aainstitutetext: School of Physics and Astronomy, Sun Yat-Sen University,
Guangzhou, 510275, China

# Towards bit threads in general gravitational spacetimes

Dong-Hui Du a and Jia-Rui Sun<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The concept of the generalized entanglement wedge was recently proposed by
Bousso and Penington, which states that any bulk gravitational region $a$
possesses an associated generalized entanglement wedge $E(a)\supset a$ on a
static Cauchy surface $M$ in general gravitational spacetimes, where $E(a)$
may contain an entanglement island $I(a)$. It suggests that the fine-grained
entropy for bulk region $a$ is given by the generalized entropy
$S_{\text{gen}}(E(a))$. Motivated by this proposal, we extend the quantum bit
thread description to general gravitational spacetimes, no longer limited to
the AdS spacetime. By utilizing the convex optimization techniques, a dual
flow description for the generalized entropy $S_{\text{gen}}(E(a))$ of a bulk
gravitational region $a$ is established on the static Cauchy surface $M$, such
that $S_{\text{gen}}(E(a))$ is equal to the maximum flux of any flow that
starts from the boundary $\partial M$ and ends at bulk region $a$, or
equivalently, the maximum number of bit threads that connect the boundary
$\partial M$ to the bulk region $a$. In addition, the nesting property of
flows is also proved. Thus the basic properties of the entropy for bulk
regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and
strong subadditivity, can be verified from flow perspectives by using
properties of flows, such as the nesting property. Moreover, in max thread
configurations, we find that there exists some lower bounds on the bulk
entanglement entropy of matter fields in the region $E(a)\setminus a$,
particularly on an entanglement island region $I(a)\subset(E(a)\setminus a)$,
as required by the existence of a nontrivial generalized entanglement wedge.
Our quantum bit thread formulation may provide a way to investigate more fine-
grained entanglement structures in general spacetimes.

##  1 Introduction

Relevant developments in the holographic nature of gravity tHooft:1993dmi ;
Susskind:1994vu  have revealed the deep connection between the spacetime
geometry and the quantum entanglement. In the AdS/CFT correspondence
Maldacena:1997re , the Ryu-Tagayanagi (RT) formula Ryu:2006bv ; Hubeny:2007xt
shows that the entanglement entropy of a boundary region $A$ is given by the
area of a bulk minimal surface homologous to $A$, i.e.

$S(A)=\min_{m\sim A}\frac{|m|}{4G_{N}}=\frac{|m_{A}|}{4G_{N}},$ (1)

where $m$ is a bulk surface homologous to $A$ (represented by $m\sim A$), and
$|m|$ represents taking the area of the surface $m$ for brevity. $m_{A}$ is
the classical minimal surface homologous to $A$ in the bulk, which is termed
as the RT surface. This formula is believed to hold at the order of
$\mathcal{O}(1/G_{N})$, in which the contribution from bulk quantum
entanglement is ignored. When the leading-order quantum correction of order
$\mathcal{O}(G_{N}^{0})$ coming from the bulk quantum entanglement is
included, the RT formula should be corrected as the Faulkner-Lewkowycz-
Maldacena (FLM) formula Faulkner:2013ana , i.e.

$S(A)=\min_{m\sim
A}\frac{|m|}{4G_{N}}+S_{\text{bulk}}(\sigma_{A})=\frac{|m_{A}|}{4G_{N}}+S_{\text{bulk}}(\sigma_{A})=S_{\text{gen}}(\sigma_{A}),$
(2)

where $\sigma_{A}$ represents the spatial homology region surrounded by the
union of surfaces $A\cup m_{A}$. $S_{\text{bulk}}(\sigma_{A})$ is the von
Neumann entropy of bulk matter fields restricted to $\sigma_{A}$. Note that in
this formula, one needs first to take the minimization for the area term among
all possible bulk surfaces $m\sim A$ to find the minimal surface $m_{A}$ and
the associated homology region $\sigma_{A}$, then to add the bulk entanglement
entropy of region $\sigma_{A}$. At the order of $\mathcal{O}(G_{N}^{0})$, the
backreaction on the classical geometry due to the quantum corrections is small
enough, so minimizing the area term gives a minimal surface that is close to
the classical RT surface with a Planck distance deviation. While it has been
shown that the entanglement entropy with any higher order quantum corrections
should be given by the known quantum extremal surface (QES) formula
Engelhardt:2014gca , that is

$S(A)=\min_{m\sim A}S_{\text{gen}}(\sigma)=\min_{m\sim
A}\left[\frac{|m|}{4G_{N}}+S_{\text{bulk}}(\sigma)\right]=\frac{|m_{\mathcal{X}}|}{4G_{N}}+S_{\text{bulk}}(\tilde{\sigma}_{A})=S_{\text{gen}}(\tilde{\sigma}_{A}),$
(3)

where $\sigma$ represents a spatial homology region with boundary
$\partial\sigma=A\cup m$ and $S_{\text{gen}}(\sigma)$ is the generalized
entropy on region $\sigma$. Note that here $G_{N}$ is the renormalized Newton
constant Susskind:1994sm  and $S_{\text{bulk}}$ is the finite part of the von
Neumann entropy of bulk matter fields.111As discussed in Bousso:2015mna , the
UV divergence of the bulk entanglement entropy is precisely canceled by the
renormalization of Newton’s constant in the area term, and the sub-leading
divergences are canceled by other geometric counter-terms. Therefore, the
generalized entropy is argued to be finite and cutoff-independent. Here we
need to take the minimization for the generalized entropy (the union of the
area term and the bulk entanglement term) among all possible bulk surfaces
$m\sim A$ to find the so-called minimal quantum extremal surface
$m_{\mathcal{X}}$, and $\tilde{\sigma}_{A}$ is a spatial homology region
bounded by $A\cup m_{\mathcal{X}}$. The quantum extremal surface
$m_{\mathcal{X}}$ is still a surface in the classical geometry, but it can
significantly deviate from the classical RT surface $m_{A}$. The so-called
entanglement wedge Czech:2012bh ; Wall:2012uf ; Headrick:2014cta
$\text{EW}(A)$ is then defined as the bulk domain of dependence of the spatial
region $\tilde{\sigma}_{A}$. It is possible for an entanglement wedge to
contain a disconnected portion by definition, called the entanglement island.
Recently, it has been shown that the QES formula plays a crucial role in
solving the black hole information paradox. The island formula
Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni , as a generalization
from the QES formula, was proposed to calculate the entanglement entropy of
the Hawking radiation and reproduce the Page curve Page:1993wv  in semi-
classical gravity, which was verified by including the replica wormholes
Penington:2019kki ; Almheiri:2019qdq  in the gravitational path integral
calculations in 2D Jackiw-Teitelboim (JT) gravity coupled to matters (for a
nice review Almheiri:2020cfm ). It turns out that the entanglement wedge
$\text{EW}(R)$ of a distant radiation region $R$ can contain an island region
$I$ which includes most part of the black hole interior after the Page time,
i.e. $\text{EW}(R)=R\cup I$. Then according to the entanglement wedge
reconstruction Jafferis:2015del ; Dong:2016eik ; Cotler:2017erl , it is
possible for an observer who is restricted to region $R$ to recover the
information on the region $I$ through sufficient complex operations.
Subsequently, the entanglement island and the entanglement entropy have been
widely studied in various gravitational spacetimes, such as AdS black hole
spacetime coupled to non-gravitational bath Almheiri:2019yqk ; Chen:2019uhq ;
Almheiri:2019psy ; Tong:2023nvi ; Lin:2024gip  and gravitational baths
Geng:2020fxl ; Anderson:2020vwi , asymptotically flat black hole spacetime
Hashimoto:2020cas ; Hartman:2020swn ; Alishahiha:2020qza ; Matsuo:2020ypv ;
Wang:2021mqq ; Yu:2021cgi ; He:2021mst ; Gan:2022jay ; Du:2022vvg ; Yu:2022xlh
; Guo:2023gfa ; Yu:2024fks , de-Sitter spacetime Balasubramanian:2020xqf ;
Sybesma:2020fxg ; Geng:2021wcq ; Azarnia:2021uch , AdS spacetime dual to the
BCFT Rozali:2019day ; Chen:2020uac ; Geng:2020qvw ; Geng:2021mic ; Hu:2022ymx
; Suzuki:2022xwv ; Geng:2024xpj  and so on. More importantly, these studies
indicated that the gravitational entropy formula may be applied in more
general spacetimes.

On the other hand, the RT formula for entanglement entropy can be equivalently
described by the bit thread formulation proposed by Freedman and Headrick
Freedman:2016zud , which can help clarify some conceptual puzzles around the
RT formula. In the bit thread formulation, the entanglement entropy of a
boundary subregion is given by the maximum flux of any flow (i.e. a
divergenceless norm-bounded vector field) out of this boundary subregion, or
equivalently the maximum number of bit threads (i.e. a set of integral curves
of the vector field) with Planck-thickness that emanate from this boundary
subregion to its complement region. With the developments of entropy formulas,
the bit threads have been further developed to contain quantum corrections by
properly modifying the divergenceless condition Chen:2018ywy ; Agon:2021tia ;
Rolph:2021hgz . Moreover, the bit threads have been generalized to the
Lorentzian setting Headrick:2017ucz , covariant setting Headrick:2022nbe  and
higher curvature gravity Harper:2018sdd . The bit threads have also been
widely studied in holography to connect with other quantum information-
theoretic concepts, such as the holographic entanglement of purification
Agon:2018lwq ; Du:2019emy ; Bao:2019wcf ; Harper:2019lff ; Du:2019vwh ;
Lin:2020yzf , holographic complexity Pedraza:2021mkh ; Pedraza:2021fgp ;
Caceres:2023ziv , holographic partial entanglement entropy Kudler-Flam:2019oru
; Lin:2021hqs ; Camargo:2022mme ; Lin:2022aqf ; Lin:2023orb ; Lin:2023rxc ;
Lin:2024dho  and multipartite entanglement Harper:2020wad ; Harper:2021uuq ;
Harper:2022sky , which reveals many quantum information-theoretic aspects on
the entanglement structures in holography. See Refs. Cui:2018dyq ;
Hubeny:2018bri ; Agon:2019qgh ; Agon:2020mvu ; Headrick:2020gyq ; Lin:2022flo
; Gursoy:2023tdx ; Lin:2023hzs ; Caggioli:2024uza  for more recent studies on
bit threads.

