Black Holes:
Information paradox and teleportation
through the wormhole
Max Oosterbeek 10168877
Thursday 29th August, 2019
MSc Physics Master Thesis
Supervised by: Ben Freivogel Second examiner: Diego Hofman
Abstract
This thesis investigates the black holes information loss paradox and the succes-sion of proposed solutions, including black hole complementarity, the AMPS firewall problem and the ER=EPR proposal of Maldacena and Susskind. The ER=EPR pro-posal equivocates wormholes with quantum entanglement. The traversable wormhole method of Gao, Jafferis, and Wall is examined. ER=EPR means that dual to the traversable wormhole is quantum teleportation through entangled black holes. The protocol for such a teleportation process is examined using a toy model for entan-gled black holes. We look at a one operator method for quantum teleportation and derive a Hamiltonian for the teleportation process. We look at the scrambling of the teleportation qubit with the black hole and what has to be done to teleport the information and retrieve it at the other black hole. Finally, we link this teleportation protocol for entangled black holes back to the method for traversable wormholes.
Contents
1 Introduction 3
2 Black holes and the information loss paradox 6
2.1 Hawking radiation . . . 6
2.2 Black hole information paradox . . . 8
2.3 Black hole complementarity . . . 12
2.4 Firewall . . . 14
2.5 ER=EPR . . . 16
3 Through the wormhole 22 3.1 Traversable wormholes . . . 22
3.2 Quantum teleportation . . . 26
3.3 Protocol revision & toy model setup . . . 27
3.4 Time dependent teleportation . . . 28
3.5 Constructing the Scrambling Matrix . . . 31
3.6 Entanglement Entropy of V . . . 31
3.7 Teleportation and Bob’s operation: Constructing Z . . . 34
3.8 Teleportation through the Wormhole . . . 37
4 Conclusions 39 A One step unitary operators 42 B Logarithm of the one step unitary operator 43 C Mathematica code 44 C.1 Logarithm of the one step operator . . . 44
C.2 Entanglement entropy . . . 44
1
Introduction
Modern theoretical physics is supported by two main pillars, quantum theory and general relativity theory, that are fundamentally at odds with each other and much work done by theoretical physicists is aimed at reconciling the two under the umbrella of a single theory. The path to get to such a theory is however unclear. Black holes provide, due to their extreme nature, an excellent arena for physicists to study the conflict between the theories and speculate about any way of reconciling them.
In this thesis we will investigate the strange nature of black holes by looking at the in-formation loss paradox and its possible solutions. We will start with the work of Stephen Hawking, who first argued that black holes emit radiation, and see how he derived Hawk-ing radiation usHawk-ing quantum field theory in curved spacetime. Next we will consider how the emission of Hawking radiation by black holes poses a paradox to physicists about the information that black holes and the Hawking radiation contain. Quantum mechanics de-mands that information is always preserved, but the emission of Hawking radiation by black holes seems to be thermal, meaning that the Hawking radiation can not contain any information about the structure that collapsed to form the black hole. A black hole can evaporate through the emission of Hawking radiation, leading to the paradox: where does the information about the structure that collapsed to form the black hole go?
We will investigate possible solutions to the black hole information loss paradox. Can the Hawking radiation still in some way contain the apparently lost information? Perhaps our descriptions of black holes using general relativity do not apply to Planck-scale black holes and it turns out that black holes do not evaporate completely but leave some kind of remnant. Or perhaps a description for a Planck-scale black hole will allow the information to come out at the very end before it evaporates. Maybe our description of black holes, based on general relativity, is incorrect or incomplete. Perhaps the no-hair theorem is incorrect and a quantum theoretical description of black holes actually would allow some, yet unknown, quantum hair to describe the information without it disappearing behind an event horizon. Or maybe black holes do not have singularities inside their event horizons but a causally disconnected ”baby universe”, and we should consider the paradox only from the perspective of a multiverse.
A more sophisticated solution to the information loss paradox was proposed by Susskind, Thorlacius and Uglum, who argued that the information is both reflected at the horizon as well as captured inside the horizon. Although the two situations seem contradicting, the event horizon ensures no observer can ever confirm that both are true at the same time, and the two situations are thus still complementary to each other. To see how this argument works we will look at Cauchy surfaces, which are slices of spacetime, before, during, and after the formation and evaporation of a black holes and relating these Cauchy surfaces to each other.
The black hole complementarity argument has, as we will see, its own problems though. We will look at an argument by Almheiri, Marolf, Polchinski, and Sully to see how the complementarity argument contains contradictions. These contradictions will force us to abandon a pivotal principle of general relativity theory: the equivalence principle. We will see how the entanglement between emitted Hawking radiation and the black hole is broken, leading to a layer of energetic particles at the horizon: the so called firewall.
The proposal of firewalls motivated Juan Maldacena and Leonard Susskind to the radi-cal proposal of ER=EPR, in which they equivocate entanglement in quantum mechanics with wormholes in general relativity theory. We will investigate this proposal by looking at entangled black holes, which thus have a wormhole between them. We will see how Maldacena and Susskind argue against the firewall proposal using this setup, and what the ER=EPR proposal means for the black hole information loss paradox.
Next we will shift our focus to the method proposed by Gao, Jafferis and Wall for the production of traversable wormholes. By introducing a double trace deformation in a setup with a BTZ black hole, a wormhole can be opened up connecting the two boundary CFTs. To check that this is true we will look at the viability for a null geodesic to go from one side to the other. This requires a very involved calculation of the averaged null energy. The result however is that indeed, the wormhole opens up and becomes traversable.
Within the paradigm of the ER=EPR proposal, the story of a traversable wormhole gets an interesting interpretation as quantum teleportation between entangled black holes. We
will look at a teleportation protocol as originally proposed by Zhao and Susskind and in-vestigate some of its details.
We will take a look at the original teleportation protocol as was proposed by Bennet et al. and consider how the protocol has to be changed to come to a teleportation protocol in a toy model for entangled black holes. If the quantum teleportation through entangled black holes story is dual to traversing a wormhole, we will expect that the teleportation takes some time. This means that we need to find a time dependent formulation of the teleportation protocol. Also, we are now dealing with black holes, which means that anything that enters a black hole will thermalize with the black hole. That is to say that all the information of the teleportation qubit will scramble. We will look at scrambling operators and how well they work. To finish the protocol requires an operation on the receiving black hole using an operator. As it turns out, constructing such an operator is difficult and we will discuss some of the potential issues we face in doing so.
2
Black holes and the information loss paradox
It was within mere months of Albert Einstein publishing the field equations of general relativity that Karl Schwarzschild found the spherically symmetric solution now named after him. The Schwarzschild metric gave rise to the conception of black holes, in a sense the theories most extreme phenomena. With the advent of quantum theories, black holes quickly became the focal point of the conflict between quantum theories and general rela-tivity, a problem still not solved today. Using quantum field theory in curved spacetime as his tool, it was Stephen Hawking that set the stage for perhaps the best illustration of this conflict. In this chapter, we will consider the argument for Hawking radiation, the paradox it creates, and the succession and development of solutions, ending up at the ER=EPR proposal of Juan Maldacena and Leonard Susskind that tries to reconcile pivotal properties of the two theories.
2.1
Hawking radiation
In the 1970’s, Stephen Hawking calculated that black holes emit radiation and thus evap-orate [1]. He considered a massless scalar field φ, obeying the Klein-Gordon equation, in a flat spacetime with a black hole in it. The mode expansion for this field in the region of the past null infinity J− is given by
φ =X
i
(fiai+ ¯fia † i),
where the fi coefficients solve the Klein-Gordon equation and contain only positive
fre-quencies. The operators ai and a †
i are the annihilation and creation operators for particles
in the ith state in the region.
