Contents

Shawn X. Cui, Patrick Hayden, Temple He, Matthew Headrick, Bogdan Stoica, and Michael Walter

Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA

Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA

Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Department of Physics, Brown University, Providence, RI 02912, USA

Korteweg-de Vries Institute for Mathematics, Institute of Physics, Institute for Logic, Language & Computation, and QuSoft, University of Amsterdam, 1098 XG Amsterdam, The Netherlands

xingshan@stanford.edu, phayden@stanford.edu, the@g.harvard.edu, headrick@brandeis.edu, bstoica@brandeis.edu, m.walter@uva.nl

Bit threads provide an alternative description of holographic entanglement, replacing the Ryu-Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information (MMI) property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglement, for any decomposition of the boundary into four regions. We also give new proofs of analogous theorems on networks.

BRX-6330, Brown HET-1764, MIT-CTP/5036

1 Introduction

One of the most important relationships between holographic gravity and entanglement is the Ryu-Takayanagi (RT) formula, which states that entanglement entropy of a region in the boundary conformal field theory (CFT) is dual to a geometric extremization problem in the bulk [1, 2]. Specifically, the formula states that the entropy of a spatial region on the boundary CFT is given by

 S(A)=14GNarea(m(A)) , (1.1)

where is a minimal hypersurface in the bulk homologous to . This elegant formula is essentially an anti-de Sitter (AdS) cousin of the black hole entropy formula, but more importantly, it is expected to yield new insights toward how entanglement and quantum gravity are connected [3, 4].

Despite the fact that the RT formula has been a subject of intense research for over a decade, there are still many facets of it that are only now being discovered. Indeed, only recently was it demonstrated that the geometric extremization problem underlying the RT formula can alternatively be interpreted as a flow extremization problem [5, 6]. By utilizing the Riemannian version of the max flow-min cut theorem, it was shown that the maximum flux out of a boundary region , optimized over all divergenceless bounded vector fields in the bulk, is precisely the area of . Because this interpretation of the RT formula suggests that the vector field captures the maximum information flow out of region , the flow lines in the vector field became known as “bit threads.” These bit threads are a tangible geometric manifestation of the entanglement between and its complement.

Although bit threads paint an attractive picture that appears to capture more intuitively the information-theoretic meaning behind holographic entanglement entropy, there is still much not understood about them. They were used to provide alternative proofs of subadditivity and strong subadditivity in [5], but the proof of the monogamy of mutual information (MMI) remained elusive. MMI is an inequality which, unlike subadditivity and strong subadditivity, does not hold for general quantum states, but is obeyed for holographic systems in the semiclassical or large limit. It is given by

 −I3(A:B:C):= S(AB)+S(AC)+S(BC) (1.2) −S(A)−S(B)−S(C)−S(ABC)≥0 .

The quantity  is known as the (negative) tripartite information, and property (1.2) was proven in [7, 8] using minimal surfaces.111MMI was also proven in the covariant setting in [9]. While MMI is a general fact about holographic states, the reason for this from a more fundamental viewpoint is not clear. Presumably, such states take a special form which guarantees MMI (cf. [10, 11]). What is this form? It was suggested in [5] that understanding MMI from the viewpoint of bit threads may shed some light on this question.

In this paper we will take up these challenges. First, we will provide a proof of MMI based on bit threads. Specifically, we show that, given a decomposition of the boundary into regions, there exists a thread configuration that simultaneously maximizes the number of threads connecting each region to its complement. MMI follows essentially directly from this statement.222V. Hubeny has given a method to explicitly construct such a thread configuration, thereby establishing MMI, in certain cases [12]. This theorem is the continuum analogue of a well-known result in the theory of multicommodity flows on networks. However, the standard network proof is discrete and combinatorial in nature and is not straightforwardly adapted to the continuum. Therefore, we develop a new method of proof based on strong duality of convex programs. Convex optimization proofs have the advantage that they work in essentially the same way on graphs and Riemannian manifolds, whereas the graph proofs standard in the literature often rely on integer edge capacities, combinatorics, and other discrete features, and do not readily translate over to the continuous case.333Conversely, when additional structure is present, such as integer capacity edges in a graph, the statements that can be proven are often slightly stronger than what can be proven in the absence of such extra structure, e.g. by also obtaining results on the integrality of the flows. The convex optimization methods offer a unified point of view for both the graph and Riemannian geometry settings, and are a stand-alone mathematical result. As far as we know, these are the first results on multicommodity flows on Riemannian manifolds.

Second, we use the thread-based proof of MMI to motivate a particular entanglement structure for holographic states, which involves pairwise-entangled states together with a four-party state with perfect-tensor entanglement (cf. [10]). MMI is manifest in this ansatz, so if it is correct then it explains why holographic states obey MMI.

It has also been proven that holographic entropic inequalities exist for more than four boundary regions [13]. For example, MMI is part of a family of holographic entropic inequalities with dihedral symmetry in the boundary regions. These dihedral inequalities exist for any odd number of boundary regions, and for five regions other holographic inequalities are also known. However, the general structure of holographic inequalities for more than four boundary regions is currently not known. It would be interesting to try to understand these inequalities from the viewpoint of bit threads. In this paper, we make a tentative suggestion for the general structure of holographic states in terms of the extremal rays of the so-called holographic entropy cone.