However, previous studies on the RT/FLM/QES formulas for holographic
entanglement entropy and their corresponding bit thread formulations were
mainly focused on the asymptotically AdS spacetimes.222The classical bit
thread description was applied in dS space Shaghoulian:2022fop ;
Susskind:2021esx , as the RT prescription can be reformulated in de Sitter
space in the framework of static-patch holography Susskind:2021esx ;
Susskind:2021dfc ; Shaghoulian:2021cef . Therefore, it is very important to
find a gravitational entropy formula and its corresponding bit thread
description that are applicable in more general spacetimes. For the entropy
formula, recently, the entanglement wedge prescription has been extended to
general spacetimes by Bousso and Penington Bousso:2022hlz ; Bousso:2023sya ,
called the generalized entanglement wedge (GEW), motivated by developments on
the gravitational entropy formulas Ryu:2006bv ; Hubeny:2007xt ;
Faulkner:2013ana ; Engelhardt:2014gca ; Penington:2019npb ; Almheiri:2019psf ;
Almheiri:2019hni ; Penington:2019kki ; Almheiri:2019qdq ; Almheiri:2020cfm
and the tensor network toy models in quantum gravity Swingle:2009bg ;
Pastawski:2015qua ; Hayden:2016cfa ; Bao:2018pvs .333Also motivated by tensor
network models, the surface/state duality was proposed in Miyaji:2015yva ;
Miyaji:2015fia , as an early attempt towards extending the holography into
more general gravitational spacetimes. It says that in general gravitational
spacetime $M$, a codimension-two convex surface $\Sigma_{A}\subset M$ (while
the region $a$ considered in GEW proposal is codimension-one) is dual to a
certain quantum state $\rho(\Sigma_{A})$, and its entanglement entropy is
given by the area of the minimal (quantum) extremal surface $m_{A}$ that is
homologous to $\Sigma_{A}$. Thus it allows us to define an entanglement wedge
$\text{EW}(\rho(\Sigma_{A}))$, i.e. the bulk domain of dependence of the
spatial region bounded by $\Sigma_{A}\cup m_{A}$, in more general
gravitational spacetimes. Another related progress was the proposal of a new
approach for bulk reconstruction called the surface growth approach, in which
the growth of general bulk extremal surfaces was used to reconstruct the bulk
geometry and matter fields Lin:2020ufd ; Yu:2020zwk ; Fang:2024mwp . It has
been shown that any bulk gravitational region $a$ posses an associated
generalized entanglement wedge $E(a)$ on a static Cauchy surface $M$ in
general spacetimes, where $E(a)$ is defined as the wedge that has the smallest
generalized entropy $S_{\text{gen}}(E(a))$ among all wedges $E\supset a$ on
$M$. Moreover, $E(a)$ may contain a disconnect entanglement island, denoted as
$I(a)$. This proposal suggests that the fine-grained (von Neumann) entropy of
bulk region $a$ is equal to $S_{\text{gen}}(E(a))$ by an entropy formula (16)
similar to the QES formula (3). While the QES prescription in AdS/CFT can be
regarded as a special case of GEW prescription. In the present paper, we find
that a direct application of GEW prescription in certain Cauchy surfaces would
imply the so-called principle of the holography of information Laddha:2020kvp
; Chowdhury:2020hse ; Raju:2020smc ; Raju:2021lwh , which reveals the
holographic nature of the boundaries of certain Cauchy slices. This prompts us
to generalize the quantum bit threads into general gravitational spacetimes.
To achieve this, we should note that the bit thread formulation was
established by the convex optimization Freedman:2016zud ; Headrick:2017ucz ,
which can be applied to more general manifolds in principle, not limited to
the AdS spacetime. As we will show in this work, the GEW proposal makes it
feasible to extend the quantum bit thread formulation to general gravitational
spacetimes. We will explore the GEW prescription from the bit thread
perspectives, as the bit threads may provide more quantum information-
theoretic aspects on the entanglement structures in general spacetimes.

The paper is organized as follows. In Section 2, we briefly review the
existing bit thread formulations corresponding to the RT/FLM/QES formulas in
the AdS/CFT correspondence. In Section 3, we introduce the GEW proposal on a
time-reflection symmetric Cauchy surface in general gravitational spacetimes,
and we find that the GEW prescription would imply the principle of the
holography of information on certain Cauchy surfaces in Section 3.1. In
Section 4, we propose the quantum bit thread formulation that is dual to the
entropy formula from the GEW proposal. We prove the dual bit thread
formulation through the convex optimization in Section 4.1. Then in Section
4.2, we first prove the nesting property of flows in Section 4.2.1, and then
we prove the basic properties of entropy for bulk gravitational regions by
using properties of flows in Section 4.2.2. In section 5, we give an intuitive
description of GEW prescription in terms of quantum bit threads, and we find
that there exists nontrivial lower bounds on the bulk entanglement entropy of
matter fields on region $E(a)\setminus a$, especially for an entanglement
island region $I(a)$. The conclusion and discussion are given in Section 6.

##  2 Review of the bit threads

Consider a time-reflection symmetric Cauchy surface $M$ with a conformal
boundary $\partial M$, where $A$ is a subregion on $\partial M$. Define a
divergenceless and norm-bounded flow $v$ on manifold $M$. The flux passing
through region $A$ is given by

$\int_{A}v:=\int_{A}\sqrt{h}\ n_{\mu}v^{\mu},$ (4)

where $h$ is the determinant of the induced metric $h_{ij}$ on $A$ and
$n_{\mu}$ is the unit normal vector with inward-pointing direction. Then the
entanglement entropy of region $A$ is given by the maximum flux of any flow
through the region $A$, that is Freedman:2016zud

$S(A)=\max_{v}\int_{A}v,$ (5)

subject to the constraints

$\displaystyle\quad|v|\leq\frac{1}{4G_{N}},\ \nabla_{\mu}v^{\mu}=0.$ (6)

The formula (5) was proved to be equivalent to the RT formula (1) by using
convex optimization and strong duality Headrick:2017ucz . As there exits a max
flow configuration $\tilde{v}$ subject to constraints (6) such that
$\tilde{v}^{\mu}=n^{\mu}/4G_{N}$ at the classical minimal surface $m_{A}$,
where the unit normal vector is outward-pointing on surface $m_{A}$. By using
Gauss’s law with the divergencelessness condition, it leads to

$S(A)=\int_{A}\tilde{v}=\int_{m_{A}}\tilde{v}=\frac{|m_{A}|}{4G_{N}}.$ (7)

The early idea of adding quantum corrections to the bit thread prescription
was discussed in Freedman:2016zud ; Chen:2018ywy  by allowing the sources and
sinks in the bulk or equivalently allowing bit threads to jump across the
classical minimal surface. Then the so-called quantum bit thread formulation
dual to the QES prescription was formally proposed in Agon:2021tia ;
Rolph:2021hgz  by using convex optimization similar to that in
Headrick:2017ucz , in which the divergenceless condition was modified properly
in order to capture the contribution from the bulk entanglement of matter
fields. The dual quantum bit thread formulation to all orders in $G_{N}$ was
given by Rolph:2021hgz

$S(A)=\max_{v}\int_{A}v,$ (8)

subject to the constraints

$|v|\leq\frac{1}{4G_{N}},\ \forall\
\sigma\in\Omega_{A}:-\int_{\sigma}\nabla_{\mu}v^{\mu}\leq
S_{\text{bulk}}(\sigma),$ (9)

where $\Omega_{A}$ represents the set of all possible homology regions for
boundary region $A$, i.e.

$\Omega_{A}:=\\{\sigma\subseteq M:\partial\sigma=A\cup m\\}.$ (10)

There exists a max flow configuration $\tilde{v}$ subject to constraints (9)
on $M$:

  1. 1.

Requiring $\tilde{v}^{\mu}=n^{\mu}/4G_{N}$ at the quantum extremal surface
$m_{\mathcal{X}}$, where $n^{\mu}$ is the unit normal vector on
$m_{\mathcal{X}}$.

  2. 2.

Saturating the divergence bound for $m=m_{\mathcal{X}}$ (thus
$\sigma=\tilde{\sigma}_{A}$), that is
$-\int_{\tilde{\sigma}_{A}}\nabla_{\mu}\tilde{v}^{\mu}=S_{\text{bulk}}(\tilde{\sigma}_{A})$.

The boundary of $\tilde{\sigma}_{A}$ has the orientation as
$\partial\tilde{\sigma}_{A}=m_{\mathcal{X}}-A$, where the unit normal vector
is inward-pointing on surface $A$ and outward-pointing on surface
$m_{\mathcal{X}}$. By using the Gauss’s law, we have

$S(A)=\int_{A}\tilde{v}=\int_{m_{\mathcal{X}}}\tilde{v}-\int_{\tilde{\sigma}_{A}}\nabla_{\mu}\tilde{v}^{\mu}=\frac{|m_{\mathcal{X}}|}{4G_{N}}+S_{\text{bulk}}(\tilde{\sigma}_{A}),$
(11)

as expected in the QES formula (3). One may note that the quantum extremal
surface $m_{\mathcal{X}}$ is generally not a minimal surface on the manifold
$M$, thus it seems to violate the norm bound condition as the classical bit
thread description shows that the norm bound can only saturate at the minimal
surface. However, in the quantum bit thread description, the threads are no
longer confined to the manifold $M$, they can jump out of the manifold at some
points and then re-enter the manifold at other points, so that it is allowed
to saturate the norm bound at a non-minimal surface. The validity of the norm
bound condition has been checked in Rolph:2021hgz  near the quantum extremal
surface.

Figure 1: The GEW prescription on a static Cauchy surface $M$. Given any bulk
region $a$ on $M$, there is an associated GEW $E(a)\supset a$, which minimizes
the generalized entropy among all possible homology wedges that contain $a$.
Moreover, $E(a)$ may contain a island region $I(a)$, such that $E(a)=W(a)\cup
I(a)$ with $W(a)\supset a$.

##  3 GEW and generalized entropy for gravitational regions

The entanglement wedge is crucial for understanding the holographic duality,
which has been extended to general gravitational spacetimes by the GEW
proposal Bousso:2022hlz ; Bousso:2023sya . For an arbitrary time-reflection
symmetric Cauchy surface $M$ with asymptotic boundary $\partial M$, i.e. a
Riemannian manifold with metric $g_{\mu\nu}$. Denoting $\partial s$ as the
boundary of a set $s\subset M$, and defining $\text{cl}\ s\equiv s\cup\partial
s$ and $\text{int}\ s=s\setminus\partial s$, then _wedge_ $a$ is defined as
any open subset of $M$ that is the interior of its closure: $a=\text{int\ cl}\
a$. For two wedges $a$ and $b$, one can define the _wedge union_ , _wedge
complement_ and _wedge relative complement_ , i.e.

$\displaystyle a\Cup b$ $\displaystyle\equiv$ $\displaystyle\text{int}\
\text{cl}(a\cup b)\leavevmode\nobreak\ ,$ (12) $\displaystyle a^{\prime}$
$\displaystyle\equiv$ $\displaystyle\text{int}\ a^{c}\leavevmode\nobreak\ ,$
(13) $\displaystyle a\setminus b$ $\displaystyle\equiv$ $\displaystyle a\cap
b^{\prime}\leavevmode\nobreak\ ,$ (14)

which are also wedges, where $a^{c}\equiv M\setminus a$. The generalized
entropy of any given wedge $a\subset M$ is defined as

$S_{\text{gen}}(a)\equiv\frac{|\partial a|}{4G_{N}}+S_{\text{bulk}}(a),$ (15)

where $G_{N}$ is the renormalized Newton constant, and
$S_{\text{bulk}}(a)=-\tr\rho_{a}\log\rho_{a}$ is the finite part of the
entanglement entropy of bulk matter fields, where
$\rho_{a}=\tr_{a^{\prime}}\rho$ is the density operator of matter fields on
wedge $a$. The generalized entropy is finite and cutoff-independent
Bousso:2015mna . In this paper, we divide the total boundary of wedge $a$ into
two parts as $\partial a=\tilde{\partial}a\cup\dot{a}$, where
$\tilde{\partial}a\equiv\partial a\setminus\partial M$ represents the part
inside the bulk and $\dot{a}\equiv\partial a\cap\partial M$ represents the
part that is overlapping with the boundary $\partial M$.

