# Relation between timelike and spacelike entanglement entropy

Wu-zhong Guo<EMAIL_ADDRESS>School of Physics, Huazhong University of
Science and Technology, Luoyu Road 1037, Wuhan, Hubei 430074, China  Song He
<EMAIL_ADDRESS>Center for Theoretical Physics and College of Physics, Jilin
University,
Changchun 130012, China  Max Planck Institute for Gravitational Physics
(Albert Einstein Institute),
Am Mühlenberg 1, 14476 Golm, Germany  Yu-Xuan Zhang
<EMAIL_ADDRESS>Center for Theoretical Physics and College of
Physics, Jilin University,
Changchun 130012, China

###### Abstract

In this study, we establish a connection between timelike and spacelike
entanglement entropy. Specifically, for a diverse range of states, the
timelike entanglement entropy is uniquely determined by a linear combination
of the spacelike entanglement entropy and its first-order temporal derivative.
This framework reveals that the imaginary component of the timelike
entanglement entropy primarily originates from the non-commutativity between
the twist operator and its first-order temporal derivative. Furthermore, we
analyze the constraints of this relation and highlight the possible extension
to accommodate more complex state configurations.

##  I Introduction

In the realm of quantum field theory (QFT) and quantum many-body systems,
entanglement entropy (EE) has become a pivotal tool for probing quantum
correlations. Traditionally, much focus has been on spatial separations, with
extensive research significantly advancing our understanding of EE
Amico:2007ag ; Calabrese:2009qy ; Rangamani:2016dms . In theories with
holographic duals, the connection between EE and minimal surfaces within the
gravitational context has been notably established by the Ryu-Takayanagi (RT)
formula Ryu:2006bv  and its extension, the Hubeny-Rangamani-Takayanagi (HRT)
formula Hubeny:2007xt , proving powerful in probing quantum black holes and
quantum gravity VanRaamsdonk:2010pw ; Maldacena:2013xja ; Almheiri:2014lwa ;
Penington:2019npb ; Almheiri:2019psf .

The system is characterized by the density matrix $\rho$, defining the reduced
density matrix as $\rho_{A}:=\text{tr}_{\bar{A}}\rho$. EE is given by the von
Neumann entropy $S(\rho_{A})=-\text{tr}(\rho_{A}\log\rho_{A})$. To evaluate
it, we introduce the Rényi entropy (RE)
$S_{n}(\rho_{A})=\frac{\log\text{tr}(\rho_{A}^{n})}{1-n}$. In QFTs, we use the
replica method to compute RE, leading to the twist operator formalism
Calabrese:2004eu . More precisely, preparing the state
$\rho=|\psi\rangle\langle\psi|$ by Euclidean path integral, the $n$-th Rényi
entropy can be computed by a path integral for $n$-copied systems glued
together along the subsystem $A$, denoted as $\Sigma_{n}$. Let us focus on
2-dimensional conformal field theories (CFTs). If $A$ is an interval,
$tr(\rho_{A})^{n}$ can be expressed as a correlator involving twist operators
for the $n$-copied CFTn, that is

$\displaystyle
tr(\rho_{A})^{n}=\langle\Psi|\sigma_{n}(\tau,x)\tilde{\sigma}_{n}(\tau^{\prime},x^{\prime})|\Psi\rangle,$
(1)

where $(\tau,x^{\prime})$ and $(\tau^{\prime},x^{\prime})$ are coordinates of
the endpoints of $A$,
$|\Psi\rangle:=|\psi\rangle_{1}\otimes...|\psi\rangle_{i}\otimes...|\psi\rangle_{n}$,
the subscripts $i$ label the $i$-th copy. In 2-dimensional CFTs, the twist
operators can be taken as local primary operators with the conformal dimension
$h_{n}=\bar{h}_{n}=\frac{c}{24}(n-\frac{1}{n})$.

The concept of EE is typically defined for a subsystem that constitutes a
spacelike region. Recently, it has been extended to timelike regions
Doi:2022iyj , referred to as timelike entanglement entropy, which is generally
complex-valued. A proper definition of timelike entanglement entropy involves
the analytical continuation of the correlator (1) to Lorentzian coordinates.
It is proposed in Doi:2022iyj  that timelike entanglement entropy should be
interpreted as the so-called pseudo entropy Nakata:2020luh , also discussed in
Murciano:2021dga . Pseudo entropy is defined by replacing the reduced density
matrix $\rho_{A}$ with a non-Hermitian transition matrix, leading to a
generally complex-valued entropy. For further advancements in this area, refer
to Mollabashi:2020yie ; Guo:2022jzs ; Mollabashi:2021xsd ; Nishioka:2021cxe ;
Goto:2021kln ; Akal:2021dqt ; Miyaji:2021lcq ; Guo:2022sfl ; Ishiyama:2022odv
; Mukherjee:2022jac ; Li:2022tsv ; He:2023eap ; Narayan:2022afv ; Doi:2023zaf
; Narayan:2023ebn ; He:2023wko ; Jiang:2023ffu ; Parzygnat:2023avh ;
Guo:2023aio ; Guo:2023tjv ; He:2023syy ; Das:2023yyl ; Chu:2023zah ;
Jiang:2023loq ; Chen:2023gnh ; He:2023ubi ; Narayan:2023zen ; Kanda:2023jyi ;
Omidi:2023env ; Shinmyo:2023eci ; Kawamoto:2023nki .

Despite notable progress in this field, there remains a significant gap in our
comprehensive understanding of the physical significance of timelike EE. A
question arises regarding its interpretation as a concept of entropy,
especially given its general complex-valued nature. A crucial objective is to
understand the origin of the imaginary component of timelike EE. While it is
possible to establish timelike EE through the analytical continuation of
spacelike EE, the extent of the intrinsic relationship between these two
aspects of entanglement entropy remains unclear. This paper aims to construct
and explore the intricate relationship between these two facets of EE,
demonstrating that timelike EE can be linked to spacelike EE and its time
derivative on the Cauchy surface at $t=0$.

From the perspective of correlators, it is natural to establish a connection
between timelike correlators and spacelike ones. For a theory with Hamiltonian
$H$, the dynamics of a local operator $O(t,\vec{x})$ is given by
$O(t,\vec{x}):=e^{iHt}O(0,\vec{x})e^{-iHt}$, which can be expressed formally
as

$\displaystyle
O(t,\vec{x})=\sum_{m}\frac{(it)^{m}}{m!}[H,...,[H,O(0,\vec{x})]],$ (2)

which means the operator $O(t,\vec{x})$ can be expressed as linear
combinations of operators $[H,...,[H,O(0,\vec{x})]]$ on the Cauchy surface at
$t=0$. Similarly, for another operator
$O^{\prime}(t^{\prime},\vec{x}^{\prime})$, a similar decomposition can be
performed. Consequently, the correlator $\langle
O(t,\vec{x})O^{\prime}(t^{\prime},\vec{x}^{\prime})\rangle$ can be formally
represented as linear combinations of spacelike correlators. However, in
general, the expression becomes too complex to be practically useful.

Since in 2-dimensional CFTs, $\sigma_{n}$ can be considered as a local
operator in the CFTn theory, one can also define its time evolution by the
Hamiltonian $H^{(n)}$. The timelike EE can be computed through the analytical
continuation of (1) with $\tau\to it+\epsilon$. By utilizing (2) for the twist
operator, one anticipates that the timelike EE can be linked to correlators on
the Cauchy surface at $t=0$. However, our findings extend beyond this
expectation. We will demonstrate in various cases that the timelike EE is
equivalent to a linear combination of solely spacelike EE and its first-order
time derivative.

##  II Derivation of the relation for vacuum state

Timelike entanglement entropy. Consider the vacuum state
$|\psi\rangle=|0\rangle$, where

$\displaystyle\text{tr}(\rho_{0,A})^{n}:=\langle\sigma_{n}(\tau,x)\tilde{\sigma}_{n}(\tau^{\prime},x^{\prime})\rangle$
(3)
$\displaystyle\phantom{\text{tr}(\rho_{0,A})^{n}:}=(w_{1}-w_{2})^{-2h_{n}}(\bar{w}_{1}-\bar{w}_{2})^{-2\bar{h}_{n}},$

with $w_{i}=\tau_{i}+ix_{i}$ ($i=1,2$).

We are interested in an interval between $(t,x)$ and $(t^{\prime},x^{\prime})$
in Minkowski spacetime with these two points being timelike. The timelike
Rényi entropy can be obtained by Wick rotation $\tau\to it+\epsilon$. More
generally, for any interval between $(t,x)$ and $(t^{\prime},x^{\prime})$, we
have

$\displaystyle
S_{n}(t,x;t^{\prime},x^{\prime}):=\frac{\log\text{tr}(\rho_{0,A})^{n}|_{\tau\to
it+\epsilon,\ \tau^{\prime}\to it^{\prime}+\epsilon^{\prime}}}{1-n}$ (4)
$\displaystyle=\frac{2h_{n}}{n-1}\log\left[\Delta
s^{2}+2i(\epsilon-\epsilon^{\prime})(t-t^{\prime})\right],$

where $\Delta s^{2}=-(t-t^{\prime})^{2}+(x-x^{\prime})^{2}$ and
$\epsilon>\epsilon^{\prime}$.

