Towards a Bit Threads Derivation of Holographic Entanglement of Purification
Ning Bao, Aidan Chatwin-Davies, Jason Pollack, Grant N. Remmen
TTowards a Bit Threads Derivation ofHolographic Entanglement of Purification
Ning Bao, a,b
Aidan Chatwin-Davies, c Jason Pollack, d and Grant N. Remmen a a Center for Theoretical Physics and Department of PhysicsUniversity of California, Berkeley, CA 94720, USA andLawrence Berkeley National Laboratory, Berkeley, CA 94720, USA b Computational Science Initiative, Brookhaven National Lab, Upton, NY 11973, USA c KU Leuven, Institute for Theoretical PhysicsCelestijnenlaan 200D B-3001 Leuven, Belgium d Department of Physics and AstronomyUniversity of British Columbia, Vancouver, BC V6T 1Z1, Canada
Abstract
We apply the bit thread formulation of holographic entanglement entropy to reducedstates describing only the geometry contained within an entanglement wedge. We arguethat a certain optimized bit thread configuration, which we construct, gives a purifica-tion of the reduced state to a full holographic state obeying a precise set of conditionalmutual information relations. When this purification exists, we establish, under certainassumptions, the conjectured E P = E W relation equating the entanglement of purifi-cation with the area of the minimal cross section partitioning the bulk entanglementwedge. Along the way, we comment on minimal purifications of holographic states,geometric purifications, and black hole geometries. e-mail: [email protected] , [email protected] , [email protected] , [email protected] a r X i v : . [ h e p - t h ] J u l ontents E P E P
175 Discussion 21A Holographic Purification of Minimal Dimension 22 Introduction
The relation between entanglement and (holographic) spacetime [1–3] is a powerful connec-tion that has been extended through many interesting developments over the past decade.Perhaps the most elegant of these extensions is the Ryu-Takayanagi (RT) formula, S [ ρ ] = A G N (cid:126) , (1)relating the entanglement entropy S [ ρ ] = − Tr ρ log ρ of the reduced density matrix ρ of aholographic large- N CFT in a boundary region to the area A of the minimal bulk surfacehomologous to that region [4–6]. This result, among others, has led to a concrete instantiationof the emergence of spacetime from entanglement in the context of AdS/CFT [7–10].Recently, the connection between boundary information-theoretic quantities and bulkareas has been extended to a conjectured “ E P = E W ” relationship between the area of bulk-anchored minimal surfaces and the entanglement of purification [11], as in Refs. [12,13]. Theentanglement of purification is an entanglement measure that characterizes the degree ofentanglement between two subsystems of a generically mixed state. It is particularly usefulwhen only partial information about a state is known. As we review below, the entanglementof purification is a bound on the entanglement between subsystems in a given purification;purifications that saturate this bound comprise a class of optimal purifications or completionsof the state in question. The E P = E W conjecture has been further extended, studied, andgeneralized in recent work [14–22].In this paper, we will seek to make progress towards proving the E P = E W conjectureusing technology from the bit threads program as described in Ref. [23]. This program,which in its original formalism [23] is formally equivalent to the RT prescription, providesuseful tools and visualizations for understanding the structure of entanglement in holographicstates. In essence, the bit threads formulation of the RT formula replaces minimization ofareas of surfaces with maximizations of constrained flows of vector fields through surfaces.These vector fields are envisioned as providing, at least heuristically, connections betweenBell pairs of maximally-entangled qubits in the boundary state.The usual conceptual flow of the bit threads program is to pass from a given state toa bit thread geometry describing (aspects of) the entanglement structure of the state. Inthis paper—and in the context of a holographic correspondence between boundary statesand bulk geometries—we wish to invert this logic: we conjecture that, under certain cir-cumstances discussed below, the existence of a bit thread geometry satisfying particular The bit threads formalism has further been extended to include multiflows and thread configurations [24]. implies the existence of a state with subsystem entanglements satisfy-ing the analogues of these constraints. Given this conjecture, we can replace the (extremelyhard) task of specifying a particular holographic state directly in the boundary CFT with amore tractable geometric construction on the level of a classical bulk geometry. Because weexpect that geometrical information is encoded redundantly in the boundary theory in themanner of an error-correcting code (see, e.g., Ref. [25]), we do not expect this constructionto uniquely specify a quantum-mechanical state. Nevertheless, we show in this paper thatit provides enough information to compute the information-theoretic quantity dual to thecross-sectional area E W and hence establish that E P = E W .The organization of this paper is as follows. In Sec. 2, we review some facts about bitthreads and the entanglement of purification. In Sec. 3, we argue that the entanglementof purification is bounded from above by the area of the entanglement wedge cross section,while in Sec. 4 we show, under certain assumptions that we clarify, that the entanglementof purification is lower-bounded by the area of the entanglement wedge cross section, estab-lishing the E P = E W relation. Finally, we conclude in Sec. 5, with a bonus related result onthe minimal dimension of holographic purifications given in App. A. We begin by introducing some definitions and briefly reviewing the concepts of entanglementof purification and bit threads.
