Bit threads and holographic entanglement of purification

The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed states, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a `bit thread' formulation for the EoP. With this formulation, we give the proofs of some known properties of EoP. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to the entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states in flows. We also derive a new lower bound for $S(AB)$, which is tighter than the one given by the Araki-Lieb inequality.


I. INTRODUCTION
Increasing evidence shows that the quantum entanglement plays an important role in the holographic descriptions of gravity [1][2][3][4][5][6][7][8][9]. In quantum entanglement, there is a key quantity, the entanglement entropy (EE), which measures how much a subsystem entangles with its complement for a pure state. According to the Ryu-Takayanagi (RT) formula [10,11] for static cases, the entanglement entropy which characterizes quantum entanglement between a given spatial region A and its complement in the boundary conformal field theory (CFT), is given by where m A is a minimal surface in the bulk homologous to A. For time-dependent cases, we just replace the minimal surface at constant time with a spacetime codimension-two extremal surface homologous to A (in what follows we will focus on the static case). As a consequence, the RT formula provides a direct evidence of the potential relations between entanglement and holography.
It hence attracts a lot of attention in the past few years.
However, there are several conceptual puzzles surrounding the RT formula. The "bit thread" formulation firstly proposed by Freedman and Headrick, provides a way to clarify these puzzles [7] (see further studies in [12][13][14][15]). The "bit thread" formulation demonstrates that the geometric extremization problem can be interpreted as a flow extremization problem. By using the Riemannian version of the max flow-min cut(MFMC) theorem, the maximum flux out of a boundary region A, optimized over all divergenceless norm-bounded vector fields in the bulk, is exactly the area of m A .
By rewriting the RT formula in terms of flows, the entanglement entropy of a boundary region can be given by the maximum flux out of it of any flow.
On the other hand, the entanglement of purification (EoP) firstly introduced in [16], a quantity which measures classical correlations and quantum entanglement for mixed states in quantum information theory, has been conjectured in [17,18] to be dual to the area of the minimal cross section of the entanglement wedge [19][20][21] holographically. Further studies could be seen in . For two non-overlapping subregions A and B on the conformal boundary, the conjecture says that where σ min AB is the minimal cross section on the entanglement wedge that is dual to ρ AB . It has been shown that the EoP also admits a bit-thread interpretation in [15]. By contrast, we use the surface-state correspondence [3,4], to give a "bit thread" formulation for the EoP by using a generalization of Riemannian MFMC theorem [43]. Following the surface-state correspondence, we restrict the bulk region to the entanglement wedge, then we define a divergenceless norm-bounded vector field on the entanglement wedge. The EoP is suggested to be given by the maximum flux of any flows through the neck of its entanglement wedge, or the maximum number of threads connecting two boundary regions through its entanglement wedge. Then the conjecture of holographic entanglement of purification is guaranteed by the generalization of Riemannian MFMC theorem.
Moreover, recalling that there is a quantity, the quantum advantage of dense code (QAoDC) [44], which reflects the increase in the rate of classical information transmission though quantum channel due to shared entanglement. It turns out that the QAoDC also admits a flow interpretation. This paper is organized as follows: In section 2, we will have a brief review of the bit threads in [7,13] and the conjecture of E P = E W [17,18]. In section 3, for part A, we start from the surface-state correspondence [3,4] to have an intuitive understanding of the purification process, and restrict the bulk region to the entanglement wedge as proposed in [38,39]. For part B, we will have a brief review of the generalization of Riemannian MFMC theorem [43]. For part C, we will define a divergenceless norm-bounded vector field on the entanglement wedge, and propose that the EoP is given by the maximum flux of any flows flowing through the entanglement wedge. By using a generalization of Riemannian MFMC theorem, its flux is precisely the area of the minimal cross section of the entanglement wedge. As examples, we consider the mixed bipartite and tripartite state cases, and give the proofs of some basic properties of EoP in flows. We find the flow interpretation of the QAoDC and we prove the monogamy relation [44] of QAoDC with the EoP for any tripartite states in terms of flows. We derive a new lower bound for S(AB) which is tighter than the one given by the Araki-Lieb inequality.