For spacelike separation, $S_{n}(t,x;t^{\prime},x^{\prime})$ is the spacelike
Rényi entropy, satisfying the symmetry
$S_{n}(t,x;t^{\prime},x^{\prime})=S_{n}(t^{\prime},x^{\prime};t,x)$. For
timelike separation with $t>t^{\prime}$, where $\Delta s^{2}<0$, we have

$S_{n}(t,x;t^{\prime},x^{\prime})=\frac{c}{12}\left(1+\frac{1}{n}\right)\log[-\Delta
s^{2}]+\frac{i\pi c}{12}\left(1+\frac{1}{n}\right).$ (5)

In this case,
$S_{n}(t^{\prime},x^{\prime};t,x)=S_{n}(t,x;t^{\prime},x^{\prime})^{*}$, and
the imaginary part is given by

$\displaystyle\text{Im}[S_{n}(t,x;t^{\prime},x^{\prime})]$
$\displaystyle=\frac{1}{2}\left[S_{n}(t,x;t^{\prime},x^{\prime})-S_{n}(t^{\prime},x^{\prime};t,x)\right]$
(6) $\displaystyle=\frac{i\pi c}{12}\left(1+\frac{1}{n}\right).$

Taking the limit $n\to 1$ yields the timelike entanglement entropy.

Figure 1: (a) An subregion $A$ on the Cauchy surface $t=0$ and its causal
domain $\mathcal{D}_{A}$. (b) In a typical scenario where $(t,x)$ and
$(t^{\prime},x^{\prime})$ are timelike, their past light cones intersect with
four points at $t=0$ with $-u<-u^{\prime}<v^{\prime}<v$.

Derivation of the relation. For an open subregion $A$ on the Cauchy surface
$t=0$ in a $d$-dimensional spacetime, the local algebra $\mathcal{R}(A)$ is
constructed using smeared local operators in $A$. The domain of dependence of
$A$, denoted by $\mathcal{D}_{A}$, is the set of spacetime points
$(t,\vec{x})$ causally influenced by or influencing points in $A$ (refer to
Fig.1). The dynamical time evolution of the theory determines operators in
$\mathcal{D}_{A}$ by those in the region $A$. Thus, one expects the algebra
$\mathcal{R}(\mathcal{D}_{A})$ associated with $\mathcal{D}_{A}$ to be equal
to the algebra $\mathcal{R}(A)$. However, the local operator
$\mathcal{O}(t,\vec{x})$ in $\mathcal{R}(\mathcal{D}_{A})$ generally has a
complex relation with those $\\{\mathcal{O}_{i}(0,\vec{y})\\}$ in
$\mathcal{R}(A)$, expressed as

$\displaystyle\mathcal{O}(t,\vec{x})=\sum_{i}\int_{A}d^{d-1}\vec{y}f_{i}(t,\vec{x};0,\vec{y})\mathcal{O}_{i}(0,\vec{y}),$
(7)

where $f_{i}(t,\vec{x};0,\vec{y})$ are functions depending on the coordinates.
For a general theory, Eq. (7) is seen as a form relation. For the operator
$\phi(t,x)$ in free scalar theory, we can explicitly write down the operator’s
relation. In this paper, we focus on the case $d=2$. The Euclidean action of a
massless free scalar theory is given by

$\displaystyle S=\frac{1}{4\pi\kappa}\int
dzd\bar{z}\partial_{z}\phi\partial_{\bar{z}}\phi,$ (8)

where $z=\tau+ix$ and $\bar{z}=\tau-ix$. For $\phi(t,x)$ we have the relation

$\displaystyle\phi(t,x)=\int_{A}d\bar{x}f_{\phi}(t,x;0,\bar{x})\phi(0,\bar{x})+\int_{A}d\bar{x}f_{\pi}(t,x;0,\bar{x})\pi(0,\bar{x}),$

where $\pi(t,x):=\frac{1}{4\pi\kappa}\dot{\phi}$ is the canonical momentum
operator, and the functions $f_{\phi(\pi)}(t,\vec{x};0,\vec{\bar{x}})$ are
given by

$\displaystyle
f_{\phi}(t,x;t^{\prime},x^{\prime})=\frac{1}{2}\left(\delta(u-u^{\prime})+\delta(v-v^{\prime})\right),$
$\displaystyle
f_{\pi}(t,x;t^{\prime},x^{\prime})=\pi\kappa\left(H(u-u^{\prime})-H(v^{\prime}-v)\right),$
(9)

where $u:=t-x$, $v=t+x$ and $u^{\prime}=t^{\prime}-x^{\prime}$,
$v^{\prime}=t^{\prime}+x^{\prime}$, $H(x)$ is Heaviside function. Refer to
Appendix A for more details.

Using the above results one could obtain the relation between timelike and
spacelike correlators, Let us consider the correlator
$\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$ with $(t,x)$ and
$(t^{\prime},x^{\prime})$ being timelike. Without loss of generality, we
assume $(t^{\prime},x^{\prime})$ is in the past lightcone of $(t,x)$ shown in
Fig. 1. $\phi(t,x)$ and $\phi(t^{\prime},x^{\prime})$ can be expressed as the
form (II). Thus the correlator is given by

$\displaystyle\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$
$\displaystyle=\frac{1}{4}\Big{(}\langle\phi(0,-u)\phi(0,-u^{\prime})\rangle+\langle\phi(0,-u)\phi(0,v^{\prime})\rangle+\langle\phi(0,v)\phi(0,-u^{\prime})\rangle+\langle\phi(0,v)\phi(0,v^{\prime})\rangle\Big{)}$
$\displaystyle+\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,-u)\pi(0,\bar{x}^{\prime})\rangle+\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,v)\pi(0,\bar{x}^{\prime})\rangle$
$\displaystyle+\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,-u^{\prime})\rangle+\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,v^{\prime})\rangle$
$\displaystyle+4(\pi\kappa)^{2}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\pi(0,\bar{x})\pi(0,\bar{x}^{\prime})\rangle,$
(10)

where $u=t-x$, $v=t+x$ and $u^{\prime}=t^{\prime}-x^{\prime}$,
$v^{\prime}=t^{\prime}+x^{\prime}$.

It is interesting that $S_{n}(t,x;t^{\prime},x^{\prime})$ (4) is same as the
two point correlator $\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$ with
$\kappa\to-\frac{2h_{n}}{n-1}$ (see (28) in Appendix A). One can also show
that

$\displaystyle\frac{1-n}{8\pi
h_{n}}\partial_{t^{\prime}}S_{n}(t,x;t^{\prime},x^{\prime})=\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle,$
(11)

and

$\displaystyle\left(\frac{1-n}{8\pi
h_{n}}\right)^{2}\partial_{t}\partial_{t^{\prime}}S_{n}(t,x;t^{\prime},x^{\prime})=\langle\pi(t,x)\pi(t^{\prime},x^{\prime})\rangle.$
(12)

Hence, by employing the formula (A), we can derive an intriguing relation that
links timelike EE with spacelike RE and their first-order time derivative.
Using the above results and taking limit $n\to 1$ we have

$\displaystyle
S(t,x;t^{\prime},x^{\prime})=\frac{1}{4}\Big{(}S(0,-u;0,-u^{\prime})+S(0,-u;0,v^{\prime})+S(0,v;0,-u^{\prime})+S(0,v;0,v^{\prime})\Big{)}$
$\displaystyle\phantom{S(t,x;t^{\prime},x^{\prime})}+\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S(0,-u;0,\bar{x}^{\prime})+\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S(0,v;0,\bar{x}^{\prime})\rangle$
$\displaystyle\phantom{S(t,x;t^{\prime},x^{\prime})}+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,v^{\prime})$
$\displaystyle\phantom{S(t,x;t^{\prime},x^{\prime})}+\frac{1}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t}\partial_{t^{\prime}}S(0,\bar{x};0,\bar{x}^{\prime}).$
(13)

The time derivative term should be interpreted as obtaining the derivative
first and then taking the limit as $t\to 0$. For example,
$\partial_{t^{\prime}}S(0,-u;0,\bar{x}^{\prime})$ should be understood as
$\partial_{t^{\prime}}S(0,-u;t^{\prime},\bar{x}^{\prime})|_{t^{\prime}\to 0}$.
Actually for the vacuum state the above equation is also correct for the RE
with replacing $S(t,x;t^{\prime},x^{\prime})$ by
$S_{n}(t,x;t^{\prime},x^{\prime})$. This is the main result of our paper.
Although we derived this formula for the vacuum state, in the following we
will demonstrate its validity for a broad class of states.