Let ρ AB ∈ L ( H AB ) be a state in the bipartite Hilbert space H AB = H A ⊗ H B . The entan-glement of purification of ρ AB is E P ( A : B ) = inf A (cid:48) B (cid:48) S ( AA (cid:48) ) , (2)where the infimum is taken over all auxiliary Hilbert spaces A (cid:48) B (cid:48) and all pure states | Ψ (cid:105) AA (cid:48) BB (cid:48) such that Tr A (cid:48) B (cid:48) | Ψ (cid:105)(cid:104) Ψ | = ρ AB . The entanglement of purification is an entanglement measurethat quantifies the minimum amount of correlation required in any purification of a mixedstate, subject to the constraint that it must preserve the factorization structure between A and B ; this is particularly useful, for example, when only a mixed subset of the full purestate is known.When H AB is a subfactor of the Hilbert space of a holographic CFT and ρ AB is thereduced state on boundary subregions A and B of a state with a well-defined dual geometry,4he entanglement of purification E P has been conjectured to be equal to the area E W ofthe entanglement wedge cross section Γ [12]. The entanglement wedge cross section Γ is asurface of minimal area anchored to the boundary of the entanglement wedge W AB , suchthat Γ partitions W AB into a region that is entirely adjacent to A and a region that is entirelyadjacent to B ; see Fig. 1. A B AB AB Figure 1: Entanglement wedge for AB , bounded by the RT surfaces γ AB . The entanglement wedge crosssection Γ is illustrated by the red dashed line. In the case of a finite-dimensional Hilbert space, dim H AB = d AB < ∞ , the infimum E P ( A : B ) can be achieved with auxiliary Hilbert spaces of dimensions d A (cid:48) = d AB and d B (cid:48) = d AB [11]. Note that while the Hilbert spaces of AdS/CFT are infinite-dimensional,we will later consider UV regularizations that render them finite-dimensional. Therefore, wecan assume that the infimum is achievable in regulated holographic settings.The state that achieves the entanglement of purification as S ( AA (cid:48) ) is of course not unique.(For example, given a state | Ψ (cid:105) that achieves E P ( A : B ) , any other state of the form U A (cid:48) ⊗ U B (cid:48) | Ψ (cid:105) also achieves E P ( A : B ) .) One should therefore think of entanglement ofpurification as picking out an equivalence class of “optimal” states on Hilbert spaces ofpossibly different dimensions that are all isometrically related. In App. A, we show that ifone holds a purification of holographic ρ AB in H CF T that achieves E P ( A : B ) , then one cancompress the purifying state on ( AB ) c to a Hilbert space with log dim H A (cid:48) B (cid:48) = S ( ρ AB ) .5 .2 Bit threads Bit threads were first introduced as a by-product of a reformulation of the RT formula interms of flows by Freedman and Headrick in Ref. [23]. Here, we introduce only the minimumresults required for our purposes, leaving the details to Ref. [23].Let M be an oriented Riemannian manifold with boundary and let C > . A flow is avector field v such that ∇ µ v µ = 0 and | v | ≤ C . Given a flow, one can of course compute theflow’s flux through any sufficiently smooth codimension-one surface m : (cid:90) m v = (cid:90) m √ h n µ v µ , (3)where h is the induced metric on m and n is the normal to m .The crux of Freedman and Headrick’s reformulation of the RT formula is the Max-Flow/Min-Cut (MFMC) Theorem [28–30], which can be stated as follows. Let A be aboundary subregion of M . Then max v (cid:90) A v = C min m ∼ A area( m ) , (4)where m ∼ A denotes that m is homologous to A . MFMC states that for any flow on M that attains the largest possible flux out of A , the value of this flux is equal to thelargest transverse flow density C multiplied by the area of a minimal surface in M that ishomologous to A . In other words, the max flow necessarily saturates any bottleneck out of A in both magnitude and direction.Letting M be a spatial slice of a static holographic geometry and C = 1 / G N (cid:126) , it shouldnow not seem surprising that one can rewrite Eq. (1) as S ( A ) = max v (cid:90) A v . (5)This was rigorously demonstrated in Ref. [23]. Bit threads are the integral curves of a max flow with transverse density | v | . As a resultof Eq. (5), one can think of S ( A ) as counting the largest number of bit threads leaving A that can pass through the RT surface for A . Due to the non-uniqueness of the max flow, onecan also compute further entropic quantities by counting bit threads of appropriately chosenflows. We will comment further on how this is done in Sec. 3 when we begin our calculationsof this type. The formulation of the RT prescription in terms of flows can also be understood [26] from the perspectiveof calibrations defined on Riemannian manifolds [27]. Upper Bound on E P Let ρ AB be a holographic CFT state on the subregions A and B whose geometric dual is theentanglement wedge of AB . In this section, we establish an upper bound on its entangle-ment of purification, E P ( A : B ) ≤ E W ( A : B ) , under an assumption regarding the relationbetween the entanglement structure of geometric purifications of the state and a bit threadflow in the entanglement wedge. To do so, it suffices to exhibit a purification for which S ( AA (cid:48) ) = E W ( A : B ) . We will show that an arbitrary purification that is everywhere holo-graphically dual to an asymptotically-AdS bulk geometry admits a factorization | Ψ (cid:105) AA (cid:48) BB (cid:48) Y for which S ( AA (cid:48) ) = E W ( A : B ) . Such a purification is guaranteed to exist by the fact thatgeometric subregions of an AdS geometry (in this case the entanglement wedge) can alwaysbe extended to a full asymptotically-AdS bulk geometry. We now specify how to determine S ( AA (cid:48) ) for this state and for a particular choice of A (cid:48) , which we fix by using the bit threadformalism to specify various conditional mutual informations. Consider two boundary subregions A and B , as well as the bit threads of some flow on ageometric purification of ρ AB . The ways in which bit threads can intersect the RT surfaceof AB (which in general consists of multiple disconnected pieces) are sketched in Fig. 2.Threads that start in AB and terminate in ( AB ) c are to be associated with the subfactorof H ( AB ) c that purifies ρ AB . That is, ( AB ) c contains many extraneous degrees of freedomthat are uncorrelated with ρ AB and hence do not purify our reduced density matrix. Thiscan be seen, for example, by looking at small boundary subregions X ⊂ ( AB ) c that are faraway from ∂ ( AB ) , as shown in Fig. 3. Such regions have zero mutual information with AB In the case where ρ AB resulted from tracing out ( AB ) c in a holographic state, then one trivial everywhere-geometric purification is the original state. More generally, on the level of the bulk geometry, a geometricpurification corresponds to gluing the entanglement wedge of AB to other subregions in such a way that theentire geometry is asymptotically AdS. In arbitrary dimensions such a gluing construction may be nontrivialfor general geometries, but it has been shown that gluing another copy of the entanglement wedge to itselfalong RT surfaces is always possible, forming a canonical purification [19, 31]; in AdS , where the Weyltensor vanishes identically, even more general gluing constructions should be fairly straightforward. Notethat it may be necessary to introduce additional copies of the CFT if, for example, AB is already a fullboundary and ρ AB is a mixed state, such as in the case of a thermal single-sided black hole. A B AB AB Figure 2: Types of bit threads that