the wormhole will be traversable from only one
side, similar to the analysis of maldacena2017diving .. In the following, we
will first describe the teleportation protocol in the SYK model, which is
realized as a quantum channel, and then calculate a left-right correlation
function, (61), that is closely related to not only the teleportation fidelity
under such a quantum channel, but also traversability in the dual picture. We
will then show that the dual of this protocol provides a realization of
traversable wormhole from the left side. Finally, we consider an encoding
method to reach nearly perfect teleportation fidelity of an arbitrary qubit
state.

Figure 11: The red line denotes the qubit that is inserted in the left side at
time $-t$. (a) Measurement is implemented at time $t=0$ in the left side, and
it creates an end-of-the-world brane. The qubit cannot escaped from the
horizon to the right side. (b) Measurement is implemented at time $t=0$ in the
left side, in the mean time, a decoding operator according to the measurement
outcome is implemented in the right side. This measurement-decoding protocol
can extract the qubit out from horizon to the right side at time $t$.

###  4.1 Teleportation protocol

Motivated by the measurement-induced entanglement wedge phase transition
studied above, we want to construct a protocol to recover the information
initially stored in the left side from the right side after measurement, when
the entanglement wedge extends up to the left boundary. Before describing our
protocol, we should note that similar teleportation protocols have been
discussed in Refs. brown2019quantum ; nezami2021quantum ; schuster2022many .
Nevertheless, in the previous literature, instead of the measurement-decoding
protocol normally associated with quantum teleportation, a two-sided operator
is implemented in the SYK model to render the wormhole traversable. While it
was argued that the two-sided coupling could be written in terms of a
measurement and decoding maldacena2017diving ; gao2021traversable ;
Milekhin:2022bzx , the precise decoding protocol was not studied in detail.
For instance, in the SYK model, the two-sided operator has the form of
$\exp(i\theta\sum_{i}\psi_{L,i}\psi_{R,i})$, which consists of a fermionic
operator in the left and right side. Thus, it is not possible to measure the
fermionic operator and get an outcome.

Here, we investigate the usual measurement-decoding quantum teleportation
protocol using a measurement operator of the form $i\psi_{L,2k-1}\psi_{L,2k}$
acting on the left side. Since it is a bosonic operator, we can get a
measurement outcome and accordingly implement the decoding operator in the
right side.

The circuit version of Fig. 11 is shown in Fig. 12. The teleportation protocol
consists of three steps:

  1. 1.

At time $t=-t_{0}$, an unknown state $|\psi\rangle$ is inserted in the left
side;

  2. 2.

At time $t=0$, a projective measurement is performed in the left side and a
decoding operator is implemented in the right side according to the
measurement outcome 151515Here, we consider measuring the whole left side for
simplicity. In general, measuring any subset of the Majorana fermions
exceeding the critical value $m^{*}$ (as suggested by the entanglement wedge
transition) will be enough. It would be interesting to determine the critical
number of measured fermions needed for a successful teleportation. We leave
this to future investigation.;

  3. 3.

At time $t=t_{0}$, the state $|\psi\rangle$ is teleported to the right side.

At time $t=0$, the measurement-decoding protocol should be understood as a
quantum channel, which can be defined by Kraus operator,

$\displaystyle K_{\textbf{l}}=(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|)\otimes U_{\textbf{l}},$ (59)

where $|L_{\textbf{l}}\rangle$ is the measurement outcome on the $k=1,...,N/2$
qubits, i.e.,

$\displaystyle-i2\psi_{L,2k-1}\psi_{L,2k}|L_{\textbf{l}}\rangle=l_{k}|L_{\textbf{l}}\rangle,\quad
k=1,...,N/2.$ (60)

This means that the decoding operator we perform on the right side explicitly
depends on the measurement outcome in the left side. Conditioned on this
measurement outcome, the full protocol is shown by the circuit in Fig. 12.

Figure 12: The circuit of the teleportation protocol. In the left side, an
unknown state $|\psi\rangle$ is inserted in the past $-t_{0}$, and a
measurement is applied at $t=0$. The projection is indicated by the triangle
with the measurement outcome l. According to the measurement outcome, a
decoding operator, denoted by $U_{\textbf{l}}$, is implemented in the right
side at the same time $t=0$. After a time evolution $t_{0}$, the state
$|\psi\rangle$ is teleported to the right side. $V$ denotes the time evolution
operator.

To quantify the teleportation fidelity, we consider the left-right correlation
function $C_{\textbf{l},Q}$ conditioned on the measurement outcome 161616This
correlation function is closely related to the state fidelity, see Appendix
F.,

$\displaystyle\begin{aligned} C_{\textbf{l},Q}&=\langle
TFD|Q_{R}(t_{0})\left(|L_{\textbf{l}}\rangle\langle L_{\textbf{l}}|\otimes
U_{\textbf{l}}\right)Q_{L}(-t_{0})|TFD\rangle,\\\
Q_{L}(-t_{0})&=(V_{L}QV^{\dagger}_{L})\otimes\mathds{1},\quad
Q_{R}(t_{0})=\mathds{1}\otimes(V_{R}^{\dagger}QV_{R}),\end{aligned}$ (61)

where $V_{L,R}$ denotes the time evolution operator in the left and right
side. $Q$ is the operator being teleported. We assume the operator is properly
normalized, $Q^{\dagger}Q=\mathds{1}$, so that the magnitude of the
correlation function through the measurement-decoding channel (summing over
all Kraus operators) is at most one.

As discussed in Section 2, in the SYK model with measurement given by (60),
the Born probability does not depend on the measurement outcome. The decoding
operator associated with the measurement is

$\displaystyle
U_{\textbf{l}}=\exp\left(-i\frac{\theta}{q}\sum_{k=1}^{N/2}il_{k}\psi_{R,2k-1}\psi_{R,2k}\right).$
(62)

where $\theta$ is a tuning parameter. Later, we will tune $\theta$ to maximize
left-right correlation. Moreover, we will see in the next subsection that the
left-right correlation function does not depend on the measurement outcome.
Therefore, we define the following left-right correlation function for the SYK
model

$\displaystyle C_{Q}=\frac{\langle
TFD|Q_{R}^{\dagger}(t_{0})\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes U_{\textbf{l}}\right)Q_{L}(-t_{0})|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes\mathds{1}\right)|TFD\rangle},$ (63)

where we have omitted the subindex denoting measurement outcome because the
result is independent of it. The denominator is the probability of having the
measurement outcome l. Further, because the probability is independent of the
outcome, we have $\langle TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes\mathds{1}\right)|TFD\rangle=1/2^{N/2}$, due to the
$2^{N/2}$ different outcomes. Thus, $C_{Q}=\sum_{\textbf{l}}C_{\textbf{l},Q}$,
and since we have properly normalized the operator $Q$, this correlation
function has maximal magnitude one. In the following, we will use large-$q$
techniques to calculate this left-right correlation function (63), and show
that by properly tuning the parameter $\theta$, the fidelity can be made close
to one.

###  4.2 Left-right correlation function in the large $q$ limit

Motivated by the left-right correlation function (63), we define the following
twisted correlation function

$\displaystyle\mathcal{G}_{Q}=\frac{\langle
TFD|Q_{R}^{\dagger}(t_{0})\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes U_{\textbf{l}}\right)Q_{L}(-t_{0})|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle L_{\textbf{l}}|\otimes
U_{\textbf{l}}\right)|TFD\rangle}.$ (64)

It is related to the left-right correlation function (63) by

$\displaystyle C_{Q}=\frac{\langle TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes U_{\textbf{l}}\right)|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes\mathds{1}\right)|TFD\rangle}\mathcal{G}_{Q}.$ (65)

In this subsection, we will calculate both the twisted correlation function
and the prefactor in the above relation.