##  III Application to more general states

Thermal state. For a CFT living on an infinite line with temperature
$1/\beta$, the EE is given by

$\displaystyle
S(t,x;t^{\prime},x^{\prime})=\frac{c}{6}\log\Big{[}\frac{\beta^{2}}{\pi^{2}\delta^{2}}\sinh\left(\frac{\pi(v-v^{\prime}-i\epsilon)}{\beta}\right)$
$\displaystyle\phantom{S(t,x;t^{\prime},x^{\prime})=\frac{c}{6}\log}\times\sinh\left(\frac{\pi(u^{\prime}-u+i\epsilon)}{\beta}\right)\Big{]},$
(14)

where $\epsilon>0$. For $(t,x)$ and $(t^{\prime},x^{\prime})$ being timelike,
we have $(v-v^{\prime})(u-u^{\prime})>0$, thus the timelike EE is

$\displaystyle
S(t,x;t^{\prime},x^{\prime})=\frac{c}{6}\log\Big{[}\frac{\beta^{2}}{\pi^{2}\delta^{2}}\sinh\left(\frac{\pi(v-v^{\prime})}{\beta}\right)$
$\displaystyle\phantom{S(t,x;t^{\prime},x^{\prime})=\frac{c}{6}}\times\sinh\left(\frac{\pi(u-u^{\prime})}{\beta}\right)\Big{]}+\frac{i\pi
c}{6}.$ (15)

The derivation of the relation (II) in the vacuum state cannot be directly
generalized to the thermal state. However, it can be demonstrated that the
relation (II) is also applicable to the thermal state. The details are
provided in Appendix.B. Specifically, assuming $-u<-u^{\prime}<v^{\prime}<v$,
the imaginary part arises from the term

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,v^{\prime}).$
(16)

By similar calculations, one can show the relation (II) is also correct for
the vacuum state of 2-dimensional CFTs on a cylinder.

States dual to pure AdS3. The general AdS3 spacetime satisfying vacuum
Einstein equation is given by the Banado geometry. By a conformal
transformation $\xi=f(u)$, $\bar{\xi}=g(v)$, the geometry can be mapped to the
Poincare patch. Using the conformal mapping one could evaluate the holographic
entanglement entropy of an interval between $(t,x)$ and
$(t^{\prime},x^{\prime})$ by RT formula. After analytic continuation the
result is

$\displaystyle
S(t,x;t^{\prime},x^{\prime})=\frac{c}{6}\log\left[\frac{(f(u^{\prime})-f(u-i\epsilon))(g(v-i\epsilon)-g(v^{\prime}))}{\delta^{2}f^{\prime}(u-i\epsilon)f^{\prime}(u^{\prime})g^{\prime}(v-i\epsilon)g^{\prime}(v^{\prime})}\right],$

where $\epsilon>0$, $\delta$ is the UV cut-off. We would expect the conformal
transformation to preserve causality. If $(t,x)$ and $(t^{\prime},x^{\prime})$
are timelike, we require that their images satisfy
$(f(u)-f(u^{\prime}))(g(u)-g(u^{\prime}))<0$. The timelike entanglement
entropy is given by

$\displaystyle S(t,x;t^{\prime},x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{c}{6}\log\left[\frac{(f(u)-f(u^{\prime}))(g(v)-g(v^{\prime}))}{\delta^{2}f^{\prime}(u)f^{\prime}(u^{\prime})g^{\prime}(v)g^{\prime}(v^{\prime})}\right]$
(17) $\displaystyle+$ $\displaystyle\frac{i\pi c}{6}.~{}$

In Appendix C, we also demonstrate that the relation (II) holds for the above
general state. We should note that the imaginary part of the timelike EE also
comes from the term (16).

Multiple intervals for holographic states. Generalizing to multiple intervals
is a highly non-trivial task. However, for states with a holographic dual, the
Ryu-Takayanagi (RT) formula allows for the evaluation of entanglement entropy
to the leading order in $G$. The holographic entanglement entropy for multiple
intervals is obtained by summing those for individual intervals. Yet,
different phases may emerge depending on the lengths of the intervals. In the
single interval case (III), we have established the validity of the relation
(II). Consequently, the timelike entanglement entropy for multiple intervals
in holographic states can also be decomposed into spacelike entanglement
entropy and its first-order time derivative.

##  IV The significance of the relation

A significant observation is that terms without time derivatives in Eq. (II)
exclusively involve entanglement between four points at $t=0$, as illustrated
in Fig. 1. Scaling the UV cutoff $\delta\to c\delta$ results in a constant
shift in both timelike and spacelike RE or EE. The constant from the left-hand
side and the right-hand side cancel each other out.

To understand the physical meaning of the relation (II), we can introduce the
operator

$\displaystyle\dot{\sigma}_{n}:=i[H^{(n)},\sigma_{n}],$ (18)

where $H^{(n)}$ is the Hamiltonian of CFTn. It can be shown that
$\dot{\sigma}_{n}$ can be taken as time derivative with respect to Lorentzian
time $t$. Similarly, we define
$\dot{\tilde{\sigma}}_{n}:=i[H,\tilde{\sigma}_{n}]$. Therefore, we would have
the correlators involving of $\dot{\sigma}_{n}$ and
$\dot{\tilde{\sigma}}_{n}$, such as
$\langle\Psi|\dot{\sigma}_{n}\tilde{\sigma}_{n}|\Psi\rangle$,
$\langle\Psi|\sigma_{n}\dot{\tilde{\sigma}}_{n}|\Psi\rangle$ and
$\langle\Psi|\dot{\sigma}_{n}\dot{\tilde{\sigma}}_{n}|\Psi\rangle$. By the
definition of Rényi entropy, we have

$\displaystyle\partial_{t}S_{n}=\frac{1}{1-n}\frac{\langle\Psi|\dot{\sigma}_{n}(t,x)\tilde{\sigma}_{n}(t^{\prime},x^{\prime})|\Psi\rangle}{\langle\Psi|\sigma_{n}(t,x)\tilde{\sigma}_{n}(t^{\prime},x^{\prime})|\Psi\rangle},$
(19)

or equally,

$\displaystyle\langle\Psi|\dot{\sigma}_{n}(t,x)\tilde{\sigma}_{n}(t^{\prime},x^{\prime})|\Psi\rangle=\partial_{t}e^{(1-n)S_{n}}.$
(20)

Applying Eq. (7) to the twist operator $\sigma_{n}$, we generally anticipate
that operators from the given theory may appear in the integral on the right-
hand side of (7). However, in all the states discussed above, it is noteworthy
that other contributions vanish, except for the four operators
$\sigma_{n}(0,x)$, $\tilde{\sigma}_{n}(0,x)$, $\dot{\sigma}_{n}(0,x)$, and
$\dot{\tilde{\sigma}}_{n}(0,x)$.

All instances indicate that the imaginary part of timelike entanglement
entropy originates from the term

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,v^{\prime}).$

For the timelike EE, we have

$\displaystyle 2\text{Im}S(t,x;t^{\prime},x^{\prime})$
$\displaystyle=\frac{1}{4}\int_{-u}^{v}d\bar{x}\lim_{n\to
1}\frac{1}{1-n}\langle\Psi|[\dot{\sigma}_{n}(0,\bar{x}),\tilde{\sigma}_{n}(0,-u^{\prime})]|\Psi\rangle$
$\displaystyle+\frac{1}{4}\int_{-u}^{v}d\bar{x}\lim_{n\to
1}\frac{1}{1-n}\langle\Psi|[\dot{\sigma}_{n}(0,\bar{x}),\tilde{\sigma}_{n}(0,v)]|\Psi\rangle.$
(21)

Refer to Appendix E for additional details. The commutator
$[\dot{\sigma}_{n}(0,\bar{x}),\tilde{\sigma}_{n}(0,\bar{y})]=0$ for
$\bar{x}\neq\bar{y}$ due to spacelike separation, yielding a delta function
$\delta(\bar{x}-\bar{y})$. For all considered states, the imaginary part of
timelike entanglement entropy remains a constant $\frac{i\pi c}{6}$. The
expected commutator is

$\displaystyle\langle\Psi|[\dot{\sigma}_{n}(0,\bar{x}),\tilde{\sigma}_{n}(0,\bar{y})]|\Psi\rangle$
$\displaystyle=\frac{2i\pi c}{3}(1-n)\delta(\bar{x}-\bar{y})+O(1-n)^{2}.$ (22)

Note that this commutator is specific to the discussed states; additional
terms at order $O(1-n)$ may exist in more general cases, represented by
$\mathcal{A}_{i}(\bar{x})\delta(\bar{x}-\bar{y})$, satisfying
$\langle\Psi|\mathcal{A}_{i}|\Psi\rangle=0$.

Lastly, the time derivative of EE can be articulated in terms of entanglement
spectra. By definition,

$\displaystyle e^{(1-n)S_{n}}=tr\rho_{A}^{n}=\sum_{i}\lambda_{i}^{n},$ (23)

where $\lambda_{i}$ are eigenvalues of $\rho_{A}$. Thus we have

$\displaystyle\partial_{t}S=-\sum_{i}\partial_{t}\lambda_{i}\log\lambda_{i}=-\int_{0}^{\lambda_{m}}d\lambda\log\lambda\
\mathcal{P}_{t}(\lambda),$ (24)

where $\sum_{i}\partial_{t}\lambda_{i}=0$, and
$\mathcal{P}_{t}(\lambda):=\sum_{i}\frac{\partial\lambda_{i}}{\partial
t}\delta(\lambda_{i}-\lambda)$. The maximal eigenvalue is denoted by
$\lambda_{m}$. The relationship between $\mathcal{P}_{t}(\lambda)$ and the
density of eigenstates
$\mathcal{P}(\lambda):=\sum_{i}\delta(\lambda_{i}-\lambda)$ is given by
$\frac{\partial\mathcal{P}}{\partial
t}=-\frac{\partial\mathcal{P}_{t}}{\partial\lambda}$ Guo:2023tys .