We organize the paper in the following manner. In Section 2, we give the necessary background on holographic entanglement entropy, flows, bit threads, MMI, and related notions. In Section 3, we state the main theorem in this paper concerning the existence of a maximizing thread configuration on multiple regions and show that MMI follows from it. In Section 4, we use bit threads and the proof of MMI to motivate the conjecture mentioned above concerning the structure of holographic states. In Section 5, we prove our main theorem as well as a useful generalization of it. Section 6 revisits our continuum results in the graph theoretic setting, demonstrating how analogous arguments can be developed there. In Section 7 we discuss open issues.

2 Background

2.1 Ryu-Takayanagi formula and bit threads

We begin with some basic concepts and definitions concerning holographic entanglement entropies. In this paper, we work in the regime of validity of the Ryu-Takayanagi formula, namely a conformal field theory dual to Einstein gravity in a state represented by a classical spacetime with a time-reflection symmetry. The Cauchy slice invariant under the time reflection is a Riemannian manifold that we will call . We assume that a cutoff has been introduced “near” the conformal boundary so that is a compact manifold with boundary. Its boundary is the space where the field theory lives.

It is sometimes convenient to let the bulk be bounded also on black hole horizons, thereby representing a thermal mixed state of the field theory. However, for definiteness in this paper we will consider only pure states of the field theory, and correspondingly for us will not include any horizons.444 may have an “internal” boundary that does not carry entropy, such as an orbifold fixed plane or end-of-the-world brane. This is accounted for in the Ryu-Takayanagi formula (2.1) by defining the homology to be relative to , and in the max flow formula (2.4) by requiring the flow to satisfy a Neumann boundary condition along , and in the bit thread formula (2.4) by not allowing threads to end on . See [6] for a fuller discussion. While we will not explicitly refer to internal boundaries in the rest of this paper, all of our results are valid in the presence of such a boundary. This assumption is without loss of generality, since it is always possible to purify a thermal state by passing to the thermofield double, which is represented holographically by a two-sided black hole.

Let be a region of . The Ryu-Takayanagi formula [1, 2] then gives its entropy as the area of the minimal surface in homologous to (relative to ):

 S(A)=14GNminm∼Aarea(m) . (2.1)

(We could choose to work in units where , and this would simplify certain formulas, but it will be useful to maintain a clear distinction between the microscopic Planck scale and the macroscopic scale of , defined for example by its curvatures.) We will denote the minimal surface by 555The minimal surface is generically unique. In cases where it is not, we let denote any choice of minimal surface. and the corresponding homology region, whose boundary is , by . The homology region is sometimes called the “entanglement wedge”, although strictly speaking the entanglement wedge is the causal domain of the homology region.

2.1.1 Flows

The notion of bit threads was first explored in [5]. To explain them, we first define a flow, which is a vector field on that is divergenceless and has norm bounded everywhere by :

 ∇⋅v=0 ,|v|≤14GN . (2.2)

For simplicity we denote the flux of a flow through a boundary region by :

 ∫Av:=∫A√h^n⋅v , (2.3)

where is the determinant of the induced metric on and is the (inward-pointing) unit normal vector. The flow is called a max flow on if the flux of through is maximal among all flows. We can then write the entropy of as the flux through of a max flow:

 S(A)=maxv flow∫Av . (2.4)

The equivalence of (2.4) to the RT formula (2.1) is guaranteed by the Riemannian version of the max flow-min cut theorem [14, 15, 16, 17, 6]:

 maxv flow∫Av=14GNminm∼Aarea(m) . (2.5)

The theorem can be understood heuristically as follows: by its divergencelessness, has the same flux through every surface homologous to , and by the norm bound this flux is bounded above by its area. The strongest bound is given by the minimal surface, which thus acts as the bottleneck limiting the flow. The fact that this bound is tight is proven by writing the left- and right-hand sides of (2.5) in terms of convex programs and invoking strong duality to equate their solutions. (See [6] for an exposition of the proof.) While the minimal surface is typically unique, the maximizing flow is typically highly non-unique; on the minimal surface it equals times the unit normal vector, but away from the minimal surface it is underdetermined.

We can further rewrite (2.4) by thinking about the integral curves of a flow , in the same way that it is often useful to think about electric and magnetic field lines rather than the vector fields themselves. We can choose a set of integral curves whose transverse density equals everywhere. In [5] these curves were called bit threads.

1. Threads end only on .

2. The thread density is nowhere larger than .

A thread can be thought of as the continuum analogue of a “path” in a network, and a thread configuration is the analogue of a set of edge-disjoint paths, a central concept in the analysis of network flows.

We denote the number of threads connecting a region to its complement in a given configuration by . We will now show that the maximum value of over allowed configurations is :

 S(A)=maxNA¯A . (2.6)

First, we will show that is bounded above by the area of any surface divided by . Consider a slab of thickness around (where again is much larger than the Planck length and much smaller than the curvature radius of ); this has volume , so the total length of all the threads within the slab is bounded above by . On the other hand, any thread connecting to must pass through , and therefore must have length within the slab at least . So the total length within the slab of all threads connecting to is at least . Combining these two bounds gives

 NA¯A≤14GNarea(m) . (2.7)

In particular, for the minimal surface ,

 NA¯A≤14GNarea(m(A))=S(A) . (2.8)

Again, (2.8) applies to any thread configuration. On the other hand, as described above, given any flow we can construct a thread configuration by choosing a set of integral curves whose density equals everywhere. The number of threads connecting to is at least as large as the flux of on :

 NA¯A≥∫Av . (2.9)

The reason we don’t necessarily have equality is that some of the integral curves may go from to , thereby contributing negatively to the flux but positively to . Given (2.8), however, for a max flow this bound must be saturated:

 NA¯A=∫Av(A)=S(A) . (2.10)

The bit threads connecting to are vivid manifestations of the entanglement between and , as quantified by the entropy . This viewpoint gives an alternate interpretation to the RT formula that may in many situations be more intuitive. For example, given a spatial region on the boundary CFT, the minimal hypersurface homologous to does not necessarily vary continuously as varies: an infinitesimal perturbation of can result in the minimal hypersurface changing drastically, depending on the geometry of the bulk. Bit threads, on the other hand, vary continuously as a function of , even when the bottleneck surface jumps.