In the large-$N$ limit, the basic twisted correlation function is given by
$Q_{L}=\sqrt{2}\psi_{L,i}$ and $Q_{R}=-i\sqrt{2}\psi_{R,i}$, where the factor
of $\sqrt{2}$ is to have a properly normalized $Q$. Teleportation of an
arbitrary Majorana string, i.e. an operator consisting of an arbitrary product
of SYK Majorana operators, can be obtained using the basic correlation
function. In fact, the disconnected diagram built from the basic correlation
function will dominate the diagrammatic expansion of correlation functions for
Majorana strings in the large-$N$ limit. The basic twisted correlation
function reads

$\displaystyle\begin{aligned} \frac{\mathcal{G}_{\psi}}{2}&=\frac{\langle
TFD|(-i\psi_{R,i}(t_{0}))\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes
U_{\textbf{l}}\right)\psi_{L,i}(-t_{0})|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle L_{\textbf{l}}|\otimes
U_{\textbf{l}}\right)|TFD\rangle}\\\ &=\frac{\langle
L_{\textbf{l}}|\psi_{L,i}(-t_{0})e^{-\beta
H_{L}/2}e^{-i\frac{\theta}{q}\sum_{k}il_{k}\psi_{L,2k-1}\psi_{L,2k}}\psi_{L,i}(t_{0})e^{-\beta
H_{L}/2}|L_{\textbf{l}}\rangle}{\langle L_{\textbf{l}}|e^{-\beta
H_{L}/2}e^{-i\frac{\theta}{q}\sum_{k}il_{k}\psi_{L,2k-1}\psi_{L,2k}}e^{-\beta
H_{L}/2}|L_{\textbf{l}}\rangle},\end{aligned}$ (66)

where we have used the properties of the thermofield double (3) to bring
operators from the right side to the left side. When it is clear we will drop
the subindex $L$ for simplicity. We define the following imaginary time
ordered correlation function,

$\displaystyle G_{\psi}(\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle\frac{\langle L_{\textbf{l}}|e^{-\beta
H}\mathcal{T}\left[e^{-i\frac{\theta}{q}\sum_{k}l_{k}i\psi_{2k-1}(\frac{\beta}{2})\psi_{2k}(\frac{\beta}{2})}\psi_{i}(\tau_{1})\psi_{i}(\tau_{2})\right]|L_{\textbf{l}}\rangle}{\langle
L_{\textbf{l}}|e^{-\beta
H}\mathcal{T}\left[e^{-i\frac{\theta}{q}\sum_{k}l_{k}i\psi_{2k-1}(\frac{\beta}{2})\psi_{2k}(\frac{\beta}{2})}\right]|L_{\textbf{l}}\rangle},$
(67)

where $\mathcal{T}$ denotes imaginary time ordering, and as above
$|L_{\textbf{l}}\rangle$ is the projected state with measurement outcomes
given by $\textbf{l}=(l_{1},...,l_{N/2})$. Owing to the large-$N$ structure,
$G_{\psi}$ does not depend on the index $i$. Note that the imaginary time
correlation function and the basic twisted correlation function differ by a
factor of two.

In the imaginary time contour (see Fig. 13 (a)), the measurement operator
leads to boundary condition at $\tau=0$ and $\tau=\beta$ given by,

$\displaystyle\psi_{2k-1}(0)=-il_{k}\psi_{2k}(0),\quad\psi_{2k-1}(\beta)=il_{k}\psi_{2k}(\beta),$
(68)

and the decoding operator (62) leads to the following boundary condition at
$\tau=\beta/2$,

$\displaystyle\left(\begin{array}[]{cccc}\psi_{2k-1}(\frac{\beta^{+}}{2})\\\
il_{k}\psi_{2k}(\frac{\beta^{+}}{2})\end{array}\right)=\left(\begin{array}[]{cccc}\cos\frac{\theta}{q}&-i\sin\frac{\theta}{q}\\\
-i\sin\frac{\theta}{q}&\cos\frac{\theta}{q}\end{array}\right)\left(\begin{array}[]{cccc}\psi_{2k-1}(\frac{\beta^{-}}{2})\\\
il_{k}\psi_{2k}(\frac{\beta^{-}}{2})\end{array}\right).$ (75)

Figure 13: (a) Imaginary time contour for the twisted correlation function. In
the left side ($\tau=0,\beta$), a projective measurement
$|L_{\textbf{l}}\rangle\langle L_{\textbf{l}}|$ with outcome l (denoted by the
two triangles) is performed, and in the right side ($\tau=\frac{\beta}{2}$),
an associated decoding operator $U_{\textbf{l}}$ is applied. (b) Imaginary
time contour for the Majorana field $\chi_{k}$. The boundary condition from
the decoding operator becomes a rotation between $s=\frac{\beta}{2}$ and
$s=\frac{3\beta}{2}$, indicated by the dashed line.

In order to take care of the boundary condition at $\tau=0,\beta$, we
introduce a new field $\chi_{k}$ as in (16),

$\displaystyle\chi_{k}(s)=\begin{cases}\psi_{2k-1}(s),\quad&0<s<\beta,\\\
il_{k}\psi_{2k}(2\beta-s),\quad&\beta<s<2\beta,\end{cases}$ (76)

and accordingly the bilocal field,

$\displaystyle
G_{\chi}(s_{1},s_{2})=\frac{2}{N}\sum_{k=1}^{N/2}\chi_{k}(s_{1})\chi_{k}(s_{2}).$
(77)

Here, $G_{\chi}$ denotes the correlation function in the presence of the
twisted boundary condition. Note that the arguments of this bilocal field $s$,
run from $s=0$ to $s=2\beta$. As above, $G_{\chi}$ will satisfy the Schwinger-
Dyson equation,

$\displaystyle
G_{\chi}=(\partial-\Sigma)^{-1},\quad\Sigma_{\chi}(s_{1},s_{2})=\frac{\mathcal{J}^{2}}{q}P(s_{1},s_{2})[2G_{\chi}(s_{1},s_{2})]^{q-1}.$
(78)

Now, however, we will have a different boundary condition at $\tau=\beta/2$
(75). In terms the correlation function, this boundary condition becomes

$\displaystyle\left(\begin{array}[]{cccc}\lim_{s_{1}/s_{2}\rightarrow\frac{\beta^{+}}{2}}G_{\chi}(s_{1},s_{2})\\\
\lim_{s_{1}/s_{2}\rightarrow\frac{3\beta^{-}}{2}}G_{\chi}(s_{1},s_{2})\end{array}\right)=\left(\begin{array}[]{cccc}\cos\frac{\theta}{q}&-i\sin\frac{\theta}{q}\\\
-i\sin\frac{\theta}{q}&\cos\frac{\theta}{q}\end{array}\right)\left(\begin{array}[]{cccc}\lim_{s_{1}/s_{2}\rightarrow\frac{\beta^{-}}{2}}G_{\chi}(s_{1},s_{2})\\\
\lim_{s_{1}/s_{2}\rightarrow\frac{3\beta^{+}}{2}}G_{\chi}(s_{1},s_{2})\end{array}\right).$
(85)

There are various symmetries that will further simplify the calculation (see
Appendix G). In particular, the twisted correlation function satisfies various
reflection conditions, which are illustrated in Fig. 14. The blue (red) dashed
lines indicate (anti-)reflection conditions. Then the calculation on the full
domain can be reduced to the fundamental domain, denoted by $A$, $B$, $C$, and
$D$. The boundary condition (85) is illustrated by the gray lines.

In the following, we will use a large-$q$ expansion to calculate the twisted
correlation function in the fundamental domain. In the large $q$ limit, and
assuming
$G_{\chi}(s_{1},s_{2})=G_{0}(s_{1},s_{2})\left(1+\frac{1}{q}g(s_{1},s_{2})\right)$,
where $G_{0}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{1}-s_{2})$ is the bare
propagator, we arrive at a Liouville equation similar to (35),

$\displaystyle\partial_{s_{1}}\partial_{s_{2}}[G_{0}(s_{1},s_{2})g(s_{1},s_{2})]=\mathcal{J}^{2}P(s_{1},s_{2})[2G_{0}(s_{1},s_{2})]^{q-1}e^{g(s_{1},s_{2})}.$
(86)

Figure 14: The full domain of twisted correlation function. The blue dashed
lines indicate the reflection boundary condition, and the red dashed line
indicates the anti-reflection boundary condition. With the help of these
boundary conditions, we can reduce the calculation to the fundamental domains
$A$, $B$, $C$ and $D$. The gray solid lines indicate the boundary condition at
$s_{1,2}=\beta/2,3\beta/2$.

The remaining problem boils down to calculating the Liouville equation in the
fundamental domains $A$, $B$, $C$, and $D$ with the boundary condition (85).
In the large-$q$ limit, this boundary condition can be simplified (as in
qi2019quantum ; gao2021traversable ), to give

$\displaystyle G_{A}(s_{1},\frac{\beta}{2})\approx
e^{i\frac{\theta}{q}}G_{C}(s_{1},\frac{\beta}{2}),\quad
G_{B}(s_{1},\frac{\beta}{2})\approx
e^{i\frac{\theta}{q}}G_{D}(s_{1},\frac{\beta}{2}).$ (87)

Because the portions of the fundamental domain $C$ and $D$ do not cross the
twist boundary condition, we assume that the solution in $C$ and $D$ has time
translational symmetry qi2019quantum . On the other hand, for the fundamental
domain regions $A$ and $B$, we need to solve the Liouville equation
supplemented with the boundary condition (87). Via a tedious calculation, we
can obtain the solution in the fundamental domains,

$\displaystyle\begin{aligned}
&G_{A}(s_{1},s_{2})=\hat{G}_{11}(s_{1},s_{2})\frac{e^{i\theta/q}}{\left(1+\frac{e^{i\theta}-1}{\sin\gamma}\frac{\sin(\alpha(s_{1}-\beta/2))\sin(\alpha(s_{2}-\beta/2))}{\sin(\alpha|s_{12}|+\gamma)}\right)^{2/q}},\\\
&G_{B}(s_{1},s_{2})=\hat{G}_{21}(s_{1},s_{2})\frac{e^{i\theta/q}}{\left(1+\frac{e^{i\theta}-1}{\sin\gamma}\frac{\sin(\alpha\beta/2)\sin(\alpha(s_{2}-\beta/2))}{\sin(\alpha(\beta-
s_{2})+\gamma)}\right)^{2/q}},\\\
&G_{C}(s_{1},s_{2})=\hat{G}_{11}(s_{1},s_{2}),\quad
G_{D}(s_{1},s_{2})=\hat{G}_{21}(s_{1},s_{2}),\end{aligned}$ (88)

where $\hat{G}_{11}$ and $\hat{G}_{21}$ are given in (38) and (39), and the
constant parameters are determined by (40). Solutions in other regions can be
obtained by the symmetries described above.