##  V Extension of the relation

In the preceding sections, we established the applicability of the relation
(II) across a diverse range of states. However, it is crucial to emphasize
that it does not hold universally. This can be demonstrated by delving into
the operator product expansion (OPE) of twist operators—a valuable tool for
computing Rényi entropies (RE) in quantum field theories (QFTs) Cardy:2007mb ;
Headrick:2010zt ; Calabrese:2010he ; Rajabpour:2011pt ; Chen:2013kpa . The OPE
of twist operators in CFTn can be formally written as

$\displaystyle\sigma_{n}(w,\bar{w})\tilde{\sigma}_{n}(w^{\prime},\bar{w}^{\prime})$
$\displaystyle=\sum_{K}d_{K}(w-w^{\prime})^{-2h_{n}+h_{K}}(\bar{w}-\bar{w}^{\prime})^{-2\bar{h}_{n}+\bar{h}_{K}}\mathcal{X}_{K},$
(25)

where $\mathcal{X}_{K}$ denotes the operators in CFTn, and $d_{K}$ are the OPE
coefficients. The computation of RE and EE involves evaluating the expectation
value $\langle\Psi|\mathcal{X}_{K}|\Psi\rangle$ and determining the
coefficients $d_{K}$. To explore EE for arbitrary intervals, analytical
continuation is employed, yielding a result expressed as an infinite
summation:

$\displaystyle S(t,x;t^{\prime},x^{\prime})$
$\displaystyle=S_{0}(t,x;t^{\prime},x^{\prime})+\sum_{k}a_{k}[-(u-u^{\prime}-i\epsilon)]^{2h_{k}}[v-v^{\prime}-i\epsilon]^{2\bar{h}_{k}},$

where $S_{0}$ denotes the EE for the vacuum state. Note that we include the
OPE coefficients and the expectation values of the operators in the
coefficients $a_{k}$. For simplicity, we consider $a_{k}$ as constants, which
is reasonable when the length of the interval is short enough.

Since $S_{0}$ satisfies the relation (II), we only need to consider the
summation terms in (V). In the following, we still assume
$-u<-u^{\prime}<v^{\prime}<v$. For the timelike separation $-u+u^{\prime}<0$,
there may be new imaginary contributions to the timelike EE from the summation
terms. One could directly check that generally, the result (V) does not
satisfy the relation (II).

However, if the summation terms are the following form

$\displaystyle\sum_{m}a_{m}\left([-(u-u^{\prime}-i\epsilon)]^{m}+[v-v^{\prime}-i\epsilon]^{m}\right),$
(26)

where $m$ are positive integers. One could check the above terms satisfy the
relation (II). It can be shown that the thermal states or the general states
dual to pure AdS3 can be written in the above form.

Let us explain why the above form is only correct for the states dual to AdS3.
It is known that these theories have a sparse light spectrum Hartman:2014oaa .
The main contributions to the OPE of twist operators come from the vacuum
family, such as the lowest contributions from the stress-energy tensor $T$ and
$\bar{T}$ Asplund:2014coa . Thus, the powers of the expansion (26) should be
integers, and $m\geq 2$. It has been shown that in this case, the OPE can be
factored into the product of holomorphic and anti-holomorphic parts.
Consequently, the entanglement entropy can be expanded in the form (26).

The relation (II) does not hold for arbitrary state. The example provided in
(V) illustrates the emergence of new imaginary contributions from the
summation terms. While the right-hand side of (II) is inadequate to account
for these additional contributions. Nevertheless, we anticipate that by
introducing further terms from other operators on the right-hand side of (II),
it is possible to establish a connection between timelike EE and its spacelike
counterpart, extending potentially beyond first-order derivatives. In cases
where OPE of twist operators is permissible, a perturbative construction of
this connection becomes feasible.

##  VI Discussion

In this paper, we establish a connection between timelike and spacelike
entanglement entropy. Our findings reveal that, within a broad class of
states, timelike entanglement entropy can be expressed as a sum over spacelike
entanglement entropy and its first derivative. This prompts a natural
question: how can our results be extended to encompass more general scenarios,
including diverse states and higher-dimensional theories?

An additional noteworthy observation is that timelike entanglement entropy
finds an interpretation through pseudo entropy. As demonstrated in Guo:2023aio
, a sum rule links pseudo Rényi entropy to Rényi entropy. An intriguing avenue
for future exploration is to investigate whether the relation (II) serves as a
specific instance of the general sum rule proposed in Guo:2023aio .

Acknowledgements We would like to thank Run-Qiu Yang, Jiaju Zhang for the
useful discussions. WZG is supposed by the National Natural Science Foundation
of China under Grant No.12005070. SH would like to appreciate the financial
support from the Fundamental Research Funds for the Central Universities, Max
Planck Partner Group, and the Natural Science Foundation of China Grants
(No.12075101, No.12235016).

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##  VII Appendix

##  Appendix A Appendix A: Canonical quantization of two-dimensional massless
free theory

In this appendix, we initially present two methods for deducing Eq.
$\eqref{fphifpi}$, subsequently substantiating the validity of relation (A).

### Method 1:

Given the action of massless free scalar, we can obtain the two-point
correlators

$\displaystyle\langle\phi(z,\bar{z})\phi(z^{\prime},\bar{z}^{\prime})\rangle=-\kappa\log\left[(z-z^{\prime})(\bar{z}-\bar{z}^{\prime})\right]$
$\displaystyle\phantom{\langle\phi(z,\bar{z})\phi(z^{\prime},\bar{z}^{\prime})\rangle}=-\kappa\log\left[(\tau-\tau^{\prime})^{2}+(x-x^{\prime})^{2}\right],$
(27)

where $z=\tau+ix$ and $z^{\prime}=\tau^{\prime}+ix^{\prime}$. We can use the
$i\epsilon$ prescription to obtain the correlator in Minkowski spacetime
through analytical continuation $\tau\to it+\epsilon$ and $\tau^{\prime}\to
it+\epsilon^{\prime}$, e.g., with $\epsilon>\epsilon^{\prime}$ we get

$\displaystyle\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=-\kappa\log\left[\Delta
s^{2}+2i(\epsilon-\epsilon^{\prime})(t-t^{\prime})\right],$ (28)

where $\Delta s^{2}=-(t-t^{\prime})^{2}+(x-x^{\prime})^{2}$.

By canonical quantization we can expand the operators $\phi(t,x)$ and
$\pi(t,x)$ as

$\displaystyle\phi(t,x)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2w_{k}}}\left(a_{k}e^{-iw_{k}t+ikx}+a_{k}^{\dagger}e^{iw_{k}t-ikx}\right),$
(29)
$\displaystyle\pi(t,x)=\frac{1}{4\pi\kappa}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\frac{-iw_{k}}{\sqrt{2w_{k}}}\left(a_{k}e^{-iw_{k}t+ikx}-a_{k}^{\dagger}e^{iw_{k}t-ikx}\right),$

where $w_{k}=|k|$ and
$[a_{k},a_{k^{\prime}}^{\dagger}]=8\pi^{2}\kappa\delta(k-k^{\prime})$. We can
obtain the correlator

$\displaystyle\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=4\pi\kappa\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{2k}e^{-ik(u-u^{\prime})}+4\pi\kappa\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{2k}e^{-ik(v-v^{\prime})},$
(30)

where $u:=t-x$, $v:=t+x$. By using
$\int_{0}^{+\infty}dke^{ikx-k\epsilon}=\frac{i}{x+i\epsilon}$ for $\epsilon>0$
we can obtain the correlator
$\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$ (28). Further, we can
obtain

$\displaystyle\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle=\frac{i}{2}\int_{0}^{+\infty}\frac{dk}{2\pi}e^{-ik(u-u^{\prime})}+\frac{i}{2}\int_{0}^{+\infty}\frac{dk}{2\pi}e^{-ik(v-v^{\prime})}$
$\displaystyle\phantom{\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle}=\frac{1}{4\pi}\frac{1}{u-u^{\prime}-i\epsilon}+\frac{1}{4\pi}\frac{1}{v-v^{\prime}-i\epsilon},$
(31)

which can be related to (28) by

$\displaystyle\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle=\frac{1}{4\pi\kappa}(\partial_{v^{\prime}}+\partial_{u^{\prime}})\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$
$\displaystyle\phantom{\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle}=-\frac{1}{4\pi\kappa}(\partial_{v}+\partial_{u})\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle.$
(32)

We can further obtain

$\displaystyle\langle\pi(t,x)\pi(t^{\prime},x^{\prime})\rangle=\frac{1}{(4\pi\kappa)^{2}}(\partial_{v^{\prime}}+\partial_{u^{\prime}})(\partial_{v}+\partial_{u})\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle.$

It is possible to compute $f_{\phi}$ and $f_{\pi}$ by using (29). Since
$f_{\phi}$ and $f_{\pi}$ are numbers, we have

$\displaystyle f_{\pi}(t,x;t^{\prime},x^{\prime})=\langle
f_{\pi}(t,x;t^{\prime},x^{\prime})\rangle=i\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle-i\langle\phi(t^{\prime},x^{\prime})\phi(t,x)\rangle,$
$\displaystyle f_{\phi}(t,x;t^{\prime},x^{\prime})=\langle
f_{\phi}(t,x;t^{\prime},x^{\prime})\rangle=-i\langle\phi(t,x)\pi(t^{\prime},x^{\prime})\rangle+i\langle\pi(t^{\prime},x^{\prime})\phi(t,x)\rangle,$

which can be evaluated by using the correlators (28) and (A). The results are

$\displaystyle
f_{\pi}(t,x;t^{\prime},x^{\prime})=2\pi\kappa\left(H(u-u^{\prime})-H(v^{\prime}-v)\right),$
$\displaystyle
f_{\phi}(t,x;t^{\prime},x^{\prime})=\frac{1}{2}\left(\delta(u-u^{\prime})+\delta(v-v^{\prime})\right).$
(33)

