Information Loss

TL;DR

This paper reviews the concept of information loss in black hole physics, discussing how quantum effects during black hole evaporation may lead to a pure state evolving into a mixed state, raising fundamental questions about quantum theory and gravity.

Contribution

It provides a comprehensive review of the arguments for and against information loss in black hole evaporation, clarifying ongoing debates in theoretical physics.

Findings

Supports the view that information is lost during black hole evaporation

Analyzes counter-arguments suggesting information preservation

Summarizes the controversy over quantum unitarity in black hole processes

Abstract

The complete gravitational collapse of a body in general relativity will result in the formation of a black hole. Although the black hole is classically stable, quantum particle creation processes will result in the emission of Hawking radiation to infinity and corresponding mass loss of the black hole, eventually resulting in the complete evaporation of the black hole. Semiclassical arguments strongly suggest that, in the process of black hole formation and evaporation, a pure quantum state will evolve to a mixed state, i.e., there will be "information loss." There has been considerable controversy over this issue for more than 40 years. In this review, we present the arguments in favor of information loss, and analyze some of the counter-arguments and alternative possibilities.

Equations

$$|{\Psi }_1⟩\otimes |{\Psi }_2⟩,$$

$$|\Psi ⟩=\sum _i{c}_i|{\Psi }_{1i}⟩\otimes |{\Psi }_{2i}⟩.$$

$${\rho }_1=\sum _{ij}{c}_i{c}_j^{\ast }⟨{\Psi }_{2j}|{\Psi }_{2i}⟩|{\Psi }_{1i}⟩⟨{\Psi }_{1j}|.$$

$$⟨\Psi |{\mathcal{O}}_1\otimes {\mathcal{O}}_2|\Psi ⟩\ne ⟨\Psi |{\mathcal{O}}_1|\Psi ⟩⟨\Psi |{\mathcal{O}}_2|\Psi ⟩.$$

$$⟨\Psi |\varphi (x_1)\varphi (x_2)|\Psi ⟩\sim \frac{U(x_1,x_2)}{(2\pi {)}^2\sigma (x_1,x_2)},$$

$$⟨\Psi |\varphi (x_1)|\Psi ⟩⟨\Psi |\varphi (x_2)|\Psi ⟩\to {\left[⟨\Psi |\varphi (x)|\Psi ⟩\right]}^2$$

Full text

I Introduction

Ever since Hawking’s discovery [1] that black holes radiate thermally and should therefore “evaporate” completely, battles have raged about the final quantum state resulting from this process. Semiclassical arguments clearly indicate that the final state should be mixed because of the entanglement of the Hawking radiation with degrees of freedom inside the black hole and the ultimate disappearance of these degrees of freedom (with respect to our universe, at least) when the black hole evaporates. It is very difficult to find a flaw in these semiclassical arguments that would invalidate this conclusion. On the other hand, it has been argued on a variety of grounds that such pure state to mixed state evolution would violate fundamental principles of physics and/or other cherished beliefs. This conflict has thus given rise to the so-called black hole information loss paradox.

Our belief is that information is lost into black holes, i.e., in the process of black hole formation and evaporation, a pure state will evolve to a mixed state111However, as discussed at the end of section IV below, we hold open the possibiity of a pure final state as a result of entanglement of the Hawking radiation with the vacuum left behind after black hole evaporation.. In classical general relativity, the singularity inside the black hole acts as a sink for any ingoing radiation, which removes it from the universe. In quantum gravity, the classical singularity will undoubtedly be replaced by some other, weirder structure, but it should still act as a sink.

The aim of this note is to review the arguments in favor of information loss and to argue that such pure to mixed evolution does not violate any fundamental principles of physics. We begin in section II with a review of entanglement in quantum mechanics and quantum field theory. As explained in section III, this gives rise to the semiclassical arguments in favor of information loss. Possible alternatives to this semiclassical picture are discussed in section IV. Arguments against information loss are analyzed in section V. Our conclusions are given in section VI.