With these results, the prefactor in (65) is now straightforward to calculate.
We can start with the following equation,

$\displaystyle\begin{aligned}
&\frac{\partial}{\partial\theta}\log\left[\frac{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle L_{\textbf{l}}|\otimes
U_{\textbf{l}}\right)|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes\mathds{1}\right)|TFD\rangle}\right]\\\ &=\frac{\langle
L_{\textbf{l}}|e^{-\beta
H}\mathcal{T}\left[\frac{1}{q}\sum_{k}l_{k}\psi_{2k-1}(\frac{\beta}{2})\psi_{2k}(\frac{\beta}{2})e^{-i\frac{\theta}{q}\sum_{k}il_{k}\psi_{2k-1}(\frac{\beta}{2})\psi_{2k}(\frac{\beta}{2})}\right]|L_{\textbf{l}}\rangle}{\langle
L_{\textbf{l}}|e^{-\beta
H}\mathcal{T}\left[e^{-i\frac{\theta}{q}\sum_{k}il_{k}\psi_{2k-1}(\frac{\beta}{2})\psi_{2k}(\frac{\beta}{2})}\right]|L_{\textbf{l}}\rangle}\\\
&=\frac{i}{q}\frac{N}{2}G_{\chi}\left(\frac{3\beta}{2},\frac{\beta}{2}\right)=\frac{i}{q}\frac{N}{4}(\sin\gamma)^{4/q},\end{aligned}$
(89)

which we can integrate to get

$\displaystyle\frac{\langle TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes U_{\textbf{l}}\right)|TFD\rangle}{\langle
TFD|\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes\mathds{1}\right)|TFD\rangle}=\exp\left[\frac{iN\theta}{4q}(\sin\gamma)^{4/q}\right].$
(90)

Thus, the difference (65) between the left-right correlation function and the
twisted correlation function is only a phase. Further note that this phase
does not depend on the operator $Q$, and therefore the magnitude of these two
correlation function is the same.

Figure 15: The magnitude of the twisted correlation function as a function of
$t_{0}$. Here the parameter is chosen to be $\beta\mathcal{=}100$,
$\theta=10^{-4}$ and $q=8$. The magnitude is close to one at a time $t_{0}$
that is related to the parameter $\theta$.

To get the teleportation fidelity, we take $s_{1}=\beta^{-}+it_{0}$ and
$s_{2}=\frac{\beta^{-}}{2}+it_{0}$, which is in region $A$, and find

$\displaystyle\mathcal{G}_{\psi}=2G_{A}\left(\beta^{-}+it_{0},\frac{\beta^{-}}{2}+it_{0}\right)\approx
e^{i\theta/q}\left(\frac{\sin\gamma}{1-\frac{\theta e^{2\alpha
t_{0}}}{4\sin\gamma}e^{i\gamma}}\right)^{2/q},$ (91)

where we have expanded the equation using $e^{-\alpha t_{0}}\ll 1$, and
$\theta\ll 1$. When the tuning parameter is chosen to be $\theta=4e^{-2\alpha
t_{0}}\sin\gamma$, the twisted correlation reaches its maximal magnitude,

$\displaystyle\mathcal{G}_{\psi}\approx\exp\left(i\frac{\pi}{q}\right)\left(\cos\frac{\gamma}{2}\right)^{2/q}.$
(92)

In the low energy limit,
$\gamma=\frac{\pi}{\beta\mathcal{J}}+\mathcal{O}\left(\frac{1}{(\beta\mathcal{J})^{2}}\right)$,
the magnitude of the twisted correlation function is close to one. This means
that the measurement-decoding channel can successfully teleport the Majorana
operator from the left side to the right side. A plot of the magnitude of the
left-right correlation function is shown in Fig. 15.

###  4.3 Traversable wormhole

To relate our results to a traversable wormhole, it will be simpler to start
with (61). We consider the sum over measurement outcome, i.e.,
$\sum_{\textbf{l}}C_{\textbf{l},Q}$. The sum over measurement outcome can be
rewritten as

$\displaystyle\sum_{\textbf{l}}|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes
U_{\textbf{l}}=\sum_{\textbf{l}}\left(\otimes_{k}|l_{k}\rangle\langle
l_{k}|\right)\otimes e^{i\frac{\theta}{2q}\sum_{k}l_{k}M_{R,k}},$ (93)

where we have introduced $M_{R,k}=-i2\psi_{R,2k-1}\psi_{R,2k}$,
$M_{L,k}=-i2\psi_{L,2k-1}\psi_{L,2k}$, and
$M_{L,k}|l_{k}\rangle=l_{k}|s_{k}\rangle$. Note that the projection operator
is an operator-valued delta function, i.e.,

$\displaystyle|l_{k}\rangle\langle l_{k}|=\delta(M_{L,k}-l_{k}).$ (94)

Using this delta function, we arrive at a two-sided operator
gao2021traversable

$\displaystyle\sum_{\textbf{l}}|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes
U_{\textbf{l}}=\sum_{\textbf{l}}\otimes_{k}\delta(M_{L,k}-l_{k})\otimes
e^{i\frac{\theta}{2q}\sum_{k}l_{k}M_{R,k}}=e^{i\frac{\theta}{2q}\sum_{k}M_{L,k}M_{R,k}}.$
(95)

After summing over all outcomes, the left-right correlation function is given
by

$\displaystyle\begin{aligned} \sum_{\textbf{l}}C_{\textbf{l},Q}&=\langle
TFD|Q_{R}(t_{0})\sum_{\textbf{l}}\left(|L_{\textbf{l}}\rangle\langle
L_{\textbf{l}}|\otimes U_{\textbf{l}}\right)Q_{L}(-t_{0})|TFD\rangle\\\
&=\langle
TFD|Q_{R}(t_{0})\left(e^{i\frac{\theta}{2q}\sum_{k}M_{L,k}M_{R,k}}\right)Q_{L}(-t_{0})|TFD\rangle.\end{aligned}$
(96)

This is the two-sided operator that makes the wormhole traversable
gao2017traversable ; maldacena2017diving ; brown2019quantum ;
nezami2021quantum ; gao2021traversable .

We can compare our results with the two-sided correlation function from an
$AdS_{2}$ gravity calculation. As depicted in Fig. 11, we consider the
following physical setup in an $AdS_{2}$ black hole: a particle created by the
operator $Q$ is released in the past $t=-t_{0}$ on the left side; at time
$t_{0}=0$, a projective measurement is performed on the left side, and
simultaneously, according to the measurement outcome, the decoding operator is
implemented in the right side with a properly chosen tuning parameter; the
particle is received in the right side at time $t=t_{0}$ with some
probability. This process is effectively quantified by the two-sided
correlation function maldacena2017diving

$\displaystyle C_{\text{grav}}=\langle
Q_{R}(t_{0})e^{i\frac{g}{N/2}\sum_{k}M_{L,k}M_{R,k}}Q_{L}(-t_{0})\rangle$ (97)

where the expectation value is evaluated in the $AdS_{2}$ black hole state.
Here, to make the comparison, we denote the two-sided deformation as $M$. In
the probe limit, where the back reaction of the particle created by $Q$ on the
geometry is neglected, the two-sided correlation function reads
maldacena2017diving

$\displaystyle
C_{\text{probe}}\propto\frac{1}{\left(2-g\frac{\Delta_{M}}{2^{2\Delta_{M}+2}}G_{N}e^{\frac{2\pi}{\beta}t_{0}}\right)^{2\Delta_{Q}}}$
(98)

where $\Delta_{M}$ and $\Delta_{Q}$ are the scaling dimension of operator $M$
and $Q$, respectively, and $G_{N}$ is Newton’s constant. It is easy to see
that when $g\sim\frac{\exp[-\frac{2\pi}{\beta}t_{0}]}{G_{N}}$, the correlation
function will diverge. Of course, this divergence is not physical and can be
cured by smearing out the particle, but, in essence, it shows the particle
traverses from the left side to the right side (see maldacena2017diving  for a
discussion).

