Inequalities of Holographic Entanglement of Purification from Bit Threads

There are increasing evidences that quantum information theory has come to play a fundamental role in quantum gravity especially the holography. In this paper, we show some new potential connections between holography and quantum information theory. Particularly, by utilizing the multiflow description of the holographic entanglement of purification (HEoP) defined in relative homology, we obtain several new inequalities of HEoP under a max multiflow configuration. Each inequality derived for HEoP has a corresponding inequality of the holographic entanglement entropy (HEE). This argument is further confirmed by geometric analysis. In addition, we may expect that, based on flow considerations, each property of HEE that can be derived from bit threads may have a corresponding property for HEoP that can be derived from bit threads defined in relative homology.


Introduction
The holographic essence of quantum gravity was elucidated in [1,2], which revealed the duality between the quantum gravity theory in a (d + 1)-dimensional space-time region and the quantum field theory on the d-dimensional boundary of this region. Specifically, in the AdS/CFT correspondence [3][4][5], the entanglement entropy for a spatial region A on conformal boundary was shown to be given by the area of minimal homologous surface [6,7], i.e., the Ryu-Takayanagi (RT) formula where m A is the minimal surface in the bulk homologous to A. This reveals the deep connections between quantum entanglement and space-time geometry. The entanglement entropy primely characterizes the quantum entanglement in pure bipartite states, and has many known properties. For example, It has been proved that the RT formula obeys all above properties of the entanglement entropy [8]. However, there is a property possessed by the holographic entanglement entropy (HEE) peculiarly [9][10][11], that is which is not obeyed by general quantum states. It gives a constraint on theories that potentially have a holographic duality. Alternatively, properties (1.2)-(1.6) of HEE can be derived by the notion of bit thread [12], an alternative description of the HEE. See further works related to the bit threads in [13][14][15][16][17][18][19][20][21][22][23] 1 . Moreover, there is a quantity called entanglement of purification (EoP) [24], which is a measure of the classical correlations and quantum entanglement for mixed bipartite states. In [25,26], it has been conjectured that EoP is dual to the area of the minimal cross section on the entanglement wedge [27][28][29] (and some related works in ). For two non-overlapping spatial subregions A and B on the conformal boundary, we have where σ min AB is the minimal cross section on the entanglement wedge. In [20,22], the bitthread formulation of the holographic entanglement of purification (HEoP) was given, and many known properties of the HEoP were proved in this formulation. In this paper, however, we would like to go step further and to explore some new properties of HEoP by using bit threads.
According to the statement in [13], we can generalize the flow description of HEE into a meaning of relative homology, while the nesting property of flows goes through as before. Thus, we could carry these flow-based proofs of properties of HEE into homology case, which means we will obtain some corresponding properties in homology case. Remember that the flow description of HEoP in [20,22] is based on the notion of relative homology. We will naturally obtain some corresponding properties for the HEoP, 2 as 1 Alternatively, in [63] the authors interpret the RT surface as special Lagrangian cycles calibrated by the real part of the holomorphic one-form of a spacelike hypersurface. 2 Recently there is a work [39], where new inequalities of HEoP are obtained from HEE, based on the wormhole geometry description of HEoP. While we start from a flow viewpoint, and arrive at a similar conclusion.
These properties of HEoP obtained from bit threads do not follow from the known properties of EoP. In this paper, we will derive these inequalities by using multiflow description of HEoP defined in relative homology. A geometric analysis is also applied and the validity of these novel inequalities is further confirmed. This paper is organized as follows: In section 2, we will briefly review the notion of bit threads. Then in section 3, we will have a discussion about generalized HEE defined in relative homology. In section 4.1, we give a multiflow description for HEoP in relative homology. Then in section 4.2, considering tripartite and quadripartite cases, we will derive out some properties of HEoP by using multiflows defined in relative homology, as inequalities (1.8)-(1.12). We notice that each property for HEoP has a corresponding property of HEE in (1.2)-(1.6). A concluding remark is given in the last section.