II Entanglement

Entanglement is a ubiquitous feature of quantum mechanics, a feature which Schrodinger [2] referred to as “the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.” Suppose that we have two quantum mechanical systems, the states of which are individually represented by Hilbert spaces ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$, respectively. Then, by the rules of quantum mechanics, the possible states of the joint system are represented by the tensor product, ${\mathcal{H}}_{1}\otimes{\mathcal{H}}_{2}$, of these Hilbert spaces. This tensor product Hilbert space is the (closure of the) span of product states of the form

[TABLE]

where $|\Psi_{1}\rangle\in{\mathcal{H}}_{1}$ and $|\Psi_{2}\rangle\in{\mathcal{H}}_{2}$. Thus, a general state in the tensor product Hilbert space is of the form

[TABLE]

We can encode the information about this state relevant to observations concerning only the first system via a density matrix $\rho_{1}:{\mathcal{H}}_{1}\to{\mathcal{H}}_{1}$ defined by

[TABLE]

A normalized state $|\Psi\rangle$ can be written in the form of a product state, eq. (1), if and only if $\rho_{1}^{2}=\rho_{1}$, in which case system 1 (and system 2) is said to be in a pure state. For such a product state, system 1 behaves just as if it is the state $|\Psi_{1}\rangle$ and the presence of system 2 is irrelevant to measurements of system 1; similarly, system 2 behaves as if it is in state $|\Psi_{2}\rangle$ and the presence of system 1 is irrelevant. However, generically a state $|\Psi\rangle$ of the form (2) cannot be written as a product state (1), in which case systems 1 and 2 are said to be entangled, and the state of system 1 (as well as system 2) on its own is said to be mixed. For entangled systems, there are nontrivial correlations between the outcomes of measurements made on the two systems. In particular, if and only if the state is entangled, one can find an observable ${\mathcal{O}}_{1}$ for system 1 and an observable ${\mathcal{O}}_{2}$ for system 2 such that

[TABLE]

Entanglement is ubiquitous because even if systems 1 and 2 are initially in a product state (1), they will generically evolve to an entangled state in the presence of any nontrivial interaction. They will then remain entangled even after they cease interacting.

Although entanglement is a ubiquitous feature of quantum mechanics in general, it is an essential feature of quantum field theory. In the case of quantum field theory, the full system consists of the quantum field observables over all of spacetime, or, equivalently—assuming deterministic evolution—the quantum field observables in a neighborhood of any Cauchy surface, $\Sigma$. We can divide the quantum field system into two subsystems by dividing the Cauchy surface into disjoint open regions $\Sigma_{1}$, $\Sigma_{2}$, with common boundary $S$, such that $\Sigma_{1}\cup\Sigma_{2}\cup S=\Sigma$. Let ${\mathcal{U}}_{1}$ and ${\mathcal{U}}_{2}$ be globally hyperbolic regions with Cauchy surfaces $\Sigma_{1}$ and $\Sigma_{2}$, respectively, as illustrated in Fig. 1.

We take system 1 to be the field observables in ${\mathcal{U}}_{1}$ and system 2 to be the field observables in ${\mathcal{U}}_{2}$. Then, for any relativistic quantum field theory, in any physically acceptable state, these two systems will be entangled. To see this explicitly, let $x\in S$ and let both $x_{1}\in\Sigma_{1}$ and $x_{2}\in\Sigma_{2}$ approach $x$. For the particular case of a free scalar field, $\phi$, the leading order behavior for $x_{1},x_{2}$ near $x$ for any physically acceptable (Hadamard) state, $\Psi$, is given by

[TABLE]

where $U(x_{1},x_{2})$ is smooth and $\sigma(x_{1},x_{2})$ is the squared geodesic distance between $x_{1}$ and $x_{2}$. Thus, the right side of (5) diverges as $x_{1},x_{2}\to x$. On the other hand, for any physically acceptable state, as $x_{1},x_{2}\to x$, we have

[TABLE]

which is not divergent. Thus, the left sides of (5) and (6) cannot be equal, and there is entanglement. Similar behavior hold for all fields (including interacting fields) for all physically acceptable states. Thus, (4) always holds, and there always is entanglement between systems 1 and 2 in any physically acceptable state.

