Bulk locality and cooperative flows

Bulk locality and cooperative flows

Veronika E. Hubeny veronika@physics.ucdavis.edu

We use the ‘bit thread’ formulation of holographic entanglement entropy to highlight the distinction between the universally-valid strong subadditivity and the more restrictive relation called monogamy of mutual information (MMI), known to hold for geometrical states (i.e. states of holographic theories with gravitational duals describing a classical bulk geometry). In particular, we provide a novel proof of MMI, using bit threads directly. To this end, we present an explicit geometrical construction of cooperative flows which we build out of disjoint thread bundles. We conjecture that our method applies in a wide class of configurations, including ones with non-trivial topology, causal structure, and time dependence. The explicit nature of the construction reveals that MMI is more deeply rooted in bulk locality than is the case for strong subadditivity.

\subheaderinstitutetext: Center for Quantum Mathematics and Physics (QMAP)
Department of Physics, University of California, Davis, CA 95616 USA

1 Introduction

Entanglement, a quintessentially quantum characteristic, is expected to play a key role in holography, not only as a powerful tool in describing a given physical system, but also as a fundamental building block in the duality itself. Understanding precisely how this works has been one of the outstanding goals in the holography program over the last few years. The basic quantity of interest is entanglement entropy, which provides a particularly natural measure of entanglement, and from which other useful quantities are built. Universal relations between these quantities allow us to gain further insights, and in the holographic context may even diagnose whether a given quantum field theory state has a holographic dual corresponding to a classical geometry. The way in which the bulk geometry implements these relations can then provide crucial insights to unraveling the emergence of spacetime in holography.

In the following, we will focus on one particular relation known as monogamy of mutual information (MMI). In order to explain its significance, we first review its construction in the broader context of other relations.111 In the interest of being reasonably self-contained, in the next few paragraphs we give a very brief review of these relations. This is well-known material; for a nice review of the relations and their proofs in the holographic context (for time reflection symmetric configurations), see e.g. Headrick:2013zda (). The entanglement entropy (EE) of a system can be thought of as a measure of mixedness of its reduced density matrix obtained by tracing out the complement in a specified bi-partitioning of the system . Explicitly, EE is defined as the von Neumann entropy


Refining the partition into several subsystems , , etc., we can consider general constraints on the respective entanglement entropies.222 It is known that even in the restricted context of holography Bao:2015bfa (), there are in fact infinitely many such relations if we allow arbitrary number of partitions, though the complete set has hitherto remained elusive. For example, among particularly useful relations are sub-additivity (SA), strong sub-additivity (SSA) and its cousin (related by purification) weak monotonicity (WM), and the above-mentioned monogamy of mutual information (MMI), which can respectively be expressed as:


where we use the shorthand , etc. While these relations look superficially similar, there is an important difference between the first three and the last one: SA, SSA, and WM are all valid for any quantum system in any state and for any partition. MMI, on the other hand, is a stronger relation which does NOT hold in general, but as initially proved in Hayden:2011ag (), it does hold for holographic states.

To digest the significance of this statement, it is useful to rewrite the above relations in a slightly more suggestive way. One can think of the triangle inequality type relation of SA (1.2) as the statement of positivity of the mutual information


which characterizes the amount of correlations (both classical and quantum) between the systems and . One can similarly view the convexity property of SSA (1.3) as the statement of positivity of the conditional mutual information


which says that the amount of correlation is monotonic under inclusion. These two statements, that the amount of correlations cannot be negative and that enlarging the system cannot decrease the amount of correlations with another system, are intuitively clear.333 Though perhaps surprisingly so; see Witten:2018zva () for an exposition of some of the subtleties.

The statement of MMI (1.5), on the other hand, translates to the statement of negativity of the tripartite information444 One could argue that ‘negative tripartite information’ (also known as interaction information) is a more natural quantity to consider, since its positivity appears in a variety of QI contexts on similar footing as other positive quantities. However, to avoid confusion, we will presently stick to the usual convention of using the standard .


which can therefore be thought of as measuring the non-extensivity of mutual information. If the correlations between the three systems are purely quantum, then they must obey the property known as monogamy of entanglement (implying that quantum entanglement between and cannot be shared by etc.), so the entanglement contributing to cannot comprise any part of contribution to and vice-versa. That in turn means that must be at least as large as the individual contributions, , implying the MMI relation (1.5) (and motivating its name). If, on the other hand, the correlations are classical, then they can be redundantly shared, allowing for the possibility of positive tripartite information.555 In fact even the party GHZ state , which in some regards is highly quantum but whose correlations are maximally redundant, violates MMI. In the holographic context, then, the fact that MMI holds (i.e. that ) is rather suggestive of sufficient quantumness of the correlations between subsystems.666 This has already been suggested by Hayden:2011ag () and is further supported by the recent exploration of Takayanagi:2018zqx () using the relative entropy of entanglement as a better measure of quantumness of correlations. This emphasizes the important point that while we tend to think of the bulk as behaving classically, this is merely an effective description: the corresponding CFT state manifests quantum features even with respect to the observables naturally associated with the bulk variables.

One of the advantages of the bulk description is that it often conveniently geometrizes highly non-trivial statements about the system, allowing them to be proved with ease, at least in a certain context.777 In everything that follows, we will focus on the leading order in the large-, large- limit of the CFT, which corresponds to a classical geometry in the bulk. (However we briefly comment on quantum and stringy correction in Section 5.) The inequalities (1.2)–(1.5) provide an excellent example of this intriguing fact. According to the Ryu-Takayanagi (RT) prescription Ryu:2006ef (); Ryu:2006bv () pertaining to static situations, the entanglement entropy of a given spatial region in the boundary CFT888 Although in our gauge theory context, the Hilbert space factorization for such a spatial partitioning strictly-speaking does not apply, this does not affect any of the results discussed below, as can be justified using the more rigorous framework of Tomita-Takesaki theory based on algebra of observables; for a recent review see e.g. Witten:2018zxz (). is computed by quarter of the proper area (in Planck units) of a minimal surface999 The surface is by definition of co-dimension-two in the full bulk spacetime, which in the static context can be more usefully viewed as spatially co-dimension-one within a constant time slice. which is homologous to (and hence anchored on the entangling surface ):


In arbitrary time-dependent situations, this relation naturally generalizes to the Hubeny-Rangamani-Takayanagi (HRT) prescription Hubeny:2007xt (), wherein the minimal surface at constant time gets simply replaced by a spacetime co-dimension-two extremal surface homologous to (and in case of multiple such candidates, being the one with the smallest area). However, in what follows we will primarily focus on the static case.