According to the GEW proposal Bousso:2022hlz ; Bousso:2023sya , the GEW $E(a)$
associated to any given wedge $a$ is defined as the wedge that minimizes the
generalized entropy among all possible wedges $\sigma\supset a$ on $M$ (as
illustrated in Figure 1). So the boundary $\partial\sigma$ is defined on
region $a^{c}=M\setminus a$ and it is homologous to $\partial a$. Therefore,
we have

$\begin{split}S_{\text{gen}}(E(a))&=\min_{\partial\sigma\sim\partial
a\atop\text{on}\ a^{c}}S_{\text{gen}}(\sigma)\\\
&=\min_{\partial\sigma\sim\partial a\atop\text{on}\
a^{c}}\left[\frac{|\partial\sigma|}{4G_{N}}+S_{\text{bulk}}(\sigma)\right]\\\
&=\frac{|\tilde{m}_{\mathcal{X}}|}{4G_{N}}+S_{\text{bulk}}(E(a)),\end{split}$
(16)

where we define $\tilde{m}_{\mathcal{X}}\equiv\partial E(a)$ as a
generalization of quantum extremal surface, distinguishing from
$m_{\mathcal{X}}$ in the QES formula. It is also possible for the GEW to
contain a disconnected portion, i.e. an entanglement island. This formula is
suggested to be used to calculate the fine-grained (von Neumann) entropy of
any codimension-one bulk region. It reduces to the QES formula (more
precisely, differing by a fixed area term $|A|$) when the bulk region $a$ is
chosen to be a near-boundary region444More precisely, this near-boundary
region $a$ can be defined as the union of the entanglement wedges of tiny
boundary regions containing slightly smeared local boundary operators. As
stressed in Bousso:2022hlz , the local CFT operators in a conformal boundary
region $A$ are dual to (quasi-) local bulk operators in the near-boundary bulk
region $a$ Banks:1998dd ; Hamilton:2006az . So the algebra generated by bulk
operators in the region $a$ is also able to encode the entire entanglement
wedge of the region $A$, therefore $E(a)=\text{EW}(A)$ in this case. Notice
that there exists an extra fixed area term $|A|$ in comparison to the QES
formula, but it does not affect us to give the correct entanglement wedge
prescription. attached to the boundary region $A$ in AdS/CFT, where
$E(a)=\text{EW}(A)$ and $\partial E(a)=m_{\mathcal{X}}\cup A$. While if we
calculate the entanglement entropy of the Hawking radiation for evaporating
black holes, we just set $a=R$, then we have $E(a)=R$ and $\partial
E(a)=\partial R$ before the Page time, while $E(a)=R\cup I$ and $\partial
E(a)=\partial R\cup\partial I$ after the Page time, as given by the island
formula.555As already mentioned in Hashimoto:2020cas , the gravitational part
of the generalized entropy should contain the area term of the boundary
$\partial R$ of the region $R$, which comes from the effect that region $R$ is
separated from its complement. This also happens in the empty flat spacetimes.
However, this term is often ignored, as it is fixed so it doesn’t affect the
minimization of the generalized entropy. The GEW proposal suggests a
generalization of entanglement wedge reconstruction such that the information
in $E(a)$ can be reconstructed from region $a$.

###  3.1 Holography of information from GEW prescription

Interestingly, we will point out that a direct application of GEW prescription
in certain Cauchy surfaces would imply the principle of the holography of
information Laddha:2020kvp ; Chowdhury:2020hse ; Raju:2020smc ; Raju:2021lwh ,
which claims that: In a theory of quantum gravity, a copy of all the
information available on a Cauchy slice is also available near the boundary of
the Cauchy slice. This redundancy in description is already visible in the
low-energy theory.

More specifically, we assume that $M$ is topologically trivial and the total
quantum state of matter fields in Cauchy slice $M$ is a pure state. We choose
the wedge $a$ to be a near-boundary bulk region $\delta_{M}$ attached to the
asymptotic boundary $\partial M$, with complement region $\delta_{M}^{c}\equiv
M\setminus\delta_{M}$. Its boundary has the orientation
$\partial\delta_{M}=m_{\delta}-\partial M$ (the unit normal vector is inward-
pointing on $\partial M$ and outward-pointing on $m_{\delta}$), where
$m_{\delta}$ represents its inner boundary. According to the GEW proposal, we
need to find the entanglement wedge $E(\delta_{M})$ that minimizes the
generalized entropy, among all wedges $\sigma\supset\delta_{M}$. The boundary
of $\sigma$ is $\partial\sigma=m-\partial M$, hence $m\sim
m_{\delta}\sim\partial M$. As $\partial M$ is fixed, we just need to vary the
inner boundary $m$ on topologically trivial manifold $M$. Finally, one can
find that the minimization leads to a trivial result, such that $m$ vanishes
as a point and the entanglement wedge of the region $\delta_{M}$ is given by
$E(\delta_{M})=M$. So the area contribution from the inner boundary $m$
vanishes, meanwhile $S_{\text{bulk}}(E(\delta_{M}))=S_{\text{bulk}}(M)=0$ as
the total state of the matter fields in $M$ is assumed to be a pure state. The
only contribution to entropy comes from the area term of the boundary
$\partial M$, which is a fixed term. That is

$\begin{split}S_{\text{gen}}(E(\delta_{M}))&=\min_{\partial\sigma\sim\partial\delta_{M}\atop\text{on}\
\delta_{M}^{c}}\left[\frac{|\partial\sigma|}{4G_{N}}+S_{\text{bulk}}(\sigma)\right]\\\
&=\frac{|\partial M|}{4G_{N}}+\min_{m\sim m_{\delta}\atop\text{on}\
\delta_{M}^{c}}\left[\frac{|\partial
m|}{4G_{N}}+S_{\text{bulk}}(\sigma)\right]=\frac{|\partial
M|}{4G_{N}}=S_{\text{gen}}(M).\end{split}$ (17)

As the entanglement wedge $E(\delta_{M})$ of the near-boundary bulk region
$\delta_{M}$ contains the whole Cauchy surface $M$, the entanglement wedge
reconstruction implies that all the information available on the Cauchy slice
$M$ is also available near the boundary of the Cauchy slice, which leads to
the principle of the holography of information. This reveals the holographic
nature of boundaries of Cauchy slices, like the conformal boundary of AdS
gravity.

In general, $M$ may have a nontrivial topology, and the quantum state of
matter fields in $M$ may be mixed. In these general situations, if we still
choose $a$ to be the near-boundary bulk region $\delta_{M}$ that is attached
to the outer asymptotic boundary (denoted as $\partial M_{o}$ here), the
entanglement wedge $E(\delta_{M})$ may have a nonvanishing interior boundary,
meanwhile the entanglement entropy of matter fields on $E(\delta_{M})$ may not
vanish. Hence $E(\delta_{M})\neq M$ in general, there are some physical
degrees of freedom outside $E(\delta_{M})$ that are not available for the
observer restricted to the near-boundary region $\partial M_{o}$. However, we
stress that it is still possible to obtain a similar result like the formula
(17) if we choose a proper near-boundary bulk region in general situations. To
show this, there are two aspects that need to be noted. On the one hand, a
topologically non-trivial $M$ can contain extra boundary $\partial M_{i}$ (may
consist of multiple separate inner boundaries). Hence the boundary of $M$ is
given by $\partial M=\partial M_{i}\cup\partial M_{o}$. We need to take
$\partial M_{i}$ into consideration as it may contain some nontrivial physical
degrees of freedom. On the other hand, a mixed bulk quantum state in $M$ means
that the matter fields in $M$ are entangled with matter fields in an extra
manifold $M^{\prime}$ (assuming that it is known and admits a semi-classical
description) with boundary $\partial M^{\prime}$. However, as long as we
consider the whole manifold $M_{\text{total}}=M\cup M^{\prime}$ (whose
topology may be complicated) with its complete boundary $\partial
M_{\text{total}}$, the total quantum state of matter fields in
$M_{\text{total}}$ would still be pure. If we choose a near-boundary bulk
region $\delta_{M_{\text{total}}}$ that is attached to the whole boundary
$\partial M_{\text{total}}$, we expect
$E(\delta_{M_{\text{total}}})=M_{\text{total}}$ to achieve the minimization of
the generalized entropy. As the area term of the inner boundary of
$E(\delta_{M_{\text{total}}})$ vanishes, meanwhile
$S_{\text{bulk}}(E(\delta_{M_{\text{total}}}))=S_{\text{bulk}}(M_{\text{total}})=0$,
with a remaining fixed area term $|\partial M_{\text{total}}|$. This leads to
a result similar to the formula (17).

##  4 Quantum bit threads in general gravitational spacetimes

In this section, we aim to find a quantum bit thread or flow description for
the entanglement entropy of any spatial region in general gravitational
spacetimes, which is dual to the entropy formula (16). Once we admit the
holographic nature of the boundaries of certain Cauchy slices, we assume that
the threads emerge from the boundaries of the Cauchy slice, as in the case of
the AdS/CFT correspondence.

In this paper, we consider an arbitrary time-reflection symmetric Cauchy
surface $M$ (a topologically trivial manifold) with asymptotic boundary
$\partial M$ (not necessary to be a conformal boundary), and we assume that
the total state of matter fields in $M$ is a pure state. Given a wedge $a$ on
the manifold $M$ with the boundary $\partial a=\tilde{\partial}a\cup\dot{a}$
(where $\dot{a}$ may be non-empty), so that $\tilde{\partial}a\sim\dot{a}$.
And $\sigma$ is a bulk homology region that contains wedge $a$, so that
$a\subset\sigma\subset M$. If $\dot{a}\neq\emptyset$, by definition we must
require $\dot{\sigma}\supset\dot{a}$, hence
$(\partial\sigma\setminus\dot{a})\sim\dot{a}$. Defining $w\equiv
M\setminus\sigma$ as the complement of the region $\sigma$, due to the purity
of the total state of matter fields on $M$, we have

$S_{\text{bulk}}(\sigma)=S_{\text{bulk}}(w),$ (18)

where the boundary of region $w$ is orientated as $\partial w=m_{w}-\partial
M$, with $m_{w}\equiv\partial\sigma$ (which is allowed to partly overlap with
$\partial M$). So the formula (16) can be written as

$\begin{split}S_{\text{gen}}(E(a))&=\min_{\partial\sigma\sim\partial
a\atop\text{on}\
a^{c}}\left[\frac{|\partial\sigma|}{4G_{N}}+S_{\text{bulk}}(\sigma)\right]\\\
&=\min_{m_{w}\sim\partial M\atop\text{on}\
a^{c}}\left[\frac{|m_{w}|}{4G_{N}}+S_{\text{bulk}}(w)\right]\\\
&=\frac{|\tilde{m}_{\mathcal{X}}|}{4G_{N}}+S_{\text{bulk}}(\tilde{w}),\end{split}$
(19)

where the minimization is reached for
$m_{w}=\partial\sigma=\tilde{m}_{\mathcal{X}}$ when $w=\tilde{w}$ with
boundary $\partial\tilde{w}=\tilde{m}_{\mathcal{X}}-\partial M$. The
generalized entanglement wedge $E(a)$ is just the bulk region
$\tilde{\sigma}=M\setminus\tilde{w}$ that is surrounded by
$\tilde{m}_{\mathcal{X}}$, with
$S_{\text{bulk}}(\tilde{w})=S_{\text{bulk}}(\tilde{\sigma})$. For clarity, we
see that the above minimization consists of two terms, i.e.