An important example of the above phenomenon is the case of Minkowski spacetime, with ${\mathcal{U}}_{1}$ and ${\mathcal{U}}_{2}$ taken to be the two Rindler wedges and $|\Psi\rangle$ taken to be the Minkowski vacuum state. The above discussion shows that there will be entanglement between the field observables in the left and right Rindler wedges. This accounts for why the family of observers in one of the Rindler wedges that follows orbits of Lorentz boosts sees a mixed state when the entire quantum field system is in the (pure) Minkowski vacuum. The Minkowski vacuum is entangled between the two wedges. That is is also a thermal state is a consequence of the particular nature of that entanglement.

Our discussion above makes it clear that the entanglement between the field in two such causally complementary regions always occurs in quantum field theory, no matter what the spacetime or the (physically acceptable) state.

III Information Loss

A spacetime of considerable interest is one in which a black hole forms by gravitational collapse. Just as in the general case above, at any “moment of time,” $\Sigma_{0}$ after the black hole has formed, in any regular state (such as the state obtained by evolution from any physically reasonable initial state before the black hole formed), there will be entanglement between the state of quantum field observables inside and outside of the horizon of the back hole. This entanglement is intimately related to the nature of the Hawking radiation emitted to infinity, allowing it to be in mixed state, and thermal.222We use the word “thermal” here in the sense that if one places a Schwarzschild black hole in a thermal bath at the Hawking temperature, then detailed balance will hold and the system will be in (possibly unstable) equilibrium. If the black hole is radiating into empty spacetime, then since some of the Hawking radiation is reflected back into the black hole by the curvature and angular momentum barriers around the black hole, the population of modes as seen by a distant observer will be reduced by this reflection. Similar behavior will occur for any other “black body of finite size.”

The mode inside the black hole that is maximally correlated with a Hawking radiation mode outside is called the partner state (see [3], [4]). If we trace the behavior of a Hawking mode and its partner mode backward in time in a free quantum field theory, we find that the entanglement of the Hawking radiation with the state of the quantum field inside the black hole arises from entanglement across the event horizon at very (i.e., transplanckian) short distance scales at early times. One would not expect the free quantum field theory description of this entanglement to be valid in this regime. However, the study of analog black holes shows that the creation of thermal radiation does not rely on such transplanckian behavior: In analog models where the dispersion relations are modified at very short wavelengths, there is no change to the prediction of Hawking radiation—and its entanglement across the horizon—at wavelengths long compared with this modification scale [5], [6]. Furthermore, the argument of Fredenhagen and Haag [7], [8] shows that Hawking radiation for black holes can be derived as a consequence of the Hadamard behavior of the state near the horizon at relatively late times, without the necessity to evolve backward all the way into a transplanckian regime.

It should be noted that there may also be considerable entanglement between the quantum field observables inside and outside of the black hole produced by much more mundane processes that do not involve short distance phenomena. For example, one can consider two physically separated matter systems that (due, e.g., to prior interactions) are highly entangled with each other. One can drop one of these matter systems into the black hole (or make it part of the matter system that collapses to form the black hole in the first place) and keep the the second matter system at a safe distance outside of the black hole. There will then be considerable entanglement between the observables inside and outside the black hole above and beyond what is predicted to occur via the Hawking effect. If one wishes to eliminate all entanglement between observables inside and outside of the black holes, one would have to eliminate entanglement arising from this means as well.