Remarkably, the proof of SSA in the static case, first obtained in Headrick:2007km () and later generalized in Hayden:2011ag (), follows almost immediately from (1.9). As we briefly review in Section 3.1 in a simple setup, one can obtain (1.3) by comparing the respective bulk minimal surface areas and using the fact that a minimal surface by definition cannot have larger area than any other surface in the same homology class. Interestingly, the argument for MMI is virtually identical Hayden:2011ag () – indeed, as indicated below, in this context the proof of MMI actually reduces to that of SSA, despite the fact that these are logically distinct relations with different physical meanings. This curious statement likewise holds in the general time-dependent setting Wall:2012uf (). In other words, the RT and HRT prescriptions do not discern the fundamental distinction between SSA and MMI. Hence their virtue, that the bulk repackaging simplifies the dual CFT description, turns into somewhat of a detriment, by obscuring a crucial aspect of these relations.101010 There are several other features, such as the bulk location of the CFT information, for which these prescriptions likewise seem more puzzling than suggestive.

To elucidate this important distinction, then, we need to go beyond the RT (or HRT) prescription. Of course the structural difference between SSA and MMI would become more apparent if we include quantum corrections, but going beyond the classical limit does not address the essence of the puzzle directly. Fortuitously, however, we in fact can see a distinction already at the leading order in large- description, as explained below. This is manifested by utilizing a different geometrical prescription for holographic entanglement entropy, namely the “bit thread” formulation of Freedman and Headrick Freedman:2016zud (). This prescription, reviewed in Section 2, captures the quantum information theoretic meaning of the relevant constructs more naturally.111111 Although, as argued in Freedman:2016zud () using the Max Flow-Min Cut theorem and shown more explicitly in Headrick:2017ucz (), the bit thread prescription for holographic entanglement entropy is mathematically equivalent to the RT prescription, it nevertheless turns out to be much better tailored to the present considerations. Here entanglement entropy is given by the maximum flux of a bulk ‘flow’ (defined as a divergence-free vector field of bounded norm), whose integral curves are the bit threads. As shown in Freedman:2016zud (), the statements of SA and SSA can be recast into statements of positivity of the number of threads obeying certain restrictions – and since this number manifestly cannot be negative, these two relations are then even more directly evident in the bit thread formulation than via RT.

However, as already emphasized in Freedman:2016zud (), the MMI relation does not lend itself to such an easy argument; in fact, Freedman:2016zud () has not been able to find a proof directly in the flow language without recourse to the equivalence with RT. The bit thread formulation, then, provides a useful window of opportunity: a framework wherein the distinction between SSA and MMI should become more directly apparent, which we can use to elucidate the role of the geometrization and ultimately the emergence of bulk spacetime. More specifically, Freedman:2016zud () demonstrates that just using the basic building blocks of the flows (and its crucial properties such as ‘nesting’) does not guarantee MMI: some other non-trivial property of flows is needed. In this paper, we set out to elucidate this property. We will find that it is deeply rooted in bulk locality.

In particular, we prove MMI directly using the language of bit threads.121212 The authors of MattMMI () (whom we thank for sharing an advance version of their draft) likewise have a proof of MMI, inspired by previously known theorems regarding multicommodity flows, using tools of the theory of convex optimization (in particular the Lagrangian duality) along similar lines as Headrick:2017ucz (), to abstractly argue for the existence of ‘cooperative flows’. Our argument is complementary to that line of reasoning: it is explicitly constructive and directly geometrical. The structure of the presentation is as follows. Our argument starts with the observation131313 We thank Matt Headrick for sharing this observation, which inspired the present work (as well as the term ‘cooperative flows’). (explained in Section 2), that under the assumption of the existence of certain ‘cooperative flows’, MMI would follow easily. The crux of the argument then is to demonstrate the existence of such cooperative flows. We achieve this by an explicitly constructive method. In Section 3 we explain the basic idea in a simple context, and in Section 4 we generalize this construction to arbitrary spatial partitions of CFT states describing arbitrary asymptotically AdS geometries. We then revisit the relation to bulk locality and discuss further implications and open questions in Section 5.

2 Bit thread proof of MMI – overview

To approximate a partitioning of the Hilbert space (with being the purifier) in the holographic context, we typically let , , , and correspond to spatial regions partitioning the space on which the theory lives.141414 For the CFT living on a single connected spacetime, this restriction corresponds to taking a pure state, so the bulk admits no eternal black hole horizons. However, it is easy to generalize to the case with black holes (or even more generally ‘multi-boundary wormholes’) by letting (as well as , , and ) include regions in multiple disconnected boundary spacetimes. Recall that the general form of MMI, written in a manifestly -cyclically symmetric form, is


where each term gives the entanglement entropy of the corresponding boundary subregion.

In this section, we outline the basic argument for demonstrating (2.1) in the holographic context using the bit thread formulation. Following Freedman:2016zud (), we can describe the entanglement entropy in terms of bulk flows. A flow is defined as divergenceless vector field of bounded norm,


whose integral curves are dubbed ‘bit threads’. The entanglement entropy of a given spatial region on the boundary is computed by maximizing the flux through over all bulk flows,


where we’re using the shorthand (with being the unit normal, and the determinant of the induced metric, on the integration surface which by homology and Stokes’ theorem can be taken at ). Any flow which maximizes the flux through a region is called a maximizer flow for the region and denoted by151515 In the interest of compactness of notation, here we deviate from Freedman:2016zud () (which uses the notation for the maximizer flow). To label multiple generic flows which are not necessarily maximizing for any region, we will use a numerical subscript, e.g. , etc. (not to be confused with a letter subscript which labels a region and signifies maximizer flow through that region). , so the entanglement entropy can be characterized more simply as the flux of a maximizer flow,


where the inequality by definition holds for any flow . Note that a maximizer flow for a given region is highly non-unique: the restriction (2.2) only fully fixes it at the minimal surface (through which it must pass perpendicularly and with unit norm). For later purposes it is also useful to note that switching the direction of given a maximizer flow for region automatically generates a maximizer flow for its complement .