can cross the RT surface, γ AB , of AB for an arbitrary flow. Theproperties that distinguish different types of bit threads are where they begin and end ( A , B , or ( AB ) c ), aswell as how many times they intersect γ AB and the entanglement wedge cross section Γ (red dashed line). and the state factorizes in the large- N limit: ρ ABX = ρ AB ⊗ ρ X . (6)As such, we conclude that it is only a subset of the degrees of freedom, H ( AB ) (cid:48) ⊂ H ( AB ) c ,that purify ρ AB ; bit threads are to be associated with this smaller set of boundary degrees offreedom. Although we have chosen notation here to suggest a division H ( AB ) (cid:48) = H A (cid:48) ⊗ H B (cid:48) ,so far we have not actually specified such a division, nor in fact required that H ( AB ) (cid:48) evenfactorize. In particular, H ( AB ) (cid:48) need not have any geometric locality, beyond simply that itis supported in ( AB ) c . This is a reflection of the freedom in choosing the flow, i.e., the factthat the location where the threads terminate in ( AB ) c can be slid around freely anywhere in In field theory, the fact that the vacuum state is entangled at all scales implies that even local operators welloutside the entanglement wedge might change the reduced state ρ AB . A finite-energy particle excitation at aparticular spacetime location in fact has (exponentially small) support on all of space. So strictly speaking,we should throughout replace any operator acting outside the entanglement wedge with a different operator,with cutoff-scale energy, that genuinely does not change the reduced state of the entanglement wedge. Or,perhaps preferably, we could instead not demand that the new reduced state be exactly the same as theold, but only the same up to exponentially small differences. That is, whenever we factorize the Hilbertspace of a CFT, we can either replace statements of factorization with appropriate statements involvingvon Neumann algebras (and/or bulk reconstruction) or edge modes, or we can embrace the existence of thecutoff and work with a latticized theory that lacks these issues. In the remainder of this paper, we take thelatter, more pedestrian approach. It should be noted that (even beyond the caveats just noted in footnote 4) this factorization is approximate,to leading order in /N , since bit threads are equivalent to the RT formula, which only computes theentropies to leading order in /N . We are helped, however, by the fact that the trace distance between theexact state and the factorized state discussed here will be small, so Fannes’ inequality [32] guarantees thatthe difference in the entanglement entropies of subsystems of the two states will also be correspondinglysmall. BX ⇢
ABX ⇡ ⇢ AB ⌦ ⇢ X Figure 3: Boundary subregions whose reduced density matrix approximately factorizes. ( AB ) c , and so they are not associated with literal, local “boundary qubits.” For example, thethreads could be evenly spaced out, or we could bunch them up together about a particularlocation on the boundary, etc. In order to fix the division of ( AB ) (cid:48) into A (cid:48) and B (cid:48) , we now construct a specific configurationof bit threads that simultaneously maximizes the number of threads crossing the RT surfaceof AB , which we denote by γ AB , and the number crossing the entanglement wedge crosssection Γ . Note, however, that such a collection of threads cannot be the flow lines of aneverywhere-continuous and -divergenceless flow with bounded norm. According to MFMC(4), if a flow v indeed maximizes the flux out of AB , then v must be normal to γ AB and have | v | = 1 / G N (cid:126) everywhere on γ AB . But then, the flux through Γ cannot be strictly equal to | Γ | / G N (cid:126) since, on the codimension-three surface where Γ intersects γ AB , v is perpendicularto the normal of Γ .Therefore, we must necessarily relax the definition of bit threads as integral curves ofa flow. We will simply take bit threads to be boundary-anchored one-dimensional objectswhose density is at most / G N (cid:126) , defining thread density, as in Ref. [14], as the length ofthreads within some small neighborhood divided by the volume of that neighborhood. Thisgeneralization was proposed in Ref. [24], in which a collection of such threads was called a“thread configuration.” Although the authors of Ref. [24] further relaxed the requirementthat bit threads be oriented, we will find it helpful to still think of our bit threads as havingan orientation.The use of this more general notion of thread configurations is very mild for our purposes,since ultimately we will only be concerned with computing entropic quantities by countingbit threads, as opposed to exploiting MFMC to identify the bottleneck of a flow. In other9ords, we will always assume that the relevant extremal surfaces are known, and we willonly need to count threads that cross these surfaces. Moreover, the only place where theassociated flow becomes multivalued is on the (codimension-three) surface Γ ∩ γ AB , so withbounded flux density, the region of this ambiguity contributes only a measure-zero fractionof the flux through Γ or γ AB .With this generalization in mind, we can now describe how to construct a thread con-figuration that saturates the number of threads crossing γ AB and Γ . Let us suppose thata UV cutoff near the boundary has been imposed, so that the areas of boundary-anchoredsurfaces are finite and well defined. First, begin by letting S ( A ) = | γ A | / G N (cid:126) thread frag-ments emanate from A and S ( B ) = | γ B | / G N (cid:126) thread fragments flow into B , where γ A and γ B denote the RT surfaces of A and B individually and where the thread fragmentsrun from the boundary to γ A and γ B for now. These are the largest numbers of threadsthat we can pipe from A to the bulk and to B from the bulk without violating the densitybound anywhere. The reader will note that we have suggestively labeled the numbers ofthreads using the same symbol as entropy, and we will proceed to work with “entropies” and“mutual informations,” but, as we discuss in the next subsection, a priori these quantitiesare purely geometric and need not be associated with the entanglement entropy of reduceddensity matrices constructed from a pure state.One of our goals is to saturate the number of threads that cross γ AB , so S ( AB ) = | γ AB | / G N (cid:126) of the fragments will have to cross γ AB and leave the entanglement wedge.Therefore, take [ S ( A ) + S ( B ) − S ( AB )] / of the thread fragments from A and the samenumber from B (i.e., the leftover fragments) and join them by having them cross Γ . At thisstage, there are thus [ S ( A ) + S ( B ) − S ( AB )] / I ( A : B ) / bit threads crossing Γ .Next, since E W ( A : B ) ≥ I ( A : B ) / (as shown in Refs. [12, 23]), we may still needto pipe some of the thread fragments through Γ . This is just a matter of ensuring that E W ( A : B ) − I ( A : B ) / of the remaining S ( AB ) fragments cross Γ before going on tointersect γ AB , after which the threads are sent to the boundary where they terminate in ( AB ) c . In particular, we can see that there are enough remaining threads to saturate Γ asfollows: S ( AB ) + I ( A : B ) / [ S ( A ) + S ( B ) + S ( AB )] ≥ max { S ( A ) , S ( B ) }≥ E W ( A : B ) . (7)In going to the second line we used the Araki-Lieb inequality, and to go to the third line weused the inequality E W ( A : B ) ≤ min { S ( A ) , S ( B ) } [12]. The result is a thread configuration10ith S ( AB ) threads crossing γ AB and E W ( A : B ) threads crossing Γ , thus saturating bothsurfaces as desired (see Fig. 4c). A B A B (a) (b)