As indicated by our discussion above, the presence of entanglement between the quantum field observables inside and outside of a black hole at some relatively early time, $\Sigma_{0}$, after black hole formation (as shown in Fig. 2 below) is completely in accord with normal behavior in quantum field theory. Assuming that one started with a pure state before the black hole was formed, the full state of the quantum field at time $\Sigma_{0}$ is still pure, even though the state of the quantum field outside of the black hole is mixed. However, when back reaction effects of the quantum field on the black hole are taken into account, the entanglement between the quantum field observables inside and outside of the black hole gives rise to a “loss of information” as follows: The Hawking radiation carries a flux of positive energy to infinity, so there must be a corresponding flux of negative energy going into the black hole333Quantum fields can have locally negative energy densities and fluxes even if the corresponding classical fields always satisfy positive energy conditions.. This flux of negative energy can be viewed as originating outside the horizon—very crudely at a scale of about a Schwarzschild radius outside—and is thus not directly tied to the entanglement. This negative energy flux will reduce the mass and area of the black hole, in full accord with conservation of total energy and the generalized second law of thermodynamics. However, the flux of Hawking radiation to infinity—and the corresponding flux of negative energy into the black hole—increases as the black hole gets smaller. As a result, one predicts that the black hole should “evaporate” completely in a finite time, resulting in a spacetime as depicted in Fig. 2.

The key point is that for a quantum field propagating in the classical spacetime of Fig. 2, the state of the quantum field on a late time surface $\Sigma_{1}$ after black hole evaporation will remain entangled with the quantum field observables inside the past black hole—even though the black hole has evaporated completely and no longer exists! Although the system was initially in a pure state, the entire system at time $\Sigma_{1}$ is described by a mixed state, so evolution from a pure state to a mixed state has occurred. Furthermore, a complete knowledge of the state at time $\Sigma_{1}$ is insufficient to determine the initial state. In this sense, information has been lost to the black hole! We note that this argument holds whether the field inside has fallen off the edge of the spacetime into a central singularity as occurs for a classical black hole, or, e.g., has gone into a baby universe created by some quantum gravity effects near what would have been a singularity in the classical treatment.

We emphasize that throughout the process of black hole formation and evaporation, the quantum field obeys deterministic, causal equations of motion. At no stage, except perhaps at the singularity, is there any breakdown of any known laws of physics, including local unitarity. The loss of information has arisen solely because the inside of the black hole has acted as a sink as seen from outside for some of the degrees of freedom of the quantum field, just as infinity is a sink for degrees of freedom of massless fields escaping from any finite region of the spacetime. We will discuss this further in section V.A below.

IV Alternatives to Information Loss

The loss of information described in the previous paragraph is sufficiently startling that it worthwhile to examine the assumptions on which it is based, and to consider the ways in which these assumptions might be wrong. As described above, information loss follows directly from applying the local evolutionary laws of quantum field theory in a classical curved spacetime of the sort depicted in Fig. 2. However, this analysis treats gravity classically, so this semiclassical analysis can at best be an approximation to the true way that nature behaves. Nevertheless, in order that the conclusion of information loss be invalidated, the semiclassical approximation must fail in a significant way during some stage of the evolution. It is useful to classify the possible ways in which it might fail into the following 4 categories444The recent work of Hawking, Perry, and Strominger [9] proposes that black holes can have “soft hair” due to BMS-like charges. However, this work does not propose any mechanism by which the presence of “soft hair” would enable the entanglement between quantum observables outside and inside the black hole can be avoided so that the final state resulting from black hole evaporation is pure. Since no concrete mechanism has been proposed, we cannot categorize this work in terms of our classification scheme below, but any such mechanism presumably would be subject to the criticisms of possibilities III and IV below.:

Possibility I: No black hole ever forms (fuzzballs): Perhaps a black hole never actually forms in the collapse, but rather—perhaps due to a quantum tunneling phenomenon—some other structure without an event horizon, such as a fuzzball [10], is formed. In that case, there would be no loss of information to black holes simply because there are no black holes.