II. REVIEW
A. Brief review of bit threads

Flows
The bit threads was first introduced in [7], which is a set of integral curves of a divergenceless norm-bounded vector field v chosen so that their transverse density equals |v| . The entanglement entropy of a boundary region is given by the maximum flux out of it, or equivalently the maximum number of bit threads that emanate from it.
To explain this, we consider a manifold M with boundary ∂M . Let A be a subregion of ∂M .
Let's define a flow from region A to its complementĀ := ∂M \A, which is a vector field v AĀ on M that is divergenceless and is norm bounded everywhere by 1/4G N : The flux of flow v AĀ through a boundary region A is given by A v AĀ : where h is the determinant of the induced metric h ij on A andn is the (inward-pointing) unit normal vector. The entanglement entropy between A andĀ is given as the flux of a max flow through A: Equivalence between (5) and the RT formula (1) is guaranteed by the MFMC of Riemannian version [7]: The left hand side is a maximum of the flux over all flows v AĀ , while the right hand side takes a minimum of the area over all surfaces m homologous to A (denoted as m ∼ A). One of the best features of this flow interpretation of the holographic entanglement entropy is that, unlike the minimal surface jumping under continuous deformations of region A [46][47][48][49], the threads do not jump. In [7], it shows that the subadditivity and strong subadditivity inequalities can be proved by making use of the formula (5).

Threads
In [13], the notion of bit threads was generalized by dropping the oriented and locally parallel conditions. As a consequence, 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 . Threads are unoriented and can even intersect with others, as long as the thread density is bounded above by 1/4G N . Given a flow v, we can choose threads as a set of integral curves whose density equals |v| everywhere. In the classical or large-N limit G N → 0, the density of threads is large on the scale of M and we can neglect the discretization error between the continuous flow v and the discrete set of threads.
For region A and its complementĀ on the boundary of manifold M . Define 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: 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 length within the slab at least R. Therefore, we have Combining formulas (7) and (9), an equality holds whereṽ AĀ denotes a max flow. Thus, S(A) is equal to the maximum number of threads connecting A toĀ over all allowed configurations: Each thread connects an EPR pair living on the boundary. In the language of entanglement distillation, the entanglement between A andĀ is distilled into a number of EPR pairs equal to S(A). Thus the maximal number of threads connecting A toĀ can be interpreted as the maximal number of EPR pairs that could be distilled out by means of the local operations and classical communication (LOCC) asymptotically.

Multiflow
The multiflow or multicommodity, is the terminology in network context [50,51]. It is a collection of flows that are compatible with each other, so they can simultaneously exist on the same geometry.
In [13], the multiflow has been defined in Riemannian setting to prove the monogamy of mutual information (MMI). Taking a Riemannian manifold M with boundary ∂M . Let A 1 , . . . , A n be nonoverlapping regions of ∂M . A multiflow is a set of vector fields v ij on M satisfying the following There are n(n − 1)/2 independent vector fields for given condition (12). Given condition (13), v ij is nonvanishing on A i and A j , by (14), their flux satisfy Define a vector field The inequality will saturate for a max flow v iī . Given v ij (i < j), we can choose a set of threads with density |v ij |. From (7), the number of threads connect A i to A j is at least the flux of v ij : On the other hand, (9) implies that the total number of threads emerging out of A i is bounded above Therefore, both inequalities (20) and (21) saturate for a max flow denoted asṽ iī with fixed i: Furthermore the inequality (19) must be individually saturated for fixed i: The above discussion focus only on the case for a fixed i. Actually, it was proved in [13] that there is a max multiflow {v ij } saturating all n bounds in (18) simultaneously. This immediately gives us a proof of MMI in terms of a max multiflow.

B. Brief review of holographic entanglement of purification
The entanglement of purification E P firstly introduced in [16], as a measure of bipartite correlations in a mixed state, is defined as follows. Let ρ AB be a density matrix on a bipartite system Let |ψ ∈ H AA ⊗ H BB be a purification of ρ AB , so that Tr A B |ψ ψ| = ρ AB . E P of ρ AB is then given by where S AA is the von Neumann entropy of the reduced density matrix obtained by tracing out the BB part of |ψ ψ|, and we minimize the entropy over all ψ and all ways of partitioning the purification into A B . For pure states ρ AB , the quantity E p is reduced to entanglement entropy found in [45].
In [17,18], it has conjectured that the entanglement of purification E P is dual to the entanglement wedge cross section E W , as E P = E W , in the sense that it obeys a same set of inequalities as E P does.
To define the entanglement wedge cross section, we consider a static classical dual gravity and wedge cross section E W is then given by which is proportional to the area of the minimal cross section σ min AB . The cross section splits r AB into two regions. One is bounded by A but not B, and the other by B but not A. Let m A , m B and m AB denote extremal surfaces respectively, then σ min AB splits r AB = r . 1).

