Towards bit threads in general gravitational spacetimes

: 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 ) ⊃ 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 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 gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂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 ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ 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.


Introduction
Relevant developments in the holographic nature of gravity [1,2] have revealed the deep connection between the spacetime geometry and the quantum entanglement.In the AdS/CFT correspondence [3], the Ryu-Tagayanagi (RT) formula [4,5] 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
where m is a bulk surface homologous to A (represented by m ∼ 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 O(1/G N ), in which the contribution from bulk quantum entanglement is ignored.When the leading-order quantum correction of order O(G 0 N ) coming from the bulk quantum entanglement is included, the RT formula should be corrected as the Faulkner-Lewkowycz-Maldacena (FLM) formula [6], i.e.

S(A) = min
where σ A represents the spatial homology region surrounded by the union of surfaces A ∪ m A . S bulk (σ A ) is the von Neumann entropy of bulk matter fields restricted to σ A .Note that in this formula, one needs first to take the minimization for the area term among all possible bulk surfaces m ∼ A to find the minimal surface m A and the associated homology region σ A , then to add the bulk entanglement entropy of region σ A .At the order of O(G 0 N ), 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 [7], that is where σ represents a spatial homology region with boundary ∂σ = A ∪ m and S gen (σ) is the generalized entropy on region σ.Note that here G N is the renormalized Newton constant [8] and S bulk is the finite part of the von Neumann entropy of bulk matter fields. 1 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 ∼ A to find the so-called minimal quantum extremal surface m X , and σA is a spatial homology region bounded by A ∪ m X .The quantum extremal surface m 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 [10][11][12] EW(A) is then defined as the bulk domain of dependence of the spatial region σA .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 [13][14][15], as a generalization from the QES formula, was proposed to calculate the entanglement entropy of the Hawking radiation and reproduce the Page curve [16] in semi-classical gravity, which was verified by including the replica wormholes [17,18] in the gravitational path integral calculations in 2D Jackiw-Teitelboim (JT) gravity coupled to matters (for a nice review [19]).It turns out that the entanglement wedge 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.EW(R) = R ∪ I. Then according to the entanglement wedge reconstruction [20][21][22], 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 [23][24][25][26][27] and gravitational baths [28,29], asymptotically flat black hole spacetime [30][31][32][33][34][35][36][37][38][39][40][41], de-Sitter spacetime [42][43][44][45], AdS spacetime dual to the BCFT [46][47][48][49][50][51][52] 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 [53], 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 [54][55][56].Moreover, the bit threads have been generalized to the Lorentzian setting [57], covariant setting [58] and higher curvature gravity [59].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 [60][61][62][63][64][65], holographic complexity [66][67][68], holographic partial entanglement entropy [69][70][71][72][73][74][75] and multipartite entanglement [76][77][78], which reveals many quantum information-theoretic aspects on the entanglement structures in holography.See Refs.[79][80][81][82][83][84][85][86][87] 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. 2 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 [92,93], called the generalized entanglement wedge (GEW), motivated by developments on the gravitational entropy formulas [4-7, 13-15, 17-19] and the tensor network toy models in quantum gravity [94][95][96][97]. 3It 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 gen (E(a)) among all wedges E ⊃ 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 2 The classical bit thread description was applied in dS space [88,89], as the RT prescription can be reformulated in de Sitter space in the framework of static-patch holography [89][90][91].
3 Also motivated by tensor network models, the surface/state duality was proposed in [98,99], 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 ΣA ⊂ M (while the region a considered in GEW proposal is codimension-one) is dual to a certain quantum state ρ(ΣA), and its entanglement entropy is given by the area of the minimal (quantum) extremal surface mA that is homologous to ΣA.Thus it allows us to define an entanglement wedge EW(ρ(ΣA)), i.e. the bulk domain of dependence of the spatial region bounded by ΣA ∪ mA, 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 [100][101][102].
S gen (E(a)) by an entropy formula (3.5) similar to the QES formula (1.3).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 [103][104][105][106], 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 [53,57], 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) \ a, especially for an entanglement island region I(a).The conclusion and discussion are given in Section 6.