$\displaystyle\frac{|\dot{a}|}{4G_{N}}+\min_{(m_{w}\setminus\dot{a})\sim(\partial
M\setminus\dot{a})\atop\text{on}\
a^{c}}\left[\frac{|(m_{w}\setminus\dot{a})|}{4G_{N}}+S_{\text{bulk}}(w)\right].$
(20)

If $\dot{a}\neq\emptyset$, we have
$(m_{w}\setminus\dot{a})\sim\dot{a}\sim(\partial M\setminus\dot{a})$, and then
$S_{\text{bulk}}(w)$ reduces to the bulk entanglement entropy on the region
that has the orientated boundary $(m_{w}\setminus\dot{a})-(\partial
M\setminus\dot{a})$. Note that this formula has the same form as the QES
formula (3) if we set $(m_{w}\setminus\dot{a})=m$ and $(\partial
M\setminus\dot{a})=A$ (differing by a fixed area term $|\dot{a}|=|A|$). For
$\dot{a}=\emptyset$, it will be similar to the case with $A=\partial M$ in the
QES formula. Here a significant difference that should be stressed is that the
minimization in formula (19) is only performed on region $a^{c}$, not the
whole manifold $M$.

Therefore, based on the existing quantum bit thread descriptions, we expect
that the entanglement entropy of a bulk region $a$ can be given by the maximum
flux of any flow from the boundary $\partial M$ to the bulk region $a$, or
equivalently the maximum number of bit threads that start from the boundary
$\partial M$ and end at the bulk region $a$. The first area term in (20) is
just dual to the part of the maximum flux directly entering the bulk region
$a$ from the boundary $\partial M$, as $\partial a$ is just overlapping with
$\partial M$ at $\dot{a}$. While the second term in (20) is dual to the part
of the maximum flux from $(\partial M\setminus\dot{a})$ to the bulk region
$a$, which can be similarly proved through the convex optimization techniques
adopted for the QES formula with $(m_{w}\setminus\dot{a})=m$ and $(\partial
M\setminus\dot{a})=A$. It suggests that the entanglement entropy of any bulk
region $a$ is equal to the maximum flux of any flow defined on
$a^{c}=M\setminus a$ that starts from $\partial M$ and then ends at region
$a$. As we will prove in the next section (at a physicist’s level of rigor),
the quantum bit thread formulation dual to the entropy formula (19) can be
formalized as

$S_{\text{gen}}(E(a))=\max_{v_{a}\atop\text{on}\ a^{c}}\int_{\partial
M}v_{a}=\max_{v_{a}\atop\text{on}\ a^{c}}\left(\int_{\partial
a}v_{a}-\int_{a^{c}}\nabla_{\mu}v^{\mu}_{a}\right),$ (21)

subjecting to the constraints

$|v_{a}|\leq\frac{1}{4G_{N}},\ \forall\
w\in\Omega:-\int_{w}\nabla_{\mu}v^{\mu}_{a}\leq S_{\text{bulk}}(w),$ (22)

where $\Omega$ represents a set of the complements of all homology regions of
region $a$, that is

$\Omega:=\\{w\subset a^{c}:\partial w=m_{w}-\partial M\\},$ (23)

where the unit normal vector is inward-pointing on the boundary $\partial M$
and outward-pointing on the surface $m_{w}$, and $m_{w}\sim\partial
a\sim\partial M$. Note that the associated vector field $v_{a}$ is defined on
region $a^{c}$, the dual bit thread program only needs the bulk metric and the
bulk entanglement entropy of any bulk region $w$ on region $a^{c}$. There
exists a max flow configuration $\tilde{v}_{a}$ which is subject to
constraints (22) on $a^{c}$, satisfying:

  1. 1.

$\tilde{v}^{\mu}_{a}=n^{\mu}/4G_{N}$ at the quantum extremal surface
$\tilde{m}_{\mathcal{X}}$, where $n^{\mu}$ is the unit normal vector on
$\tilde{m}_{\mathcal{X}}$.

  2. 2.

For $m_{w}=\tilde{m}_{\mathcal{X}}$ (hence $w=\tilde{w}$), there is
$-\int_{\tilde{w}}\nabla_{\mu}\tilde{v}^{\mu}_{a}=S_{\text{bulk}}(\tilde{w})$,
where region $\tilde{w}$ has the orientated boundary
$\partial\tilde{w}=\tilde{m}_{\mathcal{X}}-\partial M$.

Therefore we have

$S_{\text{gen}}(E(a))=\int_{\partial M}\tilde{v}_{a}=\int_{\partial
a}\tilde{v}_{a}-\int_{a^{c}}\nabla_{\mu}\tilde{v}^{\mu}_{a}=\int_{\tilde{m}_{\mathcal{X}}}\tilde{v}_{a}-\int_{\tilde{w}}\nabla_{\mu}\tilde{v}^{\mu}_{a}=\frac{|\tilde{m}_{\mathcal{X}}|}{4G_{N}}+S_{\text{bulk}}(\tilde{w}),$
(24)

where Gauss’s law is used. The dual flow description shows that a max flow
achieves both the norm and divergence bounds for the quantum extremal surface
$\tilde{m}_{\mathcal{X}}$ and its associated region $\tilde{w}$, which leads
to the entropy formula (19). Moreover, we may divide any flow $v_{a}$ into the
homogeneous part $v^{h}_{a}$ and inhomogeneous part $v^{i}_{a}$, such that

$\nabla v^{h}_{a}=0,\ \text{and}\ \forall\ w\in\Omega:-\int_{w}\nabla
v^{i}_{a}\leq S_{\text{bulk}}(w)$ (25)

on region $a^{c}$, and both the homogeneous and inhomogeneous parts can
contribute to entropy.

###  4.1 Dual quantum bit thread formulation through the convex optimization

In this section, we use the tools from the convex optimization, i.e. the
Lagrangian duality and the strong duality, to prove the equivalence between
the entropy formula (19) and the quantum bit thread formulation (21), similar
to Refs. Headrick:2017ucz ; Agon:2021tia ; Rolph:2021hgz . The Lagrangian
duality is utilized to deal with a constrained optimization problem, which
involves introducing Lagrange multipliers to enforce the constraints for a
primal program, solving for the original variables, and obtaining an
optimization problem for the Lagrange multipliers. Finally, the resulting dual
program is equivalent to the primal program under certain conditions, as a
result of strong duality.

A simple condition that implies strong duality is the Slater’s condition
Book:Convex-optimization , which is crucial to the procedure of Lagrange
duality. It requires that the primal problem admits a feasible point in the
interior of the domain such that all the inequality constraints are strictly
satisfied for this feasible point. Note that the formula (21) subject to the
constraints (22) defines a concave program, in which the constraints (22) are
concave functions of variable $v_{a}$. The Slater’s condition is satisfied, as
norm bound condition is the only non-linear constraint that needs to be
strictly satisfied and $v_{a}=0$ is a feasible point that strictly satisfies
all the inequality constraints.

To obtain the dual convex program for the max flow program, we set $v_{a}$ as
the original variable and introduce the Lagrange multipliers into each
constraint. The Lagrangian function can be organized as

$\begin{split}L[v_{a},\phi,\mu]&=\int_{\partial
M}v_{a}+\int_{a^{c}}\phi(x)\left(\frac{1}{4G_{N}}-|v_{a}|\right)+\int_{\Omega}d\mu(w)\left[\left(\int_{a^{c}}\chi(w,x)\nabla_{\mu}v_{a}^{\mu}\right)+S_{\text{bulk}}(w)\right]\\\
&=\int_{\partial
M}v_{a}+\int_{a^{c}}\phi(x)\left(\frac{1}{4G_{N}}-|v_{a}|\right)+\int_{\Omega}d\mu(w)S_{\text{bulk}}(w)\\\
&\qquad+\int_{a^{c}}\nabla_{\mu}\left[\left(\int_{\Omega}d\mu(w)\chi(w,x)\right)v_{a}^{\mu}\right]-\int_{a^{c}}v_{a}^{\mu}\partial_{\mu}\left(\int_{\Omega}d\mu(w)\chi(w,x)\right)\\\
&=\int_{\partial
M}\left(\int_{\Omega}d\mu(w)\bar{\chi}(w,x)\right)n_{\mu}v_{a}^{\mu}+\int_{\partial
a}\left(1-\int_{\Omega}d\mu(w)\bar{\chi}(w,x)\right)n_{\mu}v_{a}^{\mu}\\\
&\qquad+\int_{\Omega}d\mu(w)S_{\text{bulk}}(w)+\int_{a^{c}}\left[\phi(x)\left(\frac{1}{4G_{N}}-|v_{a}|\right)+v_{a}^{\mu}\partial_{\mu}\left(\int_{\Omega}d\mu(w)\bar{\chi}(w,x)\right)\right],\end{split}$
(26)

where $\phi$ is a non-negative scalar field on $a^{c}$, $\mu$ is a non-
negative probability measure on $\Omega$, such that $\int_{\Omega}d\mu(w)=1$.
Besides, we have performed the integral by parts in the second equality and
used the definitions of characteristic functions $\chi(w,x)$ and
$\bar{\chi}(w,x)$:

$\chi(w,x):=\begin{cases}1,\text{ for }x\in w\\\ 0,\text{ for }x\in
a^{c}\backslash w\end{cases}\text{and}\ \ \ \chi(w,x)+\bar{\chi}(w,x)=1$ (27)

Then we maximize the Lagrangian function (26) with respect to the original
variable $v_{a}$ to dualize the max flow program. Note that we impose no
restrictions on the variable $v$ here, in order to make sure the finiteness of
the maximization with respect to $v_{a}$, it demands that

$\displaystyle\psi(x)|_{\partial a^{c}}=\bar{\chi}(\partial
a^{c},x):=\begin{cases}0,\text{ for }x\in\partial M\\\ 1,\text{ for
}x\in\partial a.\end{cases}\text{and}\ \ \
\phi(x)\geq\left|\partial_{\mu}\psi(x)\right|,$ (28)

where we have defined

$\psi(x):=\int_{\Omega}d\mu(w)\bar{\chi}(w,x)\in[0,1].$ (29)

With the constraint (28), the maximization of the Lagrangian function (26)
reduces to

$\begin{split}\max_{v_{a}}L[v_{a},\phi,\mu]&=\frac{1}{4G_{N}}\int_{a^{c}}\phi(x)+\int_{\Omega}d\mu(w)S_{\text{bulk}}(w).\end{split}$
(30)

Thus the dual Lagrangian is given by

$\begin{split}\min_{\mu,\phi}\max_{v_{a}}L[v_{a},\phi,\mu]&=\min_{\mu}\left[\frac{1}{4G_{N}}\int_{a^{c}}\left|\partial_{\mu}\psi(x)\right|+\int_{\Omega}d\mu(w)S_{\text{bulk}}(w)\right],\end{split}$
(31)

which is subject to the constraint (28), in which the minimization with
respect to $\phi$ is trivial as $\phi(x)$ can reach its minimum value, i.e.
$\phi(x)=\left|\partial_{\mu}\psi(x)\right|$.