Applying (2.4) to the LHS of the MMI inequality (2.1), we have


where the inequality holds for arbitrary flows , , and . To relate this to the RHS of (2.1), we want to choose these flows (with ) to be given by


where , , , and are maximizer flows for the corresponding regions. It is easy to see that in such a case the RHS of (2.5) would reduce to


thereby proving the MMI relation.

The crux of the argument then boils down to showing that the objects defined by (2.6) are indeed flows, i.e. that they satisfy (2.2) and hence we’re allowed to equate them to the LHS terms in (2.6). While divergencelessness of ’s is guaranteed by linearity, the norm bound need not a-priori remain satisfied: By triangle inequality we’re only guaranteed that the average of any two flows and is itself a flow, whereas the expressions involving 4 flows in (2.6) can exceed unit norm (and in fact one can easily construct maximizer flows , , , and , for which each of , , and becomes 2 somewhere).

However, not all maximizer flows , , , and have this undesirable property. Given any specified regions , , and , we can find maximizer flows , , , and which guarantee that , , and simultaneously throughout the bulk. Below, we outline an explicit construction (in fact two natural ones) of maximizer flows which satisfy this requirement, thus supplying the missing step in the proof of MMI. Our construction utilizes the observation Headrick:2017ucz () that a foliation of a bulk region by minimal surfaces induces a maximally-collimated flow; we demonstrate that suitable foliations always exist which ‘comb’ the flows , , , and in such a way as to ‘cooperate’ in the requisite fashion.

3 Basics of cooperative flow construction

We start with a simple class of examples to illustrate the basic idea, and subsequently generalize our arguments to more complicated situations in Section 4. In particular, we first fix convenient dimensionality and state, but consider a generic partition within a given topology class.

3.1 Setup

Let us consider pure AdS geometry, with all regions of interest localized at a fixed time , so that we can WLOG restrict attention to bulk spatial slice, i.e. the Poincare disk.161616 We will only use the metric details of AdS for generating the explicit plots, but otherwise our arguments will be robust to deforming the geometry. Moreover, as long as we’re restricting to static context, our results will not depend on assuming any specific field equations, energy conditions, etc.. We will consider the boundary to live on space and take the partitions , , and to be simple adjoining intervals. We will denote the complementary region and assume the total state to be pure, i.e. . A representative of a generic such configuration is sketched in Fig. 1, along with the corresponding minimal surfaces relevant for the terms appearing in MMI (2.1).

Figure 1: An example of partitioning of the boundary space into regions (thick curves) , , , and the complement , as labeled. The corresponding minimal surfaces (dashed curves) for these 4 regions, along with those for and are also indicated. (In this and subsequent plots, we plot the Poincare disk in global coordinates described by constant slice of with corresponding to the boundary.)

Note that and


We can now readily confirm that the RT-based geometric proof of MMI is then manifestly identical171717 Note that our present argument based on cancelation of surfaces hinges on the fact that our ‘minimal surfaces’ are one-dimensional; in higher dimensions one has to recourse to the more refined argument of Hayden:2011ag (). to the SSA proof of Headrick:2007km (): for either option in the RHS of (3.1), is canceled by two of the terms on the RHS of (2.1), with the remaining terms equivalent to either SSA (1.3), or its purification WM (1.4). In the present configuration these two relations follow from cutting and rejoining and so as to implement homology, and observing that each surface necessarily has greater area than the globally minimal surface in its homology class, namely and in the case of (1.3), or and in the case of (1.4). In the following construction we will for definiteness181818 As we further explain in footnote 26, this is not a limitation since an analogous method will work for the opposite sign of the inequality in (3.2) as well. Moreover, MMI is preserved under purification and permutations, so the choice (3.2) merely fixes a labeling convention. pick the case


so that . (Although this implies , we will develop our proof of MMI without the reliance on this fact.)

Converting the above statements to the bit thread description Freedman:2016zud (); Headrick:2017ucz (), the entanglement entropy of each region is given by the flux of the corresponding maximizer flow. Recall that any maximizer flow for the region is then guaranteed to be maximally packed (i.e. ) on , and similarly for the regions , , and , which are maximally packed on the corresponding minimal surfaces. Elsewhere these individual flows admit some floppiness, and at the AdS boundary they are maximally floppy. Hence there is a large amount of freedom in picking the maximizer flows , , , and . However, as explained above, a generic choice would not necessarily satisfy the flow conditions (2.2) for the ’s defined in (2.6); our task is to find an explicit construction which would guarantee that , , and simultaneously throughout the bulk.

To build up some intuition, we first take a short detour to illustrate why constructing flows is not trivial, and in particular requires some global considerations. We will however start in Section 3.2 with an observation that the set up indicated in Section 3.1 actually contains a hidden simplification, which we could use to prove the desired result in this particular case. We will then explain in Section 3.3 why this method is not amenable to an easy generalization, and motivate the general construction, which will simultaneously reveal the connection to bulk locality. To skip to the construction itself, the reader may proceed directly to Section 3.4.

3.2 Cooperative flows for uncorrelated regions

In encountering the requisite expressions (2.6), one might be tempted to try reducing each line to only two terms by canceling the other two against each other. Unfortunately, we can not set, say, , since by subadditivity, such flow across and would exceed the common bottleneck ,191919 This statement holds whenever subadditivity is not saturated (i.e. when , since then ), and in the present case, the mutual information in fact diverges. (Geometrically this is associated with the UV divergence in the areas of and at the common entangling surface , which is not present for .) and similarly for the other adjoining regions. The only simultaneous flows we are allowed a-priori (without additional constraints), are the ones corresponding to nested regions, i.e.  and (at most) one of , , or . This however ensures the norm bound for only one of the ’s. To deal with the other two ’s, we need further cancellations.