Review of the bit threads
Consider a time-reflection symmetric Cauchy surface M with a conformal boundary ∂M , where A is a subregion on ∂M .Define a divergenceless and norm-bounded flow v on manifold M .The flux passing through region A is given by where h is the determinant of the induced metric h ij on A and n µ 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 [53] subject to the constraints 3) The formula (2.2) was proved to be equivalent to the RT formula (1.1) by using convex optimization and strong duality [57].As there exits a max flow configuration ṽ subject to constraints (2.3) such that ṽµ = n µ /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 The early idea of adding quantum corrections to the bit thread prescription was discussed in [53,54] 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 [55,56] by using convex optimization similar to that in [57], 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 [56] subject to the constraints where Ω A represents the set of all possible homology regions for boundary region A, i.e.
There exists a max flow configuration ṽ subject to constraints (2.6) on M : 1. Requiring ṽµ = n µ /4G N at the quantum extremal surface m X , where n µ is the unit normal vector on m X .

Saturating the divergence bound for
The boundary of σA has the orientation as ∂ σA = m X − A, where the unit normal vector is inward-pointing on surface A and outward-pointing on surface m X .By using the Gauss's law, we have as expected in the QES formula (1.3).One may note that the quantum extremal surface m 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 [56] 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) ⊃ a, which minimizes the generalized entropy among all possible homology wedges that contain a.Moreover, E(a) may contain a island region

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 [92,93].For an arbitrary time-reflection symmetric Cauchy surface M with asymptotic boundary ∂M , i.e. a Riemannian manifold with metric g µν .Denoting ∂s as the boundary of a set s ⊂ M , and defining cl s ≡ s ∪ ∂s and int s = s \ ∂s, then wedge a is defined as any open subset of M that is the interior of its closure: a = int cl a.For two wedges a and b, one can define the wedge union, wedge complement and wedge relative complement, i.e.
which are also wedges, where a c ≡ M \ a.The generalized entropy of any given wedge a ⊂ M is defined as where G N is the renormalized Newton constant, and S bulk (a) = − tr ρ a log ρ a is the finite part of the entanglement entropy of bulk matter fields, where ρ a = tr a ′ ρ is the density operator of matter fields on wedge a.The generalized entropy is finite and cutoff-independent [9].In this paper, we divide the total boundary of wedge a into two parts as ∂a = ∂a ∪ ȧ, where ∂a ≡ ∂a \ ∂M represents the part inside the bulk and ȧ ≡ ∂a ∩ ∂M represents the part that is overlapping with the boundary ∂M .
According to the GEW proposal [92,93], the GEW E(a) associated to any given wedge a is defined as the wedge that minimizes the generalized entropy among all possible wedges σ ⊃ a on M (as illustrated in Figure 1).So the boundary ∂σ is defined on region a c = M \ a and it is homologous to ∂a.Therefore, we have where we define mX ≡ ∂E(a) as a generalization of quantum extremal surface, distinguishing from m 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 region 4  , as given by the island formula. 5The GEW proposal suggests a generalization of entanglement wedge reconstruction such that the information in E(a) can be reconstructed from region a.