Now consider the mode expansion in the region of the future null infinite J+,
φ =X i (pibi+ ¯pib † i + qici+ ¯qic † i)
in which we now have a distinction between outgoing modes pi and ingoing modes qi for the
event horizon. The operators biand b †
i are annihilation and creation operators for outgoing
particles that reach the future infinity J+ and the ci and c†i operators are the annihilation
Figure 1: The Penrose diagram for the formation of a black hole, taken from [1]
the scalar field is determined by information on J−. We get for the coefficients that pi = X j (αijfj + βijf¯j) qi = X j (γijf )j + ηijf¯j)
and for the annihilation and creation operators we get that bi = X j ( ¯αijaj − ¯βija†j) ci = X j (¯γijaj − ¯ηija † j)
We are interested in finding out how many outgoing particles there are, so we want to know the expectation value of the number operator Nb,i = b
†
ibi for the outgoing mode i,
which is hNbi = h0| b † ibi|0i = X j |βij|2
because ai|0i = 0 for all i. You can calculate |βij| 2
, as in [1], by looking at the Fourier components of the ingoing and outgoing solutions and by considering the modes that are scattered of the collapsing structure. The result of the calculation is that
|αωω0| = e πω
κ |β
where κ is the surface gravity of the black hole, which for a Schwarzschild black hole of mass M is κ = 4M1 . The Bogoliubov coefficients satisfy
X ω0 (|αωω0|2− |βωω0|2) = 1 so that we get hNbi = X ω0 |βωω0|2 = 1 e2πωκ − 1
which is the Bose-Einstein distribution for a body of temperature
T = κ 2π =
1 8πM
The expectation value for the outgoing modes is non-zero. This means the black hole is radiating, what is now known as Hawking radiation.
2.2
Black hole information paradox
The emission of Hawking radiation by a black hole poses a problem to physicists. In quan-tum mechanics, the future changes to a wave functions are determined uniquely by the time evolution operator, whereas past wave functions are uniquely determined by the inverse time evolution operator. This means that information in the system must be preserved. However, Hawking argued that the radiation emitted by black holes do not contain any information about the matter that collapsed to form the black hole, which means that as the black hole evaporates, information is lost. Hawking’s calculations were semi-classical and neglected the effects of gravitational back-reaction. As exemplified by John Preskill in a 1992 paper [2], some physicists were initially skeptical of Hawking’s claims and were expecting a more thorough calculation that would include the back-reaction effects to shed light on how the radiation emitted by black holes contains the information of the mass that collapsed to form the black hole. Preskill took more than fifteen years to finally accept the paradox created by the conflicting principles of quantum theory and general relativity.
The paradox arises because Hawking found that the radiation emitted by black holes is in a mixed thermal state. The radiation depends on the geometry of the black hole outside it’s event horizon and not on the particularities of the mass that collapsed to form the black hole. This is initially not per se problematic as the radiation is entangled with the black
hole and the missing information about the original structure that formed the black hole can still be inside the black hole’s event horizon. But if the black hole completely evapo-rates the radiation is all that is left of the system. The final state of this system cannot be calculated precisely even if the initial state is entirely known. An initial pure state that col-lapsed to a black hole will end up as a mixed state, information is lost and the unitarity of the system violated. According to the AdS/CFT correspondence this is also not allowed as the black hole should be dual to a thermal system which obeys the rules of quantum theory.
Should we rewrite the principles of fundamental physics as Hawking argues or are there alternative solutions? Preskill discusses several possible alternative solutions [2]. Could the information of the original structure that collapsed in to a black hole still be encoded in the mixed state of the Hawking radiation? One could argue that the information is not lost but simply scrambled throughout the black hole and still present in some way in the mixed state of the Hawking radiation, and that corrections to the semi-classical calculations would recover the information. If we would consider not a black hole but, say, a star this would certainly still be the case. As long as we know well enough the initial state of the star and the state of the radiation emitted by the star we could, at least in principle, recover the original information if we waited long enough. But black holes are fundamentally different. The first problem with this solution is the no-hair theorem, which states that a black hole can be described completely by the use of only three classical observables: mass, electric charge, and angular momentum. All other information about the mass that collapsed to form the black hole is hidden behind the event horizon. If the Hawking radiation is entangled with the black hole, how is this information stored in the black hole?
Preskill also describes an argument made by Leonard Susskind [2]. Consider the initial state of the collapsing structure |ii. On a spacelike slice of spacetime through the Hawking radiation and the apparent horizon of the collapsing structure we could consider a state that has evolved such that the part of the system inside the horizon and the part of the system outside the horizon are uncorrelated and the two parts are both in a pure state.
|ii −→ |iiinside⊗ |iioutside But we could also consider the superposition of these states,
X
i
ci|ii −→
X
i
The inside and outside state will only be uncorrelated, and the Hawking radiation will thus be not in a thermal state but a pure state, if there is only one possible state for the inside. As the state for the inside becomes more unique, the state for the outside radiation becomes more pure. This means that the information of the collapsing structure must not fall through the horizon but must be ”stripped away” long before a singularity forms if the information is to be encoded in the Hawking radiation. There is no known way in which this can happen and it is unlikely to exist because to a freely falling observer there is nothing special about falling through the apparent horizon.
Another solution to the paradox is by arguing that the black hole will not evaporate entirely, which is based on speculation of the nature of Planck-sized black holes. It is reasonable to assume that semi-classical theory does not describe Planck-size black holes as at this size quantum effects will become significant. What does happen though at this scale is uncertain. One could argue that some stable remnant is left behind, though this too has its problems if the Planck-scale remnant is to contain all the information. The black hole could have started out at any arbitrary size which means that at Planck-scale it should still contain an arbitrarily large amount of information, with the amount of bits proportional to
Minitial2 M2
P lanck
.
As there are virtually an infinite amount of initial states that can collapse into a black hole, the must be a virtually infinite amount of different Planck-scale remnants, which causes problems in its description in quantum field theory [2]. Another issue with these remnants is that it discards the interpretation of the Hawking-Bekenstein entropy of a black hole, which relates the entropy of a black hole to its mass. But if there can be an arbitrarily large amount of information in a black hole remnant, the entropy of the black hole must be infinite.
Another alternative is to argue that the information is released from the black hole only at the very end when it is evaporating at Planck-scale. The Hawking radiation would then remain thermal, as indicated by the semi-classical calculations, for most of the evaporation, only to become pure at the very end. This does mean all the information has to be able to come out at the end. To do so the Planck-scale black hole will need to emit S ∼ M2 quanta
with other quanta, which we want to ensure that the quanta are only weakly coupled and thus able to carry the information. Thus it takes about t ∼ M4 for the Planck-scale black
hole to radiate away the information. The problem here is that M is the mass of the initial black hole and can thus be arbitrarily large, leading to arbitrarily long timescales for the evaporates of the scale black hole. With such long lived unstable Planck-scale black holes we run into the same problems as with the stable Planck-Planck-scale black holes.