A B A B combcombnucleate site absorb sitenucleate siteabsorb site (c) (d)
Figure 4: Starting with a thread configuration that saturates the number of threads intersecting γ AB , γ A ,and γ B , as shown in (a) and described in Ref. [14], the goal is to construct a configuration like (c), whichalso saturates Γ . (The parts of threads that cross the exterior component of γ AB and that lie outsidethe entanglement wedge have been suppressed in these diagrams.) Diagram (b) illustrates how this canbe done by cutting and gluing threads. First, with a UV cutoff in place, it is helpful to think of therebeing a finite number of sites (yellow dots) on both γ AB and Γ that threads must intersect. Initially, only I ( A : B ) / sites on Γ are filled by threads, which form a tube running through Γ . Divide the remaining E W ( A : B ) − I ( A : B ) / sites into a group of n hi sites above the tube and n lo sites below the tube. (In twospatial dimensions the division is unique, but in higher dimensions there may be freedom in this choice.)Then, cut the I ( A : B ) / threads that cross Γ , as well as n hi threads that intersect the exterior componentof γ AB on the side adjacent to B and n lo threads that intersect the interior component of γ AB on the sideadjacent to A . The locations of the cuts are indicated by red crosses in (a). Finally, glue adjacent cut threadstogether as shown in (b) to arrive at the maximizing configuration (c). This cutting-and-gluing procedureis equivalent to “combing” the original thread configuration, as depicted in (d). Combing means dragging n hi threads that intersect sites on the A -adjacent side of the exterior component of γ AB over to sites on the B -adjacent side, and vice-versa for n lo sites on the interior component of γ AB . The only additional subtletyis that, with a UV cutoff in place, we must think of n hi threads being nucleated from the UV where theexterior component of γ AB meets A and n hi sites being absorbed by the UV where the exterior componentof γ AB meets B . The same is true for n lo threads hitting the interior component, with A and B flipped.Although the diagrams as we have drawn them here are directly reflective of AdS /CFT , we do not believethat there are any barriers to combing and cutting-and-gluing in arbitrary dimensions. Alternatively, we can obtain our γ AB - and Γ -saturating thread configuration directly from11he thread configuration constructed in Sec. 5.2.2 of Ref. [14], via either cutting-and-gluingor combing. There, Agón et al. construct a configuration of threads that saturates thenumber of threads crossing γ AB , but not Γ (see their Fig. 11). From their configuration,one can arrive at a new configuration that saturates both quantities by cutting threads oneither side of Γ and rejoining them with neighboring threads inside the entanglement wedgeto pipe E W ( A : B ) threads through Γ ; this is illustrated in Fig. 4a-b. This is equivalent to“combing” the thread configuration by dragging the threads that intersect γ AB in oppositedirections along the interior and exterior disconnected components of γ AB , as illustrated inFig. 4d. We refer the reader to the caption of Fig. 4 for more details.Agón et al.’s construction applies specifically to the case where AB is a proper subset ofa single boundary CFT and W AB is simply connected. Nevertheless, a thread configurationthat maximizes the number of threads intersecting γ AB and Γ can always be constructedaccording to the first prescription that we gave above. This is because the inequalities S ( AB ) ≤ S ( A ) + S ( B ) , I ( A : B ) / ≤ E W ( A : B ) ≤ min { S ( A ) , S ( B ) } , and S ( AB ) ≥| S ( A ) − S ( B ) | guarantee that the right number of thread fragments can always be connectedacross γ AB or Γ and attached to A and B as we described to construct the maximizingconfiguration. A qualitatively different example is the case where W AB is disconnected, i.e., γ AB is just the union of γ A and γ B . In this case E W ( A : B ) = 0 and so there is no surface Γ to saturate. Another example is when AB is an entire CFT boundary but ρ AB is a mixedstate, such as the case of a single-sided mixed state black hole shown in Fig. 5. In this latterexample, threads that leave W AB terminate on the black hole horizon. Regulating lengths of surfaces near the boundary using a cutoff (cid:15) at finite coordinatedistance into the bulk, the length of a generic boundary region in Planck units scales as (cid:15) − ,while boundary- and bulk-anchored surfaces scale as log (cid:15) and (cid:15) , respectively. Thus, therewill always be a great excess of available bit threads anchored to A and B to saturate theflow through the RT and entanglement wedge surfaces. Viewing the bit thread configurationsas integral curves specified by a choice of vector field in the bulk, we must guarantee thatthe fields are chosen orthogonal to both the entanglement wedge cross section Γ and the RTsurface γ AB . As mentioned above, this is always possible except at Γ ∩ γ AB , which introduceserrors only of measure-zero in the net flux. See Fig. 5 for representative examples.The possible types of bit threads in a γ AB - and Γ -saturating configuration are shown inFig. 6. Here, we have only indicated threads that intersect W AB . Although for a general Alternatively, we can think of the threads as crossing through a wormhole and terminating in anotherasymptotic boundary in which the black hole is purified. AB AB AB A B A B
Figure 5: Representative examples of bit thread configurations simultaneously maximizing the flow throughthe entanglement wedge surface Γ (red dashed line) and RT surface γ AB (green line). The vector fieldsspecifying the flow are indicated on γ AB and Γ by arrows, and the individual RT surfaces for A and B aredepicted with purple dashed lines. Left: A and B are two surfaces defining a proper subregion of a singleboundary, on which the CFT state is pure. Right: A and B partition an entire boundary, on which is defineda mixed CFT state, resulting in a horizon in the bulk. thread configuration like that of Fig. 2 we could have threads that cross the RT surfacemultiple times, we can without loss of generality take such threads to be absent in a config-uration (like that of Fig. 6) that obeys the maximization condition. This can be understoodas follows. In the maximum-flux configuration, | γ AB | / G N (cid:126) threads cross γ AB and are an-chored to AB . Bit threads that pass through γ AB an even number of times contribute zeronet flow out of γ AB , while those that pass through an odd number of times contribute thesame net flow as those that pass through once. Hence, we can take the threads that passthrough γ AB to do so exactly once, without decreasing the maximum net flow out of AB .By the same token, since we are considering bit threads that simultaneously saturate theentanglement wedge cross section Γ , we can take the bit threads to cross Γ at most once,and there are E W ( A : B ) bit threads that do so.Moreover, threads that start on A and end on A (or start on B and end on B ) are notespecially meaningful for our analysis. According to the maximization condition, there mustbe | γ AB | / G N (cid:126) threads passing through γ AB and E W ( A : B ) threads passing through Γ .However, since | AB | can be large compared to E W and | γ AB | (but finite, once a UV cutoffis in place), it is possible to include some additional threads that start and terminate on A or B , while still retaining our max-flow conditions. However, such threads