$f_{\phi}$ is associated with $f_{\pi}$ by the relation

$\displaystyle
f_{\phi}(t,x;t^{\prime},x^{\prime})=-\frac{1}{4\pi\kappa}(\partial_{v^{\prime}}+\partial_{u^{\prime}})f_{\pi}(t,x;t^{\prime},x^{\prime})$
$\displaystyle\phantom{f_{\phi}(t,x;t^{\prime},x^{\prime})}=\frac{1}{4\pi\kappa}(\partial_{v}+\partial_{u})f_{\pi}(t,x;t^{\prime},x^{\prime}).$
(34)

### Method 2:

In the second method, we first leverage the Fourier transformation to find the
inverse expression of $a_{k}$ and $a^{\dagger}_{k}$ with respect to
$\phi(0,x)$ and $\pi(0,x)$,

$\displaystyle a_{k}=$
$\displaystyle\int_{-\infty}^{+\infty}dx\left(\sqrt{\frac{w_{k}}{2}}\phi(0,x)+\frac{4\pi
i\kappa}{\sqrt{2w_{k}}}\pi(0,x)\right)e^{-ikx},$ $\displaystyle
a^{\dagger}_{k}=$
$\displaystyle\int^{+\infty}_{-\infty}dx\left(\sqrt{\frac{w_{k}}{2}}\phi(0,x)-\frac{4\pi
i\kappa}{\sqrt{2w_{k}}}\pi(0,x)\right)e^{ikx}.$ (35)

Substituting Eq. (35) into Eq. (29), we then obtain the expression of
$\phi(t,x)$ in terms of $\phi(0,x)$ and $\pi(0,x)$,

$\displaystyle\phi(t,x)=\int^{+\infty}_{-\infty}d\bar{x}\Big{(}f_{\phi}(t,x;0,\bar{x})\cdot\phi(0,\bar{x})+f_{\pi}(t,x;0,\bar{x})\cdot\pi(0,\bar{x})\Big{)},$
(36)

where

$\displaystyle f_{\phi}(t,x;0,\bar{x})\equiv$
$\displaystyle\frac{1}{2}\int^{+\infty}_{-\infty}\frac{dk}{2\pi}\left(e^{-iw_{k}t+ik(x-\bar{x})}+e^{iw_{k}t-ik(x-\bar{x})}\right),$
$\displaystyle f_{\pi}(t,x;0,\bar{x})\equiv$ $\displaystyle
2\pi\kappa\int_{-\infty}^{+\infty}\frac{dk}{2\pi}\left(\frac{i}{w_{k}}e^{-iw_{k}t+ik(x-\bar{x})}-\frac{i}{w_{k}}e^{iw_{k}t-ik(x-\bar{x})}\right).$
(37)

The first integral in (37) is easy to figure out since we are attending to a
massless free scalar ($w_{k}=|k|$). It turns out that

$\displaystyle
f_{\phi}(t,x;0,\bar{x})=\frac{1}{2}\Big{(}\delta(x-t-\bar{x})+\delta(x+t-\bar{x})\Big{)}.$
(38)

On the other hand, by comparing two integrals in (37), we can find a relation
between $f_{\phi}$ and $f_{\pi}$

$\displaystyle
f_{\phi}(t,x;0,\bar{x})=\frac{1}{4\pi\kappa}\partial_{t}f_{\pi}(t,x;0,\bar{x}).$
(39)

Combining the above relation with Eq. (38), we can write down the expression
of $f_{\pi}$ immediately,

$\displaystyle
f_{\pi}(t,x;0,\bar{x})=2\pi\kappa\Big{(}H(x+t-\bar{x})-H(x-t-\bar{x})\Big{)}=\begin{cases}2\pi\kappa,\quad\bar{x}\in[x-t,x+t],\\\
0,\quad\text{otherwise}.\end{cases}$ (40)

Note that $f_{\phi}$ in Eq. (39) and $f_{\pi}$ in Eq. (40), up to a time
translation, are equivalent to $f_{\phi}$ and $f_{\pi}$ in Eq. (A),
respectively.

Substituting (A) into (36), we arrive at an expression

$\displaystyle\phi(t,x)=\frac{1}{2}\phi(0,-u)+\frac{1}{2}\phi(0,v)+2\pi\kappa\int_{-u}^{v}d\bar{x}\pi(0,\bar{x}).$
(41)

The timelike correlator $\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$
can be expanded as linear combinations of the spacetime correlators, that is

$\displaystyle\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=\int_{\Delta}d\bar{x}d\bar{x}^{\prime}f_{\phi}(t,x;0,\bar{x})f_{\phi}(t^{\prime},x^{\prime};0,\bar{x}^{\prime})\langle\phi(0,\bar{x})\phi(0,\bar{x}^{\prime})\rangle$
$\displaystyle\phantom{phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=}+\int_{\Delta}d\bar{x}d\bar{x}^{\prime}f_{\phi}(t,x;0,\bar{x})f_{\pi}(t^{\prime},x^{\prime};0,\bar{x}^{\prime})\langle\phi(0,\bar{x})\pi(0,\bar{x}^{\prime})\rangle$
$\displaystyle\phantom{phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=}+\int_{\Delta}d\bar{x}d\bar{x}^{\prime}f_{\pi}(t,x;0,\bar{x})f_{\phi}(t^{\prime},x^{\prime};0,\bar{x}^{\prime})\langle\pi(0,\bar{x})\phi(0,\bar{x}^{\prime})\rangle$
$\displaystyle\phantom{phi(t,x)\phi(t^{\prime},x^{\prime})\rangle=}+\int_{\Delta}d\bar{x}d\bar{x}^{\prime}f_{\pi}(t,x;0,\bar{x})f_{\pi}(t^{\prime},x^{\prime};0,\bar{x}^{\prime})\langle\pi(0,\bar{x})\pi(0,\bar{x}^{\prime})\rangle.$

Thus the correlator is given by

$\displaystyle\langle\phi(t,x)\phi(t^{\prime},x^{\prime})\rangle$
$\displaystyle=\frac{1}{4}\left(\langle\phi(0,-u)\phi(0,-u^{\prime})\rangle+\langle\phi(0,-u)\phi(0,v^{\prime})\rangle+\langle\phi(0,v)\phi(0,-u^{\prime})\rangle+\langle\phi(0,v)\phi(0,v^{\prime})\rangle\right)$
$\displaystyle+\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,-u)\pi(0,\bar{x}^{\prime})\rangle+\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,v)\pi(0,\bar{x}^{\prime})\rangle$
$\displaystyle+\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,-u^{\prime})\rangle+\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,v^{\prime})\rangle$
$\displaystyle+4(\pi\kappa)^{2}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\pi(0,\bar{x})\pi(0,\bar{x}^{\prime})\rangle,$
(42)

where $u=t-x$, $v=t+x$ and $u^{\prime}=t^{\prime}-x^{\prime}$,
$v^{\prime}=t^{\prime}+x^{\prime}$.

One could directly check this result by using the canonical quantization
formula (29). We can rewrite (29) as

$\displaystyle\phi(t,x)=\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{k}e^{-ik(t-x)}+a_{k}^{\dagger}e^{ik(t-x)}\right)$
$\displaystyle\phantom{\phi(t,x)=}+\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{-k}e^{-ik(t+x)}+a_{-k}^{\dagger}e^{ik(t+x)}\right),$
$\displaystyle\pi(t,x)=\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{-ik}{\sqrt{2k}}\left(a_{k}e^{-ik(t-x)}-a_{k}^{\dagger}e^{ik(t-x)}\right)$
$\displaystyle\phantom{\pi(t,x)=}+\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{-ik}{\sqrt{2k}}\left(a_{k}e^{-ik(t+x)}-a_{k}^{\dagger}e^{ik(t+x)}\right).$
(43)

Thus we have

$\displaystyle\phi(0,-u)=\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{k}e^{-iku}+a_{k}^{\dagger}e^{iku}\right)$
$\displaystyle\phantom{\phi(0,-u)=}+\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{-k}e^{iku}+a_{-k}^{\dagger}e^{-iku}\right),$
$\displaystyle\phi(0,v)=\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{k}e^{ikv}+a_{k}^{\dagger}e^{-ikv}\right)$
$\displaystyle\phantom{\phi(0,v)=}+\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{-k}e^{-ikv}+a_{-k}^{\dagger}e^{ikv}\right),$
(44)

and

$\displaystyle\int_{-u}^{v}\pi(0,\bar{x})d\bar{x}=\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{-1}{\sqrt{2k}}\left(a_{k}e^{ikv}+a_{k}^{\dagger}e^{-ikv}\right)$
$\displaystyle\phantom{\int_{-u}^{v}\pi(0,\bar{x})d\bar{x}=}-\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{-1}{\sqrt{2k}}\left(a_{k}e^{-iku}+a_{k}^{\dagger}e^{iku}\right)$
$\displaystyle\phantom{\int_{-u}^{v}\pi(0,\bar{x})d\bar{x}=}+\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{-k}e^{-ikv}+a_{-k}^{\dagger}e^{ikv}\right)$
$\displaystyle\phantom{\int_{-u}^{v}\pi(0,\bar{\pi})d\bar{x}=}-\frac{1}{2\pi\kappa}\int_{0}^{+\infty}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}\left(a_{-k}e^{iku}+a_{-k}^{\dagger}e^{-iku}\right)$

Using the above formulas, we can directly verify that Eq.(41) is correct.