Heuristically, it is useful to visualize each bit thread as defining a “channel” that allows for one bit of (quantum) information to be communicated between different regions on the spatial boundary. The amount of information that can be communicated between two spatially separated boundary regions is then determined by the number of channels that the bulk geometry allows between the two regions. Importantly, whereas the maximizing bit thread configuration may change depending on the boundary region we choose, the set of all allowable configurations is completely determined by the geometry. The “channel” should be viewed as a metaphor, however, similar to how a Bell pair can be be viewed as enabling a channel in the context of teleportation. While it is known that Bell pairs can always be distilled at an optimal rate , we conjecture a more direct connection between bit threads and the entanglement structure of the the underlying holographic states, elaborated in Section 4.

2.1.3 Properties and derived quantities

Many interesting properties of entropies and quantities derived from them can be written naturally in terms of flows or threads. For example, let , be disjoint boundary regions, and let be a max flow for their union, so . Then we have, by (2.4)

 S(A)≥∫Av ,S(B)≥∫Bv , (2.11)

hence

 S(A)+S(B)≥∫Av+∫Bv=∫ABv=S(AB) , (2.12)

A useful property of flows is that there always exists a flow that simultaneously maximizes the flux through and (or and , but not in general and ). We call this the nesting property, and it is proven in [6]. Let be such a flow. We then obtain the following formula for the conditional information:

 H(B|A):=S(AB)−S(A)=∫ABv1−∫Av1=∫Bv1 . (2.13)

We can also write this quantity in terms of threads. Let be the complement of , and let , , be the number of threads connecting the different pairs of regions in the flow .777In addition to the threads connecting distinct boundary regions, there may be threads connecting a region to itself or simply forming a loop in the bulk. These will not play a role in our considerations. Using (2.10), we then have

 S(AB)=N1AC+N1BC ,S(A)=N1AC+N1AB . (2.14)

(Note that we don’t have a formula for in terms of these threads, since the configuration does not maximize the number connecting to its complement.) Hence

 H(B|A)=N1BC−N1AB . (2.15)

For the mutual information, we have

 I(A:B):=S(A)+S(B)−S(AB)=∫A(v1−v2)=N2AB+N2BC+N1AB−N1BC , (2.16)

where is a flow with maximum flux through and , and are the corresponding numbers of threads. In the next section, using the concept of a multiflow, we will write down a more concise formula for the mutual information in terms of threads (see (3.21)).

The nesting property also allows us to prove the strong subadditivity property, , where , , are disjoint regions. (Unlike in the previous paragraph, here is not necessarily the complement of , i.e.  does not necessarily cover .) Let be a flow that maximizes the flux through both and . Then

 S(AB)≥∫ABv ,S(BC)≥∫BCv , (2.17)

hence

 S(AB)+S(BC)≥∫ABv+∫BCv=∫Bv+∫ABCv=S(B)+S(ABC) . (2.18)

2.2 MMI, perfect tensors, and entropy cones

2.2.1 Mmi

Given three subsystems of a quantum system, the (negative) tripartite information is defined as the following linear combination of the subsystem entropies:888One can alternatively work with the quantity , defined as the negative of . However, when discussing holographic entanglement entropies, is more convenient since it is non-negative.

 −I3(A:B:C) :=S(AB)+S(BC)+S(AC)−S(A)−S(B)−S(C)−S(ABC) =I(A:BC)−I(A:B)−I(A:C) . (2.19)

The quantity  is manifestly symmetric under permuting . In fact it is even more symmetric than that; defining , it is symmetric under the full permutation group on . (Note that, by purity, , , and .) Since in this paper we will mostly be working with a fixed set of 4 parties, we will usually simply write , without arguments.

We note that  is sensitive only to fully four-party entanglement, in the following sense. If the state is unentangled between any party and the others, or between any two parties and the others, then vanishes. In a general quantum system, it can be either positive or negative. For example, in the four-party GHZ state,

 |ψ⟩ABCD=1√2(|0000⟩+|1111⟩) , (2.20)

it is negative: . On the other hand, for a four-party perfect-tensor state, is positive. A perfect-tensor state is a pure state on parties such that the reduced density matrix on any parties is maximally mixed. For four parties, this implies that all the one-party entropies are equal, and all the two-party entropies have twice that value [10]:

 S(A)=S(B)=S(C)=S(D)=S0 ,S(AB)=S(BC)=S(AC)=2S0 , (2.21)

where . Hence

 −I3=2S0>0 . (2.22)

In this paper, we will use the term perfect tensor (PT) somewhat loosely to denote a four-party pure state whose entropies take the form (2.21) for some , even if they are not maximal for the respective Hilbert spaces.