would necessarilycross either γ AB or Γ an even number of times (including zero), and hence would contributenothing to the thread configuration that we are maximizing. That is, such threads will notbe relevant for any of the mutual informations we consider in our later calculations, so we13 A B AB AB Figure 6: The types of bit threads that can cross the RT surface of AB for the configuration that maximizesthe flow out of AB as well as the flux through the entanglement wedge cross section Γ (red dashed line).We have highlighted threads that have one end in A or B and cross both Γ and the RT surface in green orblue, respectively. suppress such extraneous threads from all subsequent figures and discussion.Thus, we can take all bit threads that have some segment within the entanglement wedgeto have an end in A , an end in B , or both. We will not depict possible threads that do notenter the entanglement wedge at all; according to the above discussion these should beassociated with the subfactor H ( AB ) c / H ( AB ) (cid:48) . As reviewed above, E W ( A : B ) can be computed simply by counting the number of threads, ina Γ -saturating configuration, that cross the entanglement wedge cross section. Such threadscan be categorized by whether they remain within the entanglement wedge connecting A and B , or instead start in A and leave through the RT surface γ AB , or finally terminate in B having come through γ AB . If, as suggested above, we identify threads that leave γ AB astags of the degrees of freedom in ( AB ) (cid:48) , which purify ρ AB , then the counting of threads thatleave γ AB computes various conditional mutual informations among A , B , and ( AB ) (cid:48) .Conditional mutual information (CMI) is a three-party entropic measure, defined as I ( A : B | C ) = S ( AC ) + S ( BC ) − S ( C ) − S ( ABC ) . (8)Similarly to how the number of orange-type threads connecting A and B in Fig. 6 gives14 ( A : B ) / , more generally, counting threads computes CMIs [23]. For example, I ( A : ( AB ) (cid:48) | B ) = [ S ( AB ) + S (( AB ) (cid:48) B ) − S ( B ) − S ( AB ( AB ) (cid:48) )]= [ S ( AB ) + S ( A ) − S ( B )]= S ( A ) − I ( A : B )= ( ) − ( )= ( ) . In the manipulations above, we used the fact that the geometric purification | Ψ (cid:105) AB ( AB ) (cid:48) ispure to go from the first line to the second line.Our crucial assumption will be to now postulate a factorization of ( AB ) (cid:48) into A (cid:48) and B (cid:48) according to whether threads that intersect γ AB also intersect Γ and whether they areanchored to A or B . In other words, we suppose that ( AB ) (cid:48) factorizes into A (cid:48) and B (cid:48) suchthat, referring to Fig. 6, the CMIs among A , B , A (cid:48) , and B (cid:48) are given by the following threadcounts: • The total number of threads (crossing Γ ) that pass from A to B counts I ( A : B ) / . • The number of green threads, which start in A , cross Γ , and leave the RT surface,counts I ( A : B (cid:48) | B ) / . • The number of blue threads, which start in B , cross Γ , and then leave the RT surface,counts I ( B : A (cid:48) | A ) / . • The number of magenta threads, which start in A and then leave the RT surfacewithout passing through Γ , counts I ( A : A (cid:48) | B ) / . • The number of purple threads, which start in B and leave the RT surface withoutpassing though Γ , counts I ( B : B (cid:48) | A ) / .We take these identifications to define A (cid:48) and B (cid:48) . A priori it is not clear that, for any choiceof state ρ AB , there always exists a pure state | Ψ (cid:105) ABA (cid:48) B (cid:48) whose CMIs are in fact equal to thenumbers of colored threads as described above in a given bit thread configuration. In thebroader case, with I ( A : B (cid:48) | B ) / c and I ( B : A (cid:48) | A ) / c for arbitrary c , c , such pure Earlier, we defined the geometric purification | Ψ (cid:105) AB ( AB ) c on one or more full boundaries with H ( AB ) c ∼ = H ( AB ) (cid:48) ⊗ H Y , but because Y has no entanglement with AB , ρ AB ( AB ) (cid:48) is pure and hence can be associatedwith a state vector. assume that when ρ AB is dual toan entanglement wedge and the purification is geometric, such a factorization of ( AB ) (cid:48) into A (cid:48) and B (cid:48) always exists.Another way of phrasing our assumption is as follows. Let H be the Hilbert space ofthe full CFT and | Ψ (cid:105) ∈ H be geometric. For any boundary subregion R , we can factorize H = H R ⊗ H R c . There is always a state-dependent factorization H R c ∼ = H R (cid:48) ⊗ H Y such that | Ψ (cid:105) = | ψ (cid:105) RR (cid:48) ⊗ | ζ (cid:105) Y . In this example, | ψ (cid:105) RR (cid:48) corresponds to | Ψ (cid:105) AB ( AB ) (cid:48) above. The statedependent factorization in question is precisely that of a partial entanglement distillation,which yields a factorization of exactly this form. The nontrivial assumption is that ( AB ) (cid:48) further factorizes to match the CMIs as counted by bit threads.Specifically, we can consider H A (cid:48) B (cid:48) as a subspace of H ( AB ) c because the bit threads pickout a subset of ( AB ) c that is entangled with AB , so there should exist a subspace of ( AB ) c that is unentangled with AB . This subspace will not be geometric, as any geometric subset of ( AB ) c will have some entanglement with AB , but there is no requirement in our constructionthat ( AB ) (cid:48) be a geometric boundary subregion, only that its support be contained within ( AB ) c . Therefore, there exists a unitary transformation acting on ( AB ) c that takes the state ρ ( AB ) c to ρ A (cid:48) B (cid:48) ⊗ | Φ (cid:105)(cid:104) Φ | , where | Φ (cid:105) is a state in a subspace of H ( AB ) c representing the degreesof freedom that are unentangled with AB .The best way to visualize this is as a partial entanglement distillation or an entanglementdistillation with a partial inversion. One can first distill the entanglement between AB and ( AB ) c ; this will yield a product state of Bell pairs between a subset of AB and A (cid:48) B (cid:48) , and twounentangled pure states. Now, one can simply invert the half of the entanglement distillationon the AB portion to recover ρ AB . As in App. A, it is necessary for the entanglement costof ρ AB to equal the distillable entanglement of ρ AB , as is guaranteed in Ref. [33]. This formof partial entanglement distillation is made possible holographically primarily because of theholographic sandwiching result of Ref. [34]. This (clearly state-dependent) procedure willthen yield the state described in the previous paragraph.The existence of a bit thread configuration might be interpreted as providing evidencethat the state-dependent sub-factorization ( AB ) (cid:48) = A (cid:48) B (cid:48) can be achieved in the holographiccase, but it would be nice to have a more concrete construction of such systems. We notethat the existence of a surface-state correspondence [35] would immediately imply that suchsystems exist: in this case we could directly identify local purifying degrees of freedom livingon the RT surface. We discuss this point further in Sec. 5 below.With these caveats, since E W is given by the number of threads that cross the entangle-16ent wedge cross section, our assumption then implies that E W ( A : B ) = [ I ( A : B ) + I ( A : B (cid:48) | B ) + I ( B : A (cid:48) | A )] . (9)It is also true for general pure states | Ψ (cid:105) ABA (cid:48) B (cid:48) that I ( A : B ) + I ( A : B (cid:48) | B ) + I ( B : A (cid:48) | A ) = 2 S ( AA (cid:48) ) − I ( A (cid:48) : B (cid:48) ) . (10)However, just