intellecton/papers/project_paper_6_holographic/references/HaydenPreskill2007.txt

arXiv:0708.4025v2  [hep-th]  21 Sep 2007

Black holes as mirrors:
quantum information in random subsystems

Patrick Hayden
School of Computer Science, McGill University, Montreal, Quebec, H3A 2A7, Canada

John Preskill
Institute for Quantum Information, California Institute of Technology, Pasadena CA 91125, USA

Abstract

We study information retrieval from evaporating black holes, assuming that the internal
dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has
unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has
already proceeded past the “half-way” point, where half of the initial entropy has been radiated
away, then additional quantum information deposited in the black hole is revealed in the Hawking
radiation very rapidly. Information deposited prior to the half-way point remains concealed
until the half-way point, and then emerges quickly.
These conclusions hold because typical
local quantum circuits are efficient encoders for quantum error-correcting codes that nearly
achieve the capacity of the quantum erasure channel. Our estimate of a black hole’s information
retention time, based on speculative dynamical assumptions, is just barely compatible with the
black hole complementarity hypothesis.

1
Introduction

Is the information consumed by a black hole destroyed and lost forever [1], or might it be recovered
from the Hawking radiation that is emitted as the black hole evaporates? Evidence from string
theory suggests that the information, rather than being destroyed, can be encoded in the black hole’s
internal degrees of freedom and eventually transferred to the outgoing radiation [2, 3]. However,
the issue remains controversial, and in any event the mechanism by which information escapes from
a black hole remains elusive.
Quantum information theory addresses quantitative questions about the acquisition, transmis-
sion, and processing of information in quantum systems [4]. Though quantum information theory
cannot by itself resolve the black hole information puzzle, it can provide intuition and tools that
help to sharpen our understanding of the question.
In this paper, we assume that black holes, like other thermal systems, process quantum infor-