A fourth alternative solution is to argue against the no-hair theorem and claim that the information about the collapsed structure and the entanglement with the Hawking radia-tion is preserved in some quantum hair. The no-hair theorem stems from classical general relativity theory and does per se fully limit the quantum state of the black hole. The suggestion of quantum hairs would mean that there are an infinite amount of conserved charges such that the information of an arbitrarily large black hole could be stored in their values. However, these values would depend largely on the non-local ways in which the collapsing structure was put together. The corresponding conservation laws would be so constraint this would be noticeable in low-energy physics experiments. Observations in low-energy physics however seem to support local quantum field theories [2].
The last alternative solution described by Preskill is an argument made by Hawking, Dyson, and Zeldovich. This solution to the black hole information paradox is to argue against the standard conception of black holes as having singularities. Instead, they say, the collapse of a structure to a black hole will produce a closed causally disconnected spacetime, a ”baby universe”. Our quantum theory then should not just describe our own universe but the multiverse of causally disconnected universes, the states of which are correlated by black holes, in which the information of the collapsed structure is preserved. This solution would mean that we can only ever hope of describing the evolution of a black hole on a phenomenological level instead of a description based on fundamental laws, and although it would justify the loss of information, it would not tell us how the evaporation of a black hole and the correlations of the black hole with the Hawking radiation works.
2.3
Black hole complementarity
A 1993 paper by Leonard Susskind, Larus Thorlacius, and John Uglum [3] argued for a novel solution to the black hole information loss paradox based in part on earlier work by Gerard ’t Hooft. The argument starts with the introduction of three postulates that are based on quantum theory, general relativity and statistical mechanics.
The first postulate states that quantum theory can entirely describe the collapse of a struc-ture in to a black hole and its subsequent evaporation. This means that there is a unitary scattering matrix that describes the evolution of black holes from collapse to the emission of Hawking radiation. The second postulate states that the physics outside the stretched horizon of a sufficiently massive black hole can be described to a good approximation by a semi-classical theory that accounts for the evaporation of the black hole and the Hawking radiation using quantum corrections. The third postulate states that a black hole will appear as a quantum system with discrete energy levels to a distant observer. The expo-nential of the Bekenstein entropy S(M ) describes the amount of dimensions of the states of a black hole of mass M .
Now let’s consider three kinds of Cauchy surfaces in a spacetime with a black hole. Cauchy surfaces are surfaces in spacetime that are intersected by all light-like and time-like curves exactly ones; crudely spoken, a Cauchy surface represents an instant in time. There are Cauchy surfaces P that do not intersect any singularity until they are evolved because they exist before the collapse of the black hole. Now consider a Cauchy surfaceP
P, which
contains the point P. Point P is the point in spacetime where singularity and the event horizon intersect. On one side of point P, the Cauchy surfaceP
P is inside the black hole,
whereas on the other side of point P the Cauchy surfaceP
P lies outside the black hole, as
illustrated in Penrose diagram of figure 2. Let us write P
bh and
P
out for the two disjoint
parts of Cauchy surface P
P. The third Cauchy surface to consider is the Cauchy surface
P0
that exists after the black hole has evaporated.
Let’s consider how one Cauchy surface evolves into another. The initial state |Pi of Cauchy surface P will evolve normally until it becomes Cauchy surface PP. At a later time there will be a Cauchy surface P0
Figure 2: The Penrose diagram of the collapse and evaporation of a black hole and the Cauchy surfaces in this spacetime, as presented in the paper by Susskind, Thorlacius and Uglum [3]
P
P. The state of the latter Cauchy surface,
P0 , should be a pure state related to |Pi by the scattering matrix, in line with the first postulate mentioned above.
P0 is causally disconnected from P
bh by the black hole horizon, so the state
P0 must have evolved from the other disjoint part |P
outi. But if
P0 is a pure state then |P
outi must also be
a pure state and if one of the disjointed parts of P
P is a pure state, then the state of
P
P
must be a product of two pure states X P + = X bh + ⊗ X out + .
The relation between the initial |Pi state and the latter
P0 and the latter
P0 and |P
outi must mean that the state of the disjointed part of the Cauchy surface inside the
black hole, |P
bhi is independent of the initial state |Pi. For the state inside of the black
hole to be independent of the structure that collapsed to form the black hole is a vio-lation of the equivalence principle. The event horizon of a massive black hole does not
have very large curvature and to a freely falling observer falling in to the black hole there should be nothing special about crossing the event horizon. Susskind et al. argue that this violation of the equivalence principle is unreasonable [3]. The state |P
Pi seems to
simul-taneously describe the inside and the outside of the black hole, which seems to the writers ”suspiciously unphysical”. An observer inside the black hole can never communicate any experimental results to an observer outside the black hole thus the correlations described by the state |P
Pi seems to have no operational meaning. The entire state can never be
observed by any physical observer and so there seems to be no contradiction between a distant observer retrieving the information of the collapsed body in the Hawking radiation and and a freely falling observer experiencing nothing special as the observer crosses the event horizon.
To make the situation more concrete, let’s consider the stretched horizon of a black hole. For an outside observer, the black hole can be described in terms of this stretched horizon, which has mechanical, electrical and thermal properties, and the black hole’s entropy is proportional to the surface of the stretched horizon [3]. However, to a freely falling ob-server that passes the stretched horizon, the stretched horizon will disappear. The freely falling observer however has no means to communicate this result to an observer outside of the black hole. This, Susskind et al. say, is the complementarity between between the infalling and distant observers.
2.4
Firewall
Black hole complementarity brought us closer to solving the black hole information para-dox, but does still have its own problems, as, amongst many others, discussed in the 2013 paper by Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully, which raises the possibility of a firewall surrounding a blakc hole [4]. The argument presented in the paper starts again with the three postulates of black hole complementarity as stated in the previous chapter, but now supplemented with the claim that a free falling observer experiences nothing special while falling through the event horizon explicitly as a fourth postulate.
Consider a black hole so old that it has already lost more than half of its entropy to Hawk-ing radiation. The halfway point, where the black hole has lost exactly half its entropy is called the Page time [5]. As mentioned earlier in the discussion of black hole complemen-tarity, according to the first postulate, the Hawking radiation should be in a pure state because of its connection by the scattering matrix to the state of the collapsed structure. As has been argued by Don Page [5][6], the purity of the Hawking radiation and no loss of information means that the Hawking radiation emitted before the Page time and the Hawking radiation emitted after the Page time are fully entangled. Postulate four however implies that the Hawking radiation is fully entangled with what is behind the black hole’s event horizon. But this is in violation of the principle of the monogamy of entanglement, which says that a particle that is maximally entangled with one particle cannot also be entangled to another particle.
To sharpen the issue here, let’s look at the entropy of the three subsystems, subsystem A of the Hawking modes emitted before the Page time, B an outgoing Hawking mode emitted after the Page time and C the corresponding ingoing mode. Strong subadditivity states that
SAB + SBC ≥ SB+ SABC.
After the Page time the entropy of the black hole will decrease [5], so SA > SAB. The
fourth postulate, about a freely falling observer crossing the event horizon, means that SBC = 0 and thus SABC = SA. When this is put into the equation for strong subadditivity
we get
SA ≥ SB+ SA
which could only ever be true if SB = 0, which it is not since its density matrix is
ther-mal. Argued by Page [6], SAB = SA − SB for old black holes, turning the equation for
subadditivity in to
SA≥ 2SB+ SA.