as I ( A : B ) / is given by the number of threads connecting A to B in Fig. 6, I ( A (cid:48) : B (cid:48) ) / must be identified with threads passing from one portion of ( AB ) c to another.But we argued above that these threads do not contribute to the purification of ρ AB ; theylive in the Y factor of the purifying geometry, i.e., in H ( AB ) c / H A (cid:48) B (cid:48) . Hence, without lossof generality, we may take any bit thread configuration that saturates the RT surface andremove any bit threads that do not enter W AB .Therefore, it is true here that I ( A : B ) + I ( A : B (cid:48) | B ) + I ( B : A (cid:48) | A ) = 2 S ( AA (cid:48) ) . (11)Inserting Eq. (9), we immediately have that E W ( A : B ) = S ( AA (cid:48) ) (12)for the choice of A (cid:48) defined by the bit thread configuration we have specified. By the definitionin Sec. 2, E P ( A : B ) ≤ S ( AA (cid:48) ) for any particular purification, so we have E P ( A : B ) ≤ E W ( A : B ) . (13) E P Let us take our entanglement wedge for AB and write a purification as | ψ (cid:105) AA (cid:48) BB (cid:48) Y , where wehave padded with enough unentangled ancillae that the dimension of | ψ (cid:105) has the dimensionof a full boundary CFT with some cutoff. In particular, let us choose | ψ (cid:105) to be a statethat achieves the infimum of S ( AA (cid:48) ) . (Once a cutoff is in place, which makes the boundarytheory finite-dimensional (cf. Sec. 2.1), the infimum may always be achieved.) Now, the setof CFT states | χ i (cid:105) on the entire boundary that are dual to a classical holographic bulk forman overcomplete basis for all CFT states [36]. By a slight generalization, we should be ableto express | ψ (cid:105) as some superposition | ψ (cid:105) = M (cid:88) i =1 α i | φ i (cid:105) , (14)17here the | φ i (cid:105) are a subset of the | χ i (cid:105) for which the reduced density matrix on AB is fixed, Tr ( AB ) c | φ i (cid:105)(cid:104) φ i | = ρ AB for all i . That is, we have written the purification as a superpositionover entire classical bulk geometries dual to | φ i (cid:105) , such that each has the same geometry inthe entanglement wedge.We can argue for the existence of the decomposition (14) as follows. Starting from someparticular geometric purification of ρ AB , we are free to add pairs of black holes connected bywormholes to the bulk geometry outside of W AB . As this is a local bulk modification on onlythe purifying subsystem (up to the caveats discussed in footnote 4) this action modifies theboundary state without changing the fact that it is a purification of ρ AB . (In the case where AB comprises an entire CFT boundary, with thermal density matrix dual to a black holein the bulk, this construction simply amounts to gluing on different multiboundary worm-holes behind the horizon.) Let us call the dual boundary states formed by this procedure | ξ j (cid:105) ⊂ {| φ j (cid:105)} , i.e., these geometries are a subset of those given by the set of all geometricpurifications of ρ AB . Importantly, although general purifications need not all live in thesame Hilbert space, the particular dual states | ξ j (cid:105) all live in a single Hilbert space, that ofa large- N holographic CFT, which also contains the geometric purification defined in theprevious subsection. (In the case where ρ AB describes a thermal state, the Hilbert space isinstead that of two CFTs, and includes the appropriate thermofield double states.)In particular, we are free to choose the masses and entanglement structure of such blackholes, so long as we do not change the purity of the overall state. These black holes shouldhave support over any complete basis of H ( AB ) c , by the same reasoning as in the case ofthermofield double states of differing temperatures [42, 43], as we can include all black holesfrom those of minimal size to ones that fill up a large portion of the purifying bulk region.Concretely, considering a basis | ω k (cid:105) for H ( AB ) c and writing Tr AB | ξ j (cid:105)(cid:104) ξ j | = σ j , for all j we Altering an asymptotically-AdS geometry by producing an entangled pair of black holes changes the topologyof the spacetime. This is certainly an allowed process in quantum gravity [37–40]; for example, we canconsider inserting initial excitations that collide to produce a pair of entangled black holes, i.e., the time-reversal of the process by which a pair of entangled black holes evaporate. Hence asymptotically-AdSspacetimes in which this topology-changing process takes place should still have a good dual description inthe large- N CFT; they can be produced simply by specifying appropriate initial conditions, with no needto deform the theory. However, the topology change is a fundamentally nonperturbative process from thepoint of view of a classical bulk description as a curved-space QFT, and thus we should not expect this newspacetime to be in the same code subspace [25] as one in which the process never occurs. As noted in footnote 3, this black hole gluing construction is straightforward in asymptotically-
AdS ge-ometries, due to the vanishing of the Weyl tensor. More generally, the construction of Engelhardt andWall [31] (see also Ref. [19]) allows us to glue black hole handles onto geometries in higher dimensions;though such states may not be dynamically stable [41], this will not matter for our purposes here, since wemerely require the existence of such a geometry as a locally time-symmetric solution of Einstein’s equationson a Cauchy slice. σ j = (cid:88) k c jk | ω k (cid:105)(cid:104) ω k | + (cid:88) k (cid:54) = k (cid:48) c jkk (cid:48) | ω k (cid:105)(cid:104) ω k (cid:48) | , (15)where the second sum containing the off-diagonal terms is exponentially suppressed relativeto the first sum over the diagonal terms, and c jk is (to good approximation) invertible. We expect distinct σ j , which correspond to distinct classical metrics outside of W AB , tobe orthogonal up to exponential corrections, i.e., Tr σ j σ j (cid:48) (cid:39) δ jj (cid:48) (by the same reasoningthat implies that distinct thermofield double states have exponentially small overlap [45]).Hence, a superposition of the | ξ j (cid:105) will retain the same reduced density matrix ρ AB on AB .The linear independence of the c jk weighting allows the | ω k (cid:105)(cid:104) ω k | themselves to be isolated byperforming Gauss-Jordan elimination over the black hole geometries: | ω k (cid:105)(cid:104) ω k | (cid:39) (cid:80) j a jk σ j ,where a = c − . Hence, the number of linearly independent black hole states among the | ξ j (cid:105) is at least dim H ( AB ) c , so for some α j we have | ψ (cid:105) = (cid:88) j α j | ξ j (cid:105) , (16)and therefore the | ξ j (cid:105) are a valid choice of | φ i (cid:105) in the sum in Eq. (14).It is worth emphasizing at this point that this argument suggests a subregion version ofthe fact that geometric states form an overcomplete basis of the entire Hilbert space. Thisidea is an intuitive generalization, but we have motivated it above as we are unaware of thisstatement’s appearance in extant literature.Now we wish to compute S ( AA (cid:48) ) in | ψ (cid:105) and determine its minimum value. Let us write S i = Tr ( AA (cid:48) ) c | φ i (cid:105)(cid:104) φ i | , so S i is the entanglement entropy of AA (cid:48) in the geometry i . In eachindividual geometry, we can write down a concrete bit thread configuration, where each bitthread is boundary-anchored, so we can compute S i by counting bit threads. As shown inRef. [45], we may write S ( AA (cid:48) ) as a sum (cid:80) i | α