Closely related to this phenomena, we find that the left-right correlation
function of an arbitrary Majorana string $Q$ is given by

$\displaystyle
C_{Q}\propto\left(\frac{\pi/(\beta\mathcal{J})}{1-g\frac{\beta\mathcal{J}}{4\pi
N}e^{2\alpha t_{0}}e^{i\gamma}}\right)^{2\Delta_{Q}},$ (99)

where $\Delta_{Q}=p/q$, $p$ is the number of Majorana operators contained in
the Majorana string $Q$, and $g=\frac{N}{4}\theta$. In the low energy limit,
$\gamma\rightarrow 0$, we encounter a seemly divergent expression for the
left-right correlation at $g\sim N\exp[-2\alpha t_{0}]$, which is identical to
the gravity calculation with $G_{N}=1/N$. Nevertheless, it is regularized by
$\gamma\approx\frac{\pi}{\beta\mathcal{J}}$ from the UV complete SYK model.
This shows that the teleportation protocol realizes a traversable wormhole.

###  4.4 Fidelity bound

In the previous sections, we studied the teleportation of an arbitrary
Majorana string $Q$, and found that our protocol yields a left-right
correlation that is close to one. In this subsection, we show that the left-
right correlation function gives a lower bound for state teleportation, and we
construct an encoding method that can achieve a nearly perfect teleportation
fidelity gao2021traversable . Note that the state teleportation fidelity is
related to the fidelity of the distillation of EPR pairs
yoshida2019disentangling ,

$\displaystyle\langle F_{\psi}\rangle=\frac{d}{d+1}F_{EPR}+\frac{1}{d+1},$
(100)

where $d$ is the dimension of Hilbert space of the teleported state
$|\psi\rangle$, and $\langle\cdot\rangle$ implies an average over all states.
Therefore, we will consider $F_{EPR}$ below. As shown in Appendix F, the
distillation fidelity is lower bounded by
$F_{EPR}\geq\left|\frac{1}{d^{2}}\sum_{\textbf{l},Q}C_{\textbf{l},Q}\right|^{2}$,
where $C_{\textbf{l},Q}$ is given by (61), and the sum of operators $Q$ is
over a complete basis for the $d$-dimensional Hilbert space. In the large-$N$
SYK model, as the left-right correlation function does not depend on the
measurement outcome, the bound of the distillation fidelity simplifies to

$\displaystyle F_{EPR}\geq\left|\frac{1}{d^{2}}\sum_{Q}C_{Q}\right|^{2}.$
(101)

where $C_{Q}$ is given in (63).

Now for simplicity, we consider the teleportation of an arbitrary qubit state,
for which the Hilbert space consists of two Majorana operators,
$\Psi_{1},\Psi_{2}$. In order to achieve a high fidelity, we need to embed
these two operators into $p$ SYK Majorana operators in the the following way,

$\displaystyle\begin{aligned}
&\Psi_{1}=2^{\frac{p}{2}}i^{\frac{p(p+1)}{p}}\psi^{1}\psi^{5}...\psi^{4p-3},\\\
&\Psi_{2}=2^{\frac{p}{2}}i^{\frac{p(p+1)}{p}}\psi^{3}\psi^{7}...\psi^{4p-1}.\end{aligned}$
(102)

where $p$ is an odd number and the prefactor is chosen such that these two
operators satisfy $\\{\Psi_{i},\Psi_{j}\\}=2\delta_{ij}$,
$\Psi_{i}^{\dagger}=\Psi_{i}$.

The complete basis operators for the qubit state are
$\mathds{1},\Psi_{1},\Psi_{2},i\Psi_{1}\Psi_{2}$. The fidelity bound then
becomes

$\displaystyle
F_{EPR}=\left|\frac{1}{4}\left(C_{\mathds{1}}+C_{\Psi_{1}}+C_{\Psi_{2}}+C_{i\Psi_{1}\Psi_{2}}\right)\right|^{2}=\left|\frac{1}{4}\left(1+\mathcal{G}_{\Psi_{1}}+\mathcal{G}_{\Psi_{2}}+\mathcal{G}_{i\Psi_{1}\Psi_{2}}\right)\right|^{2},$
(103)

where in the second equality, we have used the fact that the twisted
correlation function is equal to the left-right correlation function up to a
constant phase. Here, we can see an interesting distinction between the
teleportation of a Marjorana string such as $\Psi$ and the teleportation of an
arbitrary qubit state. Apart from the requirement that the magnitude of the
twisted correlation function be maximal, there is also a necessary condition
that the phases of these correlation functions should align to zero modulo
$2\pi$. This provides good reason to embed the qubit state into Majorana
strings $\Psi_{i}$ with $p$ SYK Majorana operators.

In the large-$N$ limit, the leading contribution to the twisted correlation
function is disconnected. Thus the twisted correlation function for $\Psi_{i}$
is given by the basis twisted correlation function,

$\displaystyle\mathcal{G}_{\Psi_{1}}=\mathcal{G}_{\Psi_{2}}\approx
i^{p}(\mathcal{G}_{\psi})^{p},\quad\mathcal{G}_{i\Psi_{1}\Psi_{2}}\approx-(\mathcal{G}_{\Psi})^{2p}.$
(104)

We observe that though the tuning parameter $\theta$ is fixed by the time of
state insertion, we still have freedom to tune $p$ to align the phase of the
correlation function. In particular, from (92), the fidelity is bounded by

$\displaystyle
F_{EPR}=\left|\frac{1}{4}\left(1+2e^{i2\pi(\frac{p}{4}+\frac{p}{2q})}(\cos\gamma)^{2p/q}+e^{i2\pi(p+\frac{1}{2}+\frac{p}{q})}(\cos\gamma)^{4p/q}\right)\right|^{2},$
(105)

To align the phases, we require $\frac{p}{4}+\frac{p}{2q}$ and
$p+\frac{1}{2}+\frac{p}{q}$ to be close to integers. For simplicity, we
consider $q\in 4\mathds{Z}$. We can then see that

$\displaystyle p=\begin{cases}\frac{q}{2}+1,\quad q=4(2k+1)\\\
\frac{q}{2}-1,\quad q=8k\end{cases}\quad k\in\mathds{Z},$ (106)

which is consistent with the fact that $p$ is an odd number. In the zero
temperature limit, we take $\gamma\rightarrow 0$. The fidelity bound in the
large-$q$ limit is then simply given by

$\displaystyle
F_{EPR}\geq\left|\frac{1}{4}\left(1+2e^{i\frac{\pi}{q}}+e^{i\frac{2\pi}{q}}\right)\right|^{2}=\left(\cos\frac{\pi}{2q}\right)^{4}\approx
1-\frac{\pi^{2}}{2q^{2}}.$ (107)

We can see that the embedding into $p$ Majorana operators can lead to a
fidelity for the teleportation of an arbitrary qubit state that is close to
one in the large-$q$ limit.

##  5 Discussion

In this work, we studied the effect of projective measurements on the
thermofield double state dual to an eternal black hole; in particular, we
focused on the SYK model and its dual, JT gravity coupled to matter. To
characterize the post-measurement state, we calculated the Renyi-2 mutual
information between the two sides of the SYK thermofield double state, which
showed a transition phenomenon at low temperature. We then reproduced this
mutual information calculation in the holographic JT plus matter dual. In this
gravitational dual, the measurement creates an end-of-the-world brane in the
bulk, and in particular, the matter fields dual to the measured fermions are
then modeled by a boundary CFT. The gravitational picture makes manifest that
the mutual information transition can be associated to an entanglement wedge
transition, see Fig. 10. Upon measurement of a sufficiently large fraction of
fermions, the entanglement wedge of the unmeasured side can jump to contain
large regions behind the horizon of the black hole. On the boundary side, this
change in the entanglement wedge can be understood as post-selection
teleporting information about the bulk into the unmeasured boundary. We then
explored this teleportation interpretation further, using the SYK model to
construct an explicit protocol realizing such a teleportation. Finally, we
related this telportation protocol to the physics of traversable wormholes.

In general, measurement is a nonlocal operation on a many-body quantum state,
which radically changes the entanglement structure. Thus, it is not clear when
a postselected state from an arbitrary measurement will give rise to a nice
holographic dual (e.g. a smooth, semiclassical geometry). One exception is
local projection onto a product state171717In the jargon of conformal field
theory, a projection onto a Cardy state cardy1989boundary ; miyaji2014boundary
., which can be described by a boundary conformal field theory
takayanagi2011holographic ; fujita2011aspects ; numasawa2016epr , and has been
recently studied in the context of holography numasawa2016epr ;
Antonini:2022sfm . Continuing in the same vein, the particular measurement
operator we study is the fermion parity operator for each qubit formed by two
adjacent even and odd Majorana operators (4); this is also a local projection.
This measurement procedure preserves many of the nice properties of the
original correlation function in the absence of measurement. Specifically, the
“diagonal” correlation function remains intact after measurement, indicating
the dual background $AdS_{2}$ geometry remains valid. This is consistent with
the expectation that local projective measurement leaves the dual spacetime
intact Antonini:2022sfm .