Now we argue that the result (31) is equivalent to the entropy formula (19).
Given the function $\psi(x)$ on $a^{c}$ defined in (29) with the boundary
condition (28), hence $\psi(x)=1$ at surface $\partial a$ and $\psi(x)=0$ at
surface $\partial M$. And we assume $\psi(x)$ is differentiable666Note that at
surface $\dot{a}\equiv\partial a\cap\partial M\neq\emptyset$, the function
$\psi(x)$ is non-differentiable by its boundary condition. However, the
objective function only involves the gradient, i.e. $\partial_{\mu}\psi(x)$.
At surface $\dot{a}$, the result of this non-differentiable function can be
defined as the limit of its value on a differentiable function. This argument
is similar to the case with non-differentiability at $\partial A$, referring
to Footnote 10 in Ref. Headrick:2017ucz . Moreover, it is feasible to use
differentiable functions to approximate the non-differentiable function
$\chi(w,x)$ well if we slightly smooth out the step function. Hence we can
restrict to differentiable functions in our procedures. on $a^{c}$. Then
define a one-parameter family of bulk regions as

$r(p):=\\{x\in a^{c}:\psi(x)\leq p,\ \ p\in[0,1]\\},$ (32)

whose boundary is orientated as $\partial r(p)=m(p)-\partial M$, where $m(p)$
may partly overlap with the boundary $\partial a^{c}=\partial a-\partial M$.
By the continuity, we have $\psi(x)=p$ at surface $m(p)$. And we have
$\partial a\sim m(p)\sim\partial M$ for $0\leq p\leq 1$, and $m(p)=\emptyset$
for $p<0$ or $p>1$.777The one-parameter family of bulk regions $r(p)$
corresponds to the set of the complements of all homology regions of $a$, i.e.
set $\Omega$. One may define $r^{\prime}(p):=\\{x\in a^{c}:\psi(x)\geq p,\
p\in[0,1]\\}$ as one-parameter family of bulk regions on $a^{c}$, whose
boundary is $\partial r^{\prime}(p)=\partial a-m(p)$. So that $a\cup
r^{\prime}(p)$ corresponds to the homology region of region $a$ on manifold
$M$.

As the normal derivative of a characteristic function is a surface delta
function, thus the first term in the objective function (31) can be written in
terms of the level sets as

$\begin{split}\frac{1}{4G_{N}}\int_{a^{c}}|\partial_{\mu}\psi(x)|=\frac{1}{4G_{N}}\int_{-\infty}^{\infty}dp\
|m(p)|=\frac{1}{4G_{N}}\int_{0}^{1}dp\ |m(p)|,\end{split}$ (33)

where $|m(p)|$ represents the area of $m(p)$. For the second term in the
objective function (31), it can be argued to satisfy

$\int_{\Omega}d\mu(w)S_{\text{bulk}}(w)\geq\int_{0}^{1}dp\
S_{\text{bulk}}(r(p)),$ (34)

by taking advantage of the strong subadditivity of bulk entanglement entropy,
i.e.

$\sum_{i=1}^{N}S(a_{i})\geq S(\cup_{i}a_{i})+S(\cup_{\\{ij\\}}a_{i}\cap
a_{j})+...+S(\cap_{i}a_{i}),$ (35)

as Ref. Rolph:2021hgz  did in its Argument 2. Here we choose the set of bulk
regions $a_{i}$ as an arbitrary set of $N$ bulk homology regions $w_{i}$ with
boundary $\partial w_{i}=m_{i}-\partial M$. The $n$th term on the right-hand
side of inequality (35) is the bulk region $r(n/N)$, defined as
$r(n/N):=\\{x\in a^{c}:\psi(x)\leq n/N\\}$. Thus we have

$\sum_{i=1}^{N}S_{\text{bulk}}(w_{i})\geq\sum_{i=1}^{N}S_{\text{bulk}}(r(i/N)),$
(36)

which leads to formula (34) in the limit $N\to\infty$ after dividing by $N$.
Finally, combining with (33) and (34), it gives the dual prescription

$\begin{split}\min_{\mu,\phi}\max_{v_{a}}L[v_{a},\phi,\mu]=\min_{\mu}\int_{0}^{1}dp\left(\frac{|m(p)|}{4G_{N}}+S_{\text{bulk}}(r(p))\right),\end{split}$
(37)

where $r(p)$ with its level set $m(p)$ is determined by measure $\mu$ on
$\Omega$ defined in (23). This is exactly the entropy formula (19). The
minimization is realized if $\mu$ only supports on the homology region
$\tilde{r}(p)$ that is bounded by the quantum extremal surface
$m_{\mathcal{X}}$ and the boundary $\partial M$, so that
$\partial\tilde{r}(p)=m_{\mathcal{X}}-\partial M$. Meanwhile, the entanglement
wedge time slice $E(a)$ of the region $a$ is defined as the bulk region that
is surrounded by $m_{\mathcal{X}}$, thus $\partial E(a)=m_{\mathcal{X}}$.
Therefore, the proof on the equivalence between the entropy formula (19) and
the bit thread formulation (21) subjecting to the constraints (22) is
finished.

###  4.2 Nesting property and entropy properties for bulk gravitational
regions

With the quantum bit thread formulation for bulk gravitational regions, we
will first prove the nesting property of flows. Then we are able to prove the
basic properties of entropy (such as monotonicity, subadditivity, Araki-Lieb
inequality, and strong subadditivity) for bulk gravitational regions from flow
perspectives by using the nesting property of flows.

####  4.2.1 Nesting property of flows

Let us prove the nesting property of flows on general Cauchy surface $M$.
Given any two disjoint bulk regions $a$ and $b$ (i.e. $a\cap b=\emptyset$) on
manifold $M$, we are interested in the total flux of any flow $v_{ab}$ defined
on $M$ that starts from boundary $\partial M$ and enters the union region $ab$
($ab\equiv a\cup b$ for brevity). We assume that flow $v_{ab}$ consists of two
independent components:

$v_{ab}=v_{a}+v_{b},$ (38)

where $v_{a}$ is defined on region $a^{c}$ and it represents the flow
component entering region $a$ (in both homogeneous and inhomogeneous ways),
$v_{b}$ is defined on region $b^{c}$ and it represents the flow component
entering region $b$ (in both homogeneous and inhomogeneous ways). We allow
$v_{a}$ to pass through region $b$, but it can “not end at” region
$b$,888Specifically, we mean that the total flux (including the homogeneous
and inhomogeneous parts) entering region $b$ vanishes for flow component
$v_{a}$, i.e. $(\int_{\partial b}v_{a}-\int_{b}\nabla_{\mu}v^{\mu}_{a})=0$ by
Gauss’s law. So that the corresponding bit threads of $v_{a}$ that enter
region $b$ must leave from $b$ subsequently in a homogeneous or inhomogeneous
way, this is what we mean “not end at” on region $b$. similarly for $v_{b}$.
Thus $v_{a}$ contributes to $S_{\text{gen}}(E(a))$ and $S_{\text{gen}}(E(ab))$
but not to $S_{\text{gen}}(E(b))$, meanwhile $v_{b}$ contributes to
$S_{\text{gen}}(E(b))$ and $S_{\text{gen}}(E(ab))$ but not to
$S_{\text{gen}}(E(a))$. The nesting property of flows states that there exists
a nesting max flow $v_{ab}=v_{a}+v_{b}$ that simultaneously maximizes the flux
entering union region $ab$ and the flux entering region $a$ for nesting
regions $a\subset ab$.

To prove the nesting property, first, we sum the flux that enters region $a$
and the flux entering union region $ab$, and we note that its maximum value
should be bounded by the sum of several maximum values, i.e.

$\displaystyle\max_{v_{ab}}\left(\int_{\partial M}v_{a}+\int_{\partial
M}v_{ab}\right)\leq S_{\text{gen}}(E(a))+S_{\text{gen}}(E(ab)),$ (39)

which is subject to the norm bound constraints $|v_{a}|,|v_{ab}|\leq 1/4G_{N}$
and the divergence constraints for $v_{ab}$ on the complements of homology
regions of region $a$ and region $ab$, respectively, denoted as $\Omega_{1}$
and $\Omega_{2}$, that is

$\forall\ w\in(\Omega_{1}\cup\Omega_{2}):-\int_{w}\nabla_{\mu}v_{ab}^{\mu}\leq
S_{bulk}(w),$ (40)

with $\Omega_{1}:=\\{w\subset a^{c}:\partial w=m_{w}-\partial M\\}$ and
$\Omega_{2}:=\\{w\subset(ab)^{c}:\partial w=m_{w}-\partial M\\}$, where
$\Omega_{1}\supset\Omega_{2}$ due to the relation $a^{c}\supset(ab)^{c}$.

The nesting property can be proved as we will show that the value of the term
on the left side of the inequality (39) is also lower bounded by
$S_{\text{gen}}(E(a))+S_{\text{gen}}(E(ab))$ by using the convex optimization.
Let us construct the Lagrangian
$L=L[v_{a},v_{b},\phi_{1},\phi_{2},\mu_{1},\mu_{2}]$ as follows

$\begin{split}L=&\int_{\partial M}v_{a}+\int_{\partial
M}v_{ab}+\int_{a^{c}}\phi_{1}(x)\left(\frac{1}{4G_{N}}-|v_{a}|\right)+\int_{(ab)^{c}}\phi_{2}(x)\left(\frac{1}{4G_{N}}-|v_{ab}|\right)\\\
&+\int_{\Omega_{1}}d\mu_{1}\left(\int_{a^{c}}\chi(w,x)\nabla_{\mu}v_{a}^{\mu}+S_{bulk}(w)\right)+\int_{\Omega_{2}}d\mu_{2}\left(\int_{(ab)^{c}}\chi(w,x)\nabla_{\mu}v_{ab}^{\mu}+S_{bulk}(w)\right)\end{split}$
(41)

in which $v_{a},v_{b}$ as two independent original variables, $\phi_{1}$ and
$\phi_{2}$ are two non-negative scalar fields, $\mu_{1}$ and $\mu_{2}$ are two
probability measures on $\Omega_{1}$ and $\Omega_{2}$, respectively,
satisfying $\int_{\Omega_{1}}d\mu_{1}(w)=\int_{\Omega_{2}}d\mu_{2}(w)=1$. By
performing the integral by parts, the Lagrangian can be organized as