Eq.(A) shows the timelike correlator
$\langle\phi(t,x)\phi(t^{\prime},x^{\prime})$ can be expressed as linear
combinations of spacelike correlators. We can verify the equation (A)
directly. Each term on the right-hand side of (A) can be evaluated as follows.
Using (28) we have

$\displaystyle\frac{1}{4}\left(\langle\phi(0,-u)\phi(0,-u^{\prime})\rangle+\langle\phi(0,-u)\phi(0,v^{\prime})\rangle+\langle\phi(0,v)\phi(0,-u^{\prime})\rangle+\langle\phi(0,v)\phi(0,v^{\prime})\rangle\right)$
$\displaystyle=-\frac{\kappa}{4}\left(\log(u-u^{\prime})^{2}+\log(u-v^{\prime})^{2}+\log(v+u^{\prime})^{2}+\log(v-v^{\prime})^{2}\right).$
(46)

Using (A) we have

$\displaystyle\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,-u)\pi(0,\bar{x}^{\prime})\rangle+\pi\kappa\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\phi(0,v)\pi(0,\bar{x}^{\prime})\rangle\rangle$
$\displaystyle=\frac{\kappa}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left(\frac{1}{u+\bar{x}^{\prime}-i\epsilon}+\frac{1}{-u-\bar{x}^{\prime}-i\epsilon}\right)+\frac{\kappa}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left(\frac{1}{-v+\bar{x}^{\prime}-i\epsilon}+\frac{1}{v-\bar{x}^{\prime}-i\epsilon}\right)$
$\displaystyle=0,$ (47)

where we use the fact $-u<-u^{\prime}<v^{\prime}<v$. Similarly, we obtain

$\displaystyle\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,-u^{\prime})\rangle+\pi\kappa\int_{-u}^{v}d\bar{x}\langle\pi(0,\bar{x})\phi(0,v^{\prime})\rangle$
$\displaystyle=\frac{\kappa}{4}\int_{-u}^{v}d\bar{x}\left(\frac{1}{-\bar{x}-u^{\prime}-i\epsilon}+\frac{1}{\bar{x}+u^{\prime}-i\epsilon}\right)+\frac{\kappa}{4}\int_{-u}^{v}d\bar{x}\left(\frac{1}{-\bar{x}+v^{\prime}-i\epsilon}+\frac{1}{\bar{x}-v^{\prime}-i\epsilon}\right)$
$\displaystyle=i\pi\kappa,$ (48)

where we use the Sokhotski–Plemelj theorem

$\displaystyle\frac{1}{x-i\epsilon}=i\pi\delta(x)+\mathcal{P}\left(\frac{1}{x}\right),$
(49)

where $\mathcal{P}$ denotes the Cauchy principal value. The final term is

$\displaystyle
4(\pi\kappa)^{2}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\langle\pi(0,\bar{x})\pi(0,\bar{x}^{\prime})\rangle$
$\displaystyle=-\frac{\kappa}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left[\frac{1}{(\bar{x}-\bar{x}^{\prime}+i\epsilon)^{2}}+\frac{1}{(\bar{x}-\bar{x}^{\prime}-i\epsilon)}\right]$
$\displaystyle=-\frac{\kappa}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{\bar{x}^{\prime}}\left(\frac{1}{\bar{x}-\bar{x}^{\prime}+i\epsilon}+\frac{1}{\bar{x}-\bar{x}^{\prime}-i\epsilon}\right)$
$\displaystyle=-\frac{\kappa}{4}\int_{-u}^{v}d\bar{x}\left(\frac{1}{\bar{x}-v^{\prime}+i\epsilon}-\frac{1}{\bar{x}+u^{\prime}+i\epsilon}+\frac{1}{\bar{x}-v^{\prime}-i\epsilon}-\frac{1}{\bar{x}+u^{\prime}-i\epsilon}\right)$
$\displaystyle=\frac{\kappa}{2}\log\frac{u+v^{\prime}}{v-v^{\prime}}+\frac{\kappa}{2}\log\frac{u^{\prime}+v}{u-u^{\prime}},$
(50)

where in the last step we use the Sokhotski–Plemelj theorem again. Summing
over all the terms we find the result is

$\displaystyle-\kappa\log\left[(u-u^{\prime})(v-v^{\prime})\right]+i\pi\kappa,$
(51)

which is consistent with (28).

##  Appendix B Appendix B: The relation (II) for thermal state

In this section, we would like to show the relation (II) is correct for the
thermal state. One could obtain the following results:

$\displaystyle\frac{1}{4}\left(S_{n}(0,-u;0,-u^{\prime})+S_{n}(0,-u;0,v^{\prime})+S_{n}(0,v;0,-u^{\prime})+S_{n}(0,v;0,v^{\prime})\right)$
$\displaystyle=\frac{c}{48}\left(1+\frac{1}{n}\right)\Big{(}\log\left[\frac{\beta^{2}}{\pi^{2}\epsilon^{2}}\sinh^{2}\left(\frac{\pi(u-u^{\prime})}{\beta}\right)\right]+\log\left[\frac{\beta^{2}}{\pi^{2}\epsilon^{2}}\sinh^{2}\left(\frac{\pi(u+v^{\prime})}{\beta}\right)\right]$
$\displaystyle\phantom{\frac{c}{48}\left(1+\frac{1}{n}\right)}+\log\left[\frac{\beta^{2}}{\pi^{2}\epsilon^{2}}\sinh^{2}\left(\frac{\pi(u^{\prime}+v)}{\beta}\right)\right]+\log\left[\frac{\beta^{2}}{\pi^{2}\epsilon^{2}}\sinh^{2}\left(\frac{\pi(v-v^{\prime})}{\beta}\right)\right]\Big{)}.$
(52)
$\displaystyle\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S_{n}(0,-u;0,\bar{x}^{\prime})+\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S_{n}(0,v;0,\bar{x}^{\prime})\rangle=0.$
(53)
$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,v^{\prime})=\frac{i\pi
c}{12}\left(1+\frac{1}{n}\right).$ (54)
$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,v^{\prime})$
$\displaystyle=\frac{c\pi}{48\beta}\left(1+\frac{1}{n}\right)\int_{-u}^{v}d\bar{x}\left(-\coth(\frac{\pi(\bar{x}+u^{\prime}+i\epsilon)}{\beta})+\coth(\frac{\pi(\bar{x}+u^{\prime}-i\epsilon)}{\beta})\right)$
$\displaystyle+\frac{c\pi}{48\beta}\left(1+\frac{1}{n}\right)\int_{-u}^{v}d\bar{x}\left(-\coth(\frac{\pi(\bar{x}-v^{\prime}+i\epsilon)}{\beta})+\coth(\frac{\pi(\bar{x}-v^{\prime}-i\epsilon)}{\beta})\right)$
$\displaystyle=\frac{c}{48}\left(1+\frac{1}{n}\right)\Big{[}\log\left(\sinh\frac{\pi\left(u^{\prime}-u+i\epsilon\right)}{\beta}\right)-\log\left(\sinh\frac{\pi\left(u^{\prime}-u-i\epsilon\right)}{\beta}\right)$
$\displaystyle-\log\left(\sinh\frac{\pi\left(u^{\prime}+v+i\epsilon\right)}{\beta}\right)+\log\left(\sinh\frac{\pi\left(u^{\prime}+v-i\epsilon\right)}{\beta}\right)\Big{]}$
$\displaystyle+\frac{c}{48}\left(1+\frac{1}{n}\right)\Big{[}\log\left(\sinh\left(\frac{\pi\left(-u-v^{\prime}+i\epsilon\right)}{\beta}\right)\right)-\log\left(\sinh\left(\frac{\pi\left(-u-v^{\prime}-i\epsilon\right)}{\beta}\right)\right)$
$\displaystyle-\log\left(\sinh\left(\frac{\pi\left(-v^{\prime}+v+i\epsilon\right)}{\beta}\right)\right)+\log\left(\sinh\left(\frac{\pi\left(-v^{\prime}+v-i\epsilon\right)}{\beta}\right)\right)\Big{]}$
$\displaystyle=\frac{i\pi c}{12}\left(1+\frac{1}{n}\right).$ (55)
$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t}\partial_{t^{\prime}}S_{n}(0,\bar{x};0,\bar{x}^{\prime})$
$\displaystyle=\frac{c\pi^{2}}{48\beta^{2}}\left(1+\frac{1}{n}\right)\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left[\frac{1}{\sinh^{2}\left(\frac{\pi(\bar{x}-\bar{x}^{\prime}-i\epsilon)}{\beta}\right)}+\frac{1}{\sinh^{2}\left(\frac{\pi(\bar{x}-\bar{x}^{\prime}+i\epsilon)}{\beta}\right)}\right]$
$\displaystyle=\frac{c}{24}\left(1+\frac{1}{n}\right)\Big{(}\log\left[\sinh\left(\frac{\pi(u-u^{\prime})}{\beta}\right)\right]-\log\left[\sinh\left(\frac{\pi(u^{\prime}+v)}{\beta}\right)\right]$
$\displaystyle\phantom{=\frac{c}{24}\left(1+\frac{1}{n}\right)}-\log\left[\sinh\left(\frac{\pi(u+v^{\prime})}{\beta}\right)\right]+\log\left[\sinh\left(\frac{\pi(v-v^{\prime})}{\beta}\right)\right]\Big{)}.$
(56)

The summation of the above terms is equal to the timelike Rényi entropy. The
imaginary part comes from the term (B).