In fact, one sufficient such cancellation would be easy to achieve in the case where some pair of regions has vanishing mutual information, namely when or . In the former case, we can find a common maximizer flow for both regions by channeling the flow from202020 In the interest of familiarity, here we conform to the convention of Freedman:2016zud () in drawing the bit threads for a given region as directed out of into the bulk (rather than as spacetime-outward-directed flow used in Headrick:2017ucz ()). into but not , and similarly channeling the flow from into but not . In particular, when subadditivity for and is saturated, the flow bottleneck for coincides with the union of the individual bottlenecks for and for , which means that any maximizer flow in fact saturates the norm bound on and on , so that it is by definition simultaneously a maximizer flow for these individual regions, i.e., . In such a case, using additionally the nesting of and to set , the expressions in (2.6) would simplify to


which means that all the ’s are manifestly flows.

In the alternate case of , we can similarly set . Having used up , one might worry that we can no longer use nesting to get a further simplification; however, as one can see from Fig. 1, this case was in fact just a rotated/relabeled version of the previous case, so the same arguments should still go through. And indeed, we still do have a form of nesting: , which along with purity can be used to set . In this case, (2.6) would simplify to


so once again, all the ’s are manifestly flows.

One might at this point be tempted to declare victory: after all, (3.1) implies that at least one of and vanishes, since if (3.2) holds, then , whereas for the opposite sign of the inequality, .

However, (3.1) was predicated on the simple setup sketched in Fig. 1. We can easily have both and non-zero (and even make both diverge) by taking all regions to be pairwise near (or even adjoining).212121 In the AdS case, this can be achieved by having some of the regions be composed of multiple intervals (as exemplified in Fig. 7), while in higher dimensions, this is possible even with a single simply-connected region per subsystem. Since we can generically have both and , the previous argument using (3.3) or (3.4) would not apply. On the other hand, we can still use nesting along with purification in full generality: e.g., we could use nesting of to set , and use nesting of to set . Unfortunately, substituting these into (2.6) gives


so although we can achieve full cancelation in one of the ’s, the other two fail to be flows since they exceed the norm bound. Hence we need to find a more general method of constructing the requisite maximizer flows.

3.3 Motivation for our strategy

To motivate such a construction, let us then first consider the structure of a generic flow. Since each bit thread has two endpoints, located somewhere on the boundary (i.e. within either or or or ), we can think of a most general flow as composed out of (a subset of) thread bundles joining all possible pairs of regions. In particular, in general we could have bundles connecting , , , , , and . Being a piece of a flow, each thread bundle individually satisfies the flow conditions (2.2), and if we’re decomposing a single flow, these pieces manifestly cannot overlap. On the other hand, viewed as more abstract building blocks for a collection of flows, we could in general consider overlapping thread bundles, as long as they uphold the total norm bound.222222 In terminology of MattMMI (), such objects are dubbed ‘multiflows’. In the present work (unlike MattMMI ()), we will in fact construct thread bundles which do not overlap. To use the thread bundles as building blocks of a flow for a given region , we simply consider all thread bundles emanating from .232323 We could optionally also include other thread bundles connecting pairs of regions distinct from (and not interfering with ’s bundles), but since these are irrelevant for computing , we can choose to ignore them.

So far we have merely refined the nomenclature for a given flow, but we will now explain the utility of considering such a decomposition. Recall that flows for non-nested but correlated regions and are incompatible. This however does not necessarily mean that their individual thread bundles are likewise. In particular, one can envision constructing all four maximizer flows, , , , and out of the same set of pre-specified thread bundles, by picking their orientations appropriately. For example the bundle  would contribute to in the orientation and to in the opposite orientation . (Note that by purity and nesting, we can always use the same set of threads to obtain flows for two disjiont regions and by taking .) As is intuitively clear and easy to check explicitly, keeping all the thread bundles spatially disjoint from each other would then guarantee that each of , , , and constructed from them is likewise a flow. Moreover, each is a maximizer flow if the thread bundles collectively render each of , , , and to be everywhere crossed perpendicularly by some bundle (with the correct orientation) with unit norm.

The most economical way to construct disjoint thread bundles is to require that they in fact saturate the norm bound everywhere within their domain of support, since accommodating a smaller norm for any bundle would require greater bulk volume. In other words, we wish to comb the flow configuration into maximally collimated thread bundles, in such a way that these bundles do not intersect each other. Consequently, for our configurations (cf. Fig. 1), the presence of an bundle is incompatible with simultaneous presence of a bundle. This is however not a problem, since as explained above, obviates the need for the bundle.242424 This is manifest by choosing , but our construction will actually be more thrifty, and have no overlap at all between the two flows (i.e., implement everywhere). Similarly, the alternate case would obviate the need for the thread bundle. This reasoning motivates us to try constructing a thread bundle configuration of the form sketched in Fig. 2, which exemplifies how each maximizer flow can be partitioned into several thread bundles, each remaining within its associated bulk region; for compactness of notation we label these as indicated in Fig. 2.

Figure 2: Separate components (thread bundles) of the maximizer flows , , , and , passing through separate parts of the corresponding minimal surfaces as indicated, though only a single thread for each bundle is shown explicitly. Each thread bundle is confined to its own spatial region, which we compactly label (for bundle), (for bundle), (for bundle), (for bundle), and (for bundle). The two panels illustrate a residual 1-parameter freedom in the construction, further discussed in Section 3.4.1. (a) In the bundle, the arrows indicate that the same thread can be used in both directions, labeled and . (b) The relative sizes of the thread bundles are not fixed; one can vary them at the expense of e.g. bending the flow.

Once we have constructed such a set of cooperative thread bundles from which we can assemble all four maximizer flows, it follows almost immediately that their linear combinations , , and defined via (2.6) will likewise be flows. The crux of our MMI proof then boils down to showing that a configuration envisioned in Fig. 2 is indeed always tenable, which we will now demonstrate by an explicit construction.

3.4 Flow construction

In this section we present an explicit construction for the maximizer flows , , , and which render the ’s of (2.6) likewise flows.

Starting with partitioning of AdS boundary time slice of the type sketched in Fig. 1, we first show that we can partition the maximizer flows as indicated in Table 1 (cf. Fig. 2).

0 0 0
0 0
0 0 0
0 0
Table 1: Partitioning of the 4 maximizer flows (indicated in the left column) into thread bundles confined to the 5 bulk regions (indicated in the top row). Each row corresponds to an equation delineating this partition, e.g. the first row gives , where the two RHS terms are supported entirely within the regions and , respectively, and similarly for the other rows.