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 [103][104][105][106], 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 nearboundary bulk region δ M attached to the asymptotic boundary ∂M , with complement region δ c M ≡ M \ δ M .Its boundary has the orientation ∂δ M = m δ − ∂M (the unit normal vector is inward-pointing on ∂M and outward-pointing on m δ ), where m δ represents its inner boundary.According to the GEW proposal, we need to find the entanglement wedge E(δ M ) that minimizes the generalized entropy, among all wedges σ ⊃ δ M .The boundary of σ is ∂σ = m − ∂M , hence m ∼ m δ ∼ ∂M .As ∂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 δ M is given by E(δ M ) = M .So the area contribution from the inner boundary m vanishes, meanwhile S bulk (E(δ M )) = S 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 ∂M , which is a fixed term.That is As the entanglement wedge E(δ M ) of the near-boundary bulk region δ 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 δ M that is attached to the outer asymptotic boundary (denoted as ∂M o here), the entanglement wedge E(δ M ) may have a nonvanishing interior boundary, meanwhile the entanglement entropy of matter fields on E(δ M ) may not vanish.Hence E(δ M ) ̸ = M in general, there are some physical degrees of freedom outside E(δ M ) that are not available for the observer restricted to the near-boundary region ∂M o .However, we stress that it is still possible to obtain a similar result like the formula (3.6) if we choose a proper nearboundary 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 ∂M i (may consist of multiple separate inner boundaries).Hence the boundary of M is given by ∂M = ∂M i ∪ ∂M o .We need to take ∂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 ′ (assuming that it is known and admits a semi-classical description) with boundary ∂M ′ .However, as long as we consider the whole manifold M total = M ∪M ′ (whose topology may be complicated) with its complete boundary ∂M total , the total quantum state of matter fields in M total would still be pure.If we choose a near-boundary bulk region δ M total that is attached to the whole boundary ∂M total , we expect E(δ M total ) = M total to achieve the minimization of the generalized entropy.As the area term of the inner boundary of E(δ M total ) vanishes, meanwhile S bulk (E(δ M total )) = S bulk (M total ) = 0, with a remaining fixed area term |∂M total |.This leads to a result similar to the formula (3.6).

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 (3.5).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 ∂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 ∂a = ∂a ∪ ȧ (where ȧ may be non-empty), so that ∂a ∼ ȧ.And σ is a bulk homology region that contains wedge a, so that a ⊂ σ ⊂ M .If ȧ ̸ = ∅, by definition we must require σ ⊃ ȧ, hence (∂σ \ ȧ) ∼ ȧ.Defining w ≡ M \ σ as the complement of the region σ, due to the purity of the total state of matter fields on M , we have where the boundary of region w is orientated as ∂w = m w − ∂M , with m w ≡ ∂σ (which is allowed to partly overlap with ∂M ).So the formula (3.5) can be written as where the minimization is reached for m w = ∂σ = mX when w = w with boundary ∂ w = mX −∂M .The generalized entanglement wedge E(a) is just the bulk region σ = M \ w that is surrounded by mX , with S bulk ( w) = S bulk (σ).For clarity, we see that the above minimization consists of two terms, i.e.
| ȧ| 4G N + min If ȧ ̸ = ∅, we have (m w \ ȧ) ∼ ȧ ∼ (∂M \ ȧ), and then S bulk (w) reduces to the bulk entanglement entropy on the region that has the orientated boundary (m w \ ȧ) − (∂M \ ȧ).Note that this formula has the same form as the QES formula (1.3) if we set (m w \ ȧ) = m and (∂M \ ȧ) = A (differing by a fixed area term | ȧ| = |A|).For ȧ = ∅, it will be similar to the case with A = ∂M in the QES formula.Here a significant difference that should be stressed is that the minimization in formula (4.2) 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 ∂M to the bulk region a, or equivalently the maximum number of bit threads that start from the boundary ∂M and end at the bulk region a.The first area term in (4.3) is just dual to the part of the maximum flux directly entering the bulk region a from the boundary ∂M , as ∂a is just overlapping with ∂M at ȧ.While the second term in (4.3) is dual to the part of the maximum flux from (∂M \ ȧ) to the bulk region a, which can be similarly proved through the convex optimization techniques adopted for the QES formula with (m w \ ȧ) = m and (∂M \ ȧ) = 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 \ a that starts from ∂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 (4.2) can be formalized as subjecting to the constraints where Ω represents a set of the complements of all homology regions of region a, that is where the unit normal vector is inward-pointing on the boundary ∂M and outward-pointing on the surface m w , and m w ∼ ∂a ∼ ∂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 ṽa which is subject to constraints (4.5) on a c , satisfying: 1. ṽµ a = n µ /4G N at the quantum extremal surface mX , where n µ is the unit normal vector on mX .
2. For m w = mX (hence w = w), there is − w ∇ µ ṽµ a = S bulk ( w), where region w has the orientated boundary ∂ w = mX − ∂M .
Therefore we have 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 mX and its associated region w, which leads to the entropy formula (4.2).Moreover, we may divide any flow v a into the homogeneous part v h a and inhomogeneous part v i a , such that ∇v h a = 0, and ∀ w ∈ Ω : on region a c , and both the homogeneous and inhomogeneous parts can contribute to entropy.