##  Appendix C Appendix C: States dual to pure AdS3

For the spacelike separation, we also have

$\displaystyle\partial_{t^{\prime}}S(t,x;t^{\prime},x^{\prime})=\frac{1}{6}c\left(-\frac{f^{\prime}\left(u^{\prime}\right)}{f(u-i\epsilon)-f\left(u^{\prime}\right)}-\frac{f^{\prime\prime}\left(u^{\prime}\right)}{f^{\prime}\left(u^{\prime}\right)}-\frac{g^{\prime}\left(v^{\prime}\right)}{g(v-i\epsilon)-g\left(v^{\prime}\right)}-\frac{g^{\prime\prime}\left(v^{\prime}\right)}{g^{\prime}\left(v^{\prime}\right)}\right),$
(57)
$\displaystyle\partial_{t}S(t,x;t^{\prime},x^{\prime})=\frac{1}{6}c\left(\frac{f^{\prime}(u)}{f(u-i\epsilon)-f\left(u^{\prime}\right)}-\frac{f^{\prime\prime}(u)}{f^{\prime}(u)}+\frac{g^{\prime}(v)}{g(v-i\epsilon)-g\left(v^{\prime}\right)}-\frac{g^{\prime\prime}(v)}{g^{\prime}(v)}\right),$
(58)

and

$\displaystyle\partial_{t}\partial_{t^{\prime}}S(t,x;t^{\prime},x^{\prime})=\frac{1}{6}c\left(\frac{f^{\prime}(u)f^{\prime}\left(u^{\prime}\right)}{\left(f(u-i\epsilon)-f\left(u^{\prime}\right)\right)^{2}}+\frac{g^{\prime}(v)g^{\prime}\left(v^{\prime}\right)}{\left(g(v-i\epsilon)-g\left(v^{\prime}\right)\right)^{2}}\right).$
(59)

Now we could use the above results to evaluate the right hand side of (II).
The first terms are

$\displaystyle\frac{1}{4}\left(S_{n}(0,-u;0,-u^{\prime})+S_{n}(0,-u;0,v^{\prime})+S_{n}(0,v;0,-u^{\prime})+S_{n}(0,v;0,v^{\prime})\right)$
$\displaystyle=\frac{c}{24}\Big{(}\log\left[\frac{(f(u)-f(u^{\prime}))(g(-u)-g(-u^{\prime}))}{\delta^{2}f^{\prime}(u)f^{\prime}(u^{\prime})g^{\prime}(-u)g^{\prime}(-u^{\prime})}\right]+\log\left[\frac{(f(u)-f(-v^{\prime}))(g(-u)-g(v^{\prime}))}{\delta^{2}f^{\prime}(u)f^{\prime}(-v^{\prime})g^{\prime}(-u)g^{\prime}(v^{\prime})}\right]$
$\displaystyle\phantom{\frac{c}{24}\Big{(}}+\log\left[\frac{(f(-v)-f(u^{\prime}))(g(v)-g(-u^{\prime}))}{\delta^{2}f^{\prime}(-v)f^{\prime}(u^{\prime})g^{\prime}(v)g^{\prime}(-u^{\prime})}\right]+\log\left[\frac{(f(-v)-f(-v^{\prime}))(g(v)-g(v^{\prime}))}{\delta^{2}f^{\prime}(-v)f^{\prime}(-v^{\prime})g^{\prime}(v)g^{\prime}(v^{\prime})}\right]\Big{)}.$
(60)

By using (57) we have

$\displaystyle\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S_{n}(0,-u;0,\bar{x}^{\prime})$
$\displaystyle=\frac{c}{24}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left(-\frac{f^{\prime}\left(-\bar{x}^{\prime}\right)}{f(u-i\epsilon)-f\left(-\bar{x}^{\prime}\right)}-\frac{f^{\prime\prime}\left(-\bar{x}^{\prime}\right)}{f^{\prime}\left(-\bar{x}^{\prime}\right)}-\frac{g^{\prime}\left(\bar{x}^{\prime}\right)}{g(-u-i\epsilon)-g\left(\bar{x}^{\prime}\right)}-\frac{g^{\prime\prime}\left(\bar{x}^{\prime}\right)}{g^{\prime}\left(\bar{x}^{\prime}\right)}\right)$
$\displaystyle=\frac{c}{24}\left(\log\frac{f^{\prime}(-v^{\prime})(g(-u)-g(v^{\prime}))}{(f(u)-f(-v^{\prime}))g^{\prime}(v^{\prime})}-\log\frac{f^{\prime}(u^{\prime})(g(-u)-g(-u^{\prime}))}{(f(u)-f(u^{\prime}))g^{\prime}(-u^{\prime})}\right),$
(61)

and

$\displaystyle\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S_{n}(0,v;0,\bar{x}^{\prime})$
$\displaystyle=\frac{c}{24}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left(-\frac{f^{\prime}\left(-\bar{x}^{\prime}\right)}{f(-v+i\epsilon)-f\left(-\bar{x}^{\prime}\right)}-\frac{f^{\prime\prime}\left(-\bar{x}^{\prime}\right)}{f^{\prime}\left(-\bar{x}^{\prime}\right)}-\frac{g^{\prime}\left(\bar{x}^{\prime}\right)}{g(v-i\epsilon)-g\left(\bar{x}^{\prime}\right)}-\frac{g^{\prime\prime}\left(\bar{x}^{\prime}\right)}{g^{\prime}\left(\bar{x}^{\prime}\right)}\right)$
$\displaystyle=\frac{c}{24}\left(\log\frac{f^{\prime}(-v^{\prime})(g(v)-g(v^{\prime}))}{(f(-v)-f(-v^{\prime}))g^{\prime}(v^{\prime})}-\log\frac{f^{\prime}(u^{\prime})(g(v)-g(-u^{\prime}))}{(f(-v)-f(u^{\prime}))g^{\prime}(-u^{\prime})}\right).$
(62)

The next terms are

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,-u^{\prime})$
$\displaystyle=\frac{c}{24}\int_{-u}^{v}d\bar{x}\left(\frac{f^{\prime}(-\bar{x})}{f(-\bar{x}-i\epsilon)-f\left(u^{\prime}\right)}-\frac{f^{\prime\prime}(-\bar{x})}{f^{\prime}(-\bar{x})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x}-i\epsilon)-g\left(-u^{\prime}\right)}-\frac{g^{\prime\prime}(\bar{x})}{g^{\prime}(\bar{x})}\right)$
$\displaystyle=\frac{c}{24}\int_{-u}^{v}d\bar{x}\left(\frac{f^{\prime}(-\bar{x})}{f(-\bar{x})-f\left(u^{\prime}\right)-i\epsilon
f^{\prime}(-\bar{x})}-\frac{f^{\prime\prime}(-\bar{x})}{f^{\prime}(-\bar{x})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x})-g\left(-u^{\prime}\right)-i\epsilon
g^{\prime}(\bar{x})}-\frac{g^{\prime\prime}(\bar{x})}{g^{\prime}(\bar{x})}\right)$
$\displaystyle=\frac{c}{24}\text{p.v.}\int_{-u}^{v}d\bar{x}\left(\frac{f^{\prime}(-\bar{x})}{f(-\bar{x})-f\left(u^{\prime}\right)}-\frac{f^{\prime\prime}(-\bar{x})}{f^{\prime}(-\bar{x})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x})-g\left(-u^{\prime}\right)+}-\frac{g^{\prime\prime}(\bar{x})}{g^{\prime}(\bar{x})}\right)+\frac{ic\pi}{12}$
$\displaystyle=\frac{c}{24}\log\frac{(f(u)-f(u^{\prime}))f^{\prime}(-v)}{(f(-v)-f(u^{\prime}))f^{\prime}(u)}+\frac{c}{24}\log\frac{(g(v)-g(-u^{\prime}))g^{\prime}(-u)}{(g(-u)-g(-u^{\prime}))g^{\prime}(v)}+\frac{ic\pi}{12}$
(63)

and

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S_{n}(0,\bar{x};0,v^{\prime})$
$\displaystyle=\frac{c}{24}\int_{-u}^{v}d\bar{x}\left(\frac{f^{\prime}(-\bar{x})}{f(-\bar{x}-i\epsilon)-f\left(-v^{\prime}\right)}-\frac{f^{\prime\prime}(-\bar{x})}{f^{\prime}(-\bar{x})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x}-i\epsilon)-g\left(v^{\prime}\right)}-\frac{g^{\prime\prime}(\bar{x})}{g^{\prime}(\bar{x})}\right)$
$\displaystyle=\frac{c}{24}\log\frac{(f(u)-f(-v^{\prime}))f^{\prime}(-v)}{(f(-v)-f(-v^{\prime}))f^{\prime}(u)}+\frac{c}{24}\log\frac{(g(v)-g(v^{\prime}))g^{\prime}(-u)}{(g(-u)-g(v^{\prime}))g^{\prime}(v)}+\frac{ic\pi}{12}$
(64)