III. BIT THREADS AND HOLOGRAPHIC ENTANGLEMENT OF PURIFICATION
A. Holographic entanglement of purification from surface-state correspondence To have an intuitive understanding of the purification process in holography, a recent proposed surface-state correspondence [3,4] is helpful. This is a conjectured duality between codimensiontwo space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. For a given surface with fixed boundaries, one can make a smooth deformation. In order to preserve the convexity, the deformation must be terminated when the deformed surface becomes extremal.
As an example, let us consider a mixed bipartite state ρ AB on Hilbert space which is dual to two disconnected open convex surfaces, Σ A and Σ B . To perform purification, it is useful to introduce an auxiliary system R, which is dual to an open convex surface Σ R sharing the same boundaries with Σ A and Σ B . We can then purify the mixed state ρ AB to a pure state |Ψ ABR , which is dual to a closed convex surface Σ ABR . For simplicity, we focus on the case where the Σ ABR is topologically trivial. From the definition (24), we know that the purification of a given state ρ AB is not unique. Actually, there are infinite ways. Each of them can be obtained by performing a suitable unitary transformation on the initial auxiliary system R.
Holographically, we can make a smooth unitary deformation on initial Σ R to pull it into the bulk, preserving convexity of region surrounded by Σ ABR . This deformation must terminates when it becomes an extremal surface m AB , such that the convexity can be preserved. Note that the deformation of Σ R with fixed boundaries acts non-trivially only on the quantum entanglement inside Σ R , without nontrivial action on the entanglement between Σ R and Σ AB . In other words, the unitary operation does not change the entanglement between Σ R and Σ AB . As we pull Σ R into the bulk, the inner entanglement is decreasing, and finally vanishes when it reaches the extremal surface m AB .
It gives us a special purification when the auxiliary surface Σ R reaches the extremal surface m AB .
We use R to denote the axillary system at this stage. Due to the vanishing entanglement in R , the auxiliary R is maximally entangled with ρ AB , thus ln dim(R ) = S(AB). It has the minimal possible Hilbert space dimension to purify ρ AB . In other words, all degrees of freedom of R are entangled with AB and there are no more remanent degrees of freedom inside it. In this sense, we say this is a minimal entanglement purification |Ψ ABR with minimal Hilbert dimension [38,39]. In a word, the holographic minimal entanglement purification of ρ AB is dual to a closed convex surface, which is a boundary of a boundary geometric density matrix ρ AB , as proposed in [38,39]. To get holographic entanglement entropy of purification, let us divide R into two parts, A ∪ B . Then an optimal purification |Ψ AA opti BB opti can be achieved by minimizing S(AA ) over all divisions and choosing an optimal division R = A opti ∪ B opti . It turns out that the RT surface of AA opti is exactly the σ min AB on r AB . Therefore, the minimal entropy S(AA opti ), as the maximal flux of any flow from AA opti to BB opti , is bounded above by the area of σ min AB . In this way E P (A : B) can be obtained. Assuming the surface-state correspondence, the minimal entanglement purification is a pure state living on the boundary of entanglement wedge r AB , as we have already pulled the initial boundary onto the boundary of the entanglement wedge by performing unitary transformation on auxiliary R.
In asymptotic case, we have E P = E ∞ P = E LOq . The definitions of these quantities could be found in [16], and E LOq is roughly equal to the number of EPR pairs needed to create the state ρ AB by means of LOCC. Also it has been proposed in [15] that E P could be related to the maximal number of EPR pairs which can be distilled from ρ AB using only LOCC. Recalling the interpretation of the threads that each thread connects an EPR pair on the boundary, it is natural to consider the flux of a max flow from A to B or the maximum number of threads connecting A to B on the geometry of r AB , whose value is bounded above by the area of σ min AB . Combining the assumption of E P = E W , a "bit-thread" interpretations of entanglement of purification can be achieved.
where we have imposed a Neumann boundary condition of the flow v AB on boundary region C. In this way, we restrict the flow v AB in the bulk region M , flowing between region A and B. It means that any thread emanating from the region A must end on the region B. Obviously, the flux of the maximizing flow A → B should be bounded above by the area of the neck, the minimal cross section σ min AB separating A and B on M : where σ AB is homologous to A relative to C. The generalized Riemannian MFMC theorem [43] says that the flux of maximizing flow v AB will equal to the area of the minimal cross section σ min AB :