It was proven in [7, 8] that the entropies derived from the RT formula always obey the inequality

 −I3(A:B:C)≥0 , (2.23)

which is known as monogamy of mutual information (MMI). The proof involved cutting and pasting minimal surfaces. In this paper we will provide a proof of MMI based on flows or bit threads. Since a general state of a 4-party system does not obey MMI, classical states of holographic systems (i.e. those represented by classical spacetimes) must have a particular entanglement structure in order to always obey MMI. It is not known what that entanglement structure is, and another purpose of this paper is to address this question.

2.2.2 Entropy cones

A general four-party pure state has 7 independent entropies, namely the 4 one-party entropies , , , , together with 3 independent two-party entropies, e.g. , , and . This set of numbers defines an entropy vector in . There is a additive structure here because entropies add under the operation of combining states by the tensor product. In other words, if

 |ψ⟩ABCD=|ψ1⟩A1B1C1D1⊗|ψ2⟩A2B2C2D2 , (2.24)

with etc., then the entropy vector of is the sum of those of and . The inequalities that the entropies satisfy—non-negativity, subadditivity, and strong subadditivity—carve out a set of possible entropy vectors which (after taking the closure) is a convex polyhedral cone in , called the four-party quantum entropy cone. Holographic states satisfy MMI in addition to those inequalities, carving out a smaller cone, called the four-party holographic entropy cone [13]. It is a simple exercise in linear algebra to show that the six pairwise mutual informations together with also form a coordinate system (or dual basis) for :

 I(A:B) ,I(A:C) ,I(A:D) ,I(B:C) ,I(B:D) ,I(C:D) ,−I3 . (2.25)

For any point in the holographic entropy cone, these 7 quantities are non-negative—the mutual informations by subadditivity, and by MMI. In fact, the converse also holds. Since MMI and subadditivity imply strong subadditivity, any point in representing a set of putative entropies such that all 7 linear combinations (2.25) are non-negative also obeys all the other inequalities required of an entropy, and is therefore in the holographic entropy cone. In other words, using the 7 quantities (2.25) as coordinates, the holographic entropy cone consists precisely of the non-negative orthant in .

Any entropy vector such that exactly one of the 7 coordinates (2.25) is positive, with the rest vanishing, is an extremal vector of the holographic entropy cone; it lies on a 1-dimensional edge of that cone. Since the cone is 7-dimensional, any point in the cone can be written uniquely as a sum of 7 (or fewer) extremal vectors, one for each edge. States whose entropy vectors are extremal are readily constructed: a state with and all other quantities in (2.25) vanishing is necessarily of the form , and similarly for the other pairs; while a state with and all pairwise mutual informations vanishing is necessarily a PT. It is also possible to realize such states, and indeed arbitrary points in the holographic entropy cone, by holographic states [18, 13].

The extremal rays can also be represented as simple graphs with external vertices . For example, a graph with just a single edge connecting and with capacity gives an entropy vector with and all other coordinates in (2.25) vanishing. Similarly, a star graph with one internal vertex connected to all four external vertices and capacity on each edge gives an entropy vector with and all other coordinates in (2.25) vanishing. Since any vector in the holographic cone can be uniquely decomposed into extremal rays, it is reproduced by a (unique) “skeleton graph” consisting of the complete graph on with capacity on the edge connecting and and similarly for the other pairs, plus a star graph with capacity on each edge. This is shown in Fig. 1.

Let us briefly discuss the analogous situation for states on fewer or more than four parties. For a three-party pure state, there are only 3 independent entropies (since etc.), so the entropy vector lives in . Holographic states obey no extra entropy inequalities beyond those obeyed by any quantum state, namely non-negativity and subadditivity, so the holographic entropy cone is the same as the quantum entropy cone. A dual basis is provided by the three mutual informations,

 I(A:B) ,I(B:C) ,I(A:C) . (2.26)

There are 3 types of extremal vectors, given by two-party entangled pure states etc. Thus the skeleton graph for three parties is simply a triangle, shown in Fig. 1.

Given a decomposition of the boundary into four regions, we can merge two of the regions, say and , and thereby consider the same state as a three-party pure state. Under this merging, the four-party skeleton graph on the left side of Fig. 1 turns into the three-party one on the right side as follows. The star graph splits at the internal vertex to become two edges, an edge and a edge, each with capacity . The first merges with the and edges from the four-party complete graph to give an edge with total capacity

 −12I3+12I(A:C)+12I(A:D)=12I(A:CD) . (2.27)

Similarly for the edges. The edge remains unchanged and the edge is removed. This rearrangement will play a role in our considerations of Section 4.

The situation for five and more parties was studied in [13]. For five-party pure states, there are no new inequalities beyond MMI and the standard quantum ones (non-negativity, subadditivity, strong subadditivity). There are 20 extremal vectors of the holographic entropy cone, given by 10 two-party entangled pure states, 5 four-party PTs, and 5 six-party PTs with two of the parties merged (e.g. a PT on with ). Since the cone is only 15-dimensional, the decomposition of a generic point into a sum of extremal vectors is not unique, unlike for three or four parties. For six-party pure states, there are new inequalities; a complete list of inequalities was conjectured in [13], but it has not been proven. Notable is the fact that the extremal rays are no longer only made from perfect tensors; rather, new entanglement structures come into play. For more than six parties, some new inequalities are known but a complete list has not even been conjectured.