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 (4.2) and the quantum bit thread formulation (4.4), similar to Refs.[55][56][57].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 [109], 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 (4.4) subject to the constraints (4.5) defines a concave program, in which the constraints (4.5) 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 where ϕ is a non-negative scalar field on a c , µ is a non-negative probability measure on Ω, such that Ω dµ(w) = 1.Besides, we have performed the integral by parts in the second equality and used the definitions of characteristic functions χ(w, x) and χ(w, x): χ(w, x) := 1, for x ∈ w 0, for x ∈ a c \w and χ(w, x) + χ(w, x) = 1 (4.10) Then we maximize the Lagrangian function (4.9) 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 With the constraint (4.11), the maximization of the Lagrangian function (4.9) reduces to Thus the dual Lagrangian is given by which is subject to the constraint (4.11), in which the minimization with respect to ϕ is trivial as ϕ(x) can reach its minimum value, i.e. ϕ(x) = |∂ µ ψ(x)|.Now we argue that the result (4.14) is equivalent to the entropy formula (4.2).Given the function ψ(x) on a c defined in (4.12) with the boundary condition (4.11), hence ψ(x) = 1 at surface ∂a and ψ(x) = 0 at surface ∂M .And we assume ψ(x) is differentiable6 on a c .Then define a one-parameter family of bulk regions as As the normal derivative of a characteristic function is a surface delta function, thus the first term in the objective function (4.14) can be written in terms of the level sets as as Ref. [56] 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 ∂w i = m i − ∂M .The nth term on the right-hand side of inequality (4.18) is the bulk region r(n/N ), defined as r(n/N ) := {x ∈ a c : ψ(x) ≤ n/N }.Thus we have which leads to formula (4.17) in the limit N → ∞ after dividing by N .Finally, combining with (4.16) and (4.17), it gives the dual prescription However, the objective function only involves the gradient, i.e. ∂µψ(x).At surface ȧ, the result of this nondifferentiable 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 ∂A, referring to Footnote 10 in Ref. [57].Moreover, it is feasible to use differentiable functions to approximate the non-differentiable function χ(w, x) well if we slightly smooth out the step function.Hence we can restrict to differentiable functions in our procedures. 7The one-parameter family of bulk regions r(p) corresponds to the set of the complements of all homology regions of a, i.e. set Ω.One may define r ′ (p) := {x ∈ a c : ψ(x) ≥ p, p ∈ [0, 1]} as one-parameter family of bulk regions on a c , whose boundary is ∂r ′ (p) = ∂a − m(p).So that a ∪ r ′ (p) corresponds to the homology region of region a on manifold M .
where r(p) with its level set m(p) is determined by measure µ on Ω defined in (4.6).This is exactly the entropy formula (4.2).The minimization is realized if µ only supports on the homology region r(p) that is bounded by the quantum extremal surface m X and the boundary ∂M , so that ∂ r(p) = m X − ∂M .Meanwhile, the entanglement wedge time slice E(a) of the region a is defined as the bulk region that is surrounded by m X , thus ∂E(a) = m X .Therefore, the proof on the equivalence between the entropy formula (4.2) and the bit thread formulation (4.4) subjecting to the constraints (4.5) is finished.