In particular, in Section 3.4.1 we argue that we can comb the maximizer flow from into a thread bundle252525 Recall that for chosen from the fundamental regions , , , and , the notation simply indicates the part of the which flows into . The only requirement that we impose on such a partitioning is that for all . Since this corresponds to merely reversing the orientation of the bit threads, both flows have the same norm and divergence. leading to and a thread bundle  leading to (but not an thread bundle thanks to (3.2)), and similarly for , , and , such that the thread bundles pass only through restricted disjoint bulk regions (denoted ).

Once we show (in the remainder of this section) that such a partition is possible, the rest is straightforward: Applying the definition (2.6), we can directly evaluate the ’s, and examine them within each region. The result is packaged in Table 2, which gives the explicit contributions within each region.

  0 0
  0 0
Table 2: Explicit check that the ’s defined by (2.6) are indeed flows, given the partition in Table 1. Each row gives an equation with LHS being the first column and RHS the sum of the terms in the remaining columns, as in Table 1. The columns correspond to the distinct bulk regions (cf. Fig. 2) and we see that in each region, is manifestly a flow.

The crucial observation here is that each entry in the table is of the form (i.e. a bundle of a single flow and therefore manifestly a flow). Hence even though each of the ’s is the sum of three bulk flows, the result is nevertheless of unit-bounded norm everywhere. This establishes that (i.e. each is a flow) as advertised.262626 Recall that the above construction assumed that which implies (3.2). However, as explained in footnote 18, this was merely a notational convention. If so that the flows have more stringent bottleneck than the system, we can simply reconfigure the thread bundles so as to have a component instead of , along with the corresponding rearrangement of the regions (such that adjoins , adjoins , and passes between and and between and ). Consequently in Table 2, the column would contain and in the first and 2nd rows respectively, while the rest of the table remains identical.

3.4.1 Maximizer flow partitions

We now explain why the partition indicated in Table 1 and exemplified in Fig. 2 is indeed always possible. We have already seen that the decomposition summarized in the rows of Table 1 without requiring the bulk regions , , , , and to be disjoint, is necessitated by the divergencelessness condition (2.2) (since each thread from a given region has to end in one of the other regions), along with the observation that we need not activate any thread bundles for uncorrelated regions. What remains to be shown is the compatibility between these respective partitions, namely that the thread bundles are mutually non-overlapping throughout the bulk. Since a general maximizer flow from each region necessarily saturates the norm bound on the associated minimal surface presenting a bottleneck for the flow, the first step is to verify the consistency of partitioning on the corresponding minimal surfaces.

In the present case, the surface splits into two parts, and , associated with the thread bundles ( and ) ending on and , respectively. Similarly, splits into three parts, , , and , channeling the bundles () from into , , and , respectively, and so on. Moreover, since the threads of through coincide with the threads of through , but with opposite orientation, the areas of the two minimal surface parts must likewise match, , for all and corresponding .

Written out explicitly, given , , , and , the minimal surface subdivisions need to satisfy the following constraints:


This constitutes 9 equations for 10 unknowns, and therefore we expect to have a 1-parameter family of solutions, as illustrated in Fig. 2a and Fig. 2b. One procedure of constructing a partitioning is as follows. Choose the partition of (in the present case that is just a single point ). This determines and , and hence also and . Now find the partition of (determined by a point , which immediately specifies , , , and ) so as to satisfy the remaining equation of (3.6), namely . Since if we slide sufficiently “up” along towards , we close off , whereas if we slide “down” towards , we close off , a solution clearly exists where the two terms become equal. This completes the argument that minimal surface partition envisioned in Fig. 2, i.e. satisfying (3.6), always exists.

Now that we have confirmed that a viable partitioning of the 4 minimal surfaces exits, we discuss the procedure for constructing the full maximizer flow for each region. There is of course large freedom in how we construct the requisite maximizer flows; here we present two particularly simple constructions which are easy to generalize. Although the flows are required to saturate the norm bound only on the associated minimal surface, we will find it convenient to construct them so as to remain maximally packed (and therefore equidistant) everywhere, since this utilizes the bulk geometry most economically. Geometrically, such a construction is actually quite simple, and is based on the observation explained in Headrick:2017ucz () that a foliation of a bulk region by minimal surfaces induces a maximally-collimated flow. Intuitively this is because the minimal surfaces have by definition vanishing expansion for the normal congruence (i.e., vanishing extrinsic curvature), so the bit threads which pass perpendicularly through any leaf of such foliation locally cannot change their cross-sectional density; in other words, they must remain maximally packed across any minimal surface they cross perpendicularly.272727 One might also naively worry that the flows encounter other bottlenecks which we have not taken into account. For example the thread bundle crosses both and (cf. Fig. 2), which a-priori need not have larger area than, say, . However, this is not a problem (as we explicitly demonstrate below), since the non-perpendicularly-crossed surfaces have by construction smaller thread density and therefore larger area.

3.4.2 Foliation building blocks

One strategy to obtain maximally packed thread bundle, then, is to start by considering a family of minimal surfaces which foliate the bulk space, since the normal congruence to these surfaces describes a flow. In the present case of 2-dimensional Poincare disk geometry, the foliating surfaces are simply geodesics, which indeed foliate the space as long as their endpoints vary continuously in any manner which generates a boundary foliation (which we can think of as a family of nested intervals whose endpoints collectively cover the boundary).282828 Recall that bulk minimal surfaces (here geodesics) anchored on nested regions (here intervals) are guaranteed not to intersect (since doing so would contradict global minimality), and the AdS geometry guarantees that there is a unique geodesic for each boundary interval and that a geodesic through any point and in any direction has both its endpoints on the boundary. The latter two properties do not hold universally for arbitrary static asymptotically AdS geometries, though they do hold robustly when the deformations from pure AdS are not too large. We revisit the more general case in Section 4.2. Labeling each leaf of the foliation by a parameter , we can specify the foliation by two functions, and , corresponding to the angular position of the two endpoints of the boundary interval. A foliation requires each function to be monotonic, but in opposite directions (so as to get nested intervals), such that and .

We sketch some simple examples in Fig. 3.