The last term is

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t}\partial_{t^{\prime}}S_{n}(0,\bar{x};0,\bar{x}^{\prime})$
$\displaystyle=\frac{c}{24}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\left(\frac{f^{\prime}(-\bar{x})f^{\prime}\left(-\bar{x}^{\prime}\right)}{\left(f(-\bar{x}-i\epsilon)-f\left(-\bar{x}^{\prime}\right)\right)^{2}}+\frac{g^{\prime}(\bar{x})g^{\prime}\left(\bar{x}^{\prime}\right)}{\left(g(\bar{x}-i\epsilon)-g\left(\bar{x}^{\prime}\right)\right)^{2}}\right)$
$\displaystyle=\frac{c}{24}\int_{-u}^{v}d\bar{x}\Big{[}-\frac{f^{\prime}(-\bar{x})}{f(-\bar{x}-i\epsilon)-f(-v^{\prime})}+\frac{f^{\prime}(-\bar{x})}{f(-\bar{x}-i\epsilon)-f(u^{\prime})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x}-i\epsilon)-g(v^{\prime})}-\frac{g^{\prime}(\bar{x})}{g(\bar{x}-i\epsilon)-g(-u^{\prime})}\Big{]}$
$\displaystyle=\frac{c}{24}p.v.\int_{-u}^{v}d\bar{x}\Big{[}-\frac{f^{\prime}(-\bar{x})}{f(-\bar{x})-f(-v^{\prime})}+\frac{f^{\prime}(-\bar{x})}{f(-\bar{x})-f(u^{\prime})}+\frac{g^{\prime}(\bar{x})}{g(\bar{x})-g(v^{\prime})}-\frac{g^{\prime}(\bar{x})}{g(\bar{x})-g(-u^{\prime})}\Big{]}$
$\displaystyle=\frac{c}{24}\Big{[}-\log\frac{f(u)-f(-v^{\prime})}{f(-v)-f(-v^{\prime})}+\log\frac{f(u)-f(u^{\prime})}{f(-v)-f(u^{\prime})}+\log\frac{g(v)-g(v^{\prime})}{g(-u)-g(v^{\prime})}-\log\frac{g(v)-g(-u^{\prime})}{g(-u)-g(-u^{\prime})}\Big{]}.$
(65)

Summing over all the terms one can find the relation (II) is satisfied.

##  Appendix D Appendix D: Imaginary part of timelike entanglement entropy

Suppose $-u<-u^{\prime}<v^{\prime}<v$. The imaginary part comes from the term

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,v^{\prime}).$

By definition we have

$\displaystyle\text{Im}[S(t,x;t^{\prime},x^{\prime})]=\frac{1}{2}\left[S(t,x;t^{\prime},x^{\prime})-S(t^{\prime},x^{\prime};t,x)\right].$
(66)

For the spacelike EE we have
$S(t,x;t^{\prime},x^{\prime})=S(t^{\prime},x^{\prime};t,x)$, thus the
following term does not contribute to the imaginary part.

$\displaystyle\frac{1}{4}\Big{(}S(0,-u;0,-u^{\prime})+S(0,-u;0,v^{\prime})+S(0,v;0,-u^{\prime})+S(0,v;0,v^{\prime})\Big{)}.$
(67)

Similarly, by symmetry, we can show the last term

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t}\partial_{t^{\prime}}S(0,\bar{x};0,\bar{x}^{\prime}),$
(68)

also does not contribute to the imaginary part. In the main text, we show the
first-order derivative of EE is associated with the correlator involving
$\sigma_{n}$ and $\dot{\tilde{\sigma}}_{n}$. The term

$\displaystyle\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S(0,-u;0,\bar{x}^{\prime})+\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}S(0,v;0,\bar{x}^{\prime})\rangle,$
(69)

is related to the correlator
$\langle\Psi|\sigma_{n}(0,-u)\dot{\tilde{\sigma}}_{n}(0,\bar{x}^{\prime})|\Psi\rangle$.
Since $\bar{x}^{\prime}\in(-u^{\prime},v^{\prime})$, the two operators are
spacelike, thus there is no imaginary part for this term.

Therefore, the terms contributing to the imaginary part are:

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}S(0,\bar{x};0,v^{\prime}).$
(70)

The origin of the imaginary part is the non-commutativity of
$[\sigma_{n}(0,x),\dot{\tilde{\sigma}}_{n}(0,\bar{x}^{\prime})]$ at the point
$x=x^{\prime}$.

##  Appendix E Appendix E: OPE of EE for general cases

In general cases, the OPE of twist operators cannot be factored as a product
of holomorphic and anti-holomorphic parts. This is because the operators
$\mathcal{X}_{K}(w,\bar{w})\neq\mathcal{X}_{K}(w)\mathcal{X}_{K}(\bar{w})$.
However, if we only consider contributions from the vacuum family, the
operators would be descendants of the identity operators, such as the stress-
energy tensor $T(w)$ and $\bar{T}(\bar{w})$. The OPE of $T$ and $\bar{T}$ is
trivial, thus the contributions from $T(w)\bar{T}(\bar{w})$ can be written as
the product of the holomorphic and anti-holomorphic parts $\langle
T(w)\rangle\psi\langle\bar{T}(\bar{w})\rangle_{\psi}$. It is also important
that the OPE coefficients $d_{K}$ for the operator $T(w)\bar{T}(\bar{w})$ also
factor as $d_{T\bar{T}}=d_{T}d_{\bar{T}}$. The argument can be generalized to
other higher-dimensional descendants. As a result, the OPE of twist operators
can be written as
$\log\langle\sigma_{n}(w,\bar{w})\tilde{\sigma}_{n}(w^{\prime},\bar{w}^{\prime})$
can be expressed as a summation over the holomorphic and anti-holomorphic
parts. Thus, the entanglement entropy takes the form as in Eq.(26) in the main
text.

Now let us check if the form Eq.(26) does satisfy the relation Eq.(II). We
will only need to consider the terms $\Delta
S:=S(t,x;t^{\prime},x^{\prime})-S_{0}(t,x;t^{\prime},x^{\prime})$. This can be
done by directly computing. We have

$\displaystyle\frac{1}{4}\left(\Delta S(0,-u;0,-u^{\prime})+\Delta
S(0,-u;0,v^{\prime})+\Delta S(0,v;0,-u^{\prime})+\Delta
S(0,v;0,v^{\prime})\right)$
$\displaystyle=\sum_{m}a_{m}\Big{[}\left(u^{\prime}+v+i\epsilon\right)^{m}+\left(u^{\prime}+v-i\epsilon\right)^{m}+\left(u^{\prime}-u+i\epsilon\right)^{m}+\left(u^{\prime}-u-i\epsilon\right)^{m}$
$\displaystyle+\left(-u-v^{\prime}+i\epsilon\right)^{m}+\left(-u-v^{\prime}-i\epsilon\right)^{m}+\left(-v^{\prime}+v+i\epsilon\right)^{m}+\left(-v^{\prime}+v-i\epsilon\right)^{m}\Big{]}$
(71)

It can also be shown there is no contribution from the following terms,

$\displaystyle\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}\Delta
S(0,-u;0,\bar{x}^{\prime})+\frac{1}{4}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t^{\prime}}\Delta
S(0,v;0,\bar{x}^{\prime})\rangle$
$\displaystyle+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}\Delta
S(0,\bar{x};0,-u^{\prime})+\frac{1}{4}\int_{-u}^{v}d\bar{x}\partial_{t}\Delta
S(0,\bar{x};0,v^{\prime}).$ (72)

By using

$\displaystyle\partial_{t}\partial_{t^{\prime}}\Delta
S(0,\bar{x};0,\bar{x}^{\prime})=\sum_{m}a_{m}\left[(m-1)m\left(-(x-\bar{x}+i\epsilon)^{m-2}-(x-\bar{x}-i\epsilon)^{m-2}\right)\right],$
(73)

we can obtain

$\displaystyle\frac{1}{4}\int_{-u}^{v}d\bar{x}\int_{-u^{\prime}}^{v^{\prime}}d\bar{x}^{\prime}\partial_{t}\partial_{t^{\prime}}\Delta
S(0,\bar{x};0,\bar{x}^{\prime}).$ (74)

It is easy to check that the sum of (E) and (74) is given by

$\displaystyle\sum_{m}a_{m}\left[(-1)^{m}(u-u^{\prime})^{m}+(v-v^{\prime})^{m}\right],$
(75)

which is just the $\Delta S$ for the timelike EE. As a check of the above
calculations, one could expand the thermal state in the low-temperature limit.
In this case only even $m$ contributes, thus the result is
$\sum_{m}a_{m}\left[(u-u^{\prime})^{m}+(v-v^{\prime})^{m}\right]$. By taking
the coefficients $a_{m}$ for the thermal states one could obtain the timelike
EE for thermal states, which is consistent with Eq.(III).