i | S i under certain assumptions, which we willdiscuss below. Thus, we can find a lower bound on S ( AA (cid:48) ) by simply lower-bounding S i .We can lower-bound S i using a bit thread configuration in an everywhere-classical geometry. We expect this block-diagonal assumption to be true at leading order in N by the eigenstate thermalizationhypothesis [44]. Note in particular that we are not claiming that the | ξ j (cid:105) form a complete basis for the entire CFT Hilbertspace; in particular, we will not be able to reach states that have reduced density matrices on AB orthogonalto ρ AB . E P ( A : B ) = S (Tr BB (cid:48) Y | ψ (cid:105)(cid:104) ψ | )= M (cid:88) i =1 | α i | S i + S mix + subleading ≥ min i S i + S mix + subleading= E W ( A : B ) + min i I ( A (cid:48) : B (cid:48) ) i + S mix + subleading (from threads) ≥ E W ( A : B ) , (17)where S mix is the entropy of mixing, − (cid:80) i | α i | log | α i | . An important caveat is that thesuperposition formula that we discuss above applies only when the number of terms M in thesuperposition is small relative to e O ( c ) . The necessity of this restriction can be understoodfrom the fact that any product state can be written as a superposition of a large number ofentangled geometric states, exploiting the overcompleteness of the basis, as in, e.g., Ref. [46].Therefore, an assumption must be made here to conclude the argument above: the numberof terms in the superposition must still be small relative to when the approximation breaksdown. An argument supporting this assumption follows from our reasoning for the existenceof the decomposition in Eq. (14) itself, using the construction involving black holes andsubsequent row reduction of the basis states. That construction suggests that the numberof terms in the superposition (14) will scale at most as dim H ( AB ) c . Because the total set ofstates for the entire boundary CFT scales as dim H AB ( AB ) c ∼ e O ( c ) , there will be a suppressionin the entropy-of-mixing term relative to the leading term that goes like S mix E W ( A : B ) (cid:46) log Mc (cid:46) log dim H ( AB ) c log dim H AB ( AB ) c < , (18)where the first inequality comes from E W ∼ c [47] and the upper bound on the Shannonentropy, S mix ≤ log M , the second inequality comes from the definition of the Hilbert spaceon ( AB ) c for CFTs, and the final inequality is definitionally true. When the number of terms M is small compared to dim H ( AB ) c , i.e., when | ψ (cid:105) can be formed by the superposition ofa small number of classical states, then the second inequality in Eq. (18) can be stronglysatisfied. In Eq. (18), we are working in the formalism of a UV-regulated CFT, so that allHilbert space dimensions are finite, making all the ratios well defined; holographically, thisjust corresponds to imposing a cutoff at fixed radial coordinate near the boundary, whichalso makes all of the geometric quantities we consider finite.One potentially illuminating way of thinking about this is to consider the reason why theblock-diagonal approximation of | ψ (cid:105) is appropriate for sufficiently small superpositions. The20eason is because the off-diagonal terms are suppressed by e c relative to the sum of the diag-onal terms of | ψ (cid:105)(cid:104) ψ | . This suppression only becomes weak enough to merit consideration ofthe off-diagonal terms when the number of nontrivial diagonal terms approaches e c ; notably,if the number is only e kc for k < , there should still be a relative exponential suppression ofthe off-diagonal elements by e ( k − c . This suggests that the inequality in Eq. (18) may in factbe even stronger than it appears, as it deals in terms of entropies as opposed to Hilbert spacedimensionality.Putting together Eqs. (13) and (17), we obtain E P ( A : B ) = E W ( A : B ) . (19)We have thus proven the E P = E W conjecture, under the assumption that we can associategeometric purifications with bit thread configurations in the sense of Sec. 3 and, as motivatedin this section, that our state | ψ (cid:105) realizing the optimal purification can be written as asuperposition of fewer than dim H geometric states, each of which has fixed reduced densitymatrix on AB corresponding to the entanglement wedge geometry. It should be made clear that some of the assumptions in Sec. 4 have particularly strong sup-port in three-dimensional gravity, where both a) the triviality of the Weyl tensor straight-forwardly allows for the gluing of arbitrary black hole handles and b) the validity of Cardy’sformula for density of states [42] is most compelling. We nevertheless expect some, probablymore complicated, version of these arguments to persist in higher dimensions; we leavesuch investigations to future work. It would be a highly surprising development if subregionovercompleteness of geometric states were true only in three dimensions.It is also worth stressing that the program advocated in this paper is not equivalentto the surface/state correspondence [35]. Nowhere in the present work was it necessary tolocalize Hilbert space subfactors to the RT surface or indeed to any bulk surface; the entireargument was made from the perspective of boundary Hilbert space subfactors. However,if we do assume the surface/state correspondence, then we do not need to extend the bitthreads to an entire bulk geometry, since we can then localize the purifying degrees offreedom to the boundary of the entanglement wedge. In this case, the identification of thepurification becomes trivial, with A (cid:48) (or B (cid:48) ) corresponding to the portions of γ AB between Indeed, as we have noted above, the gluing construction can be generalized using the formalism of Refs. [19,31]. (or B , respectively) and Γ . The connection between the surface/state correspondence andthe E P = E W conjecture was first pointed out in Ref. [12].We also expect that the bit threads justification for E P = E W will extend to the multi-partite generalizations of entanglement of purification studied in Refs. [15–18]. In particular,because we are classifying flows through surfaces, the same analysis should follow in thesecases. In the multipartite generalization, the property that the minimal polytope in the bulkis inscribed within the RT surfaces has the consequence that, in the bit thread construc-tion, all bit threads must cross the minimal polytope an even number of times (includingzero), which allows for the RT surfaces to be completely partitioned into the A (cid:48) i . We leaveinvestigation of the bit thread picture for multipartite entanglement of purification to futurework. Acknowledgments
We thank Charles Cao, Illan Halpern, Matt Headrick, Yasunori Nomura, Nico Salzetta, andMark Van Raamsdonk for useful discussions and comments. We are especially grateful toJamie Sully for numerous fruitful discussions and initial collaboration. N.B. is supportedby the National Science Foundation under grant number 82248-13067-44-PHPXH, by theDepartment of Energy under grant number DE-SC0019380, and by New York State UrbanDevelopment Corporation Empire State Development contract no. AA289. A.C.-D. is sup-ported by the KU Leuven C1 grant ZKD1118 C16/16/005, the National Science Foundationof Belgium (FWO) grant G.001.12 Odysseus, and by the European Research Council grantno. ERC-2013-CoG 616732 HoloQosmos. J.P. is supported in part by the Simons Foun-dation and in part by the Natural Sciences and Engineering Research Council of Canada.G.N.R. is supported by the Miller Institute for Basic Research in Science at the Universityof California, Berkeley.