Almheiri et al. suggest letting go of the fourth postulate. Freely falling observers falling through the event horizon of a black hole experience something special. They argue that the entanglement between the ingoing and outgoing modes is immediately broken leading to the infalling observer to observe a Planck density of Planck scale radiation, or, what they call, a firewall. The firewall is not necessarily visible to outside observers, but needs only
be encountered by infalling observers. The argument for the firewall arises in the context of a black hole older than its Page time, but the firewall itself should be an intrinsic property of the black hole, independent of any entanglement of the black hole with another system. Rather, for the argument for a firewall to hold, the black hole is old enough after the scrambling time, after which all subsystems in the black hole have thermalized and are maximally entangled.
2.5
ER=EPR
Spurred by the reformulation of the black hole information loss paradox by Almheiri et al. in its firewall form, Juan Maldacena and Leonard Susskind proposed a new paradigm for thinking about the conflict between quantum theory and general relativity [7]. Both quantum theory and general relativity raise apparent specific issues with the principle of locality, which states that information cannot be send from one place to another faster than the speed of light. In quantum mechanics there are the EPR pairs, named after Einstein, Podolsky, and Rosen [8]. EPR pairs are pairs of particles in an entangled state, where some information seems to be stored non-local. That is to say that there appears to be ’spooky action at a distance’; when you’d make a measurement of one of the two particle, the other appears to have also been instantaneously affected, no matter how far away that particle is. The apparent paradox is resolved by realizing that this non-locality does not mean any information can actually be send faster than the speed of light. General relativity theory gives rise to Einstein-Rosen bridges, or wormholes, that connect two points in spacetime. Wormholes can be used to send information from one point in spacetime to another point in spacetime faster than the speed of light could travel that spacetime distance through normal spacetime because the wormhole would effectively shorten the distance between the two points. Here too, there is no actual violation of locality on a local level. Although globally you could send the information from one point to the other faster than the speed of light could through spacetime if the information does not go through the wormhole, this apparent violation does not requires the information to travel through the wormhole faster than the speed of light. Locally, locality is still obeyed. The similarity between ER bridges and EPR pairs motivates the proposal that there is some connection between the two. Maldacena and Susskind propose that the two are different manifestations of the same phenomenon, with the connection being summarized as ER=EPR [7].
Figure 3: The Penrose diagram for two entangled black holes. The blue line contains the ER bridge between the two black holes. Taken from [7]
Although one could, as the writers do, speculate that this connections between ER bridges and EPR pairs exists for all entangled particles, the proposition is best exemplified and eas-iest to analyse when considering black holes, as the connections between simple entangled particles would lead to a description of a bridge of a highly quantum nature. So, consider two non-interacting maximally entangled black holes at an arbitrarily large distance apart, say black hole A at Alice’s side and black hole B on Bob’s side, as depicted in figure 3. Following ER=EPR, this would mean that, because the two black holes are entangled, there is a wormhole between them. The black holes are in a thermofield double state,
|Ψi =X n eβEn2 |Ii A|IiB with Hamiltonian H = HA+ HB
That the black holes are connected by a wormhole does not mean that the outside regions near the black holes are causally connected or that information can be transmitted between Alice outside black hole A and Bob outside black hole B (for now that is, we will return
to this point in the next chapter), thus the principle of locality is not violated. If however the observers would decide to both jump in to their respective black holes they could meet up shortly after, or, alternatively, if one of the observers would enter the black hole, that observer would be able to receive a message from the observer outside the other black hole if the message was sent early enough.
The question now is, would the infalling observer encounter a firewall? Say that the outside observer has a powerful quantum computer that can act on the black hole on her side to send a message to the infalling observer. Whether or not the infalling observer encounters a firewall entirely depends on what message the outside observer decides to send. The infalling observer would encounter a firewall as she crosses the event horizon if the outside observer at the other black hole had send a high energy shock wave early enough.
Figure 4: The laboratory experiment setup by Alice, with particles A and B to the right.
To make this more concrete, let’s look at a hypothetical laboratory experiment, depicted in figure 4, performed by Alice, in which she has prepared two condensed matter systems in the maximally entangled thermofield state, such that each side has a black hole and supports a large CFT with a gravity dual in the form of a wormhole. Bob is part of the experimental setup, he starts in the bulk outside his black hole. Now consider two maximally entangled qubits at Bob’s side at time t, qubit A inside the event horizon and qubit B outside the event horizon. Because of the thermofield double state, B is maximally entangled with qubit A0 at time −t on the left side. A is obtained by evolving A0 in the
bulk so, even though B is entangled with both A and A0, this does not lead to a violation of the principle of the monogamy of entanglement because A and A0 are not independent. We write this evolution in the bulk as
A0 −→ A,
as is depicted in the Penrose diagram of figure 4. Since A is a qubit, with for example spin, we know its commutation relations and can use the relation between A and A0 to write
[A0i, Aj] 6= 0
A0 exists on the left system’s boundary and we can evolve it in time with the left systems Hamiltonian from time −t to time t. The two systems do not interact so A0 will evolve in to A00 = ULA0U † L and we have A00 −→ A and, again through the commutation relations, that
[A00i, Aj] 6= 0.
Figure 5: The laboratory experiment performed by Alice, taking particle A0 from the left side to Bob on the right side.
Now to perform the experiment, Alice extracts qubit A0 at time −t from the left side and brings it to the right side, where she gives the qubit to Bob, as is shown in figure 5. Next, Bob jumps in to the black hole to the entanglement between qubits A and B. As he does, he runs into a particle O. Is particle O part of the firewall? No, the particle is there because Alice corrupted particle A when she extracted particle A0 and A and A0 do not commute. The corruption is only limited to qubits A and B and would happen regardless of what happened to qubit A0 after being extracted. The exact same reasoning can now be applied if Alice would instead extract A00 at time t. A would get corrupted and Bob would encounter O when falling into the black hole. But because the encounter of Bob with O is dependent on the actions of Alice, This argument contradicts the argument of Almheiri et al. about the existence of a firewall and shows why there is no firewall.
Let’s now turn to a different experiment. Consider a black hole in flat space that is older than the Page time. Alice has collected all the Hawking radiation emitted by the black hole, which she converts into a black hole with her quantum computer. The two black holes are entangled and by using her quantum computer, Alice can turn the state of the black holes in to the thermofield double state. Alice then extracts a qubit A0 from her black hole and travels back to the old black hole in time to see the emission of mode B, which is entangled with qubit A0. Because of this entanglement, it is not possible for mode B to also be entangled with qubit A behind the event horizon of the old black hole. With the entanglement between A and B broken, Alice will encounter an energetic particle if she’d jump in to the black hole. According to Almheiri et al. it did not matter that Alice carried out this experiment, she would encounter the energetic particle when crossing the event horizon regardless of whether or not she had performed the experiment. This would mean that every possible pair of qubit A and mode B should be corrupted and the black hole should have a firewall at its horizon. Maldacena and Susskind state that Almheiri et al. only succeeded in arguing that the corruption occurs on the mode B that corresponds to the qubit A0 on which Alice performs her experiment, and that they failed to general-ize this for all pairs. In the ER=EPR paradigm, the energetic particle encountered when crossing the event horizon was created by Alice when she extracted qubit A at the other side of the wormhole.
entangle-Figure 6: A depiction of the Einstein-Rosen bridges between a black hole and the Hawking radiation.
ment with the black hole in light of the ER=EPR proposal. We now have situation where the black hole is connected to all of the Hawking radiation by wormholes, as is depicted in 6. Because the interior of the black hole is connected to the Hawking radiation, the information loss paradox does not seem problematic anymore as the information can leave the black hole without having to cross the event horizon.