While a dual bulk geometry still exists, it is not left unchanged by this
local projective measurement. It is now clear that these measurements create
an end-of-the-world brane in bulk numasawa2016epr ; Antonini:2022sfm . Whereas
the Ryu-Takayanagi formula receives only minor modifications in the presence
of bulk end-of-the-world branes in the bulk, it is not well understood how the
matter entropy contribution from bulk fields—a crucial contribution in e.g.
black hole evaporation and the formation of islands penington2020entanglement
; penington2022replica ; Almheiri_2020 —is modified in the presence of a
projection on the boundary. In a broader context, what the quantum extremal
surface formula Engelhardt:2014gca  is in a system with non-unitarity in the
boundary—like that introduced by measurement—is an important open question.
Our model above suggests that the bulk matter dual to a measured boundary can
be effectively captured by a boundary CFT in two dimensional gravity. It
remains to be seen whether such a proposal could be extended to more general,
higher dimensional gravitational systems.

Apart from a scalar-type quantity to characterize entanglement, like the
entanglement entropy, the dynamics of how information “flows” in the many-body
system is often a key to understanding the physics from an information theory
perspective. For instance, studying the information flow of an evaporating
black hole has inspired tremendous progress in the black hole information
paradox. It was proposed in Ref. Antonini:2022sfm  that the information flow
induced by local projective measurement on the boundary can be largely
understood as information teleportation from the measured region to the
unmeasured region, thanks to the (pre-measurement) entanglement resource
between the two boundary subregions. Here, we have provided an explicit
protocol using the UV complete SYK model to realize this expectation.

As a consequence of this information flow, an entanglement wedge transition
shown in Fig. 10 naturally arises. It will be interesting to draw an analogy
between this measurement-induced entanglement wedge transition and the
transition between the empty set and the island in an evaporating black hole
penington2020entanglement ; penington2022replica ; Almheiri_2020 . A relation
of this sort may be natural in light of the Horowitz-Maldacena resolution of
the black hole information problem, which invokes a post-selection at the
horizon to explain how information might escape from a black hole
Horowitz:2003he . In more modern language, the excess number of black hole
states late in the evaporation process (in a semi-classical description) can
be understood via a non-isometric map akers2022quantum ; akers2022black . Of
course, projective measurements like the ones we have studied above can also
give rise to non-isometries Antonini:2022sfm . We leave exploration of this
potential connection to future work.

As stated above, local projection is a special kind of measurement that can
preserve large portions of the dual bulk geometry. There are many unknown
questions regarding “holographic measurement”. What are the neccessary
properties a measurement operation should satisfy to still give rise to a dual
bulk geometry? If a measurement can retain a portion of the dual spactime,
what kind of geometric changes might arise from the effect of measurement
(apart from the creation of an ETW brane)? An interesting extension of our
results might be to to consider a soft measurement that interpolates between
an identity operator and a projection. Since a dual description exists at both
ends of this interpolation, namely, the pre-measurement spacetime on one end
and the one including an ETW brane on the other, it seems likely that an
effective geometric description exists for this soft measurement.

The attentive reader may have realized that state preparation through
Euclidean time evolution can be regarded as a soft measurement of the
Hamiltonian operator. An interesting question that has been recently studied
in the quantum information science and many-body physics community is that of
hybrid circuit evolution, where the quantum circuits consist of not only
unitary operations but also non-unitary ones. In particular, when the non-
unitary gates are realized as a measurement interspersing the full circuit, it
also induces a transition of the entanglement structure of the final state,
the so-called measurement-induced entanglement phase transition
li2019measurement ; skinner2019measurement ; chan2019unitary ;
jian2021measurement ; Bentsen:2021ukm . Our measurement-induced entanglement
wedge transition can be viewed as the effect of “one time” measurement, in
contrast to the hybrid evolution. This provides a first step in understanding
the dynamical measurement-induced entanglement transition in a context with
holographic duality. When the system can be effectively described by an
evolution of a generic non-Hermitian Hamiltonian, several attempts have been
made to explore the dual spacetime Milekhin:2022bzx ; kawabata2022dynamical ;
garcia2022keldysh ; goto2022entanglement . Due to the non-Hermiticity in the
boundary degrees of freedom, an analysis involving complex metrics is
inevitable; nevertheless, given the larger analytical control it provides, we
believe lower dimensional gravity is a promising platform to further explore
the dynamical effects arising in the presence of measurements.

###### Acknowledgements.

We would like to thank Gregory Bentsen, Charles Cao, Raphael Bousso, Yasunori
Nomura, Ping Gao, Zhenbin Yang, Zhuo-Yu Xian 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.) .

##  Appendix A One-sided measurement in the SYK model: Renyi-2 entropy

In this appendix, we derive the Schwinger-Dyson equations and the formula for
the on-shell actions $I_{num}(m)$ and $I_{den}(m)$ introduced in Section 2.3.
These are needed ingredients to compute the SYK Renyi-2 mutual information
between the left side and the unmeasured Majorana fermions in the right side
after a projective measurement on a subset of Majorana fermions is performed
in the right side. We also provide some more detail about our numerical
analysis.

###  A.1 On-shell action for the numerator of equation (42)

Let us start by computing $I_{num}(m)$. As a first step, the boundary
conditions (45) suggest introducing bilocal fields analogous to the ones
defined in equations (13)-(14) and the corresponding Lagrange multiplier
fields $\tilde{\Sigma}_{ii}(\tau_{1},\tau_{2})$, $i=1,2,3$. The arguments now
have range $\tau_{1,2}\in(0,2\beta)$. In terms of the bilocal fields, the
action (2.3) takes the form

$\displaystyle-I=\int
d\tau_{1}d\tau_{2}\Big{[}-\frac{1}{2}\Big{(}\sum_{k=1}^{M/2}\psi_{2k-1}(\tau_{1})(\partial-\tilde{\Sigma}_{11})\psi_{2k-1}(\tau_{2})+\sum_{k=1}^{M/2}\psi_{2k}(\tau_{1})(\partial-\tilde{\Sigma}_{22})\psi_{2k}(\tau_{2})$
$\displaystyle+\sum_{j=M+1}^{N}\psi_{j}(\tau_{1})(\partial-\tilde{\Sigma}_{33})\psi_{j}(\tau_{2})\Big{)}-\frac{1}{2}\left(\frac{M}{2}\tilde{\Sigma}_{11}\tilde{G}_{11}+\frac{M}{2}\tilde{\Sigma}_{22}\tilde{G}_{22}+(N-M)\tilde{\Sigma}_{33}\tilde{G}_{33}\right)$
$\displaystyle+\frac{J^{2}}{2qN^{q-1}}\left(\frac{M}{2}\tilde{G}_{11}(\tau_{1},\tau_{2})+\frac{M}{2}\tilde{G}_{22}(\tau_{1},\tau_{2})+(N-M)\tilde{G}_{33}(\tau_{1},\tau_{2})\right)^{q}\Big{]}.$
(108)

Similar to what we did in Section 2.1, we can now introduce a new Majorana
field defined as

$\displaystyle\chi_{k}(s)=\begin{cases}\psi_{2k-1}(s),&\quad 0<s<\beta/2,\\\
i\psi_{2k}(\beta-s),&\quad\beta/2<s<\beta,\\\
-i\psi_{2k}(2\beta-s),&\quad\beta<s<3\beta/2,\\\ \psi_{2k-1}(s-\beta),&\quad
3\beta/2<s<2\beta,\\\ \psi_{2k-1}(s-\beta),&\quad 2\beta<s<5\beta/2,\\\
i\psi_{2k}(4\beta-s),&\quad 5\beta/2<s<3\beta,\\\ -i\psi_{2k}(5\beta-s),&\quad
3\beta<s<7\beta/2,\\\ \psi_{2k-1}(s-2\beta),&\quad
7\beta/2<s<4\beta.\end{cases}$ (109)

The new Majorana satisfies the conventional anti-periodic boundary conditions
for fermionic fields, i.e. $\chi_{k}(4\beta+s)=-\chi_{k}(s)$. We can then
integrate out the Majorana fermions in the path integral, obtaining the
following action for the path integral over the bilocal fields:

$\displaystyle\begin{aligned}
-\frac{I}{N}=&\frac{m}{4}\log\det(\hat{G}_{t}^{-1}-\hat{\tilde{\Sigma}})+\frac{1-m}{2}\log\det(G_{t}^{-1}-\tilde{\Sigma}_{33})\\\
&-\frac{1}{2}\int
d\tau_{1}d\tau_{2}\left(\frac{m}{2}\tilde{\Sigma}_{11}\tilde{G}_{11}+\frac{m}{2}\tilde{\Sigma}_{22}\tilde{G}_{22}+(1-m)\tilde{G}_{33}\right)\\\
&+\frac{J^{2}}{2q}\left(\frac{m}{2}(\tilde{G}_{11}(\tau_{1},\tau_{2})+\tilde{G}_{22}(\tau_{1},\tau_{2})+(1-m)\tilde{G}_{33}(\tau_{1},\tau_{2})\right)^{q},\end{aligned}$
(110)

where we introduced the free fermion propagator $G_{t}=\partial^{-1}$ and a
compact notation for the self energy

$\displaystyle\hat{\tilde{\Sigma}}(s_{1},s_{2})=\left[\begin{array}[]{cccccccc}\tilde{\Sigma}_{11}(11)&0&0&\tilde{\Sigma}_{11}(12)&\tilde{\Sigma}_{11}(13)&0&0&\tilde{\Sigma}_{11}(14)\\\
0&-\tilde{\Sigma}_{22}(11)&\tilde{\Sigma}_{22}(12)&0&0&-\tilde{\Sigma}_{22}(13)&\tilde{\Sigma}_{22}(14)&0\\\
0&\tilde{\Sigma}_{22}(21)&-\tilde{\Sigma}_{22}(22)&0&0&\tilde{\Sigma}_{22}(23)&-\tilde{\Sigma}_{22}(24)&0\\\
\tilde{\Sigma}_{11}(21)&0&0&\tilde{\Sigma}_{11}(22)&\tilde{\Sigma}_{11}(23)&0&0&\tilde{\Sigma}_{11}(24)\\\
\tilde{\Sigma}_{11}(31)&0&0&\tilde{\Sigma}_{11}(32)&\tilde{\Sigma}_{11}(33)&0&0&\tilde{\Sigma}_{11}(34)\\\
0&-\tilde{\Sigma}_{22}(31)&\tilde{\Sigma}_{22}(32)&0&0&-\tilde{\Sigma}_{22}(33)&\tilde{\Sigma}_{22}(34)&0\\\
0&\tilde{\Sigma}_{22}(41)&-\tilde{\Sigma}_{22}(42)&0&0&\tilde{\Sigma}_{22}(43)&-\tilde{\Sigma}_{22}(44)&0\\\
\tilde{\Sigma}_{11}(41)&0&0&\tilde{\Sigma}_{11}(42)&\tilde{\Sigma}_{11}(43)&0&0&\tilde{\Sigma}_{11}(44)\end{array}\right].$
(119)

In 119, we have used the short-hand notation
$\tilde{\Sigma}_{11}(ij)=\tilde{\Sigma}_{11}\left((i-1)\beta/2+s_{1},(j-1)\beta/2+s_{2}\right)$
and
$\tilde{\Sigma}_{22}(ij)=\tilde{\Sigma}_{22}\left((i-1)\beta/2-s_{1},(j-1)\beta/2-s_{2}\right)$
with $0<s_{1,2}<\beta/2$. The boundary condition resulting from the twist
operator is given by

$\displaystyle G_{t}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{1}-s_{2}),\quad
s_{1,2}\in(0,2\beta),$ (121)
$\displaystyle\hat{G}_{t}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{1}-s_{2}),\quad
s_{1,2}\in(\beta,3\beta)\text{ or }s_{1,2}(0,\beta)\cup(3\beta,4\beta).$ (122)

and $G_{t}(s_{1},s_{2})=\hat{G}_{t}(s_{1},s_{2})=0$ otherwise.

We can now derive the Schwinger-Dyson equations by varying the action (110)
for the bilocal fields:

$\displaystyle\begin{aligned}
&\hat{G}=(\hat{G}_{t}^{-1}-\hat{\Sigma})^{-1},\quad
G=(G_{t}^{-1}-\Sigma)^{-1},\\\
&\Sigma_{ii}=J^{2}\left(\frac{m}{2}(G_{11}+G_{22})+(1-m)G_{33}\right),\quad
i=1,2,3.\end{aligned}$ (123)

Here we omitted the $\tilde{}$ to indicate that the bilocal fields are on
shell. $G_{11}$ and $G_{22}$ are related to $\hat{G}$ by

$\displaystyle\begin{aligned} &G_{11}(s_{1},s_{2})=\\\
&\left[\begin{array}[]{cccc}\hat{G}(s_{1},s_{2})&\hat{G}(s_{1},3\beta/2+s_{2})&\hat{G}(s_{1},2\beta+s_{2})&\hat{G}(s_{1},7\beta/2+s_{2})\\\
\hat{G}(3\beta/2+s_{1},s_{2})&\hat{G}(3\beta/2+s_{1},3\beta/2+s_{2})&\hat{G}(3\beta/2+s_{1},2\beta+s_{2})&\hat{G}(3\beta/2+s_{1},7\beta/2+s_{2})\\\
\hat{G}(2\beta+s_{1},s_{2})&\hat{G}(2\beta+s_{1},3\beta/2+s_{2})&\hat{G}(2\beta+s_{1},2\beta+s_{2})&\hat{G}(2\beta+s_{1},7\beta/2+s_{2})\\\
\hat{G}(7\beta/2+s_{1},s_{2})&\hat{G}(7\beta/2+s_{1},3\beta/2+s_{2})&\hat{G}(7\beta/2+s_{1},2\beta+s_{2})&\hat{G}(7\beta/2+s_{1},7\beta/2+s_{2})\end{array}\right],\\\
&G_{22}(s_{1},s_{2})=\\\ &\left[\begin{array}[]{cccc}-\hat{G}(\beta-
s_{1},\beta-s_{2})&\hat{G}(\beta-s_{1},3\beta/2-s_{2})&-\hat{G}(\beta-
s_{1},3\beta-s_{2})&\hat{G}(\beta-s_{1},7\beta/2-s_{2})\\\
\hat{G}(3\beta/2-s_{1},\beta-
s_{2})&-\hat{G}(3\beta/2-s_{1},3\beta/2-s_{2})&\hat{G}(3\beta/2-s_{1},3\beta-
s_{2})&-\hat{G}(3\beta/2-s_{1},7\beta/2-s_{2})\\\ -\hat{G}(3\beta-s_{1},\beta-
s_{2})&\hat{G}(3\beta-s_{1},3\beta/2-s_{2})&-\hat{G}(3\beta-s_{1},3\beta-
s_{2})&\hat{G}(3\beta-s_{1},7\beta/2-s_{2})\\\ \hat{G}(7\beta/2-s_{1},\beta-
s_{2})&-\hat{G}(7\beta/2-,3\beta/2-s_{2})&\hat{G}(7\beta/2-s_{1},3\beta+s_{2})&-\hat{G}(7\beta/2-s_{1},7\beta/2-s_{2})\end{array}\right].\end{aligned}$
(124)

We can then discretize the imaginary time contour and numerically solve the
Schwinger-Dyson equations (123) combined with equations (119) and (124) by
iteration. Using the Schwinger-Dyson equations (123), the on-shell action
$I_{num}(m)$ can be written as

$\displaystyle-\frac{I_{num}(m)}{N}$ $\displaystyle=$
$\displaystyle\frac{m}{4}\left(\log\det[\hat{G}^{-1}\hat{G}_{t}]+2\log
2\right)+\frac{1-m}{2}\left(\log\det[G^{-1}G_{t}]+\log 2\right)$ (126)
$\displaystyle-\frac{J^{2}}{2}\left(1-\frac{1}{q}\right)\int
d\tau_{1}d\tau_{2}\left(\frac{m}{2}(G_{11}+G_{22})+(1-m)G_{33}\right)^{q},$

where we have used a normalization for the bare propagators

$\displaystyle\log(G_{t}^{-1})-\log 2=0,\quad\log(\hat{G}_{t}^{-1})-2\log
2=0.$ (127)

This is readily understood if we consider that $G_{t}$ is the same as a
propagator for a free fermion on a contour $s\in(0,2\beta)$ and $\hat{G}_{t}$
is the same as a propagator for two free fermions on two separately contours
$s\in(\beta,3\beta)$ and $s\in(0,\beta)\cup(3\beta,4\beta)$. The partition
function of a free fermionic mode is simply $2$ (accounting for the occupied
and unoccupied states).

Once the Schwinger-Dysons equations (123) are numerically solved, the on-shell
action (126) can be evaluated. In order to minimize the error due to the
discretization of the imaginary time contour, we obtain results using
different finite discretizations and then extrapolate the result in the
continuum limit (i.e. infinitely fine discretization). More explicitly, we
divide each contour $(0,\beta)$ into $L=40,60,80,100$ segments, and then
extrapolate the results in the $L\rightarrow\infty$ limit.