$\begin{split}L=&\int_{\partial
M}\left(\int_{\Omega_{1}}d\mu_{1}\bar{\chi}(w,x)+\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)n_{\mu}v_{a}^{\mu}+\int_{\partial
M}\left(\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)n_{\mu}v_{b}^{\mu}\\\
&+\int_{\partial
a}\left(1-\int_{\Omega_{1}}d\mu_{1}\bar{\chi}(w,x)-\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)n_{\mu}v_{a}^{\mu}+\int_{\partial
a}\left(1-\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)n_{\mu}v_{b}^{\mu}\\\
&+\int_{\partial
b}\left(\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)n_{\mu}v_{ab}^{\mu}+\int_{\Omega_{1}}d\mu_{1}S_{\text{bulk}}(w)+\int_{\Omega_{2}}d\mu_{2}S_{\text{bulk}}(w)\\\
&+\int_{a^{c}}\left[\phi_{1}(x)\left(\frac{1}{4G_{N}}-|v_{a}|\right)+v_{a}^{\mu}\partial_{\mu}\left(\int_{\Omega_{1}}d\mu_{1}\bar{\chi}(w,x)\right)\right]\\\
&+\int_{(ab)^{c}}\left[\phi_{2}(x)\left(\frac{1}{4G_{N}}-|v_{ab}|\right)+v_{ab}^{\mu}\partial_{\mu}\left(\int_{\Omega_{2}}d\mu_{2}\bar{\chi}(w,x)\right)\right]\end{split}$
(42)

Then we maximize the Lagrangian function with respect to the variables
$v_{a},v_{b}$, while the finiteness of the maximization demands that

$\displaystyle\psi_{1}(x)|_{\partial a^{c}}=\begin{cases}0,\text{ for
}x\in\partial M\\\ 1,\text{ for }x\in\partial a.\end{cases}\text{and}\ \ \
\phi_{1}(x)\geq\left|\partial_{\mu}\psi_{1}(x)\right|,$ (43)
$\displaystyle\psi_{2}(x)|_{\partial(ab)^{c}}=\begin{cases}0,\text{ for
}x\in\partial M\\\ 1,\text{ for }x\in\partial a.\\\ 1,\text{ for }x\in\partial
b.\end{cases}\text{and}\ \ \ \phi(x)\geq\left|\partial_{\mu}\psi(x)\right|,$
(44)

where

$\psi_{1}(x):=\int_{\Omega_{1}}d\mu_{1}(w)\bar{\chi}(w,x)\ \ \ \text{and}\ \ \
\psi_{2}(x):=\int_{\Omega_{2}}d\mu_{2}(w)\bar{\chi}(w,x).$ (45)

Therefore, we have

$\begin{split}\min_{\phi_{1},\phi_{2},\mu_{1},\mu_{2}}\max_{v_{a},v_{ab}}L&=\min_{\mu_{1},\mu_{2}}\left[\frac{1}{4G_{N}}\left(\int_{a^{c}}\left|\partial_{\mu}\psi_{1}(x)\right|+\int_{(ab)^{c}}\left|\partial_{\mu}\psi_{2}(x)\right|\right)+\int_{\Omega_{1}}d\mu_{1}S_{\text{bulk}}(w)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ \
+\int_{\Omega_{2}}d\mu_{2}S_{\text{bulk}}(w)\right]\\\
&=\min_{\mu_{1}}\left[\frac{1}{4G_{N}}\int_{a^{c}}\left|\partial_{\mu}\psi_{1}(x)\right|+\int_{\Omega_{1}}d\mu_{1}S_{\text{bulk}}(w)\right]\\\
&\qquad\qquad\qquad\qquad\ \
+\min_{\mu_{2}}\left[\frac{1}{4G_{N}}\int_{(ab)^{c}}\left|\partial_{\mu}\psi_{2}(x)\right|+\int_{\Omega_{2}}d\mu_{2}S_{\text{bulk}}(w)\right].\end{split}$
(46)

Further defining two independent sets of one-parameter family of bulk regions
as

$\displaystyle r(p_{1}):=\\{x\in a^{c}:\psi(x)\leq p_{1},\ \
p_{1}\in[0,1]\\},$ $\displaystyle r(p_{2}):=\\{x\in(ab)^{c}:\psi(x)\leq
p_{2},\ \ p_{2}\in[0,1]\\},$ (47)

with the oriented boundaries $\partial r(p_{1})=m(p_{1})-\partial M,\ \partial
r(p_{2})=m(p_{2})-\partial M$. According to Section 4.1, the objective
function in formula (46) can be written in terms of two independent level
sets, thus we obtain the following result

$\begin{split}\min_{\phi_{1},\phi_{2},\mu_{1},\mu_{2}}\max_{v_{a},v_{ab}}L&\geq\min_{\mu_{1}}\int_{0}^{1}dp_{1}\left(\frac{|m(p_{1})|}{4G_{N}}+S_{\text{bulk}}(r(p_{1}))\right)+\min_{\mu_{2}}\int_{0}^{1}dp_{2}\left(\frac{|m(p_{2})|}{4G_{N}}+S_{\text{bulk}}(r(p_{2}))\right)\\\
&=S_{\text{gen}}(E(a))+S_{\text{gen}}(E(ab)),\end{split}$ (48)

which leads to

$\displaystyle\max_{v_{ab}}\left(\int_{\partial M}v_{a}+\int_{\partial
M}v_{ab}\right)\geq S_{\text{gen}}(E(a))+S_{\text{gen}}(E(ab)),$ (49)

subject to the norm bound and divergence constraints around the formula (40).
Then combining with inequalities (39) and (49), which forces the inequalities
to take the equal sign. It means that there exists an allowed max flow
configuration $v_{ab}$, such that it simultaneously maximizes the flux
entering union region $ab$ (with the maximal value $S_{\text{gen}}(E(ab))$)
and the flux entering region $a$ (with the maximal value
$S_{\text{gen}}(E(a))$), hence we finish the proof of the nesting property of
flows.

####  4.2.2 Entropy properties for bulk gravitational regions

  * •

Monotonicity

The entropy for bulk regions $a,b$ satisfies the monotonicity property

$\displaystyle S_{\text{gen}}(E(a))\leq S_{\text{gen}}(E(b)),\ \text{if}\
a\subset b.$ (50)

Note that this property is not satisfied by the bulk entanglement entropy of
matter fields. However, the GEW proposal shows that the union of the area term
and the bulk entanglement entropy of matter fields should satisfy this
property.

Proof : For bulk regions $a,b$, the dual flow description can be formalized as

$\begin{split}&S_{\text{gen}}(E(a))=\max_{v\in\mathcal{F}}\int_{\partial M}v,\
\mathcal{F}\equiv\\{v\,|\ \forall\ w\in\Omega:-\int_{w}\nabla_{\mu}v^{\mu}\leq
S_{\text{bulk}}(w),\ \ |v|\leq\frac{1}{4G_{N}}\\},\\\
&S_{\text{gen}}(E(b))=\max_{v\in\mathcal{F}^{\prime}}\int_{\partial M}v,\
\mathcal{F}^{\prime}\equiv\\{v\,|\ \forall\
w\in\Omega^{\prime}:-\int_{w}\nabla_{\mu}v^{\mu}\leq S_{\text{bulk}}(w),\ \
|v|\leq\frac{1}{4G_{N}}\\},\end{split}$ (51)

with $\Omega:=\\{w\subset a^{c}:\partial w=m_{w}-\partial M\\}$ and
$\Omega^{\prime}:=\\{w\subset b^{c}:\partial w=m_{w}-\partial M\\}$. And the
subscripts of $v_{a},v_{b}$ are omitted for convenience. For $a\subset b$ with
$b^{c}\subset a^{c}$, we have $\Omega^{\prime}\subset\Omega$ and
$\mathcal{F}^{\prime}\subset\mathcal{F}$. Thus the set $\mathcal{F}$ has
additional constraints than the set $\mathcal{F}^{\prime}$. According to
Theorem 1 in Section 2.4 of Ref. Harper:2019lff , it leads to a constraint on
solutions of two convex maximization programs in (51), which gives the result
as $S_{\text{gen}}(E(a))\leq S_{\text{gen}}(E(b))$ for $a\subset b$.
Therefore, we always have $S_{\text{gen}}(E(a))\leq S_{\text{gen}}(E(ab))$ as
$a\subset ab$ for arbitrary bulk regions $a,b$.

  * •

Subadditivity

The entropy for any two disjoint bulk regions $a,b$ satisfies the
subadditivity (SA) property

$\displaystyle S_{\text{gen}}(E(ab))\leq
S_{\text{gen}}(E(a))+S_{\text{gen}}(E(b)).$ (52)

Proof: We can begin with a nesting max flow $v_{ab}=v_{a}+v_{b}$ starting from
boundary $\partial M$ such that it simultaneously maximizes the flux entering
the union region $ab$ and region $a$ for $a\subset ab$, as we proved before.
It directly gives the SA property:

$\displaystyle S_{\text{gen}}(E(ab))=\int_{\partial M}v_{ab}=\int_{\partial
M}v_{a}+\int_{\partial M}v_{b}\leq S_{\text{gen}}(E(a))+S_{\text{gen}}(E(b)),$
(53)

as the flux entering the union region $ab$ and region $a$ are equal to
$S_{\text{gen}}(E(ab))$ and $S_{\text{gen}}(E(a))$ respectively, while in
general the flux entering the region $b$ can not reach its maximum value
$S_{\text{gen}}(E(b))$.

  * •

Araki-Lieb inequality

The entropy for bulk regions $a,b$ satisfies the Araki-Lieb (AL) property

$\displaystyle|S_{\text{gen}}(E(a))-S_{\text{gen}}(E(b))|\leq
S_{\text{gen}}(E(ab))$ (54)

Proof : A nesting max flow $v_{ab}$ that simultaneously maximizes the flux
entering the union region $ab$ and region $a$, can also be used to prove this
property. So that we have

$\displaystyle S_{\text{gen}}(E(a))-S_{\text{gen}}(E(ab))=\int_{\partial
M}v_{a}-\int_{\partial M}v_{ab}=-\int_{\partial M}v_{b}\leq
S_{\text{gen}}(E(b)),$ (55)

because the flux of any flow entering the region $b$ can not reach its maximum
value $S_{\text{gen}}(E(b))$ in general, including the inverse flow $-v_{b}$.
Similarly one can find $S_{\text{gen}}(E(b))-S_{\text{gen}}(E(ab))\leq
S_{\text{gen}}(E(a))$, thus the AL property is proved.

  * •

Strong subadditivity

The entropy for any disjoint bulk regions $a,b,c$ satisfies the strong
subadditivity (SSA) property

$\displaystyle S_{\text{gen}}(E(b))+S_{\text{gen}}(E(abc))\leq
S_{\text{gen}}(E(ab))+S_{\text{gen}}(E(bc)).$ (56)

Proof : We can start from a nesting max flow
$v_{abc}=v_{b}+v_{ac}=v_{a}+v_{b}+v_{c}$ that simultaneously maximizes the
flux entering the union region $abc$ and region $b$ as we have $b\subset abc$.
It simply gives the SSA property:

$\begin{split}S_{\text{gen}}(E(b))+S_{\text{gen}}(E(abc))&=\int_{\partial
M}v_{b}+\int_{\partial M}v_{abc}\\\ &=\int_{\partial M}v_{ab}+\int_{\partial
M}v_{bc}\\\ &\leq S_{\text{gen}}(E(ab))+S_{\text{gen}}(E(bc))\end{split}$ (57)

as its flux entering the union region $abc$ and region $b$ are equal to
$S_{\text{gen}}(E(abc))$ and $S_{\text{gen}}(E(b))$ respectively, while in
general the flux entering the union regions $ab$ and $bc$ can not reach their
maximum values $S_{\text{gen}}(E(ab))$ and $S_{\text{gen}}(E(bc))$.