3 Multiflows and MMI

As we reviewed in Subsection 2.1.3, the subadditivity and strong subadditivity inequalities can be proved easily from the formula (2.4) for the entropy in terms of flows. Subadditivity follows more or less directly from the definition of a flow, while strong subadditivity requires the nesting property for flows (existence of a simultaneous max flow for and ). Holographic entanglement entropies also obey the MMI inequality (2.23), which was proven using minimal surfaces [7, 8]. Therefore it seems reasonable to expect MMI to admit a proof in terms of flows. However, it was shown in [5] that the nesting property alone is not powerful enough to prove MMI. Therefore, flows must obey some property beyond nesting. In this section we will state the necessary property and give a flow-based proof of MMI. The property is the existence of an object called a max multiflow. It is guaranteed by our Theorem 1, stated below and proved in Section 5.

3.1 Multiflows

It turns out that the property required to prove MMI concerns not a single flow, like the nesting property, but rather a collection of flows that are compatible with each other in the sense that they can simultaneously occupy the same geometry (we will make this precise below). In the network context, such a collection of flows is called a multicommodity flow, or multiflow, and there is a large literature about them. (See Section 6 for the network definition of a multiflow. Standard references are [19, 20]; two resources we have found useful are [21, 22].) We will adopt the same terminology for the Riemannian setting we are working in here. We thus start by defining a multiflow.

Definition 1 (Multiflow).

Given a Riemannian manifold with boundary , let be non-overlapping regions of (i.e. for , is codimension-1 or higher in ) covering (). A multiflow is a set of vector fields on satisfying the following conditions:

 vij =−vji (3.1) ^n⋅vij =0 on Ak(k≠i,j) (3.2) ∇⋅vij =0 (3.3) n∑i

Given condition (3.1), there are independent vector fields. Given condition (3.2), has nonvanishing flux only on the regions and , and, by (3.3), these fluxes obey

 ∫Aivij=−∫Ajvij . (3.5)

Given conditions (3.3) and (3.4), each is a flow by itself. However, an even stronger condition follows: any linear combination of the form

 v=n∑i

where the coefficients are constants in the interval , is divergenceless and, by the triangle inequality, obeys the norm bound , and is therefore also a flow.

Given a multiflow , we can define the vector fields

 vi:=n∑j=1vij , (3.7)

each of which, by the above argument, is itself a flow. Hence its flux on the region is bounded above by its entropy:

 ∫Aivi≤S(Ai) . (3.8)

The surprising statement is that the bounds (3.8) are collectively tight. In other words, there exists a multiflow saturating all bounds (3.8) simultaneously. We will call such a multiflow a max multiflow, and its existence is our Theorem 1:

Theorem 1 (Max multiflow).

There exists a multiflow such that for each , the sum

 vi:=n∑j=1vij (3.9)

is a max flow for , that is,

 ∫Aivi=S(Ai) . (3.10)

Theorem 1 is a continuum version of a well-known theorem on multiflows on graphs, first formulated in [23] (although a correct proof wasn’t given until [24, 25]). However, the original graph-theoretic proof is discrete and combinatorial in nature and not easily adaptable to the continuum. Therefore, in Section 5 we will give a continuum proof based on techniques from convex optimization. (This proof can be adapted back to the graph setting to give a proof there that is new as far as we know. We refer the reader to Section 6.)

MMI is an almost immediate corollary of Theorem 1.999While one may be tempted to similarly apply this theorem to -party pure states for to potentially prove other holographic inequalities, such efforts have not been successful to date (but see footnote 14 on p. 14). We set ; to match the conventional labelling, we define , , , . Given a max multiflow , we construct the following flows:

The second equality in each line follows from the condition (3.1) and definition (3.9). Each is of the form (3.6) and is therefore a flow, so its flux through any boundary region is bounded above by the entropy of that region. In particular,

 S(AB)≥∫ABu1 ,S(AC)≥∫ACu2 ,S(BC)≥∫BCu3 . (3.12)

Summing these three inequalites and using (3.10) leads directly to MMI:

 S(AB)+S(AC)+S(BC)≥∫A(u1+u2)+∫B(u1+u3)+∫C(u2+u3)=∫AvA+∫BvB+∫CvC−∫ABCvD=S(A)+S(B)+S(C)+S(D) . (3.13)

The difference between the left- and right-hand sides of (3.13) is , so (unless happens to vanish) it is not possible for all of the inequalities (3.12) to be saturated for a given multiflow. However, Theorem 2, proved in Subsection 5.2, shows as a special case that any one of the inequalities (3.12) can be saturated. For example, there exists a max multiflow such that

 ∫ABu1=S(AB) . (3.14)

3.2.1 Theorem 1

We can also frame multiflows, Theorem 1, and the proof of MMI in the language of bit threads. The concept of a multiflow is very natural from the viewpoint of the bit threads, since the whole set of flows can be represented by a single thread configuration. Indeed, for each () we can choose a set of threads with density equal to ; given (3.2), these end only on or . By (2.9), the number that connect to is at least the flux of :

 NAiAj≥∫Aivij . (3.15)

Since the density of a union of sets of threads is the sum of their respective densities, by (3.4) the union of these configurations over all is itself an allowed thread configuration. Note that this configuration may contain, in any given neighborhood, threads that are not parallel to each other, and even that intersect each other.