C. Bit threads and holographic entanglement of purification
Taking the Riemannian manifold M as a time slice of a static bulk spacetime. A and B are two non-overlapping subregions of conformal boundary ∂M , and C is the complement of AB. As discussed before, we will follow the surface-state correspondence. By performing a unitary transformation on the initial purification state |Ψ ABC , we will finally get a minimal entanglement purification |Ψ ABC of ρ AB that calculates the EoP, as sketched in FIG.3. Now let us define a vector field v AB on r AB , satisfying n · v AB = 0 on m AB .
We can set the direction of v AB as a flow from A to B (A → B), which means the flux A v AB of v AB out of A (inward-pointing on A) is non-negative: where h is the determinant of the induced metric on A andn is chosen to be a (inward-pointing) unit normal vector. Given condition (32) and the fact that v AB is non-vanishing on A and B, combining (29) and (31), we get The flow is restrained inside the entanglement wedge r AB by imposing the Neumann boundary condition (no-flux condition) on the surface m AB where C is living on. Use the generalized Riemannian

MFMC theorem introduced in previous section
where σ AB , a cross section separating A and B on r AB , is homologous to A (or B) relative to C . In this way, we show that the EoP is given by the maximum flux of any flow v AB from A to B inside the entanglement wedge r AB . Thus E P (A : B) can be written as So the formula (36) is equivalent to E P = E W , guaranteed by the generalized Riemannian MFMC theorem as shown in formula (35). We choose a set of threads with density |v AB | for the vector field v AB . From (19), the number of threads connecting A to B is at least the flux of v AB : However, the number of threads connecting A and B is bounded above by the area of σ min AB : Thus for a max flow A → B, denoted asṽ AB , the following equality holds This implies that E P (A : B) is equal to the maximal number of threads connecting A to B over all allowed thread configurations on r AB : Actually, N max AB can be interpreted as the maximal number of EPR pairs which can be distilled from ρ AB by using only LOCC as interpreted in [15].