In our view, this is an extremely radical proposal. Classically, sufficiently massive black holes form at arbitrarily low energy densities and curvatures, so semiclassical general relativity and/or quantum field theory would have to drastically fail in a regime where, a priori, one would expect these descriptions to be extremely good. If one imagines a massive shell collapsing radially inward at nearly the speed of light, we find it hard to imagine how it could halt its inward momentum at just the right moment to form a fuzzball or other structure without violating local conservation of momentum, causality, and other basic principles of physics in a low curvature regime. The effective stresses (as expressed by the Einstein curvatures of the effective spacetime metric) would have be absurdly large—vastly exceeding the effective energy densities. Since horizons (true or apparent) are global, not local, structures, the collapsing matter also would have to have non-local knowledge in order to start behaving in this radical manner in order to avoid having a horizon form.

Similar remarks apply to proposals (see, e.g., [11]) for huge local quantum backreaction effects to occur at the time when, classically, a black hole would form, thereby preventing the formation of the black hole.

Possibility II: Major departures from semiclassical theory occur during evaporation (firewalls): Perhaps a black hole forms in the expected manner but there are major departures from semiclassical theory during the evaporation process, in such a way that there is greatly diminished entanglement (or no entanglement at all) between the between the state of quantum field observables inside and outside of the black hole.

This is also a very radical proposal, since the destruction of entanglement between the inside and outside of the black hole during evaporation requires a breakdown of quantum field theory in an arbitrarily low curvature regime. In particular, significantly diminished entanglement across the event horizon would require the quantum field to be singular on the horizon—converting it to a “firewall” [12]. As in the case of fuzzball-type proposals, the laws governing the creation of firewalls would have to be drastically non-local/acausal in nature. In particular, it is difficult for us to see how the firewall proposal could be made to work unless the firewall is located at exactly the event horizon, which requires knowing the entire future history of the spacetime. Even if the teleological requirement that the firewall be formed on the true event horizon could be evaded, all other notions of “horizon” (such as “apparent horizon”) are non-local in space, so acausal rules for the formation of firewalls would still be required. Furthermore, if there is entanglement between matter that formed the black hole and matter that never fell in as mentioned in section III above, it would seem that one would need major violations of causal evolution unrelated to Hawking radiation to destroy this entanglement.

Instead of having drastic local departures from semiclassical theory at firewalls very near the horizon, one could hypothesize departures that are less singular [13] but involve significant violations of causality at low energies. However, if this were to occur, it is difficult for us to see how the laws of nature would forbid similarly large violations of causality when black holes are not present.

Possibility III: The black hole does not evaporate completely (remnants): Perhaps the evaporation process shuts off by the time the black hole has evaporated down to the Planck scale—when quantum gravity effects become dominant. The resulting “remnant” continues to contain all of the “information” that went into the black hole (i.e., the remnant is highly entangled with the exterior of the remnant), in such a way that the joint state of the remnant and exterior is pure.

This is not a radical proposal, since it requires a breakdown of the semi-classical picture only near the Planck scale, where it is expected to break down. However, a crucial issue is whether such Planck scale remnants can interact with the outside world. If they cannot, then it is not clear what “good” the remnants do, since the “information,” although still present, is inaccessible, and, in practice, the final state will still be mixed. On the other hand, if the remnants can interact with the outside world, then serious thermodynamic problems arise: The initial black hole could have been arbitrarily large, or have been fed by matter for an arbitrary long time, so the remnants would need to have arbitrarily many states in order to be entangled with all of the Hawking radiation emitted during the evaporation process. Thus, if remnants could partake in the thermodynamics of the outside spacetime, they should be (infinitely) entropically favored over all other types of matter, which suggests that they should be spontaneously produced at an arbitrarily high rate.

An additional problem with having remnants storing all of the “information” needed to restore purity with Hawking radiation is that the Bekenstein-Hawking formula strongly suggests that a Planck-sized remnant should have entropy $\sim 1$—i.e., it should have $\sim 1$ possible state—rather than arbitrarily many.