Now suppose that is a max multiflow. Summing (3.15) over for fixed yields

 n∑j≠iNAiAj≥∫Aivi=S(Ai) . (3.16)

But, by (2.8), the total number of threads connecting to all the other regions is also bounded above by :

 n∑j≠iNAiAj≤S(Ai) . (3.17)

So the inequalities (3.16) and (3.17) must be saturated, and furthermore each inequality (3.15) must be individually saturated:

 NAiAj=∫Aivij . (3.18)

Thus, in the language of threads, Theorem 1 states that there exists a thread configuration such that, for all ,

 n∑j≠iNAiAj=S(Ai) . (3.19)

We will call such a configuration a max thread configuration.

We will now study the implications of the existence of a max thread configuration for three and four boundary regions.

3.2.2 Three boundary regions

For , we have

 S(A)=NAB+NAC ,S(B)=NAB+NBC ,S(C)=NAC+NBC . (3.20)

Since , we find an elegant formula for the mutual information:

 I(A:B)=2NAB . (3.21)

Thus, at least from the viewpoint of calculating the mutual information, it is as if each thread connecting and represents a Bell pair.101010Strictly speaking, since a Bell pair has mutual information , each thread represents Bell pairs. If one really wanted each thread to represent one Bell pair, one could define the threads to have density , rather than , for a given flow . Note that (3.21) also reestablishes the subadditivity property, since clearly the number of threads cannot be negative. Similarly to (2.15), we also have for the conditional entropy,

 H(B|A)=NBC−NAB . (3.22)

As mentioned in Subsection 2.2, the three mutual informations , , determine the entropy vector in . Therefore, by (3.21) and its analogues, the thread counts , , determine the entropy vector, and conversely are uniquely fixed by it. Thus, in the skeleton graph representation of the entropy vector shown in Fig. 1 (right side), we can simply put , , as the capacities on the respective edges; in other words, the thread configuration “is” the skeleton graph.

3.2.3 Four boundary regions

For , we have, similarly to (3.20),

The entropies of pairs of regions, , , and also enter in MMI. A max thread configuration does not tell us these entropies, only the entropies of individual regions. Nonetheless, for any valid thread configuration, we have the bound (2.8). In particular, is bounded below by the total number of threads connecting to :

Similarly,

Inequalities (3.23), (3.24), and (3.25) together imply MMI.

As we did for three parties, we can look at the mutual information between two regions. However, using (3.23) and (3.24), we now merely find a bound rather than an equality:

 I(A:B)≤2NAB . (3.26)

Thus, in a four-party max configuration, each thread connecting and does not necessarily represent a “Bell pair.” To understand how this can occur, it is useful to look at a simple illustrative example, shown in Fig. 2, which is a star graph where each edge has capacity 1. It is easy to evaluate the entropies of the single vertices and pairs. One finds that they have the form (2.21) with ; in other words, this graph represents a perfect tensor. In particular, all pairwise mutual informations vanish, while . As shown in Fig. 2, there are three max thread configurations. Each such configuration has two threads, which connect the external vertices in all possible ways.

The above example highlights the fact that, unlike for , the thread counts are not determined by the entropies: in (3.23) there are only 4 equations for the 6 unknown , while the entropies of pairs of regions only impose the inequality constraints (3.24), (3.25) on the . However, based on Theorem 2 as summarized above (3.14), we know that there exists a max thread configuration that saturates (3.24) and therefore (3.26). The same configuration has . Similarly, there exists a (in general different) max configuration such that and , and yet another one such that and . In summary, is the minimal number of threads connecting and , while  is the total number of “excess” threads in any configuration:

 −I3=n∑i

These many threads are free to switch how they connect the different regions, in the manner of Fig. 2.

So far we have treated the and cases separately, but they are related by the operation of merging boundary regions. For example, given the four regions , , , , we can consider to be a single region, effectively giving a three-boundary decomposition. Under merging, not every four-party max thread configuration becomes a three-party max configuration. For example, in the case illustrated in Fig. 2, if we consider as a single region then, since , any max thread configuration must have two threads connecting to . Thus, the middle configuration, with , is excluded as a three-party max configuration.

4 State decomposition conjecture

In this section we consider the thread configurations discussed in the previous section for different numbers of boundary regions. Taking seriously the idea that the threads represent entanglement in the field theory, we now ask what these configurations tell us about the entanglement structure of holographic states. We will consider in turn decomposing the boundary into two, three, four, and more regions.

For two complementary boundary regions and , the number of threads connecting and in a max configuration is , so in some sense each thread represents an entangled pair of qubits with one qubit in and the other in . Of course, these qubits are not spatially localized in the field theory—in particular they are not located at the endpoints of the thread—since even in a max configuration the threads have a large amount of freedom in where they attach to the boundary. For three boundary regions, as discussed in Subsection 3.2.2, the max thread configuration forms a triangle, with the number of threads on the edge fixed to be and similarly with the and edges. If we take this picture seriously as a representation of the entanglement structure of the state itself, it suggests that the state contains only bipartite entanglement. In other words, there is a decomposition of the Hilbert spaces,

 (4.1)

(again, this is not a spatial decomposition) such that the full state decomposes into a product of three bipartite-entangled pure states:

 |ψ⟩ABC=|ψ1⟩A1B1⊗|ψ2⟩A2C2⊗|ψ3⟩B3C3 , (4.2)

each of which carries all the mutual information between the respective regions:

 S(A1) =S(B1)=12I(A:B) S(A2) =S(C2)=12I(A:C) (4.3) S(B3) =S(C3)=12I(B:C) .