Mixed bipartite state
In this subsection, as a warmup, we focus on a simple case: the mixed bipartite state ρ AB . To compute the EoP of a given mixed bipartite state ρ AB in terms of flows, we need firstly find the minimal purification. As shown in [39], the holographic minimal entanglement purification is the state living on the boundary of entanglement wedge r AB . Following the surface-state correspondence, we define a vector field on the geometric bulk of entanglement wedge r AB . In (36) we have shown that E P (A : B) is given by the maximum flux of any flow A → B within the bulk of r AB . The minimal surface homologous to A (or B) relative to C is by definition the σ min AB on r AB , which is not equal to m A in general 1 .
Noting that S(A) is given by the total maximum flux out of A, we suppose that E P (A : B) is given by the part of maximum flux A → B. The rest flux A → C , as we will show in the following, can be interpreted as the QAoDC [44]. Choosing a flowṽ A(B,C ) that simultaneously maximizes the flux A →Ā and the flux A → B, a vector fieldṽ A(B,C ) on r AB , satisfying Recalling the definition of the QAoDC in [44]: The QAoDC reflects the increase in the rate of classical information transmission due to shared entanglement.
It was proved in [44] that the QAoDC is non-negative, and it was shown that the QAoDC obeys a monogamy relation with the entanglement of purification for any tripartite state ρ ABC , i.e., Comparing with (42), we can write the QAoDC as We can interpret the QAoDC as the minimal flux A → C 2 or minimal number of threads A → C , as we have maximized the flux A → B (where the total flux A → BC reaches its maximum and is a fixed value S(A)). As mentioned before, the QAoDC may relate to the minimal EPR pairs that can be distilled from ρ AC using only LOCC. Its non-negative property means that the EoP is bounded above by the EE. For pure tripartite state: the difference between the maximum and the minimum of the flux A → C implies that it is nonnegative (the total flux from A into BC is fixed). Combining with ( 2 More specifically, note that the minimal cross section σ min AB divided C into A opti and B opti . When we take a flow v A(B,C ) , its fluxes or equivalently threads through the σ min AB have achieved the maximum. So the other threads emerging out of A can only end on A opti , which represent the entanglement between A and A opti while we interpret it as the QAoDC. As a consequence, we may write ∆(C > A) more precisely as ∆(A opti > A). But in this context, the symbol ∆(C > A) is enough. We do not need to differentiate them.
For pure tripartite state, we also have S(A) = ∆(BC > A). This can be used to show that the QAoDC obeys a monogamy relation: Furthermore, using (45), we can derive a new lower bound for S(AB): where the relation (34) is used. We explain that only the flow between AB and C has non-vanishing contribution to the integral on AB, while the flow between A and B does not have. However, the flux of any flow between AB and C is bounded by the area of minimal surface that separates AB and C , which is exactly the extremal surface m AB for the geometry r AB . So its flux can not exceed S(AB). Note that Subtracting (50) Comparing (49) where the formula (36) is used. For region B (as well as for region C) in the wedge r ABC , the part flux A → B is bounded by the area of minimal cross section σ min B(AC) separating B and AC. Thus, from (35) and (36), the flux of any flow A → B (or C) cannot exceed E P (B : AC) (or E P (C : AB)).
In this way, we get an inequality (54) for EoP in terms of flows, which was already derived in [38] in a different way.
As a second application, let us take a flowṽ A(C,BD ) that simultaneously maximizes the flux A →Ā and the flux A → C. We can get the monotonic property that EoP never increases upon discarding a subsystem for mixed tripartite state: Here we prove the monogamy relation of QAoDC with the EoP for tripartite state in terms of flows.

IV. CONCLUSION
In this paper, we show that entanglement of purification has a "bit thread" interpretation, with the help of recent proposed surface-state correspondence and conjecture of E P = E W .
We propose that the EoP is given by the maximum flux of two given regions on their entanglement wedge. By using the nesting property, we can choose a flow on the entanglement wedge, which allows us to compute EE and EoP simultaneously. We give a flow interpretation for the QAoDC that is proved to have a monogamy relation with the EoP for any tripartite state. We study some inequalities relations about EE, EoP and QAoDC in terms of flows. It has been showed in this paper that we can prove the known properties of EoP and QAoDC in terms of flows, and also we derive some new properties for them. In this picture, it's easy to show the monogamy relation of QAoDC with the EoP for tripartite states. Moreover, we derive a new lower bound for S(AB). This is a tighter bound than the one given by the Araki-Lieb inequality.
Here we only consider the mixed bipartite, tripartite state cases. For the case with BTZ black hole, according to the surface/state correspondence, the whole conformal boundary is dual to a mixed state. However, if we include the black hole horizon, the total system is still a pure system. . The threads connecting A to B must cross the surface σ min AB , and can not end on surface C (including the horizon). Similarly for the vector fieldṽ B(A,C ) . Where we suppose that the horizon is a part of the auxiliary system C needed to purify system AB, and |Ψ ABC is a minimal entanglement purification of ρ AB . Therefore, it still admits a flow representation of E P (A : B), as long as we suppose that the horizon is a part of the auxiliary system C needed to purify the boundary subsystem AB. As shown before, E P (A : B) is given by the maximum value of flux or the maximum number of threads from A to B (or B to A equivalently), through the geometric bulk dual of ρ AB , and these threads connecting A to B can not end on the horizon.
In this paper, we suggest that the QAoDC may also have a holographic interpretation, but it is still far to understand its information-theoretic meaning in the context of holography. Moreover, it may also admit a flow representation of holographic conditional or multipartite entanglement of purification. As the thread picture provides us a simple way to relate the information-theoretic quantities with the holographic objects, it may help us prove some nontrivial properties of these holographic objects and give some inspirations about the relations between quantum information and holography.