However, we would not expect that the monogamy of mutual information (MMI)
property holds in general gravitational spacetimes. As the bulk gravity is not
necessary to be AdS spacetime, the classical area term in the entropy formula
(16) may not satisfy the MMI property like in AdS spacetime Hayden:2011ag ;
Headrick:2013zda . Besides, the MMI property is not satisfied for general
quantum states of bulk matter fields.999In paper Bousso:2024ysg , it showed
that the MMI property is satisfied for bulk gravitational regions if the bulk
entanglement entropy can be neglected compared to the classical area term.

##  5 Constraints from GEW prescription with bit threads

In this section, we give an intuitive description of GEW prescription in terms
of quantum bit threads. We will show that the existence of a nontrivial GEW
will put constraints on the bulk entanglement entropy in certain bulk regions,
such as regions with the entanglement island.

In addition to the flow description for the entropy formula from the GEW
prescription given in Section 4, it can also be described by a set of oriented
and locally parallel discrete bit threads with Planck-thickness. The
corresponding bit threads consist of the homogeneous part (with the number of
threads denoted as $N^{h}$) as well as the inhomogeneous part (with the number
of threads denoted as $N^{i}$). The homogeneous part represents the threads
that are confined to the manifold $M$, like the description for original
classical bit threads. These threads start from $\partial M$, then pass
through $m_{\mathcal{X}}$, and finally enter the region $a$ by passing through
$\partial a$. The number of these homogeneous threads is upper bounded by the
area of the classical minimal surface that is homologous to $\partial a$.
While the inhomogeneous part represents the threads that can leave from the
manifold $M$ at some points $P_{i}$ and re-enter the manifold $M$ at other
points $Q_{i}$, whose distributions are restricted by bulk entanglement
entropies for all regions $w\in\Omega$. One may think that the homogeneous
threads are passing through “classical cuts” on manifold $M$, while
inhomogeneous threads are passing through some “quantum cuts” on geometries
emerging from the bulk entanglement VanRaamsdonk:2010pw , such as
$\text{ER}=\text{EPR}$ Maldacena:2013xja . For each “classical cut” $m_{c}$,
there is an associated “quantum cut” $m_{q}[m_{c}]$ that can be defined as a
function of $m_{c}$, whose minimal area is equal to the bulk entanglement
entropy between the matter fields on two sides of the surface $m_{c}$. In this
way, the generalized entropy is described “geometrically” as an “extended cut”
defined as $m_{\text{total}}[m_{c}]=m_{c}\cup m_{q}[m_{c}]$. Then by using a
generalization of the Riemannian max-flow min-cut theorem Headrick:2017ucz ,
the maximum number of total threads connecting the boundary $\partial M$ to
the bulk region $a$ through both “classical cuts” and “quantum cuts”, is dual
to the area of the minimal “extended cut” among all cuts
$m_{\text{total}}[m_{c}]$.

Figure 2: In a max thread configuration for GEW prescription without
entanglement island.

According to the GEW proposal, for a given bulk region $a$ on Cauchy surface
$M$, it is possible to make the total generalized entropy smaller by including
extra regions besides $a$ as a part of $E(a)$, such that
$E(a)=a\cup(E(a)\setminus a)$, where $E(a)\setminus a$ may be nonempty.
Compared to region $a$, we are able to make the bulk quantum state of matter
fields in $E(a)$ purer by including more bulk entangled partners into $E(a)$.
So the bulk entanglement entropy term in generalized entropy can be decreased,
but the risk is that the boundary area term may be increased. Thus, we need
make sure that $S_{\text{gen}}(E(a))\leq S_{\text{gen}}(a)$ by definitions. As
shown in Figure 2, for the case that $E(a)$ contains no entanglement island,
the bulk region $M$ is divided into three parts as $M=a\cup(E(a)\setminus
a)\cup(M\setminus E(a))$. Based on the ways of the inhomogeneous threads
passing through the surface $\partial a$ and $\partial E(a)$, they can be
further divided into five types. These oriented threads start from $\partial
M$, but they can directly jump into region $a$ from region $M\setminus E(a)$
(with the number of threads $N_{1}^{i}$), or jump into region $E(a)\setminus
a$ from region $M\setminus E(a)$ and then enter region $a$ by passing through
$\partial a$ (with the number of threads $N_{2}^{i}$), or enter into region
$E(a)\setminus a$ by passing through $\partial E(a)$ and then jump into $a$
from region $E(a)\setminus a$ (with the number of threads $N_{3}^{i}$). And
there are two more cases where $P_{i}$ and $Q_{i}$ are located in the same
region $M\setminus E(a)$ (with the number of threads $N_{4}^{i}$) or region
$E(a)\setminus a$ (with the number of threads $N_{5}^{i}$), these
inhomogeneous threads pass through both surfaces $\partial a$ and $\partial
E(a)$. Now consider a max flow configuration for the no-island phase,
satisfying

  1. (A1)

$\tilde{v}^{\mu}_{a}=n^{\mu}/4G_{N}$ at the quantum extremal surface
$\tilde{m}_{\mathcal{X}}=\partial E(a)$, with the unit normal vector $n^{\mu}$
on $\tilde{m}_{\mathcal{X}}$.

  2. (A2)

For $m_{w}=\tilde{m}_{\mathcal{X}}$ (hence $w=\tilde{w}=M\setminus E(a)$),
there is
$-\int_{\tilde{w}}\nabla_{\mu}\tilde{v}^{\mu}_{a}=S_{\text{bulk}}(\tilde{w})=S_{\text{bulk}}(E(a))$,
where region $\tilde{w}$ has the orientated boundary
$\partial\tilde{w}=\tilde{m}_{\mathcal{X}}-\partial M$

In other words, it requires that in a max thread configuration, the part of
oriented threads entering $E(a)$ by passing through $\partial E(a)$ reaches
its maximum number, i.e. $|\partial E(a)|/(4G_{N})$. Meanwhile, the part of
oriented threads entering $E(a)$ by jumping from $\tilde{w}$ to $E(a)$ also
reaches its maximum number, i.e. $S_{\text{bulk}}(E(a))$. While for other
surfaces $m_{w}\neq\tilde{m}_{\mathcal{X}}$ and its associated bulk region
$w$, these two parts of threads can not reach their maximum simultaneously.
Therefore, in a max thread configuration, we have

$\begin{split}S_{\text{gen}}(E(a))&=\frac{|\partial
E(a)|}{4G_{N}}+S_{\text{bulk}}(E(a))=(N^{h}+N^{i}_{3}+N^{i}_{4}+N^{i}_{5})+(N^{i}_{1}+N^{i}_{2}),\\\
S_{\text{gen}}(a)&=\frac{|\partial
a|}{4G_{N}}+S_{\text{bulk}}(a)\geq(N^{h}+N^{i}_{2}+N^{i}_{4}+N^{i}_{5})+(N^{i}_{1}+N^{i}_{3}),\end{split}$
(58)

where the number of each type of threads is non-negative. Thus the inequality
$S_{\text{gen}}(E(a))\leq S_{\text{gen}}(a)$ is satisfied automatically. It
leads to the constraint

$\begin{split}S_{\text{bulk}}(a)-S_{\text{bulk}}(E(a))\geq
N^{i}_{3}-N^{i}_{2}\geq\frac{|\partial E(a)|-|\partial
a|}{4G_{N}}.\end{split}$ (59)

Furthermore, we can find a lower bound on the bulk entanglement entropy of
region $E(a)\setminus a$ with orientated boundary $\partial(E(a)\setminus
a)=\partial E(a)-\partial a$, that is

$\begin{split}S_{\text{bulk}}(E(a)\setminus a)\geq
S_{\text{bulk}}(a)-S_{\text{bulk}}(E(a))\geq
N^{i}_{3}-N^{i}_{2}\geq\frac{|\partial E(a)|-|\partial
a|}{4G_{N}},\end{split}$ (60)

where we have assumed the Araki-Lieb inequality of the bulk entanglement
entropy for the first inequality, as $E(a)=a\cup(E(a)\setminus a)$. It puts a
constraint on the existence of a nontrivial GEW for cases without island.

Figure 3: In a max thread configuration for GEW prescription with an
entanglement island.

It is interesting to consider the case when $E(a)$ contains an entanglement
island, such that $E(a)=W(a)\cup I(a)$, as shown in Figure 3. By introducing
an extra disconnected region $I(a)$ as a part of $E(a)$, it is possible to
further purify the bulk quantum state of matter fields in $E(a)$, as we are
able to include more bulk entangled partners into $E(a)$. This makes the bulk
entanglement entropy term decrease but at the cost of increasing a boundary
area term of the island. For the island phase, we need make sure that
$S_{\text{gen}}(E(a))\leq\\{S_{\text{gen}}(W(a)),\ S_{\text{gen}}(a)\\}$ by
definitions. Recall that when there is no island region, there are six types
of threads connecting boundary $\partial M$ to region $a$, with numbers
$N^{h},N_{1}^{i},N_{2}^{i},N_{3}^{i},N_{4}^{i},N_{5}^{i}$. Then after
introducing island $I(a)$, all these six types of threads may pass through
$I(a)$. Now we consider a max flow configuration for the island phase, which
satisfies

  1. (B1)

$\tilde{v}^{\mu}_{a}=n^{\mu}/4G_{N}$ at the quantum extremal surface
$\tilde{m}_{\mathcal{X}}=\partial E(a)=\partial W(a)\cup\partial I(a)$, with
the unit normal vector $n^{\mu}$ on $\tilde{m}_{\mathcal{X}}$.

  2. (B2)

For $m_{w}=\tilde{m}_{\mathcal{X}}$ (hence $w=\tilde{w}=M\setminus E(a)$),
there is
$-\int_{\tilde{w}}\nabla_{\mu}\tilde{v}^{\mu}_{a}=S_{\text{bulk}}(\tilde{w})=S_{\text{bulk}}(E(a))$,
where region $\tilde{w}$ has the orientated boundary
$\partial\tilde{w}=\tilde{m}_{\mathcal{X}}-\partial M$.