Figure 3: Examples of flows (solid blue curves) induced by minimal surface foliations (dotted red curves). (While we use the coordinates of Fig. 1, we don’t maintain constant proper spacing between the treads – doing so would clump them too much near the boundary.) Such foliations are specified by the endpoints of the bounding interval, as functions of (see text for details). In particular, we illustrate foliations generating (a) antipodal threads, (b) threads with both endpoints pinned at a single point of the Poincare disk boundary, and (c) threads with endpoints pinned at generically-separated points.

In Fig. 3a, the endpoints vary symmetrically, e.g. . In this case the threads happen to coincide with constant-time projections of null geodesics through AdS. In Fig. 3b, we fix one endpoint and let the other span the rest of the boundary, . In this case the threads happen to be horocycles on the Poincare disk. Although this looks like all the bit threads start and end at the same point, this is merely due to the conformal rescaling. The flow lines actually follow constant- contours in Poincare coordinates (they are generated by geodesics at constant ), where it is manifest that they straddle the and parts of the boundary.292929 One can also resolve this confluence by working at finite cutoff, where we could either only require the endpoints to get within the cutoff scale from each other (e.g.  and ), or determine the endpoints by cutting off the same set of spacelike geodesics at finite radius (in which case both endpoints would vary). One can likewise separate the flow ends, as shown in Fig. 3c. For example for the endpoints given by and , we see that this case interpolates between the first two. Of course, one can also have more irregular flows, and in fact all these flows can be patched together across any common minimal surface (such at the horizontal bisectors in Fig. 3).

As we see, we can comb the flows in a large (continuously infinite) variety of ways, and from any boundary point to any other boundary point. However, such foliations cannot admit multiple thread bundles, namely multiple starting and ending points for a given flow. To incorporate this more general case, we have to use multiple foliations simultaneously, or generalize the foliating surfaces to a more localized construct. As an example of each, we consider two particularly natural constructions, shown in Fig. 4.

Figure 4: Illustration of two distinct constructions for obtaining combed maximizer flows using foliations. (a) smooth flows obtained by 5 superimposed foliations. (b) patched flows obtained by a single bulk ‘local foliation’. The boundary regions and associated minimal surfaces are as in Fig. 1, and the threads are indicated by solid curves, with distinct colors delineating the regions of Fig. 2 which contain the corresponding thread bundles.
  1. Use a complete smooth foliation (by entire minimal surfaces) for each thread bundle. This leads to all threads being smooth, but a given foliation is ‘active’ (in terms of guiding a thread) in only a subset of the bulk. Hence there are multiple simultaneous bulk foliations. See Section 3.4.3 below for explicit construction and Fig. 4a for the resulting plot.

  2. Use a single bulk foliation for (almost) the entire bulk, but by only piecewise-minimal surfaces. The resulting flows are typically not smooth. See Section 3.4.4 for explicit construction and Fig. 4b for the corresponding plot.

3.4.3 Foliation for each thread bundle, smooth flows

We first consider the case with full bulk foliation per bundle. Since we have 5 thread bundles confined to 5 separate bulk regions , we will have 5 distinct foliations. Each foliation is specified by a family of minimal surfaces (geodesics) characterized by their endpoints and for , as indicated in Fig. 5.

Figure 5: Specific foliations which generate the smooth flow configuration indicated in Fig. 4a, in particular generating (a) the flow and (b) the flow . When suitably restricted (as indicated by the solid blue curves) these give the thread bundles  and , respectively. (c) All 5 foliations (color-coded as in Fig. 2) superposed on each other.

For the , and bundles (defining the associated bulk regions), the foliations are all of the horocycle type sketched in Fig. 3b (rotated in global coordinates): the minimal surfaces ‘fan out’ from one endpoint, say , with the other endpoint spanning the rest of the boundary, . These families then contain the minimal surfaces with the given fixed endpoint: the family, shown in Fig. 5a, in particular contains and (as well as ), the family contains and , and so forth. The final family, , indicated in Fig. 5b, is of the type sketched in Fig. 3c, varying both endpoints so as to contain and .303030 While we saw that the ‘bottleneck’ equations (3.6) in general have a 1-parameter family of solutions (illustrated in the two panels of Fig. 2), in the present case that parameter is used up by choosing to be of the particular form plotted in Fig. 3c (generated by threads emanating radially outward from a shifted origin in Poincare coordinates and rotated in global coordinates), so that the and positions fix this solution uniquely, to the type sketched in Fig. 2a as opposed to Fig. 2b. In Fig. 5c we superpose all five foliations (without including the corresponding threads), to show how are the individual leaves of the foliations related to each other.

Since for a given thread bundle only a part of its corresponding foliation is used, our remaining task is to show that the utilized regions are mutually compatible. In particular for the flow associated to the bulk region (Fig. 5a), we only use the foliating minimal surfaces for normal congruence passing through and as indicated, and similarly for the other thread bundles. In Section 3.4.1 we have argued that the minimal surfaces , , , and can be partitioned in a compatible fashion, i.e. satisfying the bottleneck equations (3.6). Hence what remains is to show that the rest of the bundles can likewise accommodate themselves so that distinct thread bundles don’t overlap each other.

Let us consider, say, the compatibility of and flows (the arguments for the compatibility of the other flows being identical). We want to show that and do not intersect. The most ‘worrisome’ regions are where the thread bundles approach closest to each other, i.e. just around the minimal surface . Both flows are maximally constrained at , and they pass normally through it (consistently with belonging to both foliation families, and , cf. Fig. 5a and Fig. 5b). Since at itself we ensured that the flows are compatible by the partitioning, to see that subsequently the thread bundles bend away from each other, let us consider a leaf of the and foliations which have a common right endpoint , just ‘above’ . The corresponding left endpoints are distinct: , while , with the -leaf’s defining interval lying within the -leaf’s interval (as the reader may try to verify by a careful look at Fig. 5c). This means that the leaf bends further to the left than the leaf (recall that the leafs cannot intersect since otherwise they would contradict the assumption of minimality in their construction). Therefore, their normals (which guide their respective flows) likewise bend away from each other: (pointing towards ) more to the left and more to the right. We can make a similar argument for the behavior of the flow ‘below’ where the flows likewise bend away from each other in the downward direction. This means that the flows remain compatible around their bottleneck .

