Multipartite Entanglement and Firewalls
MMultipartite Entanglement and Firewalls
Shengqiao Luo, Henry Stoltenberg, Andreas Albrecht
University of California at Davis; Department of PhysicsOne Shields Avenue; Davis, CA 95616
Black holes offer an exciting area to explore the nature of quantum gravity. The classic workon Hawking radiation indicates that black holes should decay via quantum effects, but our ideasabout how this might work at a technical level are incomplete. Recently Almheiri-Marolf-Polchinski-Sully (AMPS) have noted an apparent paradox in reconciling fundamental properties of quantummechanics with standard beliefs about black holes. One way to resolve the paradox is to postulate theexistence of a “firewall” inside the black hole horizon which prevents objects from falling smoothlytoward the singularity. A fundamental limitation on the behavior of quantum entanglement knownas “monogamy” plays a key role in the AMPS argument. Our goal is to study and apply many-bodyentanglement theory to consider the entanglement among different parts of Hawking radiation andblack holes. Using the multipartite entanglement measure called negativity, we identify an examplewhich could change the AMPS accounting of quantum entanglement and perhaps eliminate theneed for a firewall. Specifically, we constructed a toy model for black hole decay which has differententanglement behavior than that assumed by AMPS. We discuss the additional steps that would beneeded to bring lessons from our toy model to our understanding of realistic black holes.
I. INTRODUCTION AND BACKGROUND
Treating black holes as quantum systems have posedinteresting questions about what the fundamental prop-erties of quantum mechanics should be in the contextof gravity. The so-called black hole information para-dox is the apparent contradiction during the evaporationof black holes through emission of Hawking radiation be-tween the causal structure inherent to black holes and theoverall unitarity of quantum mechanics[1]. As an illus-tration, consider a black hole that begun as an initiallypure state . Unitary quantum evolution requires thatthe evolving state should remain pure at later times as-suming no interactions with external systems are present.Inspired by the ideas of Bekenstein and Hawking, blackholes can be thought of as thermodynamic systems witha temperature and entropy[2, 3]. They will radiate like ablack body and can decrease in size and mass and pos-sibly eventually evaporate away completely. If we con-sider an initially pure black hole evolving completely intoHawking radiation, then the unitarity of quantum me-chanics would seem to imply that the final Hawking ra-diation state should be pure. However, the process of cre-ating Hawking radiation should be analogous to pair pro-duction due to quantum fluctuations. This would seem tosuggest the radiation produced should be mixed and dueto the casual structure of the black hole, there wouldn’tbe interactions between radiation emitted early and verylate in the black hole’s history capable of undoing themixed property.Black hole complementarity proposes a resolution ofthe black hole information paradox which allows a pure This is a common assumption when stating the problem but isdone for simplicity. Relaxing this assumption does not resolvethe information problem final state. The idea is that different causally discon-nected observers view different yet complementary pic-tures which disagree on the location of the informationencoded in the matter that created the black hole [4].Since the observers could not communicate, a contradic-tion would not be seen by any one observer. In recentyears, the debate has shifted; Almheriri, Marolf, Polchin-ski and Sully (AMPS) presented a simple argument thatrevealed approaches such as complementarity do notseem to be enough and that a contradiction betweenthe two complementary pictures could be observed [6].They suggested that the most modest resolution wouldbe preventing information from entering the interior ofthe black hole during late times by introducing a highenergy barrier called a firewall.The quantum information present in a black hole andits decay products can be explored by studying their en-tanglement. As demonstrated by Page [7], to achieve afinal pure state of Hawking radiation requires entangle-ment between early and late radiation in this final state.This late time entanglement constrains the type and levelof entanglement present throughout the evaporation pro-cess. In particular, AMPS argues that entanglement forlate radiation with Hawking modes behind the horizonseems to be forbidden which would result in a breakdownof the field theory vacuum [6]. This argument evokes awell known basic property of quantum mechanics calledquantum monogamy which constrains how entanglementcan be shared.Our paper focuses on the role quantum monogamyplays in this problem. In the case of maximal bipar-tite type entanglement (such as the entanglement presentin a Bell pair), quantum monogamy gives a simple con- Prior to this, Braunstein et al. came to similar conclusions anddescribed the deviation from vacuum near the horizon as an “en-ergetic curtain” [5]. a r X i v : . [ g r- q c ] F e b clusion. If system A is maximally entangled with sys-tem B then no entanglement can exist between systemA and any third system. When deviating from maximalbipartite entanglement, monogamy inequalities limit en-tanglement with a third system. We use a toy modelwhich deviates from maximal bipartite entanglement foran evaporating black hole to illustrate that we can im-pose an entanglement structure for early and late timesand show entanglement across the horizon is not strictlyforbidden. Since any realistic physical system will not ab-solutely saturate maximal entanglement, our explorationis motivated by the possibility that even extremely smalldeviations from maximal entanglement for realistic blackholes could lead to conclusions very different from thoseof AMPS.In this paper we consider the information that origi-nated in the initial state of the black hole becoming en-coded in multipartite entanglement. Other authors [8, 9]have stated that tripartite type entanglement naturallyarises from black hole evaporation. There are importantdifferences between our work and theirs. The conclusionsin [8, 9] are a consequence of how their Hilbert Spacesare constructed and partitioned which differs from whatwe do in this paper. In particular the difference is due tothe nature of their external neighborhood with which theblack hole interior remains entangled. Their construc-tion presents a resolution to the information paradox butleads to a firewall-like conclusion. Our different divisionof Hilbert spaces and construction of black hole and radi-ation states allow us to make different assumptions aboutthe quantum mechanics of black holes and leads to dif-ferent conclusions about the nature of the near horizonregion. Our work also differs from [10, 11] which involvea separate third system to entangle with while we con-sider entanglements between radiation subsystems andthe black hole without the need of an external system.We argue that our approach suggests a way to avoid fire-walls.Our paper is organized as follows: Section II presentsour list of beliefs that we assume to be true of the na-ture of an evaporating black hole. Section III introducessome basic facts about multipartite entanglement, andcontrasts multipartite vs. bipartite entanglement. Sec-tion IV sets up our toy model, referencing a toy modelfrom earlier work [10] designed to illustrate the AMPSargument. Section V gives the results of calculatingboth bipartite and multipartite entanglement measuresfor the toy model and mentions challenges with inter-preting these numbers. Section VI discusses the inter-pretation of our toy model in the context of the firewallproblem. Section VII examines how our results fit intocurrent understanding of the field theory vacuum. We A different approach to a qubit toy model can be found in [12].Their work avoids the firewall argument with different assump-tions of the form of black hole states and the division of Hilbertspaces. present conclusions in section VIII, and various technicalresults in the appendices.
II. PRINCIPLES OF BLACK HOLEEVOLUTION
How the black hole information problem is resolvedor remains a problem boils down to what you assumeis true of black hole evolution. In this section, we de-scribe what we assume for evaporating black holes andhow these beliefs are realized in our toy model. Our as-sumptions of what black hole evolution looks like far fromthe horizon mirrors postulates of black hole complemen-tarity presented in [4].Postulate 1: The process of formation and evapora-tion of a black hole, as viewed by a distant observer, canbe described entirely within the context of a standardquantum theory. In particular, there exists a unitary S-matrix which describes evolution from in-falling matterto outgoing Hawking-like radiation.Evolution in our model is explicitly unitary which isachieved by preserving the purity of the initial state ofthe entire system as the black hole evaporates.Postulate 2: Outside the stretched horizon of a massiveblack hole, physics can be described to good approxima-tion by a set of semi-classical field equations.Our toy-model constructed out of qubits is too simpleto check if it is consistent with this assumption. Theredoes not seem to be anything that directly conflicts withit though, so we assume that our model does not presentany contradictions here.Postulate 3: To a distant observer, a black hole ap-pears to be a quantum system with discrete energy lev-els. The dimension of the subspace of states describing ablack hole of mass M is the exponential of the Bekensteinentropy S ( M ).We model this with finite Hilbert spaces to describeour evolving black hole system. As it evaporates, thedimension of the subspace of states decreases to coincidewith decreasing mass.Unlike the exterior, The nature of a black hole’s inte-rior has not yet been observed. We expect and requirethe nature of the black hole’s interior to be consistentwith the assumptions of the exterior that we have listedabove. For our approach, we suggest some very non-localbehavior occurring in the interior. We expect the stan-dard effective field theory description of pair productionnear the horizon to fail somewhere (but not necessarily ina way that creates a firewall). We require the horizon toretain some memory of the black hole’s history. In orderto enforce the proper type of late time entanglement, wewish to transfer the entanglement from the infalling part-ners to entanglement with the horizon. In our toy model,our black hole interior states do not have a descriptionwith any causal or spatial structure. We expect that anextension of our ideas to a more realistic theory with aspatial interpretation of the interior would include verynon-local behavior in the interior. III. MULTIPARTITE ENTANGLEMENT
In this section we present a general discussion of multi-partite entanglement. We will use the concepts presentedhere to motivate and analyze the toy model presented inSection IV.A common notion of entanglement is defined by insep-arability of states. First consider separability for purestates. A pure state living in a product space with a bi-partition, A ⊗ B can always be written using the Schmidtdecomposition as: | Ψ A ⊗ B (cid:105) = (cid:88) i c i | i A (cid:105) ⊗ | i B (cid:105) (1)with bases | i A (cid:105) and | i B (cid:105) for Hilbert spaces A and B re-spectively and coefficients c i . A state is separable only ifthe state can be written as a single term in this sum: | Ψ A ⊗ B (cid:105) = | ψ A (cid:105) ⊗ | φ B (cid:105) (2)for some ψ A and φ B living in Hilbert spaces A and B respectively. The state, | Ψ A ⊗ B (cid:105) being separable is com-pletely equivalent to the Von Neumann entropy aftertracing out system A or B being zero. Otherwise it isinseparable or in other words entangled. For this reasonVon Neumann entropy serves as measure of entanglementin pure states with bipartitions.More generally, a state described by a density matrix, ρ A ⊗ B which describes either a mixed or pure state isseparable only if it can be written as: ρ A ⊗ B = (cid:88) i p i ρ A,i ⊗ ρ B,i (3)for some collection of density matrices, ρ A,i for subsys-tem A and ρ B,i for B and coefficients p i with (cid:80) i p i = 1. If there exists no decomposition of this form for a statethen that state is inseparable and is therefore entangled.Since there does not exist a sufficiently analogous de-composition to Schmidt decomposition for mixed states,separability can be much harder to check. Also, VonNeumann entropy is known not to be a useful indicatorof entanglement between two systems which together arein a mixed state.To understand the entanglement between systems thatare parts of an overall mixed state, negativity [13], en-tanglement of formation[14], distillable entanglement[15,16], and concurrence[14] are useful quantities to be con-sidered. Among all these entanglement measures, nega-tivity has the benefit of being generally calculable. In thispaper we will focus on calculating negativities in our toy Note that the definition of separability for pure states is consis-tent with the more general definition of separability. models. The technical definition and properties of nega-tivity can be found in Appendix B. The main propertyof negativity we are interested in is that if the negativitybetween two subsystems is non-zero then the subsystemsare inseparable i.e. entangled.To illustrate some interesting multipartite entangle-ment properties states can have, consider the W [17, 18]and GHZ [17, 19] states. The W state, when made up ofthree qubits has the form: W = 1 √ |↓↑↑(cid:105) + |↑↓↑(cid:105) + |↑↑↓(cid:105) ) . (4)For this state, tracing out any system leaves a mixedstate. The nonzero Von Neumann entropy for any singlequbit directly implies entanglement between any qubitand the remaining two quits but fails to reveal en-tanglement between any two qubits (because any twoqubits together are always in mixed state). The inter-esting property of this state is its maximal entanglement“robustness”[20] meaning it retains the most entangle-ment after “disposal” (tracing out) of one qubit for threequbits systems. After tracing out one of the qubits, thenegativity[13] between the remaining two qubits is non-zero. This means one qubit in the W state concurrentlyshares entanglement with each other qubit individually.Compare this to the GHZ state:
GHZ = 1 √ |↑↑↑(cid:105) + |↓↓↓(cid:105) ) . (5)Like the W state, for the GHZ state each qubit existsin a mixed state. However unlike the W state, aftertracing out any one qubit, the negativity of the remain-ing two qubits is zero. Furthermore, after tracing out aqubit, you are left with a density matrix in the form ofEqn. 3 meaning the remaining two qubits are also com-pletely separable. In other words, when any one qubit istraced out, no entanglement remains between the othertwo qubits; In the GHZ state, entanglement exists be-tween each single qubit and the remaining pair of qubitsbut no entanglement exists between any pair of qubits.States like these motivated us to study shared entan-glement structures in the context of the black hole infor-mation problem. We use negativity to reveal the exis-tence of entanglement between particular subsystems fora toy model of an evaporating black hole and its radia-tion.
IV. TOY MODEL SETUP
We will use a simple toy model similar to the onesused in [10]. In this section we will first describe how The converse of this statement is not true. Entangled subsystemscan have zero negativity.
FIG. 1. For each time step we illustrate the allocation of oursix qbits (labeled A-F) among the three physical roles: i) Partof the Black Hole, ii) Just radiated Hawking radiation, andiii) Other Hawking radiation. As time goes on the allocationshifts, corresponding to the decay process of the black hole. the Hilbert spaces are divided and quantum states aredescribed, then we will describe the time evolution andfollow up with specific information about the models.The first toy model in [10] illustrates the black hole in-formation problem and firewall argument by describingthe entanglement with collections of Bell pairs of qubits.Here we will refer to that toy model as the Firewall toymodel. We compare this to an analogous toy model cre-ated for this paper with a different entanglement struc-ture which we will refer to as the Multipartite toy model.For both models, the black hole and all the radiationare represented by six qubits. We keep the size of theentire Hilbert space constant, always dimension 2 . Ateach time step we reassign the degrees of freedom of theHilbert space to either black hole degrees of freedom orHawking radiation degrees of freedom. We begin theblack hole in a pure state and describe the evaporationby giving the toy model state at seven specific points intime. Each time step describes one Hawking qubit be-ing emitted from the black hole. For our analysis at eachtime step we will partition the system into 3 parts: blackhole (BH), just emitted particle (JR) and other radiation(OR) and examine the entanglement between them. Anillustration of the assigning of qubits to either describethe black hole or the decay products at each time step isshown in Fig. IV.The Firewall toy model explicitly demonstrates theconflict of trying to simultaneously enforce overall uni-tary time evolution (which results in entanglement be-tween early and late radiation) and trying to enforce “nodrama” at the horizon (which would require entangle-ment across the horizon region as seen by an infallingobserver). These two properties seem to be in conflictbecause of quantum monogamy and no cloning theoremswhich are key properties of quantum mechanics. If aqubit is maximally entangled with another qubit (thus We imagine our system is undergoing some continuous evolutionbut we are examining the system only at 7 discrete (not evenlyspaced) time steps. forming a Bell pair), then neither qubit can share anyentanglement with another system. In the Firewall toymodel examining the bipartite entanglement suffices todemonstrate the conflict.In the Multipartite toy model, we do not force thequbits to appear in parts of Bell pairs. We allow thenewly emitted Hawking qubit to share multipartite en-tanglement with the black hole system and radiation sys-tem. Multipartite entanglement offers a richer structureof entanglement sharing and we will see the conclusions offorbidden entanglement across the horizon for late timesno longer holds.The full 2 dimensional Hilbert space, U is describedby six qubits (labeled A through F ) as: U = A ⊗ B ⊗ C ⊗ D ⊗ E ⊗ F. (6)As the black hole evaporates, qubits are reassigned fromdescribing black hole degrees of freedom to radiation de-grees of freedom (they are emitted in reverse alphabeticalorder). At each time step, our tripartition looks like: U = H BH ⊗ H JR ⊗ H OR (7)using our subdivision into Black Hole ( BH ), Just Ra-diated ( JR ) and Other Radiation ( OR ). The specificqubits that describe each subsystem changes with eachtime step (as shown in Fig. IV).To get an idea of what the states look like, consider anexample state in the Firewall toy model at time 3 (twoparticles have radiated away) . The state of the systemat this time contains Bell pair-like bipartite entanglementbetween the JR particle and BH :12 [ (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) + (8) (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) ] (9)In these states, the numbers in the kets enumerate thebasis vectors of the subsystem. Here we can explicitly seeentanglement between the JR and BH subsystems. En-tanglement between the OR and JR ⊗ BH is also presentbut due to the symmetry in the JR elements, no entan-glement between JR and OR exists (they are completelyseparable).In the Multipartite toy model, we impose a more com-plicated entanglement structure at time 3: a ( | OR JR (cid:105) + | OR JR (cid:105) )( | BH (cid:105) + | BH (cid:105) ) (10)+ b ( | JR BH (cid:105) + | JR BH (cid:105) )( | OR (cid:105) + | OR (cid:105) )Here we write the state as a linear combination of stateswhere the JR system is maximally entangled with the The time steps and labels used in this paper differ than thoseused in [10]. In this paper we have chosen the Firewall andMultipartite models to have consistent labels with each other. OR system and a state where the JR system maximallyentangled with the BH system. The combination comeswith arbitrary (up to normalization) coefficients a and b .Here as long as neither coefficient is 0, there exists somelevel of shared entanglement between the three systems(under certain measures). In this case there does notexist maximal bipartite type entanglement between anytwo of these three subsystems.In both toy models, the JR qubit comes out maximallymixed (with Von Neumann entropy 1). In the firewall toymodel, a JR qubit comes out completely entangled withthe BH for early times and for late times switches tocoming out completely entangled with OR . In the mul-tipartite toy model, at every time after time step 3, theemitted JR qubits come out with shared entanglementbetween OR and BH .At the core of the black hole information problem, isdifficulty trying to enforce overall unitary evolution forblack hole evaporation. Fundamentally, unitary evolu-tion is just an inner product preserving and onto map-ping of states. Within each toy model, we only examineone example evolution for some chosen initial state. Ouroverall evolution looks something like: U ( t ) : Ψ t → Ψ t → Ψ t → ... (11)In principle, we could have created a Hamiltonian togenerate such time evolution. In this work, we are notinterested in the form of the Hamiltonian but insteadthe entanglement properties of the states as the systemevolves. Violating unitary evolution would involve evolv-ing an overall pure state into a mixed state. We enforceunitarity by writing a list of states that remain pure forthe entire history of our system.From the perspective of a stationary observer outsidethe horizon, we require the evaporation process to onlyinvolve local unitary interactions. We will take that tomean that quantum entanglement can only change whena Hawking quantum is just leaving the horizon due tointeractions that occur there. This means that new en-tanglement can be generated between BH and JR as anew Hawking quantum is created. Also, the entangle-ment between the BH and JR systems can transfer toentanglement between BH and OR and entanglementbetween OR and BH can transfer to entanglement be-tween OR and JR due to the shifting of quanta from onecategory to another (specifically, quanta used to describe BH becoming JR quanta and JR quanta becoming OR quanta). In our toy model, the density matrices of everycombination of previously emitted radiation (i.e. all ra-diation other than the just emitted particle) are forcedto remain unchanged for all subsequent time steps. Thisrestriction is somewhat stronger than is absolutely nec-essary since the only important constraint is that theeigenvalues of these density matrices do not change. Wetreat the black hole in this toy model as a “black box”.As the system evolves, we do not explicitly know whatthe quantum nature of the black hole truly is other thansome very coarse grained properties such as the Hilbert space dimension it lives in and its Von Neumann entropy.There in principle could be some very non-local behaviorin the interior.To see how this works explicitly, we will first illustratean example of going from time step 3 to time step 4 inthe Firewall toy model (which is easier to eyeball thanthe Multipartite toy model). Starting with our systemat time 3, in the state described by Eqn. 8, we evolveto time state 4, by shifting the qubits used to describeeach system. Going from time 3 to time 4, the size ofthe black hole has decreased and the number of radiatedparticles has increased. We model that by having thestates that described the black hole ( BH,
3) at time step3 become states that describe the black hole (
BH,
4) andthe newly emitted particle (
JR,
4) at time step 4. This isdone by writing the basis for (
BH,
3) in terms of (
BH,
JR, | BH, (cid:105) = 1 √ | JR, BH, (cid:105) + | JR, BH, (cid:105) ) | BH, (cid:105) = 1 √ | JR, BH, (cid:105) + | JR, BH, (cid:105) ) (12) | BH, (cid:105) = 1 √ | JR, BH, (cid:105) + | JR, BH, (cid:105) ) | BH, (cid:105) = 1 √ | JR, BH, (cid:105) + | JR, BH, (cid:105) )Here we include an additional subscript label, which enu-merates the time step (in this case for time 3 and time4). The change of basis written above prescribes whatthe next time state will be, based on the previous timestate. The choice of basis change we have made is writtenin a convenient basis and results in a particular entan-glement structure. Making different choices of this basisreassignment is effectively implementing different typesof interactions at the horizon and results in differing re-sulting entanglements.The remaining parts of the system at time 3, ( JR,
OR,
3) will also need to have a qubit reassigned to(
OR,
4) at time 4 as Eqn. 13: | JR, OR, (cid:105) = | OR, (cid:105)| JR, OR, (cid:105) = | OR, (cid:105) (13) | JR, OR, (cid:105) = | OR, (cid:105)| JR, OR, (cid:105) = | OR, (cid:105) This mapping should be thought of differently thanEqn. 12. The particle emitted at time step 3 is lumpedinto the (
OR,
4) subsystem by time step 4. There are nointeractions here (and no change in entropy) so this evo-lution can be thought of as just relabeling. In general wecould have evolved the state through trivial phase rota-tions without introducing interactions but for simplicitywe freeze the evolution when no more interactions occur.We plug in these changes of basis given by Eqn.12 andEqn.13 into the state we had previously written for time3, ( Eqn.8) and obtain the entire state for time 4:12 √ (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) (14)+ (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) (15)+ (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) (16)+ (cid:18) | JR BH (cid:105) + | JR BH (cid:105) (cid:19) | OR (cid:105) ]The overall process is unitary since the entire state re-mains pure.To reiterate, we realize locality by only allowing in-teractions to occur as the just emitted particle leavesthe horizon. During early time evolution, entanglementis created between the black hole and the just radiatedqubit. During the late time evolution, the newly emit-ted qubits become entangled with the earlier radiation.This does not occur due to interactions with earlier radia-tion but instead relabeling black hole degrees of freedomas new radiation degrees of freedom. In other words,what was previously entanglement between earlier radi-ation and the black hole becomes entanglement betweenearlier radiation and new radiation.Now that we have described the rules for how futuretime states are constructed, we will now describe keyfeatures of the Firewall and Multipartite models. Bothmodels begin with a pure quantum state only describinga black hole. Each time a state describing the next timeis generated in the Firewall model, the basis change forthe BH system takes the form of Eqn. 12. Each basisvector of BH is mapped onto a state that takes the formof a Bell pair between the new JR particle and a subspaceof the new smaller BH system. This evolution ensuresthat the final state at the end of evaporation is entirelycomprised of pairs of quanta in Bell-pairs, each exhibitingmaximal bipartite type entanglement. Our Multipartitemodel does not have this constraint imposed on its evolu-tion. The most general form that for remapping of basisvectors, | k BH, (cid:105) can take are: | k BH, (cid:105) = (cid:88) i,j a ki,j | i JR, j BH, (cid:105) (17)with the a ki,j coefficients chosen such that the basis vec-tors remain orthonormal. Particular choices for thesecoefficients will result in non-trivial multipartite entan-glement structure in the final state of radiation.A detailed list of our states for the time evolution canbe found in Appendix D. Even for this small and simplesystem, by late times the form of the states grow increas-ingly complex and it becomes difficult to intuitively seethe entanglement. When choosing these states we useda combination of intuition, trial and error and exploringthe space of coefficients with a computer. We recommendthat the reader first consider the general analysis of the properties of our toy model states presented in the fol-lowing section before looking at the details presented inthe Appendix D. V. RESULTS
The measures we use in this paper are Von Neumannentropy and negativity, each of which measures differentaspects of entanglement. First we calculated Von Neu-mann entropies for the BH , JR and OR systems throughthe black hole’s evaporation in Table I. The dashes in Ta-ble I (as well as Tables II and C) imply non-existing VonNeumann entropy values since the subsystems aren’t de-fined at those times. For example, in the first time, sincethere is no subsystem JR and OR, thus Von Neumannentropy of JR and OR are not well defined.The upper bound on Von Neumann entropy is givenby: 0 ≤ S ( ρ ) ≤ log ( D ) (18)where D is either the dimension of the measured subsys-tem or the rest of the space which forms a purificationwith the measured system, whichever is smaller. Von Neumann entropies in the Multipartite ModelTime S ( ρ BH ) S ( ρ JR ) S ( ρ OR )1 0.00 - -2 1.00 1.00 -3 2.00 1.00 1.004 2.01 1.00 2.005 1.32 1.00 2.016 0.60 1.00 1.327 - - -TABLE I. Von Neumann entropy of three subsystems com-pared to the maximum possible value of Von Neumann en-tropy in our Multipartite model at each time step. The dashesimply non-existing Von Neumann entropy values since thesubsystems aren’t defined at those times. The values fromthe first column of this table are plotted as x -s in Fig. V. For every time, when a new Hawking qubit is radiated(subsystem JR ), the emerging qubit is always maximallymixed, with the rest of our system serving as its purifi-cation. This can be seen from Table I, where the VonNeumann entropy for subsystem JR always numericallysaturates the maximum (log (2) = 1) for a Hilbert spaceof dimension 2. We enforced this property when choos-ing states for our toy model since this reflects what weexpect from the pair production process.Fig. V shows that the entropy of BH grows until thePage Time (time 4 in our system), after which it de-cays until the end of evaporation. This mirrors the wellknown curve found by Page in [7]. Von Neumann entropydemonstrates how mixed each subsystem is at each timebut since combinations of subsystem are mixed, the en-tropy will not state which subsystem is entangled withwhich. The question of whether or not a firewall exists FIG. 2. Calculated Von Neumann entropy of black hole sub-system (‘x’-s) compared to the maximum possible value ofVon Neumann entropy in our multipartite model (‘+’-s) ateach time step. The maximum values of entropy are based onthe dimension of the smaller Hilbert Space between the mea-sured subsystem and the rest of the space which together forma purication. This plot exhibits standard behaviors discussedby Page[21]. depends on entanglement. Here we consider multipar-tite entanglement across the horizon region which VonNeumann entropy of BH does not reveal but negativitycan.In table II we calculate negativity for our Multipartitemodel. Negativities in the Multipartite ModelTime BH and JR JR and OR BH and OR1 - - -2 0.500 - -3 0.500 0.000 0.5004 0.023 0.000 0.5455 0.038 0.000 0.5836 0.001 0.424 0.2337 - - -TABLE II. Negativity values between all combinations of ourthree subsystems BH , JR and OR for all times in our Multi-partite toy model. Negativity between BH and JR are nonzerofor times in which negativity values are well defined. Thisimplies entanglement across the horizon is not strongly for-bidden. At all times including the Page time, when negativ-ity values between BH and JR are defined, negativitybetween BH and JR is nonzero. As it stated earlier,nonzero values of negativity always means the subsys-tems are inseparable. Our main result is that we findentanglement between BH and JR for the entire historyof evaporation. This differs from the AMPS’s expectationthat after the Page time, entanglement between BH and JR should be forbidden. Further intuition about nega- tivity is developed in Section C, relating our results to[5] and comparing negativity values between the Multi-partite and Firewall toy models.However there are limits to what negativity can teachus. Because no upper bound for negativity in a mixedstate is known, the specific meaning of a finite nonzerovalue is unclear. Therefore, attempting to compare nega-tivity values from different time steps can be misleading.The changing dimensionality of subsystems for differenttime steps surely adds to this uncertainty. Addition-ally, special states called “PPT”states have entanglementeven though they have negativity equal to zero[13].A good way to move forward despite these confusingaspects of negativity would be to compare the multipar-tite entanglement existing across the horizon in our toymodel with the entanglement in a realistic field theorythat could give some physically meaningful point of ref-erence. In the remainder of this paper we explore whatsuch a comparison would entail. VI. DISCUSSION
Answering the question of whether or not there is a fire-wall involves asking what is seen by an observer fallinginto the black hole. The classical GR result is that aninfalling observer would not see the horizon as a speciallocation, looking no different than flat space which is inpart a consequence of the equivalence principle. Naively,a proper treatment of quantum gravity would not seemto change this result since for suitably large black holesthe energy scale set by the curvature near the horizon isfar below the Planck scale and the usual expectation isthat classical GR should apply. However, AMPS wouldargue that any attempt at black hole complementaritywould fail at late times and no vacuum-like field the-ory description can be found in the interior of the blackhole due to entanglement that already exists outside thehorizon between early and late radiation preventing en-tanglement across the horizon. In our Multipartite toymodel, we have created an example for the time evolutionwhere an infalling observer can find entanglement acrossthe horizon and a complementary description could pos-sibly exist. This is however a single example for the timeevolution, and without more knowledge of how black holeevaporation should look on the quantum level, we cannotmake any claims that this evolution would be typical .Furthermore, we don’t know how the results from ourtoy model generalize to an actual field theory description.A well-known property of the vacuum in field theory is The firewall argument can change with different assumptionsabout typical states. An example of this can be found in [22]which unlike this paper, considers an evolution that gives stateswith an entanglement structure that deviates from the Page re-sult. Also, in the ER = EP R proposal, it has been argued thattypical states do not have firewalls[23] its purity and the presence of entanglement. Althoughour work does not address restoring the purity that istypically assumed for the vacuum state, we have consid-ered a new entanglement structure that differs from thefirewall literature. Our multipartite model does not of-fer maximal bipartite type entanglement between quantaleaving the horizon and those falling in. To see how thiscould differ from the field theory vacuum state, we wouldwant to know how our proposal would change the energyper field mode vs. the ground state. Even a small changeper mode could give a divergent total effect, which wouldrestore the original arguments of AMPS . Answeringthese questions is crucial for our ideas to be importantto the actual black hole firewall problem and requiresmore technical work than we offer at this point. Still, weoffer some further reflections on this issue. VII. FIELD THEORY VACUUMENTANGLEMENT
In this section, we discuss how our results fit into cur-rent understanding of vacuum entanglement. First, foran operational description of the nature of entanglementin ground states, we will state some well known results inflat space. If an observer falling into a black hole did actu-ally encounter the vacuum (or some close approximation)in the horizon region, then we would expect the sameproperties to be present. First is the presence of entangle-ment in Unruh radiation. Unruh radiation is analogousto Hawking radiation for accelerating observers in flatspace. If two uniformly accelerated observers acceleratedin opposite directions with the same acceleration, theywould encounter pairs of entangled particles. These ob-servers could generate entanglement between each otherdespite being outside of each other’s lightcones and there-fore not causally connected. The interpretation is thatthe observers are harnessing already existing entangle-ment from the vacuum state in flat space. Through thisconstruction, a pair of observers could create Bell pairswhich could suggest Bell pair-like entanglement could beinherent to the vacuum. Also, [24, 25] have proposed var-ious experiments which could extract multipartite typeentanglement structures (like GHZ states) analogous tothe Bell pair extraction process just described. Theseappear to make the case for a more complicated multi-partite entanglement structure of the vacuum. However,because we have not made a clear connection betweenour toy model and field theory, we are unable to arguethat the multipartite entanglement considered in [24, 25]is “the same as that exhibited in our toy model. A possible extension to our work would be to create a comple-mentarity scheme that explicitly restores purity such as takingstates from or analogous to those in our multipartite model andremapping them for an infalling observer to construct a purevacuum state. We thank Steven Carlip for bringing up this point to us.
Entanglement in the ground state is also revealed us-ing the path integral formulation for quantum field theoryand is applicable to CFT’s. In this formulation densitymatrices can be constructed and represented as path inte-grals. When tracing out regions of space, the remainingdensity matrix will have a Von Neumann entropy thatscales with the area dividing the region [26]. As we havestated earlier, Von Neumann entropy is a useful measureof mixedness of a state as well as measuring entangle-ment across a bipartition for an overall pure state, butdoes not reveal all aspects of entanglement. Calculat-ing other measures of entanglement in this formulationis much more difficult although negativities have beencalculated in 1+1D CFT’s[27].In all of these constructions, it is difficult to fullyexplore all the properties of multipartite entanglementin field theory. Even in analogous and much simplerqubit systems, multipartite entanglement isn’t fully un-derstood for systems of n qubits. The main emphasis ofthe AMPS argument is that the entanglement requiredbetween early and late radiation prevents entanglementacross the horizon due to quantum monogamy and nocloning theorems. Quantum monogamy is simple to statefor maximal entanglement for a bipartite pure state (e.g.a Bell pair). However these considerations are not com-pletely general. For non-maximal multipartite entangle-ment for mixed states, quantum monogamy inequalities[28] exist for various entanglement measures which limitentanglement, but do not strictly forbid the entanglementwe are interested in.One approach to the firewall question is to ask what isexpected of the final state of the hawking radiation. Therelevance of entanglement observed in this final state tothe firewall problem can be seen by time evolving backthe radiation’s evolution to when it was just leaving thehorizon. The question can be posed as what propertiesof the final state’s entanglement say about the state ofthe quantum fields near the horizon region at an earliertime. Inspired by the discussion in [13], we ideally mightwant to translate the question of how to restore the vac-uum near a black hole’s horizon into language that ad-dresses, for example, the extent to which pure state en-tanglement can be extracted from infinitely many copiesof the state. A concrete question like this would allow usto choose measures that have a given physical interpre-tation such as “entanglement of formation”, “distillableentanglement”, etc. which could illuminate further thenature of the vacuum. However, the above translationis not straightforward and these measures are generallydifficult to calculate.
VIII. CONCLUSIONS
In our toy model we constructed a history of statesto model the evolution of a black hole. The method ofstate construction has ensured the state of the whole sys-tem remains pure, and enforced the evaporation processto only involve local unitary interactions. By introduc-ing a multipartite entanglement measure, entanglementsharing among black hole and different parts of radiationcan be measured. Our analysis has shown entanglementbetween the black hole and the qubit just radiated fromthe black hole for the entire history of evaporation. Thisdiffers from the AMPS’s expectation that after the Pagetime, entanglement among these systems should be for-bidden.Our very simple toy model explicitly shown entangle-ment across the horizon region isn’t strictly forbidden.However, it is unclear how this result translates into anactual field theory description. Because of the limitationsof our toy model, we cannot claim at this point that weknow that a more realistic model could have enough en-tanglement or even the right type of entanglement to re-store the expected properties of the field theory vacuumnear the black hole. We also do not explicitly offer ascheme to restore the vacuum’s purity which may be themore important property of the vacuum than the entan-glement. This paper is meant to show a possible loopholein the AMPS argument. For a complete and proper treat-ment, we would want to extend this toy model to onemore like a field theory. With a more realistic model,we could imagine time evolving back the hawking radia-tion and identifying what types of states you would havenear the horizon for an infalling observer, thus potentiallyachieving insights into the firewall problem.
IX. ACKNOWLEDGMENTS
We thank Veronica Hubeny, Massimiliano Rota, DonPage, Rajiv Singh, Steven Carlip and William K. Woot-ers for helpful discussions. This work was supported inpart by DOE grant DE-SC0009999.
Appendix A
In these Appendices we address several topics support-ing the main points of the paper. Appendices B and Cdefine and develop the ideas of negativity, and AppendixD gives detailed information about our toy model.
Appendix B: Negativity
Negativity is an entanglement monotone meaning thatit does not increase under local operations and classi-cal communication. As discussed in [17], negativity maythus be considered an entanglement measure. As an il-lustration we can construct mixed state systems by firstconsidering a pure state in a Hilbert space which is aproduct of three smaller spaces: U = H A ⊗ H B ⊗ H C .Tracing out system C, leaves density matrix ρ A ⊗ B .The negativity of A, B for the state, ρ A ⊗ B is given by: ε A,B ( ρ A ⊗ B ) ≡ (cid:107) ρ T A A ⊗ B (cid:107) −
12 (B1)where the trace norm (cid:107) ρ T A A ⊗ B (cid:107) is the sum of the abso-lute values of the eigenvalues λ i of ρ T A B . ρ T A B is the par-tial transpose of B respect to A . The Peres - Horodeckicriterion states that to have a state separable it is nec-essary the partial transpose of ρ has only non-negativeeigenvalues[29]. If the partial transpose of ρ has any neg-ative eigenvalues then the state is necessarily insepara-ble. The trace norm measures how much ρ T A A ⊗ B fails tobe positive[29]. Therefore if negativity is nonzero, thenthere is definitely entanglement. However, negativity be-ing zero does not imply no entanglement. There exist aclass of entangled states with zero negativity said to bePPT bound entangled states[13]. Thus it can be diffi-cult to extract precise physical meaning from the mea-sured value of negativity. It has been noted that neg-ativity places a bound on the degree to which a singlecopy of the state ρ can be used to perform quantumteleportation together with local operations and classicalcommunications[13, 18]. For our purposes in this paperwe are mostly interested in its ability to identify whenentanglement is present. Appendix C: Table of negativity for a man-madefirewall model
Negativities in the Firewall ModelTime BH and JR JR and OR BH and OR3 0.5 0.0 0.54 0.5 0.0 0.55 0.0 0.5 0.0TABLE III. Negativity values between all combinations of ourthree subsystems BH , JR and OR for all times. There is noentanglement between BH and JR at 5 because our choicesof states are known to behave as so. Thus there is firewalls forall time when negativity is defined including after page time.The dashes implies the non-existed negativity values. Forexample, in the first time, since there is no subsystem JR and OR , there are no negativity values between all combinationof subsystems. We have also calculated negativities for the Firewalltoy model which is analogous to the first toy model in[10]. This model is constructed with a final radiationstate has each pair of qubits appearing in a Bell-pair.From the form of the states, it can be explicitly seenthat entanglement exists between BH and JR until thePage Time (time 4), after which there is no entanglementpresent. This is consistent with the negativities shown inTable C, where the negativity of BH and JR is non-zero prior to time 4 and then is zero after time 4. Weremind the reader that trying gain more insight from0comparing these values with TableII can be misleadingfor the reasons listed in Sec.III. Appendix D: Time evolution of the multipartitemodel
Our method for generating time evolution is based onthe change of basis that we introduce once a qubit shiftsfrom describing a black hole system to the radiation sys-tem. Section IV, illustrates the way the new basis isused to construct the time evolution. The choice of ba-sis dictates what the entanglement will look like for thesubsequent time state. We wanted to create states thatproduced a JR qubit that was maximally mixed as wellas having non-zero negativity between JR and BH (Asexplained in IV). Finding a basis that accomplished thisgrew increasingly difficult as the states got more compli-cated and required some trial and error. All mappingsup till time 4 were done by hand, taking advantage ofsymmetries in the form of the states to find the types ofstates that matched our criteria Getting states for time 5and 6 were too difficult to do by hand, so the states werefound by varying parameters in the basis vectors witha computer program. The program is able to generaterandom orthonormal bases and do systematic mappingsfrom one time to the next. Among the many mappings itgenerated, the program can select the set of states thathave interesting negativities and have all the conditionswe want to enforce to assure the evolution to be physi-cal. In this section we give a detailed description of the toymodel state at each of the discrete time steps 1–7. Ateach time the state is a pure state, thus ensuring unitarytime evolution for the whole system. Several key featureshave already been presented in Section IV. These includethe way we enforce locality by limiting which subsystemsinteract, and how we use basis mapping as a tool to gen-erate the state at the N th time step from the state at the( N − th step. The detailed process of the time evolutionof the multipartite model is as follows:The first time state is trivial, and is written | ψ BH, (cid:105) (D1)The second time state is given by | ψ BH, (cid:105) = 1 √ | BH, (cid:105) | JR, (cid:105) (D2)+ | BH, (cid:105) | JR, (cid:105) )where | BH, (cid:105) is the 0 th basis state for the black holesubsystem at time 2,etc. Notice that there is nothing inOR subsystem at time 2.The i th black hole basis state | i BH, (cid:105) at time step 2evolve into states that describe the black hole ( BH,
JR,
3) at time step 3. This is done by writing the basis for (
BH,
2) in terms of(
BH, | BH, (cid:105) = a | BH, JR, (cid:105) + a | BH, JR, (cid:105) + b | BH, JR, (cid:105) (D3) | BH, (cid:105) = a | BH, JR, (cid:105) + a | BH, JR, (cid:105) + b | BH, JR, (cid:105) where the values of coefficients are assigned to be a = 1 √ b = 0 . (D4)We plug in this unitary change of basis into the state wehad previously written to get the third time state: | ψ (cid:105) = 1 √ a | BH, JR, (cid:105) + a | BH, JR, (cid:105) + b | BH, JR, (cid:105) ) | JR, (cid:105) (D5)+( a | BH, JR, (cid:105) + a | BH, JR, (cid:105) + b | BH, JR, (cid:105) ) | JR, (cid:105) )] . The mapping we use to go from time 3 to time 4 is | BH, (cid:105) = a ( | BH, JR, (cid:105) + | BH, JR, (cid:105) )+ b ( | BH, JR, (cid:105) − | BH, JR, (cid:105) ) | BH, (cid:105) = a ( | BH, JR, (cid:105) + | BH, JR, (cid:105) )+ b ( − | BH, JR, (cid:105) + | BH, JR, (cid:105) ) | BH, (cid:105) = a ( | BH, JR, (cid:105) − | BH, JR, (cid:105) ) (D6)+ b ( | BH, JR, (cid:105) + | BH, JR, (cid:105) ) | BH, (cid:105) = a ( | BH, JR, (cid:105) − | BH, JR, (cid:105) )+ b ( | BH, JR, (cid:105) + | BH, JR, (cid:105) ) | BH, (cid:105) = a ( | BH, JR, (cid:105) | BH, JR, (cid:105) )+ b ( − | BH, JR, (cid:105) + | BH, JR, (cid:105) )where a = 0 . b = 0 .
707 (D7)(D8)We plug this unitary change of basis into the time 3state to get the toy model state for time 4.Due to the increasing complexity of the expressions,for time steps 4 and higher we only give the basis map-pings, and do not explicitly present the outcome of thesubstitutions. The information we do provide is sufficientto fully reproduce our results.The mapping from time 4 to time 5 is1 | BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) (D9) | BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105)| BH, (cid:105) = c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) + c , | BH, JR, (cid:105) Each of the term associate with a coefficient c i,j . If co- efficients c represented as matrix, the values are assignedas the following: c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , c , (cid:107) . − . − . − . . . − . − . − . . − . − . − . . − . . − . . − . − . . − . − . − . . − . − . . − . − . − . . . . − . − . . . . . . . . − . − . − . − . . − . . − . − . − . − . . − . − . − . − . − . . − . − . . (D10)The expression of mapping from time 5 to time 6 issimilar. The coefficients matrix for mapping from time 5 to time 6 is: Since we generated the random bases for times 5 and 6 numeri-cally, the eigenvalues of the density matrix of JR for those timeswere not exactly 0 . . . c ,
00 1 c ,
00 1 c ,
00 1 c , c ,
01 1 c ,
01 1 c ,
01 1 c , c ,
10 1 c ,
10 1 c ,
10 1 c , c ,
11 1 c ,
11 1 c ,
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