Therefore, in a max thread configuration, the part of oriented threads
entering $E(a)$ by passing through $\partial E(a)$ reach its maximum number,
i.e. $(|\partial W(a)|+|\partial I(a)|)/(4G_{N})$. And the part of oriented
threads entering $E(a)$ by jumping from region $\tilde{w}$ to region $E(a)$
also reaches its maximum number, i.e. $S_{\text{bulk}}(E(a))$. Now we focus on
region $I(a)$, as the part of oriented threads entering $I(a)$ by passing
through $\partial I(a)$ reaches its maximum number, i.e. $|\partial
I(a)|/(4G_{N})$. It requires these oriented threads can only leave region
$I(a)$ by jumping across $\partial I(a)$ without passing through $\partial
I(a)$, otherwise the part of flux entering $I(a)$ by passing through $\partial
I(a)$ will not reach its maximum if these oriented threads are allowed to turn
back by passing through $\partial I(a)$ again (with negative contributions to
the flux passing through $\partial I(a)$ with orientation). Moreover, for
those oriented threads that leave region $I(a)$ by jumping across surface
$\partial I(a)$, they can only jump into region $W(a)$ instead of region
$\tilde{w}$. As the second condition requires that the part of flux jumping
from region $\tilde{w}$ to region $E(a)$ also reaches its maximum, thus it
does not allow these oriented threads to jump inversely from region $I(a)$ to
region $\tilde{w}$ (with negative contributions to the flux jumping from
region $\tilde{w}$ to region $E(a)=W(a)\cup I(a)$ with orientation). Finally,
it turns out that not all six types of threads can be assigned to the island,
there are only two types of threads that are allowed to be assigned to $I(a)$
for a max thread configuration, with numbers denoted as
$N_{1^{\prime}}^{i},N_{2^{\prime}}^{i}$ in Figure 3. Therefore, in a max
thread configuration, we have

$\begin{split}S_{\text{gen}}(E(a))&=\frac{|\partial
W(a)|}{4G_{N}}+\frac{|\partial I(a)|}{4G_{N}}+S_{\text{bulk}}(W(a)\cup
I(a))\\\
&=(N^{h}+N^{i}_{3}+N^{i}_{4}+N^{i}_{5})+(N^{i}_{1^{\prime}}+N^{i}_{2^{\prime}})+(N^{i}_{1}+N^{i}_{2}),\\\
S_{\text{gen}}(W(a))&=\frac{|\partial W(a)|}{4G_{N}}+S_{\text{bulk}}(W(a))\\\
&\geq(N^{h}+N^{i}_{3}+N^{i}_{4}+N^{i}_{5})+(N^{i}_{1}+N^{i}_{2}+N^{i}_{1^{\prime}}+N^{i}_{2^{\prime}}),\\\
S_{\text{gen}}(a)&=\frac{|\partial a|}{4G_{N}}+S_{\text{bulk}}(a)\\\
&\geq(N^{h}+N^{i}_{2}+N^{i}_{2^{\prime}}+N^{i}_{4}+N^{i}_{5})+(N^{i}_{1}+N^{i}_{3}+N^{i}_{1^{\prime}}),\end{split}$
(61)

where the number of each type of threads is non-negative. Thus the
inequalities $S_{\text{gen}}(E(a))\leq\\{S_{\text{gen}}(W(a)),\
S_{\text{gen}}(a)\\}$ are satisfied automatically. It leads to the following
two constraints

$\begin{split}&S_{\text{bulk}}(W(a))-S_{\text{bulk}}(W(a)\cup I(a))\geq
N^{i}_{1^{\prime}}+N^{i}_{2^{\prime}}=\frac{|\partial I(a)|}{4G_{N}},\\\
&S_{\text{bulk}}(a)-S_{\text{bulk}}(W(a)\cup I(a))\geq
N^{i}_{3}-N^{i}_{2}+N^{i}_{1^{\prime}}\geq
N^{i}_{3}-N^{i}_{2}-N^{i}_{2^{\prime}}\geq\frac{|\partial W(a)|+|\partial
I(a)|-|\partial a|}{4G_{N}}.\end{split}$ (62)

Then by assuming the Araki-Lieb inequality of the bulk entanglement entropy,
hence

$\begin{split}&S_{\text{bulk}}(W(a)\cup I(a))\geq
S_{\text{bulk}}(W(a))-S_{\text{bulk}}(I(a)),\\\
&S_{\text{bulk}}(a\cup(W(a)\setminus a)\cup I(a))\geq
S_{\text{bulk}}(a)-S_{\text{bulk}}((W(a)\setminus a)\cup I(a)),\end{split}$
(63)

we can find the lower bounds on the bulk entanglement entropy for region
$I(a)$ and region $E(a)\setminus a=(W(a)\setminus a)\cup I(a)$ (with
orientated boundary $\partial(E(a)\setminus a)=\partial W(a)+\partial
I(a)-\partial a$) respectively as

$\begin{split}&S_{\text{bulk}}(I(a))\geq\frac{|\partial I(a)|}{4G_{N}},\\\
&S_{\text{bulk}}(E(a)\setminus a)\geq\frac{|\partial W(a)|+|\partial
I(a)|-|\partial a|}{4G_{N}}.\end{split}$ (64)

The constraints (60) for no-island phase and (64) for island phase are
convincing, as we only utilized definitions of the GEW such that
$S_{\text{gen}}(E(a))\leq\\{S_{\text{gen}}(W(a)),\ S_{\text{gen}}(a)\\}$, and
the Araki-Lieb inequality for the bulk quantum state of matter fields. They
put constraints on the existence of a nontrivial GEW in general gravitational
spacetimes, particularly the existence of an entanglement island, which may
also be interpreted as the physical conditions for inducing nontrivial quantum
phase transitions in GEW prescription.

One may recall the studies on the entropy bounds, from the Bekenstein Bound
Bekenstein:1980jp , to the spherical entropy bound Susskind:1994vu , the
spacelike entropy bound Bousso:2002ju  and the covariant entropy bound
Bousso:1999xy ; Bousso:1999cb , which put upper bounds on the bulk
entanglement entropy of matter fields in certain regions. The first three
entropy bounds suffered from several problems as stated in Bousso:2002ju ,
while the covariant entropy bound has been proven to hold in a wide variety of
situations Flanagan:1999jp ; Wall:2010cj ; Wall:2011hj ; Bousso:2014sda . In
this paper, we are interested in the conjecture of spacelike entropy bound,
which states that the entanglement entropy of matter fields contained in any
codimension-one spatial region $V$ will not exceed the boundary area of region
$V$ in static Cauchy slices, that is

$S_{\text{bulk}}(V)\leq\frac{|\partial V|}{4G_{N}}.$ (65)

In some special situations, the spacelike entropy bound may be valid for
region $I(a)$, thus it forces that

$S_{\text{bulk}}(I(a))=\frac{|\partial I(a)|}{4G_{N}},$ (66)

by combining with constraint (64). However, we stress that this entropy bound
can be violated even for isolated, spherical, weakly gravitating matter
systems Bousso:2002ju . As for the covariant entropy bound, it focuses on the
entropy on the light-sheet $L(\partial V)$ generated by null geodesics
originating from codimension-two spacelike surface $\partial V$, rather than
the codimension-one spacelike region $V$, hence it may not be directly related
to our results. However, under certain conditions, the covariant bound indeed
implies a spacelike bound by the spacelike projection theorem Bousso:1999xy ;
Bousso:1999cb ; Bousso:2002ju .

##  6 Conclusion and discussion

In the paper, we showed that the GEW prescription implies the principle of the
holography of information in topologically trivial Cauchy surfaces with a pure
total state of matter fields, which indicates the holographic nature of the
boundaries of these Cauchy slices. It prompted us to assume that the bit
threads emerge from the boundaries of these Cauchy slices, like in AdS/CFT
correspondence. Then we proposed the quantum bit thread formulation that is
dual to the entropy formula from the GEW proposal by using the tools from
convex optimization. In this way, we succeeded in extending the bit thread
description to a static Cauchy slice in more general gravitational spacetimes,
not limited to the AdS spacetime. By using the properties of flows, we proved
the basic properties of the entropy for bulk gravitational regions, such as
the monotonicity, subadditivity, Araki-Lieb inequality and strong
subadditivity. We did not expect that the MMI property is satisfied for bulk
regions in general gravitational spacetimes, but the MMI property may hold in
some special gravitational spacetimes (such as AdS spacetime), then it may be
possible to show the MMI property in these special gravitational spacetimes by
introducing the so-called multiflow configurations Cui:2018dyq ;
Hubeny:2018bri ; Agon:2021tia . It would also be interesting to investigate
whether there are certain conditions that lead to the existence of the MMI
property from the GEW perspective. Furthermore, we found that the bulk
entanglement entropy of matter fields in the region $E(a)\setminus a$ must be
lower bounded by the area of the orientated boundary, as required by the
existence of a nontrivial GEW on a static Cauchy slice in general
gravitational spacetimes. In particular, the bulk entanglement entropy of
matter fields on an entanglement island should be lower bounded by the
boundary area of the entanglement island.

Note that we only investigated the bit thread formulation in static scenarios
in the present paper. The generalizations to the Lorentzian and covariant
settings like in Headrick:2017ucz ; Headrick:2022nbe  would be worth studying,
although a general time-dependent extension of GEW proposal Bousso:2022hlz ;
Bousso:2023sya  is still under study. Furthermore, though the GEW proposal
provides a potential pattern of holographic encoding in general spacetimes,
the fine-grained description of the entanglement structures still needs
further studies. Since the bit thread description may help reveal more
detailed structures of the entanglement entropy by connecting it with the
information-theoretic contents as did in the AdS/CFT correspondence. For
example, when the bulk quantum entanglement can be neglected, the entropy
formula (16) becomes an RT-like formula. Meanwhile, the quantum bit thread
description will reduce to the classical one that only contains the
homogeneous threads, thus it would be easier to deal with. It may be feasible
to introduce the corresponding bit thread description for other information-
theoretic concepts (such as the entanglement of purification, partial
entanglement entropy and multipartite entanglement) into general spacetimes.

Moreover, although we only considered a topologically trivial manifold $M$ and
the total state of matter fields in $M$ is a pure state, we expect our bit
thread description could be suitably extended to more general scenarios where
$M$ has a non-trivial topology and the total state of matter fields in $M$ can
be a mixed state. Note that there may exist a “hole” inside $M$, whose
interior is not accessible for an outside observer on $M$. It allows the
existence of an extra manifold $M^{\prime}$ with another boundary $\partial
M^{\prime}$ behind the “hole”, where $M$ and $M^{\prime}$ are smoothly joined
together along the boundary of the “hole”, such as a wormhole geometry. In
fact, the topology of manifolds may be even more complicated. In addition, it
is also possible for the total state of matter fields in $M$ to be a mixed
state, as the matter fields in manifold $M$ may entangle with the matter
fields in another manifold $M^{\prime\prime}$ with boundary $\partial
M^{\prime\prime}$, even though manifold $M^{\prime\prime}$ may not connect
with manifold $M$ through any classical geometry. In short, it means that some
threads are allowed to connect region $a$ to these extra boundaries in
homogeneous and inhomogeneous ways, thus we need also take this extra part of
threads into consideration when maximizing the flux of any flow getting into
region $a$. According to the entropy formula (16), we should minimize the
generalized entropy among all homology regions of the region $a$, including
the geometry and the entanglement entropy of matter fields on these extra
manifolds. As long as we consider the whole manifold $M_{\text{total}}$ with
its boundary $\partial M_{\text{total}}$, the state of the matter fields in
$M_{\text{total}}$ would be pure. By replacing $M$ with $M_{\text{total}}$
(hence $a^{c}\equiv M_{\text{total}}\setminus a$) and then maximizing the
number of total threads from boundary $\partial M_{\text{total}}$ to region
$a$ in above formulation (21), our bit thread description would be applicable.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant No. 11675272.

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