A Holographic Purification of Minimal Dimension
Consider a purification | ψ (cid:105) AA (cid:48) BB (cid:48) of a state ρ AB —in a holographic theory with a UV cutoff—that realizes the infimum of S ( AA (cid:48) ) over all purifications of ρ AB . In this Appendix we arguethat | ψ (cid:105) AA (cid:48) BB (cid:48) can be compressed in the dimensionality of H A (cid:48) ⊗ H B (cid:48) to a new purifica-tion | φ (cid:105) AA (cid:48) BB (cid:48) such that dim( H A (cid:48) ⊗ H B (cid:48) ) = e S ( ρ AB ) , the minimal possible dimension of apurification of ρ AB .We begin by noting that a distillation of Bell pairs between AA (cid:48) and BB (cid:48) for | ψ (cid:105) AA (cid:48) BB (cid:48) is an LOCC procedure, so it preserves S ( AA (cid:48) ) . If we view A , A (cid:48) , B , and B (cid:48) as each being22ome large tensor product over a number of qubits, where we imagine that each qubit iseither in a Bell state or unentangled (i.e., ignoring issues of multipartite entanglement), thenthere should be no pairs of qubits, one in A (cid:48) and one in B (cid:48) , that are in a Bell state. Inparticular, this is true because, were such pairs of qubits to exist, one could decrease S ( AA (cid:48) ) by simply excising such pairs of qubits from | ψ (cid:105) AA (cid:48) BB (cid:48) , in contradiction with the hypothesisthat | ψ (cid:105) AA (cid:48) BB (cid:48) realizes the infimum of S ( AA (cid:48) ) . That is, we should have I ( A (cid:48) : B (cid:48) ) = 0 .More generally, if there were multipartite entanglement among subfactors of A , A (cid:48) , B , and B (cid:48) , the purifying system could be re-factored to eliminate any contribution to I ( A (cid:48) : B (cid:48) ) .For illustration, supposing that there were tripartite entanglement between some subfactors a ⊂ A , a (cid:48) ⊂ A (cid:48) , and b (cid:48) ⊂ B (cid:48) , then redefining A (cid:48) to include a (cid:48) and b (cid:48) eliminates the possiblecontribution to I ( A (cid:48) : B (cid:48) ) , while also lowering S ( AA (cid:48) ) . Again, if this were possible the initialpartition in fact could not have been optimal.We now perform the distillation procedure, which results in a tensor product of threepure states: | ψ (cid:105) AA (cid:48) BB (cid:48) ∈ H A ⊗ H B ⊗ H A (cid:48) ⊗ H B (cid:48) LOCC → | φ (cid:105) AA (cid:48) BB (cid:48) = | φ (cid:105) A A (cid:48) ⊗ | φ (cid:105) B B (cid:48) ⊗ | φ (cid:105) bell , (20)where H A = H A ⊗ H A bell H B = H B ⊗ H B bell H A (cid:48) = H A (cid:48) ⊗ H A (cid:48) bell H B (cid:48) = H B (cid:48) ⊗ H B (cid:48) bell (21)and where | φ (cid:105) bell ∈ H A bell ⊗ H B bell ⊗ H A (cid:48) bell ⊗ H B (cid:48) bell | φ (cid:105) A A (cid:48) ∈ H A ⊗ H A (cid:48) | φ (cid:105) B B (cid:48) ∈ H B ⊗ H B (cid:48) . (22)Note that, at this point, none of the Hilbert space factors are “prunable,” in the sense thatnone of the factors of | φ (cid:105) AA (cid:48) BB (cid:48) lives exclusively in H A (cid:48) or H B (cid:48) (and we recall that we arenot allowed to delete any data about ρ AB , lest we jeopardize the recoverability of our originaldensity matrix).Next, let us distill twice more on | φ (cid:105) A A (cid:48) and | φ (cid:105) B B (cid:48) : | φ (cid:105) A A (cid:48) LOCC → | φ (cid:105) A ⊗ | φ (cid:105) A (cid:48) ⊗ | φ (cid:105) A A (cid:48) , bell | φ (cid:105) B B (cid:48) LOCC → | φ (cid:105) B ⊗ | φ (cid:105) B (cid:48) ⊗ | φ (cid:105) B B (cid:48) , bell , (23)23here H A = H A ⊗ H A , bell H B = H B ⊗ H B , bell H A (cid:48) = H A (cid:48) ⊗ H A (cid:48) , bell H B (cid:48) = H B (cid:48) ⊗ H B (cid:48) , bell (24)and where | φ (cid:105) A A (cid:48) , bell ∈ H A , bell ⊗ H A (cid:48) , bell | φ (cid:105) B B (cid:48) , bell ∈ H B , bell ⊗ H B (cid:48) , bell | φ (cid:105) A ∈ H A | φ (cid:105) B ∈ H B | φ (cid:105) A (cid:48) ∈ H A (cid:48) | φ (cid:105) B (cid:48) ∈ H B (cid:48) . (25)Note that these distillations will also not affect S ( AA (cid:48) ) , since the LOCCs in Eq. (23) areacting only on subfactors of the Hilbert space that are unused in calculating S ( AA (cid:48) ) , as thelatter depends only on | φ (cid:105) bell . See Fig. 7 for an illustration of the Hilbert space decompositionand entanglement distillation in Eqs. (20) through (25). B A B discard A B B B B bell discard A A , bell B B , bell A A A A Figure 7: Hilbert space decomposition and entanglement distillation of AB and its purification. Now, we are in a position to “prune” the Hilbert space factors H A (cid:48) and H B (cid:48) , sincethese have been rendered superfluous. Let us then enumerate the total remaining Hilbertspace dimensions of the “primed” factors. We note that the remaining “primed” factors ofthe Hilbert space now each contain a state that is maximally mixed and purified by someanalogous unprimed factor. This means that the Hilbert space dimension of the remaining24rimed factors (i.e., less H A (cid:48) ⊗ H B (cid:48) ) equals e S ( ρ AB ) , because the state on the remainingprimed factors is simply a maximally-mixed purification of a state σ AB obtained by tracingout all the remaining primed factors from our state we have after all the LOCC steps, forwhich S ( σ AB ) = S ( ρ AB ) .We now note several additional facts. First, holographically, ρ AB is recoverable to leadingorder from σ AB via a suitable unitary transformation, which follows from the fact that S ( σ AB ) = S ( ρ AB ) to leading order by the Hayden-Swingle-Walter sandwiching result [34](for a proof sketch, see Sec. 2.3 of Ref. [33]). It was also necessary that there existed noentanglement between the reduced states on H A (cid:48) bell and H B (cid:48) bell , in order to guarantee themaximal mixing condition that led to our conclusion of minimality of dimension. Havingstarted with a state | ψ (cid:105) AA (cid:48) BB (cid:48) with no mutual information between A (cid:48) and B (cid:48) (which weargued was a consequence of | ψ (cid:105) AA (cid:48) BB (cid:48) achieving the infimum of S ( AA (cid:48) ) ), the Bell pairdistillation gains no advantage by introducing correlation between A (cid:48) bell and B (cid:48) bell , so thiscondition should follow.In addition, in order for ρ AB to be recoverable after our LOCC steps, it is necessaryfor the entanglement cost of ρ AB to equal the distillable entanglement of ρ AB . Luckily, thisequality was shown for holographic states (like ρ AB ) in Ref. [33].In summary, given a purification of the holographic state on AB , such that the purificationminimizes S ( AA (cid:48) ) , we can exhibit a purification, with the same S ( AA (cid:48) ) , that also attainsthe minimal possible Hilbert space dimension. References [1] G. ’t Hooft, “Dimensional reduction in quantum gravity,” in
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