We can now make a more general argument, applying to the entire remainder of and thread bundles of the flow. Consider any bulk point defined by the intersection of the -foliation and -foliation leafs (which, except for points lying on , is uniquely specified by and . Inside ’s entangling region, and simultaneously, in order for the leafs to intersect, the corresponding intervals cannot be nested, so . Moreover, the leafs cannot have multiple intersections. This means that at the intersection point, the normal for the leaf must bend more towards the region than the normal to the leaf.

Identical analysis for the remaining places where the thread bundles touch on one of the minimal surfaces then shows that all thread bundles bend away from each other. Namely, once they are compatible on the minimal surfaces (i.e. satisfy the bottleneck equations (3.6)), then they are compatible in the rest of the bulk — in other words they generate cooperative flows.

3.4.4 Single local foliation for all thread bundles, patched flows

The previous construction required not just one, but five complete overlapping foliations to argue for cooperation (in particular to verify that the endpoints of foliating geodesics were such as to channel the thread bundles away from each other, cf. Fig. 5c). Since we ended up using only parts of these foliations, it might seem a bit more economical to consider just a single set of curves within the bulk which would generate all the thread bundles in one go. Of course such a set cannot truly be a foliation in the strict sense that for every bulk point there exists a unique leaf of the foliation which contains , because the topology of the thread bundles would necessitate some junctions in the leaves. However, what really matters is not that we cover the entire bulk, but rather that the leaves don’t intersect. Once we exclude the junctions, this weaker (uniqueness) requirement is certainly possible to satisfy. We will call such partial foliations (which we could think of as foliations of the bulk with some points removed) local foliations.

One (impractical) way to achieve a single local foliation of the bulk would be to take the set of foliations constructed in Section 3.4.3, cut them off at their respective thread bundle’s edges (which are determined by the minimal surface partitions), and join the leafs through the remaining bulk gaps. However, this offers no particular (either constructive or logical) simplification. It also does not take the opportunity to avoid what is perhaps the most awkward feature of the previous construction, namely that the thread bundles between adjoining regions in Section 3.4.3 appear to start and end at the same boundary point. Here we remedy this by a more convenient choice of local foliation, indicated in Fig. 6.

Figure 6: A local foliation designed to obtain the thread bundles indicated in Fig. 4b. The 6 black line segments along with the 4 minimal surfaces break up the bulk into 9 different regions, each foliated by its own family of geodesics.

In the homology region313131 By ‘homology region’ for , we mean the bulk region bounded by the given boundary region and its associated bulk minimal surface , cf. footnote 42 for a more covariant definition. for each of the boundary regions , we use the foliation generated by symmetrically nested boundary intervals as in Fig. 3a. All thread bundles  emanate from ’s midpoint, and automatically reach the corresponding minimal surface perpendicularly. On the 4 minimal surfaces we pick a partition satisfying (3.6), which specifies 6 bulk points labeled , , , , , and as shown in Fig. 6 (the subscripts indicate where the given thread bundles separate, cf.Fig. 2). Consider the triangle formed by , , and , and take its ‘center’ (to be specified in footnote 32) point , and similarly for the junction. Using these 6+2 points, construct 6 (geodesic) line segments joining the vertices of each triangle to its center point. These segments partition the central quadrilateral region (bounded by the 4 minimal surfaces ) into 5 bulk regions, one for each thread bundle, thus partitioning the full Poincare disk into 9 regions.

To construct the five thread bundles, we proceed analogously to the previous construction, but with the crucial difference that we don’t specify these geodesics by their endpoints on the boundary, but rather by their location along these interior line segments. Concretely, the foliation for thread bundle  is given by the family of geodesics between and the interior bulk point lying in the segments , with at and at . Since all geodesics ‘fan out’ from , this is part of a foliation of the type indicated in Fig. 3b. Moreover, our choice of guarantees323232 The central point is defined as any point which lies within the triangle formed by the threads passing through the three pairs of vertices in the construction of Section 3.4.3. (Hence we retain a lot of freedom in this construction.) Since these threads bend away from their tangents at the minimal surfaces , they cannot intersect the line segments extending these tangents (pieces of which correspond to the segments and ). that the corresponding threads from reach and vice-versa, without leaving this region.

The other three thread bundles between adjoining regions (namely , , and ), are constructed in identical fashion. The remaining bundle, , can actually be generated as a part of the foliation indicated in Fig. 3c. And by the same argument as above, these threads joining and cannot leave the hexagonal region . This finishes the alternate construction of the thread bundles generating a cooperative flow.

Notice that unlike the previous case discussed in Section 3.4.3 which left 8 bulk regions unpopulated by any thread bundle, we now have only two unpopulated regions (the triangles mentioned above).333333 The reader might at this point wonder why the latter appears to have manifestly smaller volume (in fact a finite one) than the former (which has infinite volume owing to some of the regions adjoining the boundary), despite the fact that in both cases we kept all our thread bundles maximally collimated. This apparent discrepancy stems from the usual issue of the UV divergences when dealing with the entire bulk spacetime. In particular, the threads of the present construction include those which ‘hug’ the boundary and therefore utilize greater bulk volume than the threads of Section 3.4.3. In fact, we can reduce this further to just a single unpopulated region, by using only two intersecting line segments, and say one joining midpoint of to midpoint of . The , , , and thread bundles would remain as before, but the thread bundle now gets split into two kinked sub-strands, one adjoining the and thread bundles and the other adjoining the and thread bundles. The kinks appearing along the segments pose no particular problem (the norm bound is satisfied everywhere) other than the flow field being strictly-speaking undefined at the segment. We can however locally smooth out the corners to avoid this.

The overall lesson is that there are many ways to construct cooperative flows generated by local foliations (using minimal surface segments). The resulting thread bundles are automatically disjoint and maximally collimated. The gaps they leave behind in the bulk have no physical significance – they (as well as precise position of the thread bundles themselves) may be viewed analogously to a gauge choice. On the other hand, the freedom of satisfying the bottleneck equations (e.g. how far along the surface can the thread bundle shift) may have a more direct significance in terms of multipartite entanglement (cf. MattMMI ()’s interpretation of ) which would be interesting to explore further. It is worth noting that these thread bundles constitute a special case of a multiflow discussed in MattMMI (), where the more general ’s of MattMMI () in our construction all saturate the norm bound individually in some regions and vanish entirely in others. We therefore provide an explicit construction of a special type of a ‘max multiflow’ (in MattMMI ()’s language).