So far the picture only includes bipartite entanglement. For four boundary regions, however, with only bipartite entanglement we would necessarily have , which we know is not always the case in holographic states. Furthermore, as we saw in Subsection 3.2.3, even in a max thread configuration, the number of threads in each group, say , is not fixed. We saw that there is a minimal number of threads connecting and , plus a number of “floating” threads that can switch which pairs of regions they connect. This situation is summarized by the skeleton graph of Fig. 1, which includes six edges connecting pairs of external vertices with capacities equal to half the respective mutual informations, plus a star graph connecting all four at once with capacity . The star graph has perfect-tensor entropies. This suggests that the state itself consists of bipartite-entangled pure states connecting pairs of regions times a four-party perfect tensor:

 |ψ⟩ABCD =|ψ1⟩A1B1⊗|ψ2⟩A2C2⊗|ψ3⟩A3D3⊗|ψ4⟩B4C4⊗|ψ5⟩B5D5⊗|ψ6⟩C6D6 (4.4) ⊗|PT⟩A7B7C7D7 .

This is the simplest ansatz for a four-party pure state consistent with what we know about holographic entanglement entropies. In this ansatz the MMI property is manifest.

The conjectured forms (4.2), (4.4) of the state for three and four regions respectively are not independent of each other. The four-region conjecture implies the three-region one, either by taking one of the regions to be empty or by merging two of the regions. A four-party perfect tensor, under merging two of the parties, factorizes into two bipartite-entangled states. For example, if we write , then the bipartite-entangled factors in (4.4) clearly take the form (4.2), while the perfect-tensor factor splits into bipartite-entangled pieces:

 |PT⟩A7B7C7D7=|ψ′1⟩A7C′7,1⊗|ψ′2⟩B7C′7,2 (4.5)

for some decomposition , as follows from the fact . Conversely, the three-region decomposition strongly suggests the four-region one, since a generic four-party pure state does not decompose into the form (4.2) upon merging of two of the parties; for example, the four-party GHZ state (2.20) becomes the three-party GHZ state. Rather, the four-party state must have a very particular form for this to work, and the perfect tensor is the simplest one with this property.

We remind the reader that, throughout this paper, we have been working in the classical, or large-, limit of the holographic system, and we emphasize that the state decomposition conjectures stated above should be understood in this sense. Thus we are not claiming that the state takes the form (4.2) or (4.4) exactly, but rather only up to corrections that are subleading in . If we consider, for example, a case where at leading order, such as where and are well-separated regions, the three-party decomposition (4.2) would suggest that . However, even in this case could still be of order , so we should not expect this decomposition to hold approximately in any norm, but rather in a weaker sense.

Support for these conjectures comes from tensor-network toy models of holography [26, 27, 10]. Specifically, it was shown in [10] that random stabilizer tensor network states at large indeed have the form (4.2), (4.4) at leading order in . More precisely, these decompositions hold provided one traces out many degrees of freedom in each subsystem. In other words, there are other types of entanglement present (such as GHZ-type entanglement), but these make a subleading contribution to the entropies. We believe that it would be interesting to prove or disprove the state decomposition conjectures (4.2), (4.4), as well as to sharpen them by clarifying the possible form of the corrections.

Finally, we note that it is straightforward to generalize (4.2) and (4.4) to more than four regions. Namely, we can conjecture that for parties, decomposes into a direct product of states, each realizing an extremal ray in the -party holographic entropy cone. A new feature that arises for , as mentioned in Subsection 2.2.2, is that a generic vector in the holographic entropy cone no longer admits a unique decomposition into extremal rays. Therefore the amount of entropy carried by each factor in the state decomposition cannot be deduced just from the entropy vector, but would require some more fine-grained information about the state. Another new feature that arises for is that the extremal rays no longer arise only from perfect tensors; rather, new entanglement structures are involved. It would be interesting to explore whether the thread picture throws any light on these issues.

5 Proofs

In this section, we give proofs of our main results on the existence of multiflows in Riemannian geometries. We are not claiming mathematical rigor, particularly when it comes to functional analytical aspects. To simplify the notation, we set throughout this section.

5.1 Theorem 1

For convenience, we repeat the definition of a multiflow and the statement of Theorem 1.

Definition 1 (Multiflow).

Given a Riemannian manifold with boundary111111As mentioned in footnote 4, the case where has an “internal boundary” is also physically relevant. In this case, not included in the decomposition into regions , all flows are required to satisfy the boundary condition on , and homology relations are imposed relative to . The reader can verify by following the proofs, with replaced by , that all of our results hold in this case as well. , let be non-overlapping regions of (i.e. for , is codimension-1 or higher in ) covering (). A multiflow is a set of vector fields on satisfying the following conditions:

 vij =−vji (5.1) ^n⋅vij =0 on Ak(k≠i,j) (5.2) ∇⋅vij =0 (5.3) n∑i
Theorem 1 (Max multiflow).

There exists a multiflow such that for each , the sum

 vi:=n∑j=1vij (5.5)

is a max flow for , that is,

 ∫Aivi=S(Ai) . (5.6)

Our proof of Theorem 1 will not be constructive. Rather, using tools from the theory of convex optimization, specifically strong duality of convex programs,121212We refer the reader to [28] for an excellent guide to this rich subject, but we also recommend [6] for a short physicist-friendly introduction summarizing the concepts and results applied here. we will establish abstractly the existence of a multiflow obeying (5.6). The methods employed here will carry over with only small changes to the discrete case, as shown in Section 6.

Proof of Theorem 1.