Flow
The bit threads [12], which are a set of integral curves of a divergenceless norm-bounded vector field v with transverse density equal to |v|. The threads of a given vector field are oriented and locally parallel. Consider a manifold M with a conformal boundary ∂M , where A is a subregion on ∂M and its complement isĀ := ∂M \A. Define a flow v AĀ with direction from A toĀ on M , that is divergenceless and is norm bounded by 1/4G N : As we set the direction for v AĀ flowing from A toĀ, it means its flux given by A v AĀ is non-negative: where h is the determinant of the induced metric h ij on A and n µ is the (inward-pointing) unit normal vector. Then the entanglement entropy between A andĀ is suggested to given by the maximum flux through A among all flows: Equivalence between (2.3) and the RT formula (1.1) is guaranteed by the Riemannian MFMC theorem [12]: The left-hand side is a maximum of the flux over all flows v, while the right-hand side takes a minimum of the area over all surfaces m homologous to A (written as m ∼ A). The flow interpretation of the holographic entanglement entropy, unlike the minimal surface captured by RT formula jumping under continuous deformations of region A [64][65][66][67], varies continuously. And the subadditivity and the strong subadditivity inequalities of HEE can be proved by making use of the properties of flows [12].
A v AĀ m AĀ Figure 1. The vector field v AĀ defined on manifold M . The entanglement entropy S A is equal to the maximum flux from A toĀ, or equivalently the maximum number of threads connecting A withĀ.

Threads
In [16], the notion of bit threads was generalized. Instead of being oriented and locally parallel, threads are unoriented and can even intersect with others. The notion of transverse density is replaced by density, defined at a given point on a manifold M as the total length of the threads in a ball of radius R centered on that point divided by the volume of the ball, where R is chosen to be much larger than the Planck scale G 1/(d−1) N and much smaller than the curvature scale of M . In the classical or large-N limit G N → 0, we can neglect the discretization error between the continuous flow v and the discrete set of threads as the density of threads is large on the scale of M .
For region A and its complementĀ on the boundary ∂M . Defining a vector field v AĀ , we can construct a thread configuration by choosing a set of integral curves with density |v AĀ |. The number of threads N AĀ connecting A toĀ is at least as large as the flux of v AĀ on A: (2.5) Generally, this inequality does not saturate as some of the integral curves may go fromĀ to A which have negative contributions to the flux but positive ones to N AĀ . Consider a slab R around m, where R is much larger than the Planck length and much smaller than the curvature radius of M . The volume of this slab is R·area(m), the total length of the threads within the slab should be bounded above by R · area(m)/4G N . Moreover, any thread connecting A toĀ must have a length within the slab at least R. Therefore, we have Particularly, for the minimal surface m A , we have Combining formulas (2.5) and (2.7), equality holds Thus, S(A) is equal to the maximum number of threads connecting A toĀ over all allowed thread configurations.

Multiflow
The multiflow or multicommodity is the terminology in the network context [68,69]. It is a collection of flows that are compatible with each other, existing on the same geometry simultaneously. It was defined in Riemannian setting to prove the monogamy of mutual information (MMI) in [16]. Consider a Riemannian manifold M with boundary ∂M , and let A 1 , . . . , A n be non-overlapping regions of ∂M , a multiflow is then defined as a set of vector fields {v ij } on M satisfying the following conditions: There are n(n − 1)/2 independent vector fields for the given condition (2.9). Given condition (2.10), v ij is nonvanishing only on A i and A j , by (2.11), their flux satisfy 14) The flux of flow v iī should be bounded above by the entropy of A i : The inequality will saturate for a max flow. Given v ij (i < j), we can choose a set of threads with density |v ij |. From (2.5), the number of threads connect A i to A j is at least the flux of v ij : On the other hand, (2.7) implies that the total number of threads emerging out of A i is bounded above by S(A i ): Therefore, both inequalities (2.17) and (2.18) saturate for a max flow with fixed i: Furthermore, the inequality (2.16) must be individually saturated: (2.20) The above discussion focuses only on the case for a fixed i. Remarkably, it has been proved in [16] that there exists a so-called max multiflow {v ij } saturating all n bounds in (2.15) simultaneously. Or equivalently, there exists a so-called max thread configuration in the language of threads, as the formula (2.19) holding for all i.

Relative homology and generalized HEE
The notion of relative homology was introduced to generalize the MFMC theorem in [13]. To get an intuition for the generalized MFMC (gMFMC) theorem, we consider a manifold M with a conformal boundary ∂M and A is a subregion on the boundary. Specifically, let R be a bulk surface attached to the boundary, for instance as Figure 2. For region A, we can define the surfacem homologous to A relative to R (written asm ∼ A rel R), where surfacẽ m is allowed to begin and end on R. 3 On flow side, this corresponds to imposing a Neumann condition (no-flux condition) on R, thereby This means no flux through R. Therefore, the flow is restricted within region M with boundary ∂M = A ∪ B ∪ C ∪ D ∪ R. Finally, we can arrive at the gMFMC theorem wherem A is the minimal surface homologous to A relative to R. The flow description of HEE in [12] is specifically based on the homology relative to R = ∂A. As proposed in [13], we will have a generalized HEES This notion originates from HEE, but defined in a meaning of general relative homology. We stress that, differing from HEE, this quantity has no clear quantum information meaning generally in holography. However, there is another quantity with the form (3.3) arousing our interests in holography, i.e. HEoP. Similar to HEE, the HEoP can also be regarded as a special case in (3.3) with the homology relative to R = m AB [20,22]. The formula (3.2) establishes the equivalence between the flow objects and geometric objects in holography. We have learned that we could derive the properties of holographic objects from the properties of flows, such as nesting of flows. Remember the flow-based proofs of the Araki-Lieb (AL) inequality, the subadditivity (SA) and the strong subadditivity (SSA) for the HEE [12], also multiflow-based proof of the monogamy of the mutual information (MMI) [16,17]. Note that when the min cut is defined in homology relative to boundary subregion R, the dual flow (or multiflow) is subjected to a no-flux boundary condition on R. The nesting property of flows in relative homology goes through as before [13]. Thus these flow-based proofs could be carried into the relative homology cases in like manner. Namely, based on the flow viewpoints, we argue that these properties will hold for quantityS generally (including HEoP case), as  It seems that these inequalities could be intuitively obtained geometrically by making use of the minimality of relative homology surface. For example, as AB ∼m A ∪m B ∼m AB rel R, due to the minimality ofm AB among all surfaces homologous to AB relative to R, we naturally obtain |m AB | ≤ |m A | + |m B |. As to A ∼m A ∼m AB ∪m B rel R, due to the minimality of m A among all surfaces homologous to A relative to R, we have |m A | ≤ |m AB | + |m B |. In the following, we will focus on the multiflow-based proofs of these inequalities for HEoP.

Multiflow description of HEoP as relative homology
The bit thread formulation of the HEoP has been given in [20,22], which is based on the gMFMC theorem. In this section, instead, we would like to apply the notion of multiflow defined in relative homology, to give a multiflow description for HEoP. Taking a manifold M with non-overlapping regions A 1 , . . . , A n on boundary ∂M . Consider the entanglement wedge We could define a multiflow {v ij } on the entanglement wedge, subject to a Neumann boundary condition on m A 1 A 2 ...An . Namely, This means no flux through m A 1 A 2 ...An or no threads connecting to m A 1 A 2 ...An . In this way, we restrict the multiflow inside geometry r A 1 A 2 ...An . All following discussions will be based on such a multiflow configuration.
We set the direction of v ij as a flow from A i to A j , which means the flux A i v ij out of A i (inward-pointing on A i ) is non-negative: where h is the determinant of the induced metric on A and n is chosen to be a (inward-pointing) unit normal vector. Given condition (4.2) and (4.3) the fact that v ij is non-vanishing only on A i and A j , combining (4.1) and (4.4), we get Similarly as before, we can define a minimal cutm A i on r A 1 A 2 ...An homologous to region A i relative to m A 1 A 2 ...An , which is exactly the minimal cross section σ min A iĀi Here, the quantityS is just E P as proposed in [20,22]. The dual flow is defined as (4.8) The flux of flow v iī should be bounded above by the area of minimal cross section σ min A iĀi . Combining with the conjecture of E P = E W , we have In addition, the number of threads connect A i to A j is at least the flux of v ij : Let us consider a max multiflow (or equivalently max thread configuration), as introduced in section 2.3, the inequality in (4.9) saturates simultaneously for all i, thus n j =i Furthermore, all inequalities (4.10) individually saturated: The bipartite case has been shown in [20,22]. In what follows, we will study such a max multiflow configuration (or equivalently max thread configuration) for tripartite and quadripartite cases, to derive out some new inequalities of the HEoP. As we will show that for each property (considering only AL, SA, SSA, MMI) of HEE that can be derived from bit threads, there is a corresponding property for HEoP.

Tripartite case
Consider a max multifow (or max thread configuration) for tripartite case with completely connected phase 4 , as shown in figure 3. By (4.11) and (4.12), we have (4.13) 4 Note that there are two other disconnected phases of region rABC . Subject to the no-flux condition on mABC , the disconnected phase means no-flow (or no-thread) passing between disconnected regions. Here, we just focus on completely connected case. Similar consideration is adopted for quadripartite case.  (4.15) The first inequality can also be obtained from the second inequality by alternating labels A, B and C. The inequalities (4.15) exactly correspond to the inequalities (3.4) for HEoP case, i.e. the AL inequality and the SA. This property of HEoP has already been derived in [20,39]. It does not follow by the known properties of EoP. Moreover, this can also be intuitively obtained from geometric side by comparing with the cuts defined in relative homology, as shown in figure 3. As AB ∼ σ min A(BC) ∪ σ min B(AC) ∼ σ min (AB)C ∼ C rel m ABC , due to the minimality of σ min (AB)C among all surfaces homologous to AB relative to m ABC , we naturally obtain the area relation |σ min

Quadripartite case
Consider a max multifow (or max thread configuration) for quadripartite case as sketched in figure 4. From (4.11) and (4.12), we have Noting that for a max multifow configuration, the flux through the union regions AB, AC and BC (or equivalently the number of threads emerging from these union regions) cannot reach its maximum value in general. Thus    Here we obtain the inequalities (4.18), (4.19) and (4.20) by multiflows, which respectively correspond to (3.5), (3.6) and (3.7) for HEoP case. As far as we know, these properties of the HEoP are also new, and are out of the known properties of the EoP. It is worth to explore whether these properties are held by EoP for generic quantum states, or only valid for the holographic quantum states that have classical gravity duality.

Conclusion
In this paper, we obtain some new properties of the HEoP that corresponds, respectively, to AL, SA, SSA and MMI by utilizing the multiflow description of the HEoP. Thus, starting from flow viewpoints, we arrive at a similar conclusion as in [39], and we give the flow-based derivations of these new inequalities. Intuitively, these inequalities can also be obtained from geometric side by comparing with the cuts defined in relative homology for tripartite and quadripartite cases. But usually it could be more complicated and less obvious to find the inequalities of HEoP hiding in more partite cases from geometric side. Moreover, as argued in section 3, a possible conjecture is that: For each property of HEE that can be derived from bit threads, there is a corresponding property for HEoP that can be obtained from bit threads defined in relative homology. We explain that all the flowbased proofs of the properties of HEE, can be carried into the relative homology cases in a similar manner, thus we could finally obtain some corresponding properties for HEoP. This is remarkable if it is true. We leave the proof of this conjecture as a challenge for the future.