Possibility IV: The information comes out in a final burst: Perhaps the evaporation process proceeds as in the semiclassical analysis until the black hole reaches the Planck scale. However, perhaps all of the information that had been stored within the black hole then emerges in a final burst, so the final state is pure.

As in the case of remnants, this is not a radical proposal, since it requires a breakdown of the semi-classical picture only near the Planck scale, where it is expected to break down. However, at first glance, this proposal may seem absurd, since it requires an arbitrarily large amount of “information” to be released from an object of Planck mass and size. Clearly, this would not be possible if the large amounts of “information” in the sense relevant here were required to carry off correspondingly large amounts of energy, e.g., if it were necessary for the black hole to emit some sort of burst of “ordinary particles” entangled with the Hawking radiation to carry this information. However, this is not necessarily the case because “vaccum fluctuations” can contain arbitrarily large amounts of “information.” Indeed, recently, Hotta, Sch utzhold, and Unruh [4] have considered the model of a mirror in $(1+1)$-dimensions that accelerates in such a manner as to emit Hawking-like radiation and then becomes inertial. This is clearly a “unitary” process, but during the accelerating phase the quantum field emission from the mirror is thermal in analogy with the emission from a black hole, i.e., the analog of Hawking radiation is thermal, but the full state of the quantum field must be pure at all times. The purity of the full state can be understood as a consequence of entanglement of the Hawking radiation with “partner particles” that are present outside of the mirror and eventually “bounce off” the mirror after it becomes inertial. However, these “partner particles” are locally indistinguishable from vacuum fluctuations! The state of the quantum field as seen by an observer at late retarded times (after the mirror has become inertial) would be indistinguishable from the ordinary vacuum state. But the vacuum fluctuations seen by such an observer would be correlated with the Hawking radiation in such a way as to produce a pure state.

It would seem more difficult to have analogous behavior in the black hole case, where the highly non-classical “burst” region should be of Planck scale in time and space. We are currently investigating models that may yield such analogous behavior for the black hole case. At the present time, we consider this to be a potentially viable alternative.

V Arguments Against Information Loss

With the possible exception of “possibility IV” above, we feel that it is fair to say that the above alternatives to information loss are neither plausible nor palatable. Why, then, have people been driven to consider such alternatives? There are three basic arguments that have been given against information loss. We now proceed to analyze these arguments.

V.1 Violation of Unitarity

In scattering theory, the word “unitarity” has two completely different meanings: (i) Conservation of probability. (ii) Evolution from pure states to pure states. Failure of (i) would represent a serious breakdown of quantum theory (and, indeed, of elementary logic). However, it is (ii)—not (i)—that is being proposed by the semiclassical picture.

Failure of (ii) would be expected to occur in any situation where the final “time” is not a Cauchy surface. Such a failure of unitarity is entirely innocuous. For example, we get evolution from a pure state to a mixed state for a massless Klein-Gordon field in Minkowski spacetime if the final “time” is chosen to be a hyperboloid rather than a hyperplane, as illustrated in Fig. 3.

This is because the state of the quantum field on the hyperboloid is entangled with the state of the quantum field on the portion of future null infinity that lies to the past of the cross-section of null infinity corresponding to its intersection with the hyperboloid. In this case, at the “time” represented by the hyperboloid, information has been “lost” to null infinity, leaving the field on the hyperboloid in a mixed state.

The situation illustrated in Fig. 3 is not an artificial example but rather illustrates phenomena that occur around us all of the time. If an atom in your living room emits a photon, the state of that atom will be in entangled with the photon. If that photon escapes out the window and is not reflected/absorbed by clouds or any other intervening matter in the universe, it will be “lost forever” as far as you are concerned. The state of your living room and any additional portion of the universe that you observe will be mixed. “Information” will have been lost. An initial pure state in any experimentally accessible region around you will have been converted into a mixed state.