###  A.2 On-shell action for the denominator of equation (42)

In a similar fashion, the denominator of equation (42) can be written in terms
of a path integral

$\displaystyle\overline{(\text{Tr}[\Psi(m)])^{2}}=\int D\psi e^{-I},$ (128)

where the action is again given by equation (2.3). We can then rewrite the
action in terms of bilocal fields:

$\displaystyle\begin{aligned}
-\frac{I}{N}=&\frac{m}{4}\log\det(\hat{G}_{i}^{-1}-\hat{\tilde{\Sigma}})+\frac{1-m}{2}\log\det(G_{i}^{-1}-\tilde{\Sigma}_{33})\\\
&-\frac{1}{2}\int
d\tau_{1}d\tau_{2}\left(\frac{m}{2}\tilde{\Sigma}_{11}\tilde{G}_{11}+\frac{m}{2}\tilde{\Sigma}_{22}\tilde{G}_{22}+(1-m)\tilde{G}_{33}\right)\\\
&+\frac{J^{2}}{2q}\left(\frac{m}{2}(\tilde{G}_{11}(\tau_{1},\tau_{2})+\tilde{G}_{22}(\tau_{1},\tau_{2}))+(1-m)\tilde{G}_{33}(\tau_{1},\tau_{2})\right)^{q}\end{aligned}$
(129)

where again $G_{i}=\partial^{-1}$. The boundary conditions (46) now lead to

$\displaystyle G_{i}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{1}-s_{2}),\quad
0<s_{1,2}<\beta\text{ or }\beta<s_{1,2}<2\beta,$ (130)
$\displaystyle\hat{G}_{i}(s_{1},s_{2})=\frac{1}{2}\text{sgn}(s_{1}-s_{2}),\quad
0<s_{1,2}<2\beta\text{ or }2\beta<s_{1,2}<4\beta.$ (131)

and $G_{i}(s_{1},s_{2})=\hat{G}_{i}(s_{1},s_{2})=0$ otherwise. The Schwinger-
Dyson equations are given by

$\displaystyle\hat{G}=(\hat{G}_{i}^{-1}-\hat{\Sigma})^{-1},\quad
G=(G_{i}^{-1}-\Sigma)^{-1},$ (132)
$\displaystyle\Sigma_{ii}=J^{2}\left(\frac{m}{2}(G_{11}+G_{22})+(1-m)G_{33}\right),\quad
i=1,2,3.$ (133)

with $\hat{\Sigma}$ ($\hat{G}$) given by (119) and (124). The on-shell action
can then be written as

$\displaystyle-\frac{I_{den}(m)}{N}$ $\displaystyle=$
$\displaystyle\frac{m}{4}\left(\log\det[\hat{G}^{-1}\hat{G}_{i}]+2\log
2\right)+\frac{1-m}{2}\left(\log\det[G^{-1}G_{i}]+2\log 2\right)$ (134)
$\displaystyle-\frac{J^{2}}{2}(1-\frac{1}{q})\int
d\tau_{1}d\tau_{2}\left(\frac{m}{2}(G_{11}+G_{22})+(1-m)G_{33}\right)^{q},$

where the normalization is

$\displaystyle\log(G_{i}^{-1})-2\log 2=0,\quad\log(\hat{G}_{i}^{-1})-2\log
2=0.$ (135)

We can then numerically solve the Schwinger-Dyson equations and evaluate the
on-shell action $I_{den}(m)$ like we did for $I_{num}(m)$. Once we have the
numerically-obtained value of both on-shell actions (126) and (134), we can
finally compute the Renyi-2 entropy—given by
$S_{L}^{(2)}(\Psi(m))=I_{num}(m)-I_{den}(m)$ in the saddle point
approximation—and the Renyi-2 mutual information—given by
$I_{LR}^{2}(\Psi(m))=2S_{L}^{(2)}(\Psi(m))$. The numerical results obtained
following the procedure described in this appendix are reported in Fig. 6.

##  Appendix B The $m=1$ case: Kourkoulou-Maldacena state

The setup studied by Kourkoulou and Maldacena kourkoulou2017pure  can be seen
as a special case of our setup involving a one-sided measurement on a TFD
state. In particular, when all the Majorana fermions on one side are projected
using the measurement operator (4) (namely, $m=1$), the resulting post-
measurement state on the other side is pure and given by a Kourkoulou-
Maldacena state. For completeness, in this appendix we calculate the Renyi-2
entropy of a subset of Majorana fermions in the Kourkoulou-Maldacena state
zhang2020entanglement  obtained in the $m=1$ case of the analysis of Section 2
and Appendix A. Without loss of generality, we assume the Majorana fermions on
the right side are projected.

We can then divide the Majorana fermions on the left side into two subsets
$A=\\{\psi_{i}|i=1,...,N_{A}\\}$ and $A^{c}=\\{\psi_{i}|i=N_{A}+1,...,N\\}$
and define $\lambda=N_{A}/N$. The Renyi-2 entropy of the subset $A$ is given
by

$\displaystyle
e^{-S_{A}^{(2)}(\lambda)}=\frac{\overline{\text{Tr}_{A}\left[\left(\text{Tr}_{\bar{A}}[\rho_{\text{KM}}]\right)^{2}\right]}}{\overline{\left(\text{Tr}[\rho_{\text{KM}}]\right)^{2}}},\quad\rho_{\text{KM}}=\text{Tr}_{R}|\Psi\rangle\langle\Psi|,$
(136)

where $|\Psi\rangle=|\Psi(m=1)\rangle$ is defined in equation (8) and we
traced over the right side (which after the measurement is in a product state
with the left side). Like in our previous analysis, the numerator and
denominator of equation (136) can be represented using a Euclidean path
integral. The corresponding action for the numerator is the same as (2.3) on a
contour $\tau\in(0,2\beta)$, but the boundary conditions are modified to take
into account the insertion of a twist operator only for the first $N_{A}$
Majorana fermions on the left and the fact that all Majorana fermions are
measured on the right:

$\displaystyle\begin{aligned}
&\psi_{i}(0)=-\psi_{i}(2\beta),\quad\psi_{i}(\beta_{-})=\psi_{i}(\beta_{+}),\quad
i=1,...,N_{A},\\\
&\psi_{N_{A}+j}(0)=-\psi_{N_{A}+j}(\beta_{-}),\quad\psi_{N_{A}+j}(\beta_{+})=-\psi_{N_{A}+j}(2\beta),\quad
j=1,...,N-N_{A},\\\
&\psi_{2k-1}(\frac{\beta_{-}}{2})=i\psi_{2k}(\frac{\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{\beta_{+}}{2})=-i\psi_{2k}(\frac{\beta_{+}}{2}),\quad
k=1,...,N/2,\\\
&\psi_{2k-1}(\frac{3\beta_{-}}{2})=i\psi_{2k}(\frac{3\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{3\beta_{+}}{2})=-i\psi_{2k}(\frac{3\beta_{+}}{2}),\quad
k=1,...,N/2.\end{aligned}$ (137)

The first line gives the anti-periodic boundary conditions for Majorana
fermions in $A$ (for which a twist operator is inserted in the left); the
second line gives the anti-periodic boundary conditions for Majorana fermions
in $A^{c}$ (for which no twist operator is inserted); the last two lines are a
consequence of the measurement of all Majorana fermions in the right side.

On the other hand, the action for the denominator is again given by equation
(2.3), while the boundary conditions now reflect the fact that no twist
operators are inserted in the left side:

$\displaystyle\begin{aligned}
&\psi_{j}(0)=-\psi_{j}(\beta_{-}),\quad\psi_{j}(\beta_{+})=-\psi_{j}(2\beta),\quad
j=1,...,N,\\\
&\psi_{2k-1}(\frac{\beta_{-}}{2})=i\psi_{2k}(\frac{\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{\beta_{+}}{2})=-i\psi_{2k}(\frac{\beta_{+}}{2}),\quad
k=1,...,N/2,\\\
&\psi_{2k-1}(\frac{3\beta_{-}}{2})=i\psi_{2k}(\frac{3\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{3\beta_{+}}{2})=-i\psi_{2k}(\frac{3\beta_{+}}{2}),\quad
k=1,...,N/2.\end{aligned}$ (138)

Figure 16: Renyi-2 entropy per Majorana fermion of a subset $A$ of fermions in
the left side after all fermions in the right side have been measured. The
left system is in the Kourkoulou-Maldacena state. $\lambda$ denotes the
fraction of left fermions included in subsystem $A$. Different colors denote
different values of $\beta$. The Renyi-2 entropy is symmetric around
$\lambda=0.5$ due to the purity of the Kourkoulou-Maldacena state and the
$O(N)$ symmetry of the large-$N$ SYK model. Increasing $\beta$ we increase the
amount of Euclidean time evolution preparing the Kourkoulou-Maldacena state,
and therefore the entanglement among fermions.