3
Through the wormhole
The possibility of a traversable wormhole has been of interest to physicists for decades but these wormholes required exotic material that violates the null energy condition and were considered unfeasible [9][10]. A new method for generating traversable wormholes was introduced by Gao, Jafferis, and Wall in 2016 [11]. In this method, the wormhole is stabilized by an interaction that couples the boundaries of the black holes. Within the framework of ER = EP R, where entangled particles are connected by a wormhole, the method of Gao, Jafferis, and Wall raises the interesting picture of teleportation through entangled black holes with a gravity dual of a traversable wormhole. In this chapter, we will start by examining the method of Gao, Jafferis, and Wall for traversable wormholes, after which we will delve in to the corresponding protocol for teleportation through entangled black holes as proposed by Zhao and Susskind [12]. We will look in detail at some of the elements of this protocol to see how it works.
3.1
Traversable wormholes
The argument of Gao, Jafferis, and Wall starts with an uncharged, non-rotating, eternal BTZ black hole with a horizon of radius rh in an Anti-de Sitter space of radius l, described
by the metric ds2 = r 2− r2 h l2 dt 2 + l2 r2− r2 h dr2+ r2dφ2
Following the AdS/CFT correspondence, dual to the gravity picture are two entangled CFTs in a thermofield double state. To open up the wormhole, we introduce a double trace deformation
δH(t) = − Z
dφh(t, φ)OR(t, φ)OL(−t, φ)
with coupling constant h(t, φ), which is equal to zero when t < t0, and scalar operators OR
and OL. The scalar operators O are dual to bulk scalar fields ψ. To verify that the above
coupling does indeed lead to the existence of a wormhole we must calculate that there is a null geodesic from one side to the other. To simplify this calculation, we will consider the BTZ black hole in Kruskal coordinates, as shown in figure 7,
e2rht= −U
V r
Figure 7: Kruskal diagram of a BTZ black hole with Kruskal coordinates U and V. The metric for the BTZ black hole in Kruskal coordinates is
ds2 = − 4l 2 (1 + U V )2dU dV + r2 h(1 − U V )2 (1 + U V )2 dφ 2.
The singularities are now at U V = 1. The boundaries at r = ∞ are now at U V = −1 and, crucially, the two horizons are now at U = 0 and V = 0. These coordinates simplify the calculation as we can now pick a null geodesic at V = 0.
To see whether the null geodesic makes it from one side to the other, we need to look at the averaged null energy
Z ∞
U0
hTU UidU,
where Tµν is the stress-energy tensor. For the wormhole to open, we will require the
averaged null energy to be negative. We can use point splitting to obtain the stress-energy tensor, as in [13],
hTU Ui(U ) = lim U0→U∂U∂U
0Gh(U, U0).
Gh is the propagator between the two bulks under the double trace deformation, given in
the usual coordinates in terms of Heisenberg fields by Gh = hψHR(t, r, φ)ψ
H R(t
0
, r0, φ0)i.
As is calculate by Gao et al. in [11], Gh can be expressed to leading order in h in terms of
the bulk to boundary correlation function
K∆(t, r, φ) = hψR(t, r, φ)O(0, 0)i = r∆ h 2∆+1π − pr2− r2 h rh cosh rht + r rh cosh rhφ !−∆
and the retarded bulk to boundary correlation function K∆r(t, r, φ) = |K∆(t, r, φ)|θ(t)θ p r2 − r2 h rh cosh rht − r rh cosh rhφ ! ,
where θ is the Heavyside step function, as Gh = 2 sin π∆
Z
dt1K∆(t0+ t1− iβ/2)K∆r(t − t1) + (t ⇐⇒ t0),
where β = 2πl2/r
h is the inverse temperature of the BTZ black hole. Using this expression
for Gh we get, as done in [13], for the stress-energy tensor that
hTU Ui(U ) = −h l 212−2∆∆ sin π∆Γ(1 − ∆) π3/2Γ(3 2 − ∆) lim U0→U∂U Z U U0 dU1 F1 1 2; 1 2, ∆ + 1; 3 2 − ∆; U1−U 2U1 , U1−U U1(1+U0U1) U1−∆+1/2(U − U1)∆−1/2(1 + U0U1)∆+1 , in which F1 is the Appell hypergeometric function. We can now calculate the averaged
null energy, as done in [11], as
Z ∞ U0 hTU UidU = − h l Γ(2∆ + 1)2 24∆(2∆ + 1)Γ(∆)2Γ(∆ + 1)2 2F1 1 2 + ∆, 1 2 − ∆; 3 2 + ∆; 1 1+U2 0 (1 + U2 0)∆+1/2 .
where 2F1 is Gauss’s hypergeometric function. Figure 8 shows the graph, taken from [11],
of RU
0TU UdU as a function of ∆, including a line taking the difference between
R
Uf TU UdU
and RU
0TU UdU , to show what happens if the interaction is stopped at Uf.
Figure 8: R
The averaged null energy is, as can be seen in figure 8 negative and the averaged null en-ergy condition is thus violated, which is a necessity for a wormhole to become traversable [14][15][16].
Figure 9: Penrose diagram for a traversable BTZ wormhole. The shock wave, represented by the yellow line, shifts the apparent horizon, in green, such that a signal can reach the opposed boundary, represented by the red and yellow line. Image taken from [17].
Within the paradigm of ER=EPR, the above argument about traversable wormholes gets an interesting interpretation. In the section discussing the ER=EPR proposal we saw that the existing that wormholes between entangled black holes were not traversable and ob-servers could only communicate with eachother by crossing the event horizon. If however, two observers jump in the two black holes and meet inside, and we then use the above method to make the wormhole traversable, we could verify from the outside that the ob-servers have met.
It is interesting to consider what happens at the EPR side of the equation when there are traversable wormholes at the ER side; from traversable wormholes we now switch our considerations to quantum teleportation through entangled black holes.
3.2
Quantum teleportation
The standard protocol for quantum teleportation was devised by Bennet et al in 1993 [18]. The protocol is used by Alice to send information encoded on a qubit to Bob. They start out by creating an entangled pair of qubits, one in the possession of Alice and one in the possession of Bob, say |ΨABi = |00i + |11i . Alice has another qubit that contains
the information that she wants to teleport, |ΨTi = α |0i + β |1i. The initial state of the
combined system then is
|Ψinitiali =
1 √
2(|00i + |11i)AB⊗ (α |0i + β |1i)T = 1 √
2(α |000i + α |110i + β |001i + β |111i) (3.1) To do the teleportation means we want to end up with a system in which Bob’s qubit is unentangled with the other two qubits and containing the information of the teleport qubit, that is to say
|Ψf inali = (...)AT ⊗ (α |0i + β |1i)B (3.2)
To this end we introduce the 4 Bell states, which are
φ+= √1 2(|00i + |11i) φ− = √1 2(|00i − |11i) ψ+ = √1 2(|01i + |10i) ψ− = √1 2(|01i − |10)i
The computational basis states can be expressed in terms of Bell states as
|00i = √1 2(φ ++ φ− ) |11i = √1 2(φ +− φ− ) |01i = √1 2(ψ + + ψ−) |10i = √1 2(ψ +− ψ− )
Rewriting the initial state of the system in terms of the Bell states gives us |Ψinitiali = 1 2(α(φ ++φ− )AT⊗|0iB+α(φ +−φ− )AT⊗|1iB+β(ψ ++ψ− )AT⊗|0iB+β(ψ +−ψ− )AT⊗|1iB) = 1 2(φ +
AT(α |0i + β |1i)B+ φ−AT(α |0i − β |1i)B+ ψ+AT(α |1i + β |0i)B+ ψAT− (−α |1i + β |0i)B)
In the first step of the protocol, Alice makes the system go into one of the four terms of the above equation by making a measurement in the Bell state basis. The system is now in a state
|Ψi = (...)AT ⊗ (γ |0i + δ |1i)B
To get to the desired |Ψf inali Bob has to be able to relate γ and δ to α and β. To do so,
in the second step of the protocol Alice sends Bob the result (φ+, φ−, ψ+ or ψ−) of her
measurement on the AT subsystem to Bob through a classical channel. Considering that there are four possible results of the measurement, Alice must send two bits of information through the classical channel. Lastly, Bob uses this information to perform the right trans-formation on his qubit to retrieve the teleportation qubit’s state. If the result of Alice’s measurement is |φ+iAT no transformation is required, that is to say, Bob should use an identity operator. The transformations corresponding to the other results are constructing using the Pauli matrices. If Alice measured |φ−iAT, Bob needs to use the σ3 operator, for
|ψ+i
AT Bob needs to use the σ1 operator, and for |ψ −i
AT Bob needs to use −iσ2 as the
operator.
After Bob has used the appropriate operator the system is now in a state
|Ψi = (...)AT ⊗ (α |0i + β |1i)B
Thus, the qubit on Bob’s side is now in the state of the original teleportation qubit, and the state has been teleported, whereas the two qubits on Alice’s side are now in an entangled Bell state.
3.3
Protocol revision & toy model setup
The AB system starts out entangled. The A system is entangled with the T qubit. A chan-nel or coupling is used between AT and B. The teleportation qubit is retrievd in system
B. This short summary of quantum teleportation will remain unchanged when considering teleportation through a wormhole. There are two major difference though.In teleportation through a wormhole, there are two entangled black holes, one on Alice’s side and one on Bob’s side. Before a qubit can be teleported between the two black holes Alice first has to throw the qubit in to the black hole on her side. The qubit will thermalize with the black hole, that is to say that, the information encoded in the qubit will be scrambled throughout the black hole.
Secondly, the qubit will need to traverse the wormhole to reach Bob. Whereas the stan-dard quantum teleportation is done instantaneously when Alice measures her AT system, traversing a wormhole takes time. This means we will want a time dependent teleporta-tion protocol or, in other words, a Hamiltonian. We will first look at the time dependent teleportation protocol before turning to the scrambling.
In this chapter we are considering explicit and specific cases of the simplified model for entangled black holes as described by Susskind and Zhao[12], which will be summerized in the following. A black hole will be represented simply as a collection of qubits, or quantum particles with more than two states like qutrits, and an entangled pair of black holes will be described by a thermofield double state
|T F Di =X
I
|IiA|IiB = (|00i + |11i)⊗N (3.3)
The initial state of the teleportation setup then is |Ψinitiali =
X
Im
φ(m) |miT |IiA|IiB (3.4)
where |φiT =P
mφ(m) |miT is the information carrying qubit to be teleported. The goal
of the teleportation process is to end up with a final state where one of the qubits on Bob’s side has the state |φiB =P
nφ(n) |niB.
3.4
Time dependent teleportation
is to say, we want to find a unitary operator U (t) such that U (t∗) |Ψinitiali = |Ψf inali ,
where t∗ is the time it takes to traverse the wormhole. The unitary operator should
include a projection operator P , to represent Alice’s measurement on the AT system, and the transformation S that Bob has to do on his qubit to retrieve the teleportation qubit state. Projection operators are operators P = |φi hφ| such that they are idempotent
P2 = |φi hφ|φi hφ| = |φi hφ| = P and can be used to project a state onto some subspace,
P |Ψi = |φi hφ|Ψi . The one step unitary operator is given by [11][12]
U =X
θ
PATθ ⊗ Sθ
B (3.5)
The index θ represents the state of the two qubits as measured by Alice and the transmission of this state by a classical channel between Alice and Bob.
The initial state of the three qubit system |Ψinitiali in the computational basis is given by
the vector
|Ψinitiali =
1 √
2(α, β, 0, 0, 0, 0, α, β)ABT (3.6) There are four possible final states due to there being four possible final states of the AT subsystem. Let’s consider as the final state of the AT subsystem the φ+ state. The final
state is then given by the vector |Ψf inal−φ+i =
1 √
2(α, 0, β, 0, 0, α, 0, β)ABT
The unitary transformation that takes the initial state to this final state is found to be
Uφ+ = 1 2 1 0 1 0 1 0 1 0 1 0 1 0 −1 0 −1 0 0 1 0 −1 0 −1 0 1 1 0 −1 0 1 0 −1 0 0 1 0 −1 0 1 0 −1 1 0 −1 0 −1 0 1 0 0 1 0 1 0 −1 0 −1 0 1 0 1 0 1 0 1
The matrices Bob has to use when Alice measured the AT -subsystem to be one of the other Bell states can be found in Appendix A.
This unitary matrix was found by trial and error using constructions with the identity matrix and the Hadamard matrices
H2 = 1 2 1 1 1 −1 !
and H8 = H2⊗ H2⊗ H2. By interchanging the rows of
H8.(I4x4⊗ H2) = 1 2 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 −1 0 1 0 −1 0 0 1 0 −1 0 1 0 −1 1 0 1 0 −1 0 −1 0 0 1 0 1 0 −1 0 −1 1 0 −1 0 −1 0 1 0 0 1 0 −1 0 −1 0 1
we can construct all four of the unitary transformations that take the initial states to the four possibble final states.
To obtain a time evolution U (t) = eiHt~ for quantum teleportation we want to find a
Hamiltonian H such that U (t∗) = e
iHt∗
~ = U . We thus want to find the logarithm of
U . First U must be diagonalized using the diagonalization matrix D and its inverse D−1. The diagonalization matrix D can be constructed by using the eigenvectors of the unitary matrix U as the columns. By taking the logarithm of the diagonalized matrix and using the diagonalization matrix to return to the original basis we obtain the Hamiltonian.
iHt∗
~ = log(U ) = Dlog(U
0
)D−1 = Dlog(D−1U D)D−1
The resultant matrix can be found in Appendix B due to its size. Matrix U is again retrieved when taking the exponent of the matrix iHt∗
~ , thus confirming it is the correct
result. By dividing out it∗
3.5
Constructing the Scrambling Matrix
As mentioned in the protocol revision section, Alice needs to throw an information car-rying qubit in to the black hole on her side before the qubit can be teleported between the two entangled black holes. The information of the qubit will be scrambled throughout the black hole as the qubit thermalizes. In our toy model, this scrambling of information will be mimicked with a scrambling operator V . Before we can construct an operator that scrambles information between many qubits we must first look at an operator that scram-bles the information in a system with just two qubits.
To scramble the information in a two qubit system we use a unitary matrix taken from the circular unitary ensemble. The distribution of the circular unitary ensemble is the Haar measure on U (4) [19]. Such a matrix is a random matrix which can act as a scrambling operator representing a random interaction between the two qubits. An example of such a matrix is
V =
−0.293752+0.202461i 0.0805433 −0.272692i 0.588225 −0.61393i 0.219368 +0.144323i −0.123609+0.449596i −0.398763+0.74209i 0.195532 +0.0816254i 0.159594 +0.0500926i
0.101986 +0.478288i 0.0787814 −0.100683i −0.087298−0.0246889i −0.657034+0.55188i −0.186435+0.617564i 0.271697 −0.345369i −0.133075+0.454143i 0.379092 −0.151963i
Next we need to use the two qubit scrambler to construct a scrambling operator that can be used to thermalize the information carrying qubit with Alice’s black hole. This operator can be constructed by randomly picking two qubits from Alice’s black hole and using the two qubit scrambler to let these qubits interact. Repeat this step a number of times to make the operator scramble the information until the qubit has thermalized with the black hole. Combining all these scrambling operations into one gives us the scrambling operator V .
3.6
Entanglement Entropy of V
We want to know how well our scrambling operator works by looking at how much the qubits have become entangled. A good measure for the entanglement between qubits is the entropy of entanglement, which is the Von Neumann entropy of the reduced density matrix. If we split a qubit system Ψ into two subsystems, say, A and B, then the reduced
density matrix of A is constructed by ”tracing out” system B,
ρA= TrBρAB =
X
i
hi|B(|Ψi hΨ|) |iiB
The Von Neumann entropy of the reduced density matrix ρA is then given by
S(ρA) = − Tr ρAlog ρA
For a pure state S is equal to zero. In our toy model, we can split up our qubits such that system A is the qubit that originally contained the information and system B is the rest of our qubits. This way we will end up with a subsystem A with a Hilbert space of dimension 2. For a two dimensional Hilbert space the maximum entropy can be easily derived. Expressing the entropy in terms of the eigenvalues λ1 and λ2 of the reduced
density matrix and using that their sum equals 1 we get that
S(λ1) = −λ1log λ1− (1 − λ1) log(1 − λ1).
Setting the derivative of the entropy to zero to find its maximum we have dS(λ1) dλ1 = log 1 λ1 − 1 = 0 λ1 = λ2 = 1 2.
Thus a perfect scrambling operator would bring our system in to a maximally entangled state with an entanglement entropy of Smax(λ1) = log 2 ≈ 0.693147. More generally, for
a maximally entangled state, Smax = log N where N is the amount of dimensions of the
Hilbert space of the subsystem.
Unfortunately, due to the complexity of the calculation and the computational limits of Wolfram Mathematica, only scrambling operators that represent a single qubit interaction can be evaluated. Figures 11a and 11b show the distribution of the entanglement entropy of the 1-interaction scrambling operators in three qubit and four qubit systems respectively. The three qubit system has an entanglement entropy distribution with a mean value of µ = 0.33 and a standard deviation of 0.18. After only one interaction the system is still likely to be some way off from being maximally entangled, but conversely is also some way
to better entanglement. In the sections that follow a scrambling operator V was used that was constructed using 10 scrambling interactions between qubits.
Figure 10: The distribution of the entropy for the entanglement of a subspace of dimension 2 as a function of one of the eigenvalues λ1 of the reduced density matrix. The other
eigenvalues is given by 1 − λ1.
(a) (b)
Figure 11: The distribution of the entanglement entropy after one interaction using the 2 qubit scrambling operator, on the left for a three qubit system with 10000 scrambling operations, on the right in a four qubit system with 11000 scrambling operations. A normal distribution with the right values for the mean and standard deviation has been added on the left graph to show the entropy is not normally distributed.
3.7
Teleportation and Bob’s operation: Constructing Z
Using the scrambling operator V on the initial state Ψinitial in equation 3.4 will give
V |Ψinitiali =
X
m,I
φ(m)V |mIiAT|IiB
In the first step of the original standard protocol Alice measures the two qubits of the AT subsystem and in the second step transmits the result of the measurement of these two qubits by a classical channel to Bob. In the time dependent teleportation protocol of chapter 4, this information was encoded in equation 3.5 by θ. However, in the current setup, the information of the qubit will be scrambled throughout the black hole and we become free in selecting two qubits from the AT system to measure as our equivalent of step 1 of the standard protocol. In the scrambled AT subsystem, the two qubits we pick will again be labeled θ and the remaining N − 1 qubits will be labeled α. By multiplying by 1 = P
θα|θαi hθα|, the scrambling operator acting on the initial state V |Ψinitiali can
thus be written as in [12] as V |Ψinitiali =
X
mIθα
φ(m) |θαi hθα| V |mIiAT|IiB = X
mIθα
φ(m) |θαi VmIθα|IiB
with
VmIθα = hθα| V |mIiAT
After the information carrying qubit has thermalized with the black hole on Alice’s side, Alice makes a measurement of the state of the two qubits designated with θ and sends the result of her measurement to Bob via a classical channel. Once Bob has received the information he can use an operator Zθ on his subsystem B,
ZθV |Ψinitiali =
X
mIαβn
φ(m)VmIθα|θαiAT ZI,nβθ |nβi
where
ZI,nβθ = hnβ| Zθ|IiB,
β are N − 1 qubits on Bob’s side and n is the remaining qubit, and we have multiplied with 1 =P
nβ|nβi hnβ|. What now needs to be shown [12] is that Z θ
I,nβ can be chosen in
such a way that the final state is X
Figure 12: An illustration of the relation between Zθ
I,βn and V †θβ
In , taken from [12]
where W[αβ is a unitary operator and
P
αβWαβ represents the maximally entangled state
of the 2(N − 1) α and β qubits.
Susskind and Zhao suggest [12]
ZI,nβθ =X
γ
VnI†θγWγβ
the connection of which is illustrated in figure 12 as quantum circuits. Let’s simplify by using Wγβ = δγβ such that
ZI,βnθ = VIn†θβ.
V , being the scrambling operator acting on Alice’s side of the black hole, acts on the N + 1 qubits. Z however acts on the N qubits on Bob’s side. Zθ is constructed by projecting V on to θ.
There is some ambiguity in the protocol as described by Zhao and Susskind about the exact construction. One possibility is to project using V† such that
VIn†θβ = hθ| ⊗ hβ| V†|Ii ⊗ |ni
|θi here represents the state of the two measured qubits on Alice’s side, which can be |00i , |01i , |10i or |11i . Thus we get for example that
VIn†00β = h00| ⊗ (hβ| V†|Ii ⊗ |ni) = Znβ00,I
Another possibility is that we should first project V and then take either the conjugate transpose or the inverse (the projected V is not unitary, so they are not the same).
Znβ00,I = h00| ⊗ (hβ| V |Ii ⊗ |ni)−1
Above we see |θi interpreted as a single state picked from all possible 2 qubit states in the computational basis, but perhaps the protocol should use some combination of the possible states, for example,
VIn†θβ = 1
2(h00| + h01| + h10| + h11|) ⊗ (hβ| V
†|Ii ⊗ |ni) = Zθ,I nβ
I have tried several constructions of Znβθ,I, using qubits and using qutrits, with small and large qubit systems, with projection operators added and without projection operators, using different combination of states for |θi, but unfortunately without any success. To evaluate the results of applying my constructions of Znβθ,I, I split the final system such that the qubit that should have the teleportation information and that should not be entangled with the other qubits is a subsystem. I then calculated the reduced density matrix and used that to calculate the entanglement entropy. For none of my constructions did the entanglement entropy become zero or anything close to it.
There are a number of issues that have arisen from this pursuit. The protocol as described by Susskind and Zhao does not explicitly mention how the measurement of the two θ qubits happens. But this is an important step in the original standard protocol that ”forces” the information of the teleportation qubit to the other side of the system. In the new protocol the measurement is made in a basis that will not leave the θ qubits maximally entangled. How does the information go from Alice’s side to Bob’s side? Is the ”unscrambling” by Zθ
sufficient?
Zθ seems to be constructed as an ”unscrambling” operator encoded with the information of the measurement of θ, but that information is not in the scrambling operator V , which acts before the measurement is made. Zθ cannot alone be responsible for the teleportation process as it acts only on Bob’s side of the system. Alice’s side of the system must also be affected somehow to make the information disappear on that side as to not violate the no-cloning theorem. If the Identity operator would be used instead of a scrambling operator, the measurement on Alice’s side would have to be done explicitly and it would need to leave the θ qubits maximally entangled.
3.8
Teleportation through the Wormhole
In the protocol as described in the previous section, the system is evolved from time t = 0 to t = t∗, where t∗is the scrambling time, as V acts on Alice’s side of the system. Bob’s side
all the while stays unchanged. Using precursors U (t) we can change this by introducing
Sθ(t) = U†(t)ZθU (t).
By using Sθ(t) instead of ZθBob can act at any past time −t. Acting with Sθ(t) sufficiently
far away in the past will create a shock wave, as displayed in figure 13. If the shock wave is made of positive energy it will shift the path of the teleportation qubit such that it will end up at the singularity. If however the shock wave has negative energy the teleportation qubit will make it to the other side, as is in accordance with the method of Gao et al. [11].
Figure 13: Penrose diagram depicting the the shock wave and how the shock wave changes the path of the teleportation qubit, shown in red. Image taken from [12]
Let’s now boost the above situation such that the teleportation qubit enters the black hole at a time t = −t∗, Alice measures the θ qubits at time t = 0 and the teleportation
information appears on Bob’s side at t = t∗, as shown in figure 14. The Penrose diagram of
this situation is very similar to the geometry that comes from the method of Gao, Jafferis, and Wall [11]. Zhao and Susskind conclude from this that the gravitational dual to their
teleportation protocol must thus be similar to the setup of Gao et al.
Comparing the Penrose diagram in figure 14 of the Zhao and Susskind protocol to the Penrose diagram of Bao et al. [17] in figure 9 we see indeed the similarities between the two geometries.
Susskind and Zhao mention the one operator protocol as we have discussed in section 3.4, in which they combine the Sθ(t) operator with a projection operator to turn it in to a one
step operator, as we have done in equation 3.5. This would allow the possibility to remove the classical channel from the protocol, turning it in to a purely quantum procedure just like the method for traversable wormholes of Gao, Jafferis, and Wall.
4
Conclusions
We have seen the derivation of Hawking radiation by using quantum field theory in curved spacetime and considered the origin of the black hole information loss paradox: Where does the information of the structure that collapsed to form the black hole go when the black hole evaporates? We discussed some of the early proposed solutions to the paradox but they turned out to have issues of their own. For the Hawking radiation to have the information requires some mechanism for the information to be stripped away as it enters the black hole. The information could stay behind in some Planck-scale remnant, but this would require an infinite amount of different remnant, which would clash with QFT. The remnant solution would also discard the interpretation of the entropy of a black hole as the remnant could have an infinite entropy. For the information to come out at the end when the black hole is Planck-scale would take a great amount of time, giving the same kind of problem as the stable remnant solution. The information could be stored in some yet unknown quantum hair, though this would constrain conservation laws to such a degree that we should have noticed this already. The fifth solution discussed was that of the information going to a disconnected ”baby universe” spacetime, which is an unsatisfying solution that offers no real explanation.
The black hole complementarity solution showed better equipped to solve the paradox. It is somewhat a combination of the first and last solution described above. It says that the information is both ”stripped” and put into the Hawking radiation as well as falls behind the event horizon never to return. But, as argued, the event horizon prevents both destinies of the information to be confirmed, with any observer only ever to confirm one of the two, and thus the inside and outside descriptions are complementary. The black hole comple-mentarity solution does have its own issues though. As argued, the entanglement between the ingoing and outgoing particles will break leading to a Planck-scale Firewall. This does have the undesirable consequence that we have to discard the equivalence principle, which is the central principle of general relativity.
ER=EPR is an interesting proposal in response to the firewall proposal. ER=EPR equiv-ocates Einstein-Rosen bridges, also known as wormholes, with Einstein-Podolsky-Rosen pairs, also known as entanglement. This way the information loss paradox is solved by
circumventing the event horizon by connecting the Hawking radiation with the black hole through wormholes. It remains to be seen what value ER=EPR ultimately has in solving the conflict between quantum theory and general relativity theory, but it does offer an interesting dual to the traversable wormholes of Gao, Jafferis, and Wall. They proposed a method to make a wormhole traversable. We checked that this method works by calculat-ing the averaged null energy of a null geodesic gocalculat-ing from one side to the other side of the wormhole.
The dual to traversable wormholes is quantum teleportation through entangled black holes, which we investigated further. We looked at the original quantum teleportation protocol and considered how it should be revised. Wormholes take time to traverse and thus re-quires a time dependent operator. We saw that indeed one step teleportation operators are possible and that we can extract a Hermitian Hamiltonian evolution operator from them. This means that a time dependent teleportation protocol is possible. Using a protocol with only one operator also means there is no longer an explicit classical channel, instead a fully quantum teleportation protocol might be possible. A qubit thrown in to a black hole will thermalize with the black hole. We have looked at the scrambling process and, although we were limited in how much we could calculate, saw that indeed the scrambling operators used lead to entanglement by checking the entanglement entropy.
To finish the protocol requires the construction of an ’unscrambling’ operator. The con-struction of the Zθ operators proved to be troublesome. We discussed some of the issues
of this construction and why the proposed protocol might not be complete. There is some ambiguity in the protocol as proposed by Zhao and Susskind that is left to the interpreta-tion of the reader. The method to construct the Zθ operator still seems unclear. Should we first take the inverse of the scrambling operator V and then project onto θ of should we first construct VθβIn and then take its inverse. The role of the measurement of the θ
qubits is also dubious. The protocol as described does not mention any details about the measurement and nothing about the use of a projection operator and it remains unclear whether or not it should be used. The protocol leaves enough room for further research, both in to the construction of Zθ and whether or not a projection operator should be used.
teleporta-tion. We used precursors to turn Zθ in to Sθ(t) so that the operator can be used at a time −t and reconstructed a physical teleportation process, the geometry of which does indeed look like that of traversable wormholes.
A
One step unitary operators
The one step unitary operator Bob needs to use when Alice has measured the AT subsystem to be in the state |φ−i, |ψ+i and |ψ−i respectively,
Uφ− = 1 2 1 0 1 0 1 0 1 0 1 0 1 0 −1 0 −1 0 0 1 0 −1 0 −1 0 1 1 0 −1 0 1 0 −1 0 0 1 0 −1 0 1 0 −1 −1 0 1 0 1 0 −1 0 0 1 0 1 0 −1 0 −1 0 −1 0 −1 0 −1 0 −1 Uψ+ = 1 2 0 1 0 −1 0 1 0 −1 1 0 −1 0 −1 0 1 0 0 1 0 1 0 −1 0 −1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 −1 0 −1 0 0 1 0 −1 0 −1 0 1 1 0 −1 0 1 0 −1 0 Uψ− = 1 2 0 1 0 −1 0 1 0 −1 1 0 −1 0 −1 0 1 0 0 1 0 1 0 −1 0 −1 0 1 0 1 0 1 0 1 −1 0 −1 0 −1 0 −1 0 1 0 1 0 −1 0 −1 0 0 −1 0 1 0 1 0 −1 1 0 −1 0 1 0 −1 0