The pure state to mixed state evolution predicted by the semiclassical analysis of black hole evaporation is of an entirely similar character. It is a prediction of quantum (field) theory in any situation where the final “time” is not a Cauchy surface, not a violation of quantum theory.

We find it ironic that some researchers who may have been seeking to “save quantum mechanics” by trying to evade the semiclassical arguments that a pure state evolves to a mixed state are, in fact, effectively attempting to “destroy quantum mechanics” by seeking to modify quantum mechanics (see, e.g., [14]) and/or pursuing truly drastic alternatives (such as Possibility II of the previous section) that really are violations of quantum (field) theory in a regime where it should be valid. We remain firm in our belief in the validity of quantum theory in regimes away from the Planck scale.

V.2 Failure of Energy Conservation

It is commonly claimed that any evolution law taking a pure state to a mixed state would necessarily result in unacceptable violations of energy conservation. The reference most frequently cited in support of this claim is a paper of Banks, Peskin, and Susskind [15]. However, examination of this reference shows that they considered only a “Markovian” type of evolution law, namely, that given by the Lindblad equation [16]. It is true that evolution governed by the Lindblad equation does not conserve energy, although the violations of energy conservation can be adjusted so as to be negligible at laboratory scales [17]. However, Markovian evolution would not be an appropriate model for black hole evaporation, since the black hole clearly should retain a “memory” of what energy it previously emitted.

The widespread belief that pure state to mixed state evolution must be accompanied by violations of energy conservation appears to be closely associated with a widespread belief that any quantum mechanical decoherence process requires energy exchange and therefore a failure of conservation of energy for the system under consideration. The latter belief is true in the case where the “environment system” is taken to be a thermal bath of oscillators [18]. However, it is not true in the case where the environment system is a “spin bath” [19] where excitation of the degrees of freedom of the environmental system does not require energy.

Based on this spin bath idea, Unruh [20] has provided a simple example of a quantum mechanical system that interacts with a “hidden spin system” in such a way that an initial pure state of the quantum mechanical system will evolve to a final mixed state, but exact energy conservation holds in the process. As a result of interactions, the quantum mechanical system becomes entangled with the spin system, but at late times the interactions become negligible, and the spin system carries no energy.

Unruh’s model is not a realistic model for black hole evaporation, and it yields a relatively small amount of “information loss.” Nevertheless, it provides a proof that there is no problem of principle with maintaining exact energy conservation in quantum mechanics with an evolution wherein an initial pure state evolves to a final mixed state. We see absolutely no reason why evolution from a pure state to a mixed state cannot occur as suggested by the semiclassical analysis while maintaining exact energy conservation.

V.3 AdS/CFT

During the 1980s and most of the 1990s, the main arguments given against information loss were “violation of unitarity” and “failure of energy conservation,” as described in the above subsections. However, since the late 1990s, these arguments have largely been supplanted by the assertion that evolution from a pure state to a mixed state in the process of black hole formation and evaporation cannot occur since it would violate the “AdS/CFT correspondence.”

The AdS/CFT correspondence is the assertion that quantum gravity (at least on asymptotically anti-deSitter spacetimes) is “dual” to a (non-gravitational) conformal field theory defined on the boundary of anti-deSitter spacetime. The one sentence version of AdS/CFT argument against information loss is that since the conformal field theory—being an ordinary quantum field theory in a fixed classical spacetime—presumably does not admit pure state to mixed state evolution, such evolution must also not be possible in quantum gravity, including when black holes form and evaporate.

The AdS/CFT correspondence is a conjecture. Our difficulty in assessing the validity of the AdS/CFT argument against information loss is not so much that this conjecture has not been proven, but rather that it has not been formulated in sufficient detail and with sufficient precision to make a clear argument. In particular, relatively little is explicitly known about the conjectured “dictionary” between the “bulk observables” in the asymptotically AdS spacetime and the CFT observables defined in the boundary theory555Of course, a key reason why relatively little of the conjectured dictionary is known is that there is very little, if any, understanding of what “bulk observables” are supposed to be in quantum gravity.. However, as we shall now explain, the precise nature of this correspondence dictionary is crucial to using AdS/CFT in arguments against information loss.