As discussed in Subsection 3.1, for any multiflow, is a flow and therefore obeys

 ∫Aivi≤S(Ai) . (5.7)

What we will show is that there exists a multiflow such that

 n∑i=1∫Aivi≥n∑i=1S(Ai) . (5.8)

This immediately implies that (5.7) is saturated for all .

In order to prove the existence of a multiflow obeying (5.8), we will consider the problem of maximizing the left-hand side of (5.8) over all multiflows as a convex optimization problem, or convex program. That this problem is convex follows from the following facts: (1) the variables (the vector fields ) have a natural linear structure; (2) the equality constraints (5.1), (5.2), (5.3) are affine (in fact linear); (3) the inequality constraint (5.4) is convex (i.e. it is preserved by taking convex combinations); (4) the objective, the left-hand side of (5.8), is a concave (in fact linear) functional. We will find the Lagrangian dual of this problem, which is another convex program involving the constrained minimization of a convex functional. We will show that the objective of the dual program is bounded below by the right-hand side of (5.8), and therefore its minimum is bounded below:

 d⋆≥n∑i=1S(Ai) . (5.9)

We will then appeal to strong duality, which states that the maximum of the original (primal) program equals the minimum of the dual,

 p⋆=d⋆ . (5.10)

We thus obtain

 p⋆≥n∑i=1S(Ai) , (5.11)

showing that there is a multiflow obeying (5.8).

To summarize, we need to (a) derive the dual program and show that strong duality holds, and (b) show that its objective is bounded below by . We will do these in turn. Many of the steps are similar to those in the proof of the Riemannian max flow-min cut theorem, described in [6]; the reader who wishes to see the steps explained in more detail should consult that reference.

(a) Dualization:

The Lagrangian dual of a convex program is defined by introducing a Lagrange multiplier for each constraint and then integrating out the original (primal) variables, leaving a program written in terms of the Lagrange multipliers. More specifically, an inequality constraint is enforced by a Lagrange multiplier which is itself subject to the inequality constraint . In integrating out the primal variables, the objective plus Lagrange multiplier terms (together called the Lagrangian) is minimized or maximized without enforcing the constraints. The resulting function of the Lagrange multipliers is the objective of the dual program. The requirement that the minimum or maximum of the Lagrangian is finite defines the constraints of the dual program (in addition to the constraints mentioned above). If the primal is a minimization program then the dual is a maximization one and vice versa.

In fact it is not necessary to introduce a Lagrange multiplier for each constraint of the primal program. Some constraints can be kept implicit, which means that no Lagrange multiplier is introduced and those constraints are enforced when integrating out the primal variables.

Our task is to dualize the program of maximizing over all multiflows, i.e. over sets of vector fields obeying (5.1)–(5.4); as discussed above, this is a convex program. We will choose to keep (5.1) and (5.2) implicit. For the constraint (5.3), we introduce a set of Lagrange multipliers (), each of which is a scalar field on . Note that is only defined for since, given the implicit constraint (5.1), the constraint (5.3) only needs to be imposed for . For the inequality constraint (5.4) we introduce the Lagrange multiplier , which is also a scalar function on and is subject to the constraint . The Lagrangian is

 L[{vij},{ψij},λ] =n∑i=1∫Ai√hn∑j=1^n⋅vij (5.12) +∫M√g[n∑i

Rewriting the first term slightly, integrating the divergence term by parts, and using the constraint (5.2), the Lagrangian becomes

 L[{vij},{ψij},λ] =n∑i

We now maximize the Lagrangian with respect to (again, only imposing constraints (5.1), (5.2) but not (5.3), (5.4)). The requirement that the maximum is finite leads to constraints on the dual variables , . Since the Lagrangian, as written in (5.13), is ultralocal in , we can do the maximization pointwise. On the boundary, for a given , at a point in or , can take any value. Therefore, in order for the maximum to be finite, its coefficient must vanish, leading to the constraints

 ψij=−1 on Ai ,ψij=1 on Aj . (5.14)

When those constraints are satisfied, the boundary term vanishes. In the bulk, the term is unbounded above as a function of unless

 λ≥|∇ψij| , (5.15)

in which case the maximum (at ) vanishes. (As a result of (5.15), the constraint is automatically satisfied and can be dropped.) The only term left in the Lagrangian is .

All in all, we are left with the following dual program:

 Minimize ∫M√gλ with % respect to {ψij},λ subject to λ≥|∇ψij| ,ψij=−1 on Ai ,ψij=1 on Aj , (5.16)

where again, is defined only for .

Strong duality follows from the fact that Slater’s condition is satisfied. Slater’s condition states that there exists a value for the primal variables such that all equality constraints are satisfied and all inequality constraints are strictly satisfied (i.e. satisfied with replaced by ). This is the case here: the configuration satisfies all the equality constraints and strictly satisfies the norm bound (5.4).

(b) Bound on dual objective:

It remains to show that, subject to the constraints in (5.16), the objective is bounded below by .

First, because on and on , for any curve from a point in to a point in , we have

 (5.17)

where is the proper length element, is the unit tangent vector, and in the second inequality we used the Cauchy-Schwarz inequality. Now, for each , define the function on as the minimum of over any curve from to :

 ϕi(x)=infC fromAi to x∫Cdsλ . (5.18)

By virtue of (5.17),

 ϕi(x)+ϕj(x)≥2(i≠j) . (5.19)

Define the region as follows:

 Ri:={x∈M:ϕi(x)<1} . (5.20)

It follows from (5.19) that for . Given that , this implies that the dual objective is bounded below by the sum of the integrals on the s: