JHEP08(2014)126 Springer July 11, 2014 July 24, 2014 : : August 21, 2014 : September 25, 2013 : Revised Accepted 10.1007/JHEP08(2014)126 Published Received doi: Published for SISSA by b [email protected] , . 3 1308.4209 The Authors. c Black Holes, Spacetime Singularities

and John Preskill Almheiri et al. have emphasized that otherwise reasonable beliefs about black , a [email protected] Cambridge, MA 02139, U.S.A. Institute for Quantum InformationPasadena, and CA Matter, 91125, Caltech, U.S.A. E-mail: Department of Mechanical Engineering, MIT, b a Open Access Article funded by SCOAP ArXiv ePrint: the computational complexity of decodingviolation the Hawking unobservable. radiation may Final-state renderof projection the standard causality models quantum illustrate mechanics how might be inviolable circumvented principles in aKeywords: theory of quantum gravity. the model seems to requirehole carefully S-matrix, tuned dynamics for to aare ensure generic exact exponentially unitarity small final-state of in boundary the theneed condition black black not hole the detect entropy; any deviations furthermore deviations fromperformed observers from inside inside unitarity standard black old quantum holes black mechanics. holes Though could measurements potentially produce causality-violating phenomena, Abstract: hole evaporation are incompatible with theproperty monogamy of of quantum quantum entanglement, mechanics. ahole general evaporation We proposed investigate by the Horowitz and final-state Maldacena,cloning projection pointing of out model quantum that this states of model andration black admits polygamous process entanglement, to allowing be unitarity of reconciled the with evapo- smoothness of the black hole event horizon. Though Seth Lloyd Unitarity of black holeprojection evaporation models in final-state JHEP08(2014)126 7 13 8 ], the notion that the inside and outside 24 18 4 , 3 5 – 1 – 10 14 in the radiation, violating the linearity of quantum 19 9 cloned 16 11 15 7 23 1 25 ] nearly 40 years ago that black holes evaporate. The crux of the puzzle is this: 1 ] that unitarity fails in quantum gravity. Efforts to rescue unitary led to the 2 This puzzle has spawned many audacious ideas, beginning with Hawking’s bold pro- 6.6 Computational power of the HM model 6.1 Unitarity constraint 6.2 Chronology violation 6.3 Entanglement-verifying measurements 6.4 The AMPS6.5 experiment Acausal signaling? 4.1 Entanglement across4.2 the horizon A generic final state 3.1 Relaxing the3.2 no-cloning principle: Relaxing black entanglement hole monogamy: complementarity easing the AMPS puzzle mechanics. posal [ formulation of black holeof complementarity a [ black hole are not really two separate subsystems of a composite quantum system, if a pure quantumblack state hole collapses contains spacelike to surfaceshorizon form crossed and a by nearly both black all the hole, ofprocess collapsing the the body is geometry emitted inside unitary, Hawking the of then radiation event also the the outside be evaporating quantum the information encoded event horizon. encodedthe (perhaps If in infalling in this the quantum a collapsing state highly matter is scrambled must form) in the outgoing radiation; hence 1 Introduction The quantum physics ofdiscovered black [ holes has caused great puzzlement since Stephen Hawking 7 Discussion 5 Detecting postselection when approaching6 the singularity Measurements inside black holes 3 Features of the model 4 Conditions for unitarity Contents 1 Introduction 2 The Horowitz-Maldacena proposal JHEP08(2014)126 ) ), a to be 1 B firewall ], and its propagates ], is that in 6 ] that these , to be highly 8 emitted by a 8 5 B ]. All of these B B . (For example, . If monogamy as 21 R B B R ], which asserts that R 12 , inaccessible. Thus black with , and 11 A B B , A ]. In contrast (as indicated in figure ].) Assumptions (1) and (3) require 16 10 , while assumption (2) requires , , ], which provides a pleasingly unitary picture of – 2 – R 9 7 15 , of 4 B R should be regarded as a complementary description of B in the black hole interior. Taken together, then, the three R ]. Or, clinging to a revised version of the complementarity are maximally entangled, then neither can be correlated with A 18 B to be transferred to entanglement of A and ] and the entanglement verification described by AMPS [ ]. Another possibility is that modifications of assumption (3) allow the A with 16 , 17 , B 8 15 , 4 ]. ], possibly connected to the black hole interior via a wormhole [ 14 20 , , Like Hawking’s original black hole information loss puzzle, the AMPS puzzle has also When the notion of black hole complementarity was initially formulated, it was argued Black hole complementarity seeks to reconcile three reasonable beliefs: (1) An evap- 13 19 could be the radiation so far emitted by an old black hole which has already radiated [ entanglement of away from the blackprinciple, hole one [ can assertA that ideas will need to be fleshed out further before they can be accurately assessed. the horizon, while in thehole latter case complementarity no needs such to delay be makes system reconsidered. spawned audacious ideas. AMPSthat themselves an advocated old relaxing blackand assumption hole no (and (2), interior [ perhaps arguing also a young one) has a singular horizon (a of entanglement were reallynot violated, verify there the is violation. nodescribed in obvious The [ reason key why differencethe this former between observer case the could the cloningHawking observer verification radiation, needs by experiment to which wait time for it quantum is information too to late be to revealed catch in up the with its putative clone behind be verified in anytrustworthy conceivable semiclassical experiment, approximations. at Thisthe least observation cloning bolstered within is the the contention operationally domainsingle that fictitious of observer [ falling applicability into of could the be black hole, in when causal still contact a with safe distance all from three the of singularity, the systems if quantum systems any other system. Thisin tension [ between unitarity and monogamy had beenthat noted earlier the cloning of quantum states occurring during black hole evaporation could not R away more than half of itshighly initial entangled entropy with [ a subsystem entangled with a subsystem assumptions violate the principle of monogamy of entanglement [ orating black holefalling scrambles observer quantum encounters information nothinghole. without unusual destroying (3) upon it. An crossingeffective observer quantum who the (2) field stays event theory. A outside horizon Butthree freely a Almheiri assumptions of black are et a hole incompatible. al. black (AMPS) detects Theyblack recently consider hole no argued the which violations [ Hawking is of radiation nearly relativistic maximally entangled with an exterior system map. Black hole complementaryeventual set realization in the AdS/CFT stage duality for [ black the hole holographic evaporation principle in [ duality asymptotically regarding AdS the spacetimes, black though hole the interior implications remain of unclear. this but rather two complementary views of the same system, related by a complex nonlocal JHEP08(2014)126 , M out H ], and ⊗ H 24 , in H 23 , where out ⊗ H in ⊗ H M H ] by Horowitz and Maldacena were highly entangled with both 22 B is the Hilbert space of outgoing . Speaking fancifully, information . If out B H B R much more outlandish than these other R is the Hilbert space of infalling negative in , and H B , B – 3 – time a priori A A singularity

postselected teleportation radius = 0 = radius , the observer should be able to verify the violation of entanglement monogamy. . The AMPS puzzle: an infalling observer, while still a safe distance from the singularity, B R HM proposed imposing a final-state boundary condition requiring a particular quantum Here we suggest another possible response to the AMPS puzzle, based on the final- and is the Hilbert spaceenergy of Hawking the radiation infalling behindpositive matter, the energy Hawking horizon, radiation and outsidea the freely horizon. falling What observer appears crossingand to the be the horizon the HM is vacuum boundary to a maximally condition entangled projects state onto of a particular maximally entangled state of state at the spacelike singularityfrom inside the the black black hole hole, interiorresiding which by in allows the information collapsing to matterinside escape propagates the from black past hole, infinitytime to where from the the it spacelike singularity is singularity infinity. to scrambled More the concretely, and HM horizon, reflected, consider and the then forward composite propagates in system time backward from in the horizon to future tarity is realized, andresolve there the is AMPS no puzzle, need thefairly for assessed, HM firewalls. but proposal we requires Just doproposals. further as not development with regard before it other it as proposed can ways be to (HM). In this scenario, theing S-matrix matter relating and the asymptotic asymptotic outgoingunitarity state incoming can of state be the of temporarily emitted the violated radiation collaps- dating during can violations the be black of unitary; hole however monogamy evaporation ofallowing process, assumptions entanglement accommo- and (1), the (2), no-cloning and principle (3) [ to be reconciled. A type of black hole complemen- is in causal contact withA each of the systems state projection model of black hole evaporation proposed [ Figure 1 JHEP08(2014)126 ]; the dynamics 30 – 28 and other systems are ] may suffice to enforce in H 31 ]. In particular, unless appropriate 27 , 26 ], it may not be unreasonable to impose 25 – 4 – is the black hole entropy. Such exponentially small violations BH S where 2 / , which encodes the black hole S-matrix. While the horizon crosser sees nothing in BH S − e ⊗ H After reviewing the HM model in section 2, we comment on its relevance to the AMPS Even if it turns out that the HM model is not realized in nature, the model is still quite A careful analysis of the entanglement-verifying measurements discussed by AMPS Our attitude is that these potential bugs in the HM proposal may actually be welcome, The HM proposal has the appealing feature that the new physics responsible for evad- M controversy in sectionmatrix 3. might fail In incondition section the at 4 HM the we model, singularity,in examine the concluding the how deviations that, black from unitarity for hole exact of a entropy. unitarity generic the are In final-state black exponentially boundary section hole small 5 S- we argue that observers with limited access to radiation; hence thecausality. computational complexity of this task [ instructive. It cautionssuch us as that the inviolable no-cloning consequences principlequantum of and gravity. monogamy standard Perhaps of that quantum entanglement, is mechanics, need the not proper be lesson respected to in be drawn from the AMPS puzzle. reveals that imposing unitaritythe of HM the model black is holeblack physically S-matrix hole sensible; might may produce in not causality violating particular sufficethe modifications measurements to of black performed the ensure Hawking inside hole, radiation that an outside This enabling old backward observers signaling can who be stay achieved, however, outside only to by rapidly send decoding signals the Hawking backward in time. model, since nonperturbative quantumare gravity corrections beyond of the that scopeboundary order of are the condition, expected analysis. that and Wedetected deviations also by argue, from infalling again observers standard as assuming quantum they a approach theory generic the final-state are singularity. unlikely to be appropriately tuned. A deeper understandingwhether of black hole quantum evaporation gravity really may fulfills befinal-state these needed boundary conditions, to condition but decide we at find thelike that singularity, for the a deviations generic fromof exact unitarity unitarity scale could well be regarded as a success for our semiclassical analysis of the HM process. helping to steerfocus us on toward delineating a sufficientthe deeper conditions evaporation understanding for process the of is proposal quantum unitary to gravity. if work. the Therefore, In interactions brief, between we we find that like singularities in cosmologicalfinal-state spacetimes boundary conditions [ atBut spacelike the singularities proposal in has black otherconstraints hole less are spacetimes pleasing imposed features as on [ well. thewith dynamics, effective postselected closed quantum mechanics timelike canmay curves be need and afflicted to other be causality carefully adjusted paradoxes to [ protect the unitarity and causality of the evaporation out of the ordinary, aninfalling observer who matter stays and outside the the state black of hole finds the that outgoing the radiationing state are information of related loss the by occurs a atbadly. unitary the Furthermore, map. singularity, if where we we are expect willing semiclassical to physics impose to initial-state fail boundary conditions at space- H JHEP08(2014)126 . † 2 V . † is a † from (2.1) (2.2) , first V U † V C i U ψ | can be ex- 〉 , then up to , i B )

ψ C | 〉 ∗ i can apply to system ψ U U V ψ | | VU and A , then perform an C occurs with proba- Φ( | i VU A ) or the projected state 1 d BC ∗ U = = is projected to a maximally BC Φ( A . | 〉 by classical communication, i of system i i ψ ψ AB C | | Φ ψ | | A ) would be non-unitary and hence V of system I U ], which is illustrated in figure 〉
VU denote orthonormal bases, . To teleport the state U ⊗ 35 Φ V } BC | BC T i or i B ) V i Φ i | V V applies the unitary transformation {| Φ = ( C h Φ( , -dimensional systems | | i } d Φ AB A Φ | i 〈 ) i – 5 – 〉 C V {| V U

ψ is also known, then a party at is prepared, and then | ⊗ = I ( , a party at A BC BC . If either the initial state of i i ψ ψ C of | = | 〉 is the transpose of i ≡ , where i
) ) . If the outcome of the measurement is

ψ B T | i V V 〉 becomes BC i V i ) AB Φ( Φ( C | | C | VU ⊗ | V A in system i Φ( ⊗ Φ | i i . To recover | ) indicates that the measurement outcome IV i ψ ∗ ) | ∗ =1 U B d i /d U Φ( P matrix, and h Φ( | U I d | 1 d 〉 √ ]. AB Φ ⊗ ψ 〈 34 × . Once known, this outcome can be transmitted to | A = – 2 . Quantum teleportation. To convey a quantum state d i , we first prepare the entangled state /d were not maximally entangled, then either Φ 32 C | , Connections between the AMPS puzzle and the HM model have also been discussed 20 to AB bility 1 and if the initialto entangled recover state the of state of unphysical; in that case the teleportation process would have imperfect fidelity. entangled measurement on normalization the state of where the factor 1 pressed as Here unitary A 2 The Horowitz-Maldacena proposal The HM proposal isAny based maximally on quantum entangled teleportation pure [ state of two mechanics. We examine measurements performed insidein a black particular hole whether in section such 6, measurementssome discussing can concluding enable comments. acausal signaling. Section 7 contains in [ Figure 2 a maximally entangled state entangled state the infalling Hawking radiation need not detect any deviations from standard quantum JHEP08(2014)126 , out will ⊗ in out i Φ H | out can be made and the infalling out outside the black . M S ⊗H time out i in i | H M singularity is the negative energy Hawking state in H , quantum information is teleported from the of 3 – 6 – in = that is emitted as the black hole evaporates. The i specified by the final-state boundary condition is i | in out H out ⊗ H M , the source for the black hole’s classical geometry, to the H M H time in is unitary, then the infalling matter state and the outgoing radia- S S is teleported out of a black hole. Outgoing Hawking radiation is maximally M M is the number of distinguishable microstates for a black hole with specified , where d in singularity ⊗ . The Horowitz-Maldacena model, in which quantum information carried by the collapsing M i ) ∗ is a system of infalling Hawking quanta behind the horizon. We use a microcanonical If the entangled state of Loosely speaking, the basis state The initial maximally entangled state used in the protocol is the Unruh state In the HM proposal depicted in figure S in Φ( because the evaporating blackgranted, hole though, is that not the static notionprecise. of and a has maximally no entangled Killing state of vector. We take it for | behind the horizon pairedhole. with the “Negative positive energy” energyexterior is Hawking geometry state really becomes a spacelike misnomer,hole behind because is the the really horizon; timelike momentum. hence Killing “energy” In vector inside of any the case the black this description of the Unruh state is not precise this state is maximallyappropriate entangled if rather we than wish thermal.sharply defined to energy; (The of consider microcanonical course, thesee ensemble an observer a is formation with thermal access and state.) tomaximally evaporation a By entangled small of a subsystem state suitable a of to basis the black choice, identity. hole we set with the unitary matrix specifying this communication to convey the outcome of the entangled measurement. which looks like the vacuum stateH to a freely fallingdescription, observer who summing crosses the over horizon. all Here microstates with approximately the same energy, so that collapsing matter system outgoing Hawking radiation system dimension total mass. Because themaximally final-state entangled boundary state condition is specifies accepted that only at one the particular singularity, there is no need for classical Figure 3 matter system entangled with infalling radiation, and aradiation final-state to boundary a condition maximally projects entangled state which encodes the unitary S-matrix JHEP08(2014)126 is also avail- M H , which may not be M H and is required to rigorously sat- S out H on the same spacelike slice. From the is the black hole S-matrix, presumed to S out H – 7 – ; thus M i illustrates, cloning of quantum states can occur in we will revisit the sufficient conditions for unitarity ψ | 3 4 S = is perfectly isolated from out in i H ϕ | . appearing in the final-state boundary condition playing the roll of the 5 S indicates, the HM model supports a characteristic flow of information in space- ], and we assume the singularity is always spacelike and unavoidable in realistic 3 36 Thus black hole complementarity is realized in the HM model in the sense that observ- Though analytically extended non-Schwarzschild black hole geometries can have time- radiation is merely the“time” same and as in the a state different of basis. the infalling matter,ables except inside viewed at and a outsideFrom later the the horizon perspective acting of on the the causally same ordered spacelike information slice flow, do this not commute. failure of commuta- S-matrix relating the asymptotic incominganomalous and phenomena outgoing can states. occur, But whichmechanics. would at be intermediate For disallowed times in example, standardpostselected as unitary quantum quantum figure mechanics. Theable, quantum albeit information encoded in in perspective a of highly the scrambled causally form, ordered in straightened process, the cloned state in the outgoing 3.1 Relaxing the no-cloningOnce principle: straightened, black the hole overallunitary complementarity process matrix clearly preserves quantum information, with the out” the bends in theinfalling flow. radiation This system alternative descriptionprecisely can true; be therefore strictly justified inin only section a if more the general setting.consistent causal But ordering, for and now consider we some will of assume the that consequences. the information flow admits a time, which ensures thetially unitarity encoded of the in blackspacelike the hole singularity, collapsing evaporation then process. matter backwardnally flows in Information forward forward time ini- in from in timepropagation the time from backward singularity the from in to horizon time, past to the therea infinity is future horizon, conventional an to infinity. causal and equivalent the ordering; Despite description fi- the of the the apparently information same acausal flow process can with be “pulled tight” to “straighten 3 Features of the model As figure stable [ collapse scenarios. Because theof final-state the boundary condition infalling accepts matterwith any access system, quantum to state observers only approaching aor local the any subsystem, singularity, departure need particularly from notfurther those the experience in usual section a laws reversal in of the quantum arrow mechanics. of time We will discuss this point be a highly nonlocalisfy scrambling conservation unitary of transformation. energysymmetries, and which other are exact expected gauge to charges, be but broken it in need quantumlike gravity. not rather respect than global spacelike singularities, the interior geometries of these solutions are un- tion state are related by JHEP08(2014)126 , ), S m log m ( ] will occur, O quanta. On 16 B 2 1 3 ]. As depicted in out out out 10 , should be maximally 9 B 1 2 3 S S S to be maximally entangled B 3 2 1 M M M = out 2 – 8 – 3 1 ], then the “information mirror” phenomenon [ ], we consider the case of an “old” black hole which has already 3 behind the horizon, the Unruh partners of the M 37 17 , A 8 2 M of the previously emitted radiation. In standard quantum mechanics R 1 M denote some Hawking quanta which have been recently emitted by this old B is the black hole mass. We note that if additional mass is thrown into the black . Information flow for a black hole that maintains its mass by accreting a steady stream , and in particular the final-state boundary condition will respect the requirement , let m 5 4 We may also consider a process in which we continually feed a black hole with additional figure black hole. If the horizonentangled looks with smooth system to infallingthe observers, other then hand, unitarity ofwith the a evaporation subsystem process requires 3.2 Relaxing entanglement monogamy:Now, following easing AMPS [ the AMPSemitted puzzle more than half of itsbecome initial maximally entropy, so entangled that with its its microscopic degrees previously of emitted freedom radiation have [ in which, for aby black the black hole hole highly returnswhere entangled in with the emitted its radiationhole surroundings, after after information a it absorbed Schwarzschild initially time apparent forms, horizon, to not avoid at firewallsof the we the global require hole. horizon a whose smooth position Unruh depends vacuum on at the the future history black hole, if thea overall quantum state circuit is whosefigure pure. bounded The width information isthat processing determined information can cannot by escape be thecircuit from described black scrambles the by hole rapidly evaporating [ entropy black as hole in before it falls in. If this matter to maintain its mass forfinally a long allowing time compared the to evaporation itsrather natural to evaporation than time, proceed before being tothe an completion. outgoing arbitrary In radiation, unitary that mustthe transformation case have entropy mapping the of a the S-matrix the special infalling radiation structure should matter enforced never to by exceed the the requirement Bekenstein-Hawking entropy that of the of infalling matter. tivity is expected, becauseobservables, but the at outside a observables later act “time”. on the same system as the inside Figure 4 JHEP08(2014)126 ; . A A 6 and HR BR B and but not AB R R B onto a maximally HA inside the black hole; is entangled with in the previously emitted R R R , B (b) time A is entangled with ], which must be achieved in order B , the final-state boundary condition 31 M via entanglement swapping. onto a maximally entangled state, then the BR H the Unruh partner of HA – 9 – H ]. In standard entanglement swapping, an entangled cannot be entangled with each of the two systems is maximally entangled as well. This phenomenon is 39 , and the outcome of this measurement must be commu- B , 38 BR [ HA are maximally entangled Unruh vacuum states. Now, for a R was premised on the assumption that the postselected infor- 3 . We note by HR from the Hawking radiation [ 5 entangled R B (a) time entangled entanglement. Furthermore, measurements in postselected quantum me- and of to complete the swapping protocol. No such communication is necessary in BR AB BR . (a) An old black hole is maximally entangled with system entanglement. entanglement swapping . To relieve this tension AMPS proposed that R This picture is oversimplified. For one thing, we have ignored the computational com- In the HM model, however, smoothness of the horizon can be reconciled with unitar- AB chanics raise some daunting conceptual puzzles. We will return to4 these issues in section Conditions for unitarity The discussion in section mation flow in spacetime has a consistent causal ordering, ensuring the unitarity of the entanglement, while an infallingthe observer could pass safely through the horizon, verifying plexity of extracting to verify the called measurement is performed on nicated to the HM model, becausecan the occur. boundary In condition principle an dictates observer that outside only the one black hole possible could outcome successfully verify the hence both particular fixed state of theat infalling the matter singularity system projectspose the that infalling the radiation boundary onto conditionresulting the projects corresponding postselected state. state If of we sup- entanglement is monogamous — and hence the infalling observer encounters a firewall at theity, horizon. as shown in figure Figure 5 Hawking radiation. After theno black longer hole entangled emits with the theare black Hawking maximally hole. quanta entangled (b) Unruh Entanglement vacuumentangled transfer states. state in The the creates final-state HM maximal projection model. entanglement of of JHEP08(2014)126 , i = | for Ψ and | Θ (and h out in in in be very H which is H ⊗ H . Neither , M out S out M ⊗ in is likely to be H i | ⊗ H Ψ ) we say that an | Θ in h H in H ⊗ M is maximally entangled, out ]. For most of the rest of our 40 ⊗ H in is a canonical maximally entangled H i Φ | below, it seems natural to conjecture that denotes the dimension of , where . i in 4.2 Φ – 10 – |H| | H ) I ⊗ and U (where out d H is “full rank” if the marginal density operator on is unitary, then the black hole S-matrix will be . Actually, the compatibility of the mode-by-mode thermal = | S out out , and the final-state boundary condition projects onto out compensates perfectly for the non-maximally entanglement of out nonzero (possibly degenerate) eigenvalues. Any full-rank bipartite ⊗ H | ⊗ H |H ⊗ d in Θ in in = h i we will revisit how this entanglement is affected by horizon-crossing H | H ), where Φ | 6 I in ) ) has ⊗ U |H ⊗ is invertible (though not necessarily unitary). S out 1 I H − U =( U : a freely falling observer crossing the apparent horizon should see a smooth ( | out out Φ might disrupt this ordering; will the black hole S-matrix still be unitary in that case? h ⊗ ⊗ However, as we will discuss in section We have another reason to demand a high degree of entanglement for the state If the initial state used in postselected teleportation is the full-rank entangled state Assuming in in in i i out ⊗ Ψ Ψ of the global statehard (required to resolve for decisively; unitarity) relateddiscussion, is issues we were a will discussed just delicate in assumethough quantitative [ that issue in the which section initial state weagents of find who interact with both thermal when its partnerhas is nearly definite traced energy, out. thenthe we global presume If that state the the of statemicrocanonical reduced this ensemble that density in Unruh operator collapses a vacuum on narrow to energy isentangled band, form nearly state whose a on purification maximally is black mixed aentanglement hole nearly — (required maximally for it smoothness is of the essentially horizon) the with the near maximal entanglement | vacuum state rather than ato seething closely firewall. resemble Smoothness the athas Unruh the sharply state, horizon defined requires in frequency the which with state ruh a respect partner mode to behind localized Schwarzschild time the outside is horizon, the entangled such with horizon that its which Un- the reduced density operator of either mode is the postselected state isvery in close some to sense maximally entangled. generic,nearly In maximally which that entangled means as case, that well. unitarity demands that the initial state norentanglement the of the postselected stateresulting is in maximally overall entangled, unitarity. butmaximally We the entangled see non-maximal to that ensure thefinal-state the state condition unitarity at is of adjusted the postselected appropriately. apparent teleportation, horizon as need long not as be the entangled state can be expressed as ( state, and | M One important criterion forused unitarity as concerns a the resource entangled in state postselected of teleportation. entangled state of hence also H 4.1 Entanglement across the horizon evaporation process. In general, though, interactions among the systems JHEP08(2014)126 ), 2 out / 3 , and ]. Put m , as in ⊗ H in ( will not M in O H 30 , in H H ). The vast 29 3 m ⊗ H and ( collapse rapidly; O M in 1 H H M H behind the horizon, as in ), have entropy is sampled uniformly with m M ( in out H V O ⊗ H (b) and in M in H denotes the dimension of the Hilbert and the other systems. In the tele- H in M |H| ) and evaporation time H 2 , where the states in 2 m is exactly maximally entangled, and consider ( M O out – 11 – ⊗ H 1, where (a) acting on 1 ⊗ H 6 can be decoded from the outgoing Hawking radiation M  1 in H
out ) M H 2 = H m has entropy ( (a) M in figure in O H m U U (b), also cause trouble because in effect the state of − 6 M ). Black holes created rapidly, in time 3 = exp | m ( M O |H , as in figure / | 1 out . Entangling interactions between the infalling radiation and the collapsing matter (a), M H . (a), compromise the fidelity of teleportation, because in effect (a). Intriguingly, if the final-state projection is chosen generically, or equivalently if |H H 6 6 For the purpose of analyzing whether quantum information initially carried by the A black hole with mass Let’s assume that the state of and in matter into two subsystems, hence space rapidly collapsing matter system require a time and hence have manythe fewer creation and possible evaporation microstatesevaporation of occurs than a while generic generic the blackthe black black case holes. hole hole where may the is Analysis be black still of subtle, hole being because forms assembled. rapidly. substantial We Let’s divide focus the instead Hilbert on space of the infalling figure the unitary transformation respect to the invariantthough Haar not measure, quite then exactly, the unitary. evaporation process is very,majority very of ways nearly, of making a black hole look like the time-reversed evaporation process and be projected onto aH maximally entangled state. Likewise,used entangling in interactions the between teleportation protocol will not be maximallythe entangled. consequences of entangling interactions between portation circuit, quantuminteractions information of effectively such chronology flows violatingbe backward systems dangerous, in with inducing chronology closed timemore respecting timelike in curves, prosaically, systems and entangling can hence interactions failurefigure behind of the unitarity horizon [ between postselected teleportation. 4.2 A generic finalUnitarity state may fail due to interactions between Figure 6 or between the infalling radiation and the outgoing radiation (b), may compromise the fidelity of JHEP08(2014)126 is out in ⊗ 1 of the N i 1 ⊗ H ) M M U . A similar H that defines
; (4.1) Ψ( | )
2 U ) . In the resulting 2 m / , and U ( 3 1 O ), or approximately N m 4 ( − O m ] ( ], but one might instead O 41 is the black hole entropy. starts out in a fixed state, 2 + 27 | / 2 is almost maximally mixed , . It follows that for a Haar- in 1 1 M 26 BH can be decoded in the outgoing N N |H S ρ 1 ρ ]. − we find [ M 42 out [ = ln U
) is maximally entangled with a reference exp in BH 1 / with infalling matter behind the horizon. S 0 | ≈ 〈 M in | | 1 〉 were sampled from a unitary 2-design rather H in 0 | 2 log(1 M is pure, that | 0 〈 U U M , and m |H – 12 – |H 1 out 0 | 1 N 〈 s log M is almost maximally entangled with a subsystem of ≤ is the normalized Haar measure on the unitary group, ⊗ H 1 . Tracing out the radiation system we obtain a mixed O 1 1 1 in N k N N dU 1 H . We assume that subsystem max N ⊗ H is extremely close to maximally mixed. ρ , and by averaging over 7 1 1 M − is maximally entangled with a reference system N N ) H -norm, 1 U M ( L 1 ) on N is very nearly maximally entangled with a subsystem of the outgoing Hawking U ρ 1 ( k 1 N N is obtained, which depends on the unitary transformation ρ dU by circuits with depth out Z as shown in figure  denotes the 1 . The HM model for a generic final-state boundary condition. Subsystem can isolate this subsystem which almost purifies 1 ⊗ H N 1 is the maximally mixed state on N out k · k , it is convenient to ask what happens when Nearly perfect unitarity is gratifying, but exact unitarity is what we yearn for. To Since the overall state of H H 1 out max N ensure exact unitarity, we must restrictthe the HM form model, of as the wellThis initial as necessary and the fine-tuning final interactions in entangled of the statesregard model in it has as been a criticized tantalizing hint [ about quantum gravitational dynamics. Surely, that generic typical final-state projection, an arbitrary initialHawking state of radiation with aconclusion fidelity would deviating still from apply ifthan one the the by Haar unitary measure, just a exp samplingachieved task exactly which by (unlike sampling a from relativelywith Haar small measure) error can quantum be circuit with size Thus the typical state on means that the referencethe system outgoing Hawking radiation, andon correspondingly that a unitary decoding map acting here ρ system e.g., its vacuum state.on After the final-state projection,the a postselected random state pure of state marginal state projected onto a Haar-randompostselected state state, determined by theradiation. unitary transformation H Figure 7 collapsing matter system JHEP08(2014)126 1 M could well , which de- 2 8 / . If the subsys- S BH S − e becomes nearly uncorrelated in1 H in2 H . in ⊗ H 1 is maximally entangled with a subsystem of the in M 1 S could be induced by the Hamiltonian dynamics H H M – 13 – in in1 H H and . What do infalling observers see? 1 M in M H H Hilbert space. To clarify this claim, consider figure , specified by the scrambling unitary transformation and in in H H M H is larger than half the size of is discarded, then the complementary subsystem ]. Indeed, the deviation from exact unitarity might be undetectable even in in in2 H H 31 . The collapsing matter subsystem of , if 1 in2 M On the other hand, because the final-state projection throughly scrambles the quan- Entangling interactions of H tum information encoded intimelike the curves collapsing behind matter, the the horizonto weird may only a consequences be portion of undetectable of by such the picts infalling closed the observers time-reversed with evolution from access the singularity into the black hole interior. Here as the matter fallssuitably from augmented the or horizon reversedof to by the the the black singularity. final-state hole projectionhave If S-matrix. a so, in well order these The defined to interactions resultingfind causal must ensure that information order, be unitarity the flow or quantum behind inin information the principle other encoded interact horizon words in with does if the its we not time-reversed earlier infalling try self radiation encoded to could in define the a collapsing causal matter. order we asymptotic infalling matter andis the exactly asymptotic outgoing unitary, though, radiation.departures infalling Even from observers if conventional inside this quantum the S-matrix interactions theory black between in hole the might HM still model, experience arising from entangling regarded as a success rather than a failure of5 the HM model. Detecting postselection whenUp approaching until the now singularity we have focused on the unitarity of the black hole S-matrix relating the practice”, even if weradiation disregard the [ complexity ofprinciple decoding until the the highly very last scrambledmethods stage Hawking of no the longer black apply. holea evaporation Since rather process, assuming crude when a guess, semiclassical generic finding final-state such boundary an excellent condition approximation is to just exact unitarity might be final-state projections come so closedynamical to constraints achieving we unitarity demand. enhancesbe the Violations artifacts plausibility of of of unitarity the the scalingnonperturbative semiclassical like quantum framework gravitational used correctionsmore, in of the information that formulation loss order of at are the such expected. HM a model, tiny Further- as scale would be exceedingly difficult to detect “in Figure 8 infalling radiation system tem with JHEP08(2014)126 . . r − in2 (5.1) H obeys , then in in1 , 1 . Suppose ⊗ H M in at the singu- 1 ρ H M S H , we find that the , r | − in1 ) | k in2 − | · |H n 1 |H ( M 1 2 . The conclusion is that, for |H = in1 | s H in1 ≤ |H 1 ), and hence has very limited time to k 2 m ( max in1 O ρ ), which falls into the black hole and interacts ⊗ M ) , and log – 14 – , which the observer might be able to access. Av- H S , of the infalling radiation system n ( S is larger than half the full system 1 in1 = M | , presumed inaccessible to our infalling observer, and H ρ in2 in in is pure, we find that the density operator H − H at all; instead it is nearly maximally entangled with |H in ) -norm from an uncorrelated product state by at most 2 2 1 S of ( in1 L ⊗ H H in2 in1 1 , log , 1 k H M M H = ρ | k 1 dS M | ] 2 Z , we obtain the same result by averaging over a unitary 2-design rather 41 deviates in the 4.2 denotes the maximally entangled state of in1 , 1 over the normalized invariant Haar measure on the unitary group, and assuming , if the discarded system meets the singularity in proper time max in1 M S S ρ m In the HM model we can imagine a meter (which can be regarded as a late-arriving In standard measurement theory, we usually suppose that the measured system in- Translated into the language of the HM model, this statement means that when quan- is hardly entangled with 1 with its environment cause theThe measurement meter alternatives to provides decohere a in recordmeasurement. a which particular can basis. be consulted later on to verifycomponent the of the outcome infalling of matter the system perform highly nongeneric measurements.resolved To address by more the fully HM howcondition model, the affects AMPS we puzzle measurements must is straddle examine performed the more inside black deeply hole a how horizon. black the final-state hole, boundary orteracts measurements unitarily which with a suitable “meter” and that subsequent interactions of the meter 6 Measurements inside blackSo holes far we have argued thatdynamics the under HM generic model conditions. yields But unitary (or the very AMPS nearly puzzle unitary) concerns black infalling hole observers who mass perform complex decoding operationshorribly on when the subjected infalling to radiation.larity, the but This highly she observer nonlocal might may scrambling not suffer standard transformation have time quantum to mechanics. discern any other troubling violations of the rules of tum information encoded in“reflected” a at small the subsysteminformation singularity of escapes by the the collapsing a notice matterinfalling generic radiation. of Hilbert final-state An an space boundary infalling is observer observer condition, who with crosses the access the event reflected to horizon much of a less black than hole with half of the generic M Specifically, if log state of As in section than Haar measure. the inequality [ where determined by the scramblingwe unitary discard the subsystem retain the complementary subsystem eraging that the overall state of is a subsystem of the collapsing matter, which is maximally entangled with a subsystem, JHEP08(2014)126 in PT H U and M H is applied to the meter U – 15 – . The meter eventually reaches the singularity, and in H . At the singularity, the meter is paired with a different 9 , this process is equivalent to one in which the operator 9 ; an entangling unitary transformation in H -out system are transposed, while those of the meter are not. In general, and teleported out of the black hole. . Here “PT” denotes “partial transpose”, meaning that the initial and M in in H H ] prevents backward signaling, hence protecting causality. 31 As indicated in figure It seems that either point of view — (1) that measurements performed inside black hole As we will discuss, though, measurements performed inside black holes which are not We have seen that unitary interactions inside the black hole between Not necessarily. If the overall evolution is unitary, then the meter, like the rest of the subsystem of acts on the metersubsystem and of the infallingfinal states matter of subsystem the that is paired with the measured 6.1 Unitarity constraint Consider a measurement of theprepared infalling outside Hawking the radiation horizon behind and the droppedwith horizon. into a the A subsystem black meter hole. of is and It that is programmed subsystem to as interact in figure radiation [ are undone by the final-state condition,by or decoding (2) the that measurement Hawking outcomes radiation canHM — be model. raises recovered intriguing questions about the viability of the agent behind the horizonward light may cone; be furthermore, able thisremain to causality outside violation send the may causality-violating be blackhard signals detectable hole. to by into reconcile observers her Such with who causal; causality back- a a violation dual possible boundary in resolution is description the that of bulk the the computational spacetime dynamics complexity might of which decoding be is the unitary Hawking and meaning of such measurements. Andbe if measured, the even degrees in of principle, freedom in inside the what black sense hole canpermanently cannot the erased black can hole be interiorteractions reconciled be between with said system to unitarity and exist? of meter the obey black suitable hole constraints. S-matrix In if that the case, in- however, an can threaten the unitaritydemand of that the such black interactions holeat are S-matrix. in the One effect singularity. way undoneblack In to by protect hole that the unitarity would final-state case, is boundary be however,radiation to condition records permanently after of erased, the measurements and black performed could hole inside not has the be evaporated. extracted from We the might Hawking then question the operational meter can beIndeed, extracted this decoding and of decoded thecompletely meter by and might disappeared. performing be In done ain long this principle, complex after sense, which the a quantum can black recordin computation. hole be of standard has consulted the measurement evaporated later measurement theory. survives, on at to least verify the measurement outcome, just as is subjected to themeasurement final-state outcome boundary is condition. destroyed? Does this mean thatinfalling the matter, record will of be teleported the outoutgoing of Hawking the radiation, black albeit hole. in A a relic highly of scrambled the form. meter In survives in principle, the the scrambled with the infalling radiation system JHEP08(2014)126 ), in 2 H (the U (6.1) (6.2) a meter meter PT U acting on M out inside the black U in H = out . (the unitary transformation ]. , PT . 26 U out (the unitary transformation H meter
meter in |
b | H b ih meter U a is an orthonormal basis for the meter. ih a + } meter is also a complete set of orthogonal projec- . There is no equivalent circuit, without b + | } b | ⊗ ,...N – 16 – ⊗ M T 3 if the measured state is in the support of Π ) , a in ,...N 2 a a M 3 , Π , = is unitary, then the transformation 2 (Π is the entangling unitary that realizes an orthogonal , a,b X = 1 out a,b PT U X = = 1 , b U = has a unitary partial transpose, the resulting circuit has a U , a is a complete set of orthogonal projectors acting on the mea- meter U ). The partial transpose of this unitary, 2 T chronology violating } meter ) U i N b a M {| (Π . { ,...N 3 , in 10 2 , is modulo a = 1 M + , a b . Measurement of the infalling Hawking radiation inside a black hole. The meter used a in Π { meter In this case, even if For example, suppose that ), then falls into the black hole and interacts with between system and meter is replaced by its partial transpose 1 measurement, a meter interacts firstU with a subsystem of as shown in figure peculiar property — it is tors. We see, therefore,hole that need orthogonal not measurements violate performedsome unitarity. on other If grounds. such measurements are to6.2 be forbidden, it Chronology must violation beWe on also wish to consider measurements which straddle the black hole horizon. In such a addition is also unitary, since Here sured subsystem, and Hence the state of the meter shifts by behind the horizon is compatibleof with the unitarity criterion of for the unitarity S-matrix. of This the is S-matrix just formulatedmeasurement a in performed special [ case on the infalling radiation: teleported out of theU black hole. In the equivalent causally order process,the the partial unitary interaction transpose ofmatrix a is unitary not operator unitary. is not But unitary, if and therefore the resulting S- Figure 9 in the measurement interacts with the measured system behind the horizon; then the meter is JHEP08(2014)126 , can in H R occurs first and is 1 U B (b) A out 1 that was emitted before the agent fell U H out 2 H meter meter U , while from the perspective of the radiation the 2 meter meter in U – 17 – outside the horizon. out BR M in (a) M (a). , while from the perspective of the radiation the interactions occur in the 11 2 occurs first and is followed by U 1 U . (a) A measurement inside a young black hole may influence the Hawking radiation . A measurement straddling the black hole horizon. From the meter’s perspective the meter meter In the case of a young black hole (one not entangled with its surroundings), this Even if the S-matrix relating the asymptotic infalling matter to the asymptotic out- influence the state of thein, Hawking as radiation in in figure ability to influence theso Hawking disturbing. radiation by The tossing previously a emitted meter Hawking into radiation the is black highly hole mixed is because not it is “pulled tight” because fromfollowed the by meter’s perspective theopposite interaction order. going radiation is unitary, thisspacetime. chronology violation For may example lead an to agent acausal falling effects in into the the bulk black hole, by interacting with outside, but the effect ofan the measurement old is black not hole immediately modifies detectable. the (b) state A measurement of inside postselection, which describes thetime information flow direction and for admits a both globally the defined radiation forward and the meter. The information flow cannot be Figure 11 interaction interactions occur in the opposite order. Figure 10 JHEP08(2014)126 (6.3) (6.4) (6.5) are the Z today can . The circuit and BR , so the state of 12 X , we can verify the BR i + superpositions of two φ | is maximally entangled behind the horizon may ± , where in B B H . Z , .

! ⊗ , 1 AB AB A i i i − b Z , we prepare the meter qubit in the 1 0 0 11 10 ⊕ Z

⊗ ± | ± | and a, a = Z B AB AB in the Bell basis: = 1. i i X Z i 7→ | 00 01 ⊗ | | AB ⊗ – 18 – A ,Z 2 2 a, b | 1 1 X Z √ √ ! . But if a meter tossed into the black hole interacts R = = 0 1 1 0 . To measure

X AB AB (a) by saying that the meter which falls in becomes rapidly i i CNOT : = 1 and = ⊗ ± ± 11 φ ψ X | X | X is the “phase bit”, distinguishing the ⊗ X X and ⊗ Z , perform two successive CNOT gates from the qubits to be measured X (b), interactions of an infalling meter with . Two meter qubits are used in the measurement, one each for the i ⊗ } 0 | 1 Z 11 , behind the horizon, the meter becomes entangled with 0 A ∈ { is the “parity bit” of the qubit pair, distinguishing the strings 00 and 11 from the a, b, Z is no longer pure. Thus the outcomes of measurements performed on A quantum circuit for measuring in the Bell basis is shown in figure Our goal is to measure the qubit pair However for an old black hole (one highly entangled with its previously emitted radia- = 1 eigenstate ⊗ on a complete basis is where measurement of Z strings 01 and 10. strings of the samestate parity. by If checking both a two-qubit state is expecteduses to a be coherent two-qubit quantum gate, the controlled-NOT (CNOT) gate, whose action of the two commutingPauli Pauli matrices operators Z These four orthonormal states can be usefully characterized as the simultaneous eigenstates detail how one can usea a maximally quantum entangled computer basis. tothe perform For discussion measurements simplicity can which we easily project will be onto consider extended measurement to of higher qubit dimensional pairs; systems. depend on whether orwill not analyze we such decide causality to violating toss effects the in meter more6.3 into detail the in the black Entanglement-verifying following hole measurements subsections. tomorrow.To We prepare for the ensuing discussion of the AMPS experiment, we will first discuss in tion) as in figure produce genuinely detectable acausalcondition effects dictates outside that the thewith black recently the hole. previously emitted emitted radiation Herewith system system the system final-state BR the process depicted in figure distributed among the black hole’sthe degrees meter, of having freedom. been Muchwith emitted later, the in a radiation the scrambled that Hawking version had radiation, of already could been wind emitted up before being the entangled meter fell in. entangled with the black hole microstates. An observer outside the black hole can interpret JHEP08(2014)126 ), i |−i 1 | . The to the onto a + behind R i X 0 B | ( HA ⊗ 2 1 √ X 1 eigenstate = i − + | = is inside the horizon, X A 1 or +1. to the . Hence we read out the sum − i . After commuting two CNOT Z 〉 is the Unruh partner of + is A Z | | 0 = 1 eigenstate is the Unruh partner of H X X A 〉 ⊗ + X . | X B , while is performed first, followed by the measurement (and does nothing if the meter is in the state B i = Z 1 | – 19 – ⊗ = 1 eigenstate . The pair of CNOT gates applies is a subsystem of the early radiation, maximally Z X X R 〉 Z | 0 . , our Bell measurement circuit has the desired sequential form. The 〉 + X 12 AB | , we prepare the meter qubit in the is outside the horizon of a black hole, and qubit X B ⊗ . Quantum circuit for measuring a qubit pair in the maximally entangled Bell basis. In . In the equivalent circuit on the right, the meter interacts first with one of the two X X ⊗ If qubit first, then falls through the horizon to interact with ), hence flipping the meter from the X i 0 which the black holeentangled initially formed, with the recentlythe emitted horizon, radiation also maximallyfinal-state entangled boundary with condition at themaximally singularity entangled inside state. the black For hole conceptual projects clarity and notational simplicity, we suppose of the qubit pair 6.4 The AMPS experiment Now we will discuss entanglement-verifying measurementsas performed considered on by an AMPS. old For black a hole, particular pure quantum state of the infalling matter from B gates as in figure two-qubit meter falls to theis singularity thoroughly and scrambled is with teleportedhole, other out but qubits of in in the principle the black it Hawking hole. can radiation The be emitted meter decoded by the and consulted black later on to verify the entanglement measured qubits if the meter is| in the state (or not) depending on whether the eigenvalue of then the Bell measurement should be performed sequentially; the meter interacts with to the meter, andmodulo then two read of out the two themeasure measured meter qubits by without measuring collectingperform any additional two information. successive To CNOTthen gates read out from the the meter meter by to measuring the qubits to be measured, and Figure 12 the circuit on theof left, the measurement of measured qubits, thenverify with entanglement the that other. straddles a black This hole latter event circuit horizon. may be used by an infalling agent to JHEP08(2014)126 , . = AB Z AB i + ⊗ entan- φ Z entangle- . BRenda ⊗| and ABby BR , obtaining to transform R HR AB AB i basis. In that BR Z , obtaining the + 14 φ . Provided ABby ⊗ Z | AB B B Z before ABby, then neither first, and then falls into the B . B HA i many times (each time with a fresh + φ | and ABby, who measures BR ]. In principle, then, a single agent could many times (each time with a fresh meter), 2 rather than probability 1. onto 31 BR / can be extracted efficiently from the Hawking AB followed by an entangled measurement of R HA – 20 – BR basis and BRenda’s meter in the every time. i X + φ | later on. every time. Each of ABby’s meters could be extracted from the A i , we can use the quantum circuit identity in figure + φ | 13 are all single-qubit systems, that the Unruh state is . These two measurements do not commute, so their order makes a R B with probability 1, hence successfully verifying the expected before BRenda, both BRenda and ABby successfully verify the entan- i X + B with probability 1, hence successfully verifying the expected ⊗ φ | , and i = 1 is obtained with probability 1 A + B X , φ X | A ⊗ , H X How should we interpret this failure? Has the system measured by each party been As shown in figure If both BRenda and ABby measure, then both must interact with If there we no BRenda, ABby could perform Bell measurement on If there were no ABby, BRenda could perform Bell measurement on Let us suppose for now that the system until ABby’s (scrambled) metermeter is with emitted BRenda’s in and thereading execute Hawking out a radiation, ABby’s CNOT then meter gatecase unite both in from ABby’s verifications ABby’s the succeed meter withfollowed to probability by ABby, BRenda’s 1. then before Furthermore, BRenda if can BRenda measure measures multiple first, times after ABby’s interaction with modified due to the measurementment performed meter by the used other by party?the each Or other party, is party’s rather action? it than that We the the believeters the measure- measured used correct system, interpretation by is has BRenda thatwhile been the and measurement disturbed the ABby me- by become measured entangled systems when themselves BRenda are measures not before actually ABby, modified. Indeed, we can wait first, but this equivalent circuitBRenda’s meter contains which an maximally additional entangles the CNOTfirst, two BRenda’s gate meters. and from Hence ABby’s ABby’s meters, when each BRenda metertherefore considered measures the to individually, readout become of maximally the mixed; meter1 yields a or random outcome. The expected outcome performs just halfmeasures of the Belldifference. measurement — BRenda measures the circuit in which BRenda measures first to an equivalent circuit in which ABby measures Hawking radiation after the black hole has evaporated, allowinginteracts us with to confirm herglement results. as described above.verification succeeds. But To if understand BRenda what interacts happens, with it suffices to suppose that each party meter), obtaining the outcome outcome ment. Just to be sure,obtaining ABby the could outcome measure acting independently: BRenda, whonever measures needs to enterblack the hole black to hole; interact with ABby interacts with the outcome glement. Just to be sure, BRenda could measure and that the boundary condition projects radiation, despite the complexity ofperform this an task entangled [ measurementTo of illustrate our essential point, though, it will suffice to consider two different agents that JHEP08(2014)126 A 〉 c ⊕ Z | 0 b c R B b ⊕ a b A . (ABby performs her measurement = 1 with probability 1. BRenda H B X a a R 〉 + X Z ⊗ | A ⊗ times, each time with a new meter, after X B k Z B = = X – 21 – ⊗ A 〉 Z X | 0 c ⊕ R b c . To see what happens, we may apply the circuit identity R before ABby measures or on the meters used in BRenda’s later measurements. Z R B ⊗ Z BR A B outside the horizon first, then falling into the black hole to interact with ⊗ b ⊕ Z B B a b Z . 13 H times, once for each time one of ABby’s CNOT gates moves from after BRenda’s . The quantum circuit identity used to establish the equivalence of the two circuits . Entanglement verifying measurements for an old black hole. In the circuit on the left, 〉 k + X | a a 14 Likewise, ABby might measure , each time with a fresh meter. When she does so, although BRenda’s first measurement, naturally concludes that ABby’s measurementeffect disturbed on BRenda’s the first state meter, of but had no BRenda measures figure shown in figure B performed before ABby’s, yieldsperformed a after random ABby’s, outcome, successfully all of verify Brenda’s later measurements, Figure 14 inside the horizon later.)and there In is the an additional equivalent CNOT circuit gate on from the ABby’s right, meter to ABby BRenda’s. measures before BRenda, Figure 13 BRenda measures by interacting with JHEP08(2014)126 X (6.6) succeeds once, the i additional R + Z k meter; these φ | ⊗ Z B Z basis and BRenda’s meter Z X basis and BRenda’s qubits in the . X
meter to ABby’s meter qubits and BRenda’s meter +1) X k k ( ⊗ |−i , but the circuit also includes + B basis the outcome would be uniformly random. X – 22 – +1) k Z ( four times after BRenda measures ⊗ or i B X + X | X ⊗ 2 1 ABby BRenda A √ X X . Using this procedure, all the verifying measurements succeed 15 X 4. However, once ABby’s meter has been emitted in the Hawking meter qubits to BRenda’s meter qubit in order to disentangle the / meter and one from Brenda’s k Z . If ABby measures basis, as in figure Z This discussion requires only minor modification when both BRenda and ABby perform basis. meter to BRenda’s gates maximally entangle ABby’s metermaximally and mixed, BRenda’s. and for Hence both eachwith parties meter, only the individually, verification probability is of 1 theradiation, entangled the state meters can be disentangled by reversing these CNOTs, and then measuring complete Bell measurements. Nowidentities both we parties may carry again two-qubitwhere meters, transform ABby and the measures using first, scenario circuit thisof where time Brenda’s. BRenda by measures moving This two first procedure of to generates ABby’s the CNOT two one gates additional through CNOT two gates, one from ABby’s of the qubits were measuredTo in complete the the verifying measurements properly,from we must each perform of a compensating ABby’s CNOT meters, and only then measurein each the of ABby’s meterswith in certainty. the qubit is In this state, the marginal density operator of each meter qubit is maximally mixed; if any CNOT to before BRenda’s CNOTmeasurements gate. occur before In the BRendaCNOTs equivalent touches circuit acting we on obtain, the allqubit. meters, of ABby’s one Thus, from after eachoutside all of the of black ABby’s hole, ABby’s meter the qubits meter quantum to qubits state BRenda’s are of meter decoded ABby’s from the Hawking radiation Figure 15 measurements are completedBRenda’s by meter performing qubit, before CNOT measuringZ gates ABby’s from qubits in each the of ABby’s meter qubits to JHEP08(2014)126 , N B BR is the black m , ABby’s meter should ) (where B m log in a region of low curvature before ABby sends it, then m t A behind the horizon, a protocol ( Z O A X if ABby does not measure later on, and i + φ | – 23 – ’s Unruh partner . These disentangled measurements successfully B 16 , and then signal BRenda by either holding the robot Z AB ABby BRenda X entanglement with certainty. AB and tomorrow. If BRenda reads her meter immediately after interacting with BR . If ABby performs a complete Bell measurement after BRenda performs a complete Bell AB entanglement (and hence unitarity of the black hole S-matrix) in the HM model. There is a limit on how far backward in time the signal can propagate, if we are to Though ABby sends her message by manipulating entanglement (and hence a smooth black hole horizon) can be reconciled with verifiable describe this protocol semiclassically. Afterenter BRenda the interacts with black hole within lesshole than mass), the if scrambling time itinside is the to black “catch hole. up”ABby On can with send the a other signal hand, which is we received may by chain BRenda a together time many such protocols; if or dropping it intoquantum the information black manages hole.already to in Of escape a course, sense from “causality ifallows violating”, behind a black but the stronger hole the type postselection black evaporation of inblack is hole the causality hole. unitary, HM violation, horizon, model then detectable which potentially by observers is who stay outside the is uniformly random if ABbybackward does light measure cone later which on. BRenda Thus may ABby receive. can send afor signal into backward her signaling canto be perform executed Bell outside measurement on the horizon. ABby can program a robot BR Nevertheless the implications of themeasurement scenario performed are by perplexing, BRenda because todaymeasure the can outcome depend of on a whetherthe or not outcome ABby of decides her to Bell measurement yields 6.5 Acausal signaling? This scenario indicates how postselection might resolveAB the AMPS puzzle, because verifiable Figure 16 measurement, the measurements are completedmeters. by performing this circuit on ABby’s and BRenda’s in the appropriate basis,verify the as in figure JHEP08(2014)126 before it ]. Acausal , or equiva- 7 Nt B (a measurement AB . In principle, she could R entanglement — she will per- ). AB BR – 24 – to initiate the protocol; the decoding time, then, could B ]. To decipher ABby’s message, BRenda needs to perform 31 ] have made observations related to ours regarding measure- ; before that she needs to extract ) Page time, and long compared to the scrambling time. Even alone, rather than a joint measurement on 3 33 BR m B ( O ].) On the other hand, since the signaling protocol involves decoding the Hawk- well before she touches 43 R ), which might be further extended by feeding the black hole a steady diet of infalling 3 Bousso and Stanford [ We expect, at least in the case of an asymptotically AdS bulk spacetime, to be able Another possible way to prevent acausal signaling is to invoke the complexity of de- This way of enforcing causality in the HM model severely limits ABby’s ability to One way to prevent acausal signaling would be to place further restrictions on ABby’s m ( 6.6 Computational powerIf of acausal the signaling HM in the model “time-travel bulk paradoxes” spacetime really threaten is the possible,that it consistency if is of natural we to the demand wonder theory. overall whether unitarity We of emphasize, the though, black hole S-matrix (or of the boundary field ing radiation, the correspondingcomplex. process One in possibly the consistentpleted either point dual the of field black view theory holesignaling is has might from that completely inside be by evaporated the astoundingly the or black time a hole. firewall the has decoding arisen, isments preventing inside com- black holes, albeit from a different perspective and in a different language. on the boundary ofsignaling outside spacetime, the where black thestrict hole causality field horizon, in theory if the is itwould field correspond unitary can theory. to occur, and (Local precursor might operators, local observablestheory which be used [ [ might hard in be the to highly nonlocal, bulk reconcile in signaling with the protocol boundary decode be comparable to the so, high decoding complexity may make the signaling protocolto infeasible. describe the dynamics of an evaporating black hole using a dual field theory defined possible, it might be reasonableor to in claim other that words that the the black hole horizon interior is does reallycoding not a the really singular Hawking radiation exist, firewall. [ a joint measurement on form a measurement on which, by the way, will damage the entangled statemeasure of the black holerecord interior, or that at can any survive rate after limits her the ability black to hole create evaporates. a measurement If no such measurements were actions behind the horizon,ABby is going unable beyond to the applylently the enforcement if unitary of this transformation transformation unitarity. to iswill the In not Unruh reversed partner be by particular, of able the if to final-statethe alter boundary other the condition, hand, outcome then ABby of ABby will a also measurement BRenda be performed unable previously. to On verify the is sent (ignoring the timein needed principle for can each be party parametricallyfundamental to small limitation execute compared on her to how part the farO of scrambling backward the time). a protocol, signal Then which can thematter be only to sent is maintain the its black mass hole as lifetime it radiates. parties acting together should be able to send a signal which is received time JHEP08(2014)126 – 25 – ]. Hence general final-state projection models allow very 44 In the HM model, observables inside the horizon fail to commute with observables Our main point is that quantum correlations can be polygamous in the Horowitz- It is therefore important to recognize that the HM model admits only a restricted kind Another natural question concerns the computational complexity of simulating the generic final-state boundary condition might be regarded as aoutside success the of horizon the acting model. ordered on information the flow, same the timeobservables, but slice, outside at because observables a in act later the on “time”. corresponding the Other causally same features of system black as hole the complementarity inside are also flow moving only forwardobservers in who time, stay at outside least thedynamical black for hole. models, the These purpose but constraints oftuning as are the describing nearly best model. the fulfilled we by On viewpoint generic thespacetime can of other background, tell hand, achieving since they the unitarity can HM up model be to is formulated rigorously exponentially on fulfilled small a semiclassical only corrections using by a fine ciled. In thepostselected HM quantum teleportation, model, due quantum tosingularity. a information Loosely boundary escapes speaking, condition quantum from imposed informationity at the flows to the black forward the spacelike in hole singularity, time backwardin interior from in time via past time from infin- from the thethis horizon singularity to flow to future of the infinity. information horizon, If is then suitable essentially forward dynamical equivalent constraints to are a satisfied, manifestly unitary causally ordered horizon, and (3) validitythat of local these effective three fieldcorrelations assumptions theory can are outside be inconsistent, a polygamous, black since contrary to hole. together standard They they quantum argued mechanics. implyMaldacena that final-state projection quantum model, permitting these three assumptions to be recon- 7 Discussion The AMPS puzzle has deepenedthat the mystery falls surrounding into the fateassumptions: a of (1) quantum black unitarity information of hole. black hole evaporation, (2) AMPS smoothness of investigated the the black hole compatibility event of three reasonable quantum gravity computers canquantum solve computers, problems we which know areHM no model beyond should reason the exceed why reach that the of of computational standard other quantum power gravity of models. the (unitary) complexity class NP, theusing class a of classical problems computer). for which a solution canof be postselection, efficiently if verified we requirean the open model question to be whetherquantum compatible quantum computer, with gravity so unitarity. can it Though is it be at is simulated still least efficiently possible with in a principle standard based on current knowledge that inconsistent. HM model with ais quantum known computer. to Quantum behard computation PP-complete with computational [ final-state problems projection to be solved “efficiently” (in particular, PP contains the theory), the allowed acausal bulk phenomena are highly constrained and not obviously JHEP08(2014)126 ] for other 46 , 45 , 40 ] may inoculate the HM model 31 – 26 – Even if the HM model turns out to be wrong in detail, we believe that the picture of Like all other resolutions of the AMPS puzzle proposed so far, the HM model will need In postselected quantum mechanics, cloning of quantum states is possible, and because However, this signaling protocol only works if Hawking radiation can be rapidly de- If a black hole has an interior, it should be possible to perform measurements inside the dacena, Joe Polchinski, and Douglasregarding Stanford for final-state inspiring projection discussions andother models. correspondence participants at JP theFuzz, appreciates or August many Fire”, 2013 and helpful KITP weand interactions workshop also Douglas with benefited “Black Stanford. from Holes: The comments research Complementarity, on of the SL manuscript was from supported Don in Marolf part by DARPA, by AFOSR, hole physics. Acknowledgments We gratefully acknowledge very valuableTegmark for discussions helpful with discussions, Alexei and JP Kitaev. thanks Raphael SL Bousso, thanks Daniel Max Harlow, Juan Mal- This picture reminds usand that the in global particular physicsthe of that the no-cloning fundamental black principle properties hole andtheory interior of monogamy of could quantum standard of be gravity. entanglement quantum And subtle, if mightsingularities, mechanics nature be we really such may indulges relaxed anticipate in as in postselection deep at consequences a future in complete spacelike quantum cosmology as well as black to be developed furtherstrive before to it expunge can thepersuasively be why dynamical the conclusively fine fine assessed. tuning tuning is In the somehow particular, natural. model weinformation seems flow should in to black hole require, spacetimes provided or by the to model explain is interesting and valuable. argument for the existencethe of HM a model; conceivably, firewallreasons though, at the other the horizon than black couldarguments those hole nevertheless supporting originally horizon fail the promulgated to within existence be of by the firewalls.) smooth AMPS. context for of (See [ physics of the HMmodel model occurs may provide inside the helpfulshould black hints work, hole, about if particularly such how at a a the description dual singularity; exists. description the ofmonogamy the of black quantum hole interior entanglement can be relaxed, we know no logically compelling horizon can enable causality-violating signaling outside the horizon. coded; the high computationalagainst complexity acausal of signaling decoding outside [ physics the outside horizon. the In horizontheory that as could event, in we not AdS/CFT know duality, be where no the accurately reason field why captured theory the is by unitary a and local. dual The boundary truly novel field behaves like a rapidly scrambling quantum system interacting withblack its hole. surroundings. Furthermore, recordsfrom of such the measurements Hawking should radiation insuch measurements emitted principle in be after the recoverable HM the model, black finding hole that actions evaporates. performed behind We the have black hole analyzed realized; in particular, from the viewpoint of an observer who stays outside, the black hole JHEP08(2014)126 ]. ] , SPIRE , IN ][ ]. ]. ] , SPIRE IN [ hep-th/9409089 [ ]. SPIRE ]. IN (2004) 71 , , ][ (1993) 1291 48 (1999) 1113 arXiv:0907.1190 SPIRE ]. ]. [ IN SPIRE 71 ]. 38 [ IN (1995) 6377 ][ ]. SPIRE gr-qc/9310026 SPIRE SPIRE , 36 IN IN [ IN ][ ]. Black holes: complementarity or ][ (1974) 30 Better late than never: information SPIRE hep-th/9306069 IN (2013) 101301 [ ][ 248 IBM J. Res. Dev. Phys. Rev. Lett. , 110 , The stretched horizon and black hole ITP, UC Santa Barbara Conference on Quantum – 27 – Int. J. Theor. Phys. arXiv:0909.1038 J. Math. Phys. ]. Nature ˙ [ Zyczkowski, , , hep-th/9305040 (1993) 3743 arXiv:0708.4025 , [ quant-ph/0310037 arXiv:1207.3123 SPIRE [ [ Gedanken experiments involving black holes IN limit of superconformal field theories and supergravity ), which permits any use, distribution and reproduction in hep-th/9308100 [ D 48 [ (1998) 231 [ ]. N Monogamy of quantum entanglement and other correlations Black holes as mirrors: quantum information in random ]. Phys. Rev. Lett. 2 , (2009) 224001 (2007) 120 , 21–26 Jun 1993. SPIRE (2013) 062 26 SPIRE ]. IN 09 IN Phys. Rev. CC-BY 4.0 (1994) 966 (1976) 2460 (2004) 022309 02 ][ The large- , ][ Black hole explosions Breakdown of predictability in gravitational collapse This article is distributed under the terms of the Creative Commons The information paradox: a pedagogical introduction Is entanglement monogamous? The world as a hologram Dimensional reduction in quantum gravity JHEP Average entropy of a subsystem Black hole information A 69 D 49 D 14 , JHEP , ]. SPIRE gr-qc/9305007 quant-ph/0307120 hep-th/9711200 IN retrieval from black holes Aspects of Black Holes subsystems Phys. Rev. Class. Quant. Grav. [ [ Adv. Theor. Math. Phys. [ firewalls? Phys. Rev. [ Phys. Rev. complementarity J. Preskill, private communication at the P. Hayden and J. Preskill, M. Koashi and A. Winter, S.D. Mathur, S.L. Braunstein, S. Pirandola and K. D.N. Page, B.M. Terhal, D.N. Page, J.M. Maldacena, A. Almheiri, D. Marolf, J. Polchinski and J. Sully, L. Susskind and L. Thorlacius, G. ’t Hooft, L. Susskind, S.W. Hawking, L. Susskind, L. Thorlacius and J. Uglum, S.W. Hawking, [9] [7] [8] [4] [5] [6] [2] [3] [1] [15] [16] [12] [13] [14] [10] [11] any medium, provided the original author(s) and source are credited. References supported in part byMatter NSF, ARO, (IQIM) and is DOE. anBetty The Moore NSF Institute Foundation. Physics for Frontiers Quantum Center Information and with supportOpen from Access. the GordonAttribution License and ( by the ARO under a MURI program, and by Jeffrey Epstein. The research of JP was JHEP08(2014)126 ] , , , (2004) 026 (1982) 802 ]. (1983) 2960 ]. 03 299 (1988) 1351 (2013) 212 (1990) 1796 (2011) 025007 ]. SPIRE D 28 SPIRE 60 (2004) 008 10 (2013) 085 Quantum IN IN JHEP [ arXiv:1211.7070 , D 41 D 84 02 [ ][ SPIRE 06 , Nature (2006) 061302 arXiv:1405.4710 IN , , [ JHEP 96 , An apologia for firewalls Phys. Rev. JHEP JHEP , , (2013) 124023 , (1982) 271 ]. ]. Phys. Rev. Phys. Rev. , , Phys. Rev. Lett. (2013) 064023 , D 87 (1993) 1895 A 92 SPIRE SPIRE IN arXiv:1005.2219 70 IN ][ [ ]. ]. D 88 Phys. Rev. Lett. ][ , Phys. Rev. How the result of a measurement of a component SPIRE SPIRE Phys. Lett. , IN IN , – 28 – ][ ][ Phys. Rev. , The black hole final state (2011) 040403 Phys. Rev. Lett. , A single quantum cannot be cloned arXiv:1310.7457 106 Wave function of the universe [ Cool horizons for entangled black holes arXiv:1306.0533 An infalling observer in AdS/CFT Comment on ‘The black hole final state’ ]. [ Quantum channels in quantum gravity Black hole entanglement and quantum error correction ]. ]. ]. Measurements without probabilities in the final state proposal ]. ]. Quantum computation vs. firewalls Internal structure of black holes SPIRE SPIRE SPIRE SPIRE arXiv:1211.6913 arXiv:1304.6483 IN SPIRE SPIRE [ [ IN IN IN Teleporting an unknown quantum state via dual classical and ][ IN IN (2013) 781 ]. ][ ][ ][ (2014) 044038 ][ ][ 61 Phys. Rev. Lett. Closed timelike curves via post-selection: theory and experimental Nonviolent nonlocality , Complementarity is not enough D 89 (2013) 107 (2013) 018 Communication by EPR devices Almost certain escape from black holes ]. ]. ]. ]. ]. ]. 10 09 SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE IN IN arXiv:1207.5192 arXiv:1007.2615 arXiv:1301.4504 IN IN hep-th/0311269 quant-ph/0406205 hep-th/0310281 IN IN arXiv:1211.6767 [ Einstein-Podolsky-Rosen channels [ [ Phys. Rev. mechanics of time travel[ through post-selected telportation [ of the spin of[ a spin-1/2 particle can turn out to be 100 demonstration [ [ [ [ [ JHEP Fortschr. Phys. JHEP [ [ C.H. Bennett et al., E. Poisson and W. Israel, R. Bousso, R. Bousso and D. Stanford, M. Rangamani and M. Rota, S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti and Y. Shikano, D. Harlow and P. Hayden, Y. Aharonov, D.Z. Albert and L. Vaidman, S. Lloyd et al., D. Gottesman and J. Preskill, S. Lloyd, W.K. Wootters and W.H. Zurek, D. Dieks, J.B. Hartle and S.W. Hawking, E. Verlinde and H. Verlinde, J. Maldacena and L. Susskind, G.T. Horowitz and J.M. Maldacena, S.B. Giddings, K. Papadodimas and S. Raju, A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, [35] [36] [32] [33] [34] [30] [31] [28] [29] [26] [27] [23] [24] [25] [20] [21] [22] [18] [19] [17] JHEP08(2014)126 , , ] ] ]. ] , , arXiv:1308.2665 [ . arXiv:0808.2096 [ ]. arXiv:1308.3697 quant-ph/0606161 [ [ ]. ]. (1993) 4287 ]. (2013) 084035 SPIRE IN “Event-ready-detectors” Bell 71 ]. (2008) 065 ][ SPIRE SPIRE D 88 IN 10 IN A decoupling approach to the quantum ][ Exact and approximate unitary 2-designs: (2014) 041102 (2009) 012304 ][ 112 JHEP Negative energy, superluminosity and holography quant-ph/0702005 , A 80 [ Phys. Rev. – 29 – Phys. Rev. Lett. , , arXiv:1307.4706 quant-ph/0412187 A multiparticle generalization of entanglement swapping [ [ (2008) 7 hep-th/9903228 [ Phys. Rev. 15 , quant-ph/9708004 Gauge/gravity duality and the black hole interior [ Phys. Rev. Lett. Fast scramblers , (2005) 3473 (2013) 171301 (1999) 084006 (1998) 822 111 A 461 Quantum computing, postselection, and probabilistic polynomial-time Firewalls from double purity Frozen vacuum A 57 D 60 Open Syst. Inf. Dyn. ]. ]. ]. , ˙ Zukowski, A. Zeilinger, M.A. Horne and A.K. Ekert, SPIRE SPIRE SPIRE IN IN IN [ [ constructions and applications Phys. Rev. Proc. Roy. Soc. Phys. Rev. Lett. capacity [ experiment via entanglement swapping Phys. Rev. R. Bousso, R. Bousso, J. Polchinski, L. Susskind and N. Toumbas, S. Aaronson, D. Marolf and J. Polchinski, P. Hayden, M. Horodecki, A. Winter and J. Yard, C. Dankert, R. Cleve, J. Emerson and E. Livine, M. S. Bose, V. Vedral and P.L. Knight, Y. Sekino and L. Susskind, [45] [46] [43] [44] [40] [41] [42] [38] [39] [37]