One way of formulating the notion of “information loss” in black hole formation and evaporation is the statement that the bulk observables at late times are not the complete set of bulk observables. The bulk observables at late times thereby comprise only part of the independent degrees of freedom of the bulk system, and the (pure) state of the complete bulk system (which includes early time observables) is mixed when restricted to the late time observables. That the bulk state is mixed when restricted to late time observables is not, by itself, in conflict with the assertion that the complete set of bulk observables is in 1-1 correspondence with the complete set of CFT observables. A conflict arises only when one adds additional assumptions about the correspondence, such as (1) the correspondence is sufficiently “local in time” that the late time bulk observables are in 1-1 correspondence with the late time CFT observables, and (2) the CFT observables at one time comprise all of the observables of the CFT system (so that the late time CFT observables are the complete set of CFT observables). Assumptions of this sort are implicitly made in all AdS/CFT arguments against information loss.

Assumptions (1) and (2) may appear reasonable, but it is far from clear that they are valid—even assuming the validity of some version of the AdS/CFT conjecture. In particular, if the CFT observables can be obtained by taking limits to the boundary of bulk observables, then (1) would hold automatically but one would not expect (2) to hold666For example, one obtains a CFT by taking a boundary limit of a free scalar field on AdS, but this CFT is a “generalized free field,” which does not obey assumption (2) [21].. Indeed, if (2) held, then the bulk observables at all times would be determined by the bulk observables near infinity at one time, in direct conflict777This conflict between predicted behavior in quantum gravity if AdS/CFT together with assumptions (1) and (2) hold and known classical behavior in general relativity would raise issues that, in our opinion, are far more significant than the issues raised by the “information paradox.” This remark also applies to the boundary unitarity arguments of [22], which can be made independently of the AdSCFT conjecture. with known classical behavior of general relativity, as expressed in the “gluing theorems” [23]. On the other hand, if the CFT observables cannot be obtained as limits to the boundary of bulk observables, then it is far from obvious that (1) will hold.

It is our hope that the AdS/CFT ideas can be developed further so as to make a mathematically precise argument regarding information loss in black hole formation and evaporation. A properly developed argument that AdS/CFT is in conflict with information loss would necessarily contain an explanation of how information is regained—and where the semiclassical behavior is violated—not just an assertion that it must happen somehow or other. With such an argument in hand, one would have to choose between AdS/CFT and information loss—and one would be in a position to do so intelligently. At the present time, we see no necessity of rejecting either alternative.

VI Conclusions

We have argued that entanglement between the observables outside and inside of a black hole is a natural and inevitable consequence of the laws of quantum mechanics and quantum field theory. The proposals to evade or destroy this entanglement typically require drastic violations of the local laws of physics in regimes where, a priori, one would expect them to be valid. We have also argued that loss of information in black hole formation and evaporation does not violate any fundamental principles of physics and is not, in any way, a radical proposal. Thus, our strong inclination is to believe that there is loss of information in the process of black hole formation and evaporation.

Acknowledgements

The research of W.G.U. was supported by the Natural Sciences and Engineering Research Council of Canada, the Canadian Institute for Advanced Research, the Perimeter Institute, and the Templeton Foundation. The research of R.M.W was supported by NSF grant PHY 15-05124 to the University of Chicago. We thank numerous colleagues—far too many to name individually here—for many extremely stimulating discussions on this topic during the past 40 years. Of particular note were numerous interactions that we both had at the KITP workshop “Black Holes: Complementarity, Fuzz, or Fire?” in August, 2013. We also thank Don Marolf for several comments on a draft of this manuscript.