4 Generalizations

In Section 3 we constructed cooperative flows in the simplest class of examples: spatial slice of pure AdS with boundary partitioning given by 4 adjoining intervals. This starting point was chosen primarily for ease of illustration, and our arguments in fact apply much more broadly. In this section we examine various natural generalizations. These entail considering more general partitions (still restricted to the form relevant for MMI), more general state (which modifies the asymptotically AdS bulk geometry and allows for time dependence), or even a different theory (including changing the number of dimensions and number of background spacetime components on which the field theory lives).

We conjecture that the basic construction outlined above can be adapted to any of the above-mentioned situations, describing a generic classical bulk spacetime satisfying physical energy conditions. More specifically:

Conjecture 4.1.

In any holographic CFT state whose gravitational dual corresponds to a classical bulk spacetime with entanglement entropy computed by the HRT prescription, a cooperative flow for any boundary spatial partition exists and can be constructed by maximally collimated thread bundles confined to non-intersecting regions of a bulk Cauchy slice.

Rather than attempting a full analysis (which we leave for future work), we provide some basic arguments and simple checks of our conjecture, intended more as a plausibility argument than a proof. Instead of considering the most general case from the outset, we address how our method can be adapted for each of these generalizations individually. In particular, we consider generalizing the number of spatial regions in Section 4.1, generalizing the bulk geometry (within a class of time-reversal symmetric asymptotically AdS spacetimes) in Section 4.2, generalizing to higher dimesions in Section 4.3, and finally allowing for genuine time-dependence in Section 4.4.

4.1 Multiple sub-partitions

Recall that the case of 4 adjoining intervals , , , and discussed in the previous section has either or , so the more immediate proof of Section 3.2 suffices to guarantee the existence of cooperative flows: in particular, we can pick and to be arbitrary and specify the remaining maximizer flows accordingly, without any need to split them into disjoint thread bundles. We now demonstrate explicitly that in the more general situation with and simultaneously , the ‘combed thread bundle’ method of Section 3.4 indeed applies. We will first briefly outline a specific construction and then indicate what happens in full generality.

Figure 7: (a) An example of partitioning of the boundary space into 4 subsystems , , , , analogously to Fig. 1, but now consisting of 5 simple regions as labeled. (b) Corresponding thread bundle construction analogous to Fig. 4a, which delineates the associated bulk regions connecting the requisite boundary regions as specified in the text.

The simplest type of configuration which has all pairwise mutual informations non-vanishing is shown in Fig. 7a. Here we take the system to be larger than the one, automatically implementing . We also take with adjoining , so that (where the inequality follows by monotonicity of mutual information (SSA), and the RHS is the usual property of adjoining regions). This means that in decomposing the flows into thread bundles, we will necessarily have , , , , , as well as in order to accommodate the non-vanishing mutual information between the corresponding intervals. On the other hand, by the same argument as above, which (again by monotonicity along with non-negativity) implies that and . That in turn means that we do not need to turn on the or thread bundles.

There are two additional thread bundles to consider: and , but turning on both of these maximally collimated thread bundles simultaneously would not produce cooperative flows, since the thread bundles necessarily intersect and therefore in the common region exceed the norm bound. However, it is easy to see that we can always choose one of these thread bundles to vanish: If , then we can set (so in this case cooperative flows exist trivially).343434 In particular, by argument of Section 3.2, we can set and , which gives and , manifesting all as flows. On the other hand, if , then the complementary regions must be uncorrelated: which means that , so that we can set . Let us then consider this latter case.

The first thing to check is that the requisite thread bundles can fit through the bottlenecks in a disjoint fashion. We have 5 minimal surfaces (, , , , and ) and 7 bulk regions containing the 7 thread bundles which form the building blocks for the flows, which we wish to prove to be mutually non-overlapping (, , , , , , and ), cf. Fig. 7b (which shows the actual thread construction). These thread bundles must partition the minimal surfaces so as to satisfy a set of bottleneck equations analogous to (3.6). These consist of 5 equations (one for each interval ) of the form


where denotes one of the 5 simple boundary intervals and denotes all the thread bundles with cross , and 7 equations (one for each thread bundle z) of the form


where and is a pair of simple boundary intervals joined by a thread bundle . Since each thread bundle gives a partition for two minimal surfaces (associated with the regions the thread bundle connects), we have total of 14 partitions. Hence we have 12 equations for 14 unknowns, which gives a 2-parameter family of solutions; we leave it as an easy exercise for the reader to write out the explicit equations and confirm that a solution must necessarily exist.

So far, we have ascertained that the 5 minimal surfaces can be partitioned so as to accommodate all the thread bundles correctly. We can now employ the same arguments as in Section 3.4 to verify that by using minimal surface foliations, the remainder of the thread bundles must bend away from each other. Either of our explicit constructions is generalizable, but it’s most convenient to adopt the first one, using 7 global foliations. Since the reasoning presented at the end of Section 3.4.3 was local in the sense of involving only two adjoining thread bundles, we can employ exactly the same argument in the present case. The explicit thread bundle construction is given in Fig. 7b. As for the simple case, we can guide the bundles using only the two foliation families depicted in Fig. 3b (for bundles ) and Fig. 3c (for bundles  and ). This choice again fixes the solution to the bottleneck equations uniquely. We could also introduce more freedom by using only the separated endpoint form of the bundles, or even more generally, a hybrid of the two methods. This proves the existence of a cooperative flow for the 5-region case, where the methods of Section 3.2 (utilizing vanishing mutual information) would have failed.

It should be evident by now that we can generalize our construction to arbitrarily large set of boundary intervals , , , and , with arbitrary sizes and ordering. To characterize the system of equations we have to solve, it is convenient to represent any such configuration by a graph, with nodes corresponding to the boundary regions and links to the thread bundles joining them. In order for the thread bundles to be compatible, they cannot intersect, which means that the graph must be planar.353535 While planarity is not a fundamental requirement in full generality (e.g., it can easily be violated in the