We can then introduce again the bilocal fields, derive and solve numerically
the Schwinger-Dysons equations, evaluate the on-shell action for the numerator
and denominator, and finally compute the Renyi-2 entropy in the saddle point
approximation. This procedure is completely analogous to the one carried out
in Section 2 and Appendix A, to which we refer for technical details. The
results of our numerical evaluation of the Renyi-2 entropy as a function of
the fraction $\lambda_{A}$ of fermions in subsystem $A$ are shown in Fig. 16
for different values of $\beta$.

A general feature for all curves is that the entropy increases with $\lambda$
for $\lambda<1/2$, and decreases for $\lambda>1/2$. This is a consequence of
the fact that the Kourkoulou-Maldacena state—obtained in the left side by
measuring all fermions in the right side—is a pure state. For this reason and
due to the $O(N)$ symmetry of the SYK model at large $N$, the Renyi-2 entropy
is symmetric with respect to $\lambda=1/2$. Notice also that the Renyi-2
entropy is lower for smaller values of $\beta$. This behavior can be easily
understood. For $\beta=0$, the Kourkoulou-Maldacena state is simply given by
the eigenstate of the measurement operator, which is a product state of the
Majorana fermions (and therefore any subsystem of Majorana fermions has
vanishing entropy). Increasing $\beta$, the Kourkoulou-Maldacena state is
given by the same product state evolved by an amount $\beta$ of Euclidean
time. Such Euclidean time evolution introduces entanglement among different
Majorana fermions, increasing the Renyi-2 entropy kourkoulou2017pure ;
Antonini:2021xar .

##  Appendix C Two-sided measurement and mutual information in the SYK model

In this appendix we consider the case where the projective measurement is
performed on a subset of fermions in both sides of a SYK thermofield double.
Similar to the analysis of Section 2, we measure the fermionic parity on the
left and right side. The measurement operators are given by

$\displaystyle\begin{aligned} &M_{R,k}=-i2\psi_{R,2k-1}\psi_{R,2k},\quad
k=1,...,M/2\\\ &M_{L,k}=-i2\psi_{L,2k-1}\psi_{L,2k},\quad
k=(N-M)/2+1,...,N/2,\end{aligned}$ (139)

where again we assume $M$ and $N$ are both even, and we take the large-$N$
limit with fixed $m=M/N$. The eigenvalues of the measurement operators are
$r_{k},l_{k}=\pm 1$. Assume we measure the TFD state and observe the
measurement outcomes $\textbf{l}=(l_{(N-M)/2+1},...,l_{N/2})$ and
$\textbf{r}=(r_{1},...,r_{M/2})$ for the left side and right side,
respectively. Then the post-measurement states for the measured Majorana
fermions on the left and right side are given by $|L_{\textbf{l}}(m)\rangle$
and $|R_{\textbf{r}}(m)\rangle$, where

$\displaystyle\begin{aligned}
&M_{R,k}|R_{\textbf{r}}(m)\rangle=r_{k}|R_{\textbf{r}}(m)\rangle,\quad
k=1,...,M/2,\\\
&M_{L,k}|L_{\textbf{l}}(m)\rangle=l_{k}|L_{\textbf{l}}(m)\rangle,\quad
k=(N-M)/2+1,...,N/2.\end{aligned}$ (140)

The resulting full (unnormalized) post-measurement state for the two-sided
system is given by

$\displaystyle|\Phi_{\textbf{l,r}}(m)\rangle=\left(\mathds{1}^{(1-m)}_{R}\otimes|R_{\textbf{r}}(m)\rangle\langle
R_{\textbf{r}}(m)|\right)\otimes\left(\mathds{1}^{(1-m)}_{L}\otimes|L_{\textbf{l}}(m)\rangle\langle
L_{\textbf{l}}(m)|\right)|TFD\rangle,$ (141)

where $\mathds{1}^{(1-m)}_{L,R}$ denotes the identity operator acting on the
unmeasured Majorana in the left and right sides.

We are interested in computing the mutual information between the unmeasured
Majorana fermions in the two sides. Because the state
$|\Phi_{\textbf{l,r}}(m)\rangle$ is pure, the mutual information satisfies

$\displaystyle
I_{LR}(\Phi_{\textbf{l,r}}(m))=S_{L}(\Phi_{\textbf{l,r}}(m))+S_{R}(\Phi_{\textbf{l,r}}(m))-S_{LR}(\Phi_{\textbf{l,r}}(m))=2S_{L}(\Phi_{\textbf{l,r}}(m)),$
(142)

where
$\Phi_{\textbf{l,r}}(m)=|\Phi_{\textbf{l,r}}(m)\rangle\langle\Phi_{\textbf{l,r}}(m)|$
is the associated unnormalized density matrix. Like in Section 2.3, we focus
on the Renyi-2 entropy

$\displaystyle
e^{-S_{L}^{(2)}(\Phi(m))}=\frac{\overline{\text{Tr}_{L}\left[(\text{Tr}_{R}[\Phi(m)^{2}])\right]}}{\overline{\left(\text{Tr}[\Phi(m)]\right)^{2}}}.$
(143)

Since our analysis does not depend on the measurement outcomes, we omitted the
subscripts in equation (143) and in the following calculation we set for
simplicity $l_{k}=r_{k}=1$ for $k=1,...,M/2,(N-M)/2+1,...,N/2$.

(a) $1\leq i\leq M$ (b) $M<i\leq N-M$ (c) $N-M<i\leq N$

Figure 17: Imaginary time contour $\tau\in(0,2\beta)$ for the Euclidean path
integral computing the numerator of equation (143) when $M<N/2$. The twist
operator inserted in the left system is indicated by the dashed line. The
insertion of a measurement operator in the left and right sides is indicated
by the triangles. When the projective measurement operator is inserted in the
left side, the effect of the twist operator is trivial and we therefore omit
the dashed line. (a) Boundary conditions (146) for Majorana fermions
$\psi_{i}$, $1\leq i\leq M$. (b) Boundary conditions (147) for Majorana
fermions $\psi_{i}$, $M<i\leq N-M$. (c) Boundary conditions (148) for Majorana
fermions $\psi_{i}$, $N-M<i\leq N$.

The path integral representation for the numerator of equation (143) is

$\displaystyle\overline{\text{Tr}_{L}\left[(\text{Tr}_{R}[\Phi(m)^{2}])\right]}=\int
D\psi e^{-I},$ (144)

with the action

$\displaystyle-I=-\int
d\tau\sum_{j}\frac{1}{2}\psi_{j}(\tau)\partial_{\tau}\psi_{j}(\tau)+\int
d\tau_{1}d\tau_{2}\frac{J^{2}}{2qN^{q-1}}\left(\sum_{j}\psi_{j}(\tau_{1})\psi_{j}(\tau_{2})\right)^{q}.$
(145)

The imaginary time runs from $\tau=0$ to $\tau=2\beta$. We now need to specify
the correct boundary conditions taking into account both the measurement
operators and the insertion of a twist operator in the left side. To this end,
we need to discuss two separate cases, i.e. $M<N/2$ and $M>N/2$.

####  M<N/2

For $M<N/2$, the measurement on the right side is performed on Majorana
fermions $\psi_{i}$ with $i\leq M$, while on the left side it is performed on
Majorana fermions $\psi_{i}$ with $i>N-M$. For the Majorana fermions
$\psi_{i}$ with $M<i\leq N-M$, there is no measurement in either side. A
graphical representation of the boundary conditions is given in Fig. 17.

For $1\leq i\leq M$, the boundary conditions must take into account the
measurement operator inserted at $\tau=\beta/2,3\beta/2$ as well as the twist
operator inserted at $\tau=0,\beta,2\beta$ (see Fig. 17 (a)):

$\displaystyle\begin{aligned} &k=1,...,M/2\\\
&\psi_{2k-1}(\frac{\beta_{-}}{2})=i\psi_{2k}(\frac{\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{\beta_{+}}{2})=-i\psi_{2k}(\frac{\beta_{+}}{2}),\\\
&\psi_{2k-1}(\frac{3\beta_{-}}{2})=i\psi_{2k}(\frac{3\beta_{-}}{2}),\quad\psi_{2k-1}(\frac{3\beta_{+}}{2})=-i\psi_{2k}(\frac{3\beta_{+}}{2}),\\\
&\psi_{2k-1}(0)=-\psi_{2k-1}(2\beta),\quad\psi_{2k}(0)=-\psi_{2k}(2\beta),\end{aligned}$
(146)

where we used the shorthand notation $\beta_{\pm}$ defined in Section 2.3.

For $M<i\leq N-M$, there is no measurement operator and the twist operator is
again inserted at $\tau=0,\beta,2\beta$ (see Fig. 17 (b)):

$\displaystyle i=M+1,...,N-M,\quad\psi_{i}(0)=-\psi_{i}(2\beta).$ (147)

For $N-M<i\leq N$, the boundary conditions must take into account the
insertion of the measurement operator at $\tau=0,\beta,2\beta$. The twist
operator at the same point is now trivial because $\langle