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.

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 ∩ b = ∅) on manifold M , we are interested in the total flux of any flow v ab defined on M that starts from boundary ∂M and enters the union region ab (ab ≡ a ∪ b for brevity).We assume that flow v ab consists of two independent components: 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, 8 similarly for v b .Thus v a contributes to S gen (E(a)) and S gen (E(ab)) but not to S gen (E(b)), meanwhile v b contributes to S gen (E(b)) and S gen (E(ab)) but not to S 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 ⊂ 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. which is subject to the norm bound constraints |v a |, |v ab | ≤ 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 Ω 1 and Ω 2 , that is with where The nesting property can be proved as we will show that the value of the term on the left side of the inequality (4.22) is also lower bounded by S gen (E(a)) + S gen (E(ab)) by using the convex optimization.Let us construct the Lagrangian in which v a , v b as two independent original variables, ϕ 1 and ϕ 2 are two non-negative scalar fields, µ 1 and µ 2 are two probability measures on Ω 1 and Ω 2 , respectively, satisfying By performing the integral by parts, the Lagrangian can be organized as Then we maximize the Lagrangian function with respect to the variables v a , v b , while the finiteness of the maximization demands that where Therefore, we have Further defining two independent sets of one-parameter family of bulk regions as with the oriented boundaries ∂r(p 1 ) = m(p 1 ) − ∂M, ∂r(p 2 ) = m(p 2 ) − ∂M .According to Section 4.1, the objective function in formula (4.29) can be written in terms of two independent level sets, thus we obtain the following result which leads to subject to the norm bound and divergence constraints around the formula (4.23).Then combining with inequalities (4.22) and (4.32), 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 gen (E(ab))) and the flux entering region a (with the maximal value S gen (E(a))), hence we finish the proof of the nesting property of flows.

• Monotonicity
The entropy for bulk regions a, b satisfies the monotonicity property 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 with 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 (3.5) may not satisfy the MMI property like in AdS spacetime [110,111].Besides, the MMI property is not satisfied for general quantum states of bulk matter fields. 9 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 ∂M , then pass through m X , and finally enter the region a by passing through ∂a.The number of these homogeneous threads is upper bounded by the area of the classical minimal surface that is homologous to ∂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 ∈ Ω.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 [113], such as ER = EPR [114].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 total [m c ] = m c ∪ m q [m c ]. Then by using a generalization of the Riemannian max-flow min-cut theorem [57], the maximum number of total threads connecting the boundary ∂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 total [m c ].
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 ∪ (E(a) \ a), where E(a) \ 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 gen (E(a)) ≤ S 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 ∪ (E(a) \ a) ∪ (M \ E(a)).Based on the ways of the inhomogeneous threads passing through the surface ∂a and ∂E(a), they can be further divided into five types.These oriented threads start from ∂M , but they can directly jump into region a from region M \ E(a) (with the number of threads N i 1 ), or jump into region E(a) \ a from region M \ E(a) and then enter region a by passing through ∂a (with the number of threads N i 2 ), or enter into region E(a) \ a by passing through ∂E(a) and then jump into a from region E(a) \ a (with Furthermore, we can find a lower bound on the bulk entanglement entropy of region E(a) \ a with orientated boundary ∂(E(a) \ a) = ∂E(a) − ∂a, that is where we have assumed the Araki-Lieb inequality of the bulk entanglement entropy for the first inequality, as E(a) = a ∪ (E(a) \ a).It puts a constraint on the existence of a nontrivial GEW for cases without island.It is interesting to consider the case when E(a) contains an entanglement island, such that E(a) = W (a) ∪ 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 gen (E(a)) ≤ {S gen (W (a)), S gen (a)} by definitions.Recall that when there is no island region, there are six types of threads connecting boundary ∂M to region a, with numbers 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 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 [55,79,80].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) \ 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 [57,58] would be worth studying, although a general time-dependent extension of GEW proposal [92,93] 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 (3.5) 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 ′ with another boundary ∂M ′ behind the "hole", where M and M ′ 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 ′′ with boundary ∂M ′′ , even though manifold M ′′ 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 (3.5), 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 total with its boundary ∂M total , the state of the matter fields in M total would be pure.By replacing M with M total (hence a c ≡ M total \ a) and then maximizing the number of total threads from boundary ∂M total to region a in above formulation (4.4), our bit thread description would be applicable.
attached to the boundary region A in AdS/CFT, where E(a) = EW(A) and ∂E(a) = m X ∪ 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 ∂E(a) = ∂R before the Page time, while E(a) = R ∪ I and ∂E(a) = ∂R ∪ ∂I after the Page time

v
ab ≤ S gen (E(a)) + S gen (E(ab)), (4.22) 8 Specifically, we mean that the total flux (including the homogeneous and inhomogeneous parts) entering region b vanishes for flow component va, i.e. ( ∂b va − b ∇µv µ a ) = 0 by Gauss's law.So that the corresponding bit threads of va 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.

∂M≤
Ω := {w ⊂ a c : ∂w = m w − ∂M } and Ω ′ := {w ⊂ b c : ∂w = m w − ∂M }.And the subscripts of v a , v b are omitted for convenience.For a ⊂ b with b c ⊂ a c , we have Ω ′ ⊂ Ω and F ′ ⊂ F. Thus the set F has additional constraints than the set F ′ .According to Theorem 1 in Section 2.4 of Ref.[63], it leads to a constraint on solutions of two convex maximization programs in(4.34), which gives the result as S gen (E(a)) ≤ S gen (E(b)) for a ⊂ b.Therefore, we always have S gen (E(a)) ≤ S gen (E(ab)) as a ⊂ ab for arbitrary bulk regions a, b.• SubadditivityThe entropy for any two disjoint bulk regions a, b satisfies the subadditivity (SA) propertyS gen (E(ab)) ≤ S gen (E(a)) + S gen (E(b)).(4.35)Proof:We can begin with a nesting max flow v ab = v a + v b starting from boundary ∂M such that it simultaneously maximizes the flux entering the union region ab and region a for a ⊂ ab, as we proved before.It directly gives the SA property:S gen (E(ab)) = ∂M v ab = ∂M v a + ∂M v b ≤ S gen (E(a)) + S gen (E(b)),(4.36)asthe flux entering the union region ab and region a are equal to S gen (E(ab)) and S gen (E(a)) respectively, while in general the flux entering the region b can not reach its maximum value S gen (E(b)).•Araki-Lieb inequalityThe entropy for bulk regions a, b satisfies the Araki-Lieb (AL) property|S gen (E(a)) − S gen (E(b))| ≤ S gen (E(ab)) (4.37)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 haveS gen (E(a)) − S gen (E(ab)) = ∂M v a − ∂M v ab = − ∂M v b ≤ S gen (E(b)), (4.38) because the flux of any flow entering the region b can not reach its maximum value S gen (E(b)) in general, including the inverse flow −v b .Similarly one can find S gen (E(b)) − S gen (E(ab)) ≤ S 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 S gen (E(b)) + S gen (E(abc)) ≤ S gen (E(ab)) + S gen (E(bc)).(4.39) 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 ⊂ abc.It simply gives the SSA property: S gen (E(b)) + S gen (E(abc)) = S gen (E(ab)) + S gen (E(bc)) (4.40) as its flux entering the union region abc and region b are equal to S gen (E(abc)) and S gen (E(b)) respectively, while in general the flux entering the union regions ab and bc can not reach their maximum values S gen (E(ab)) and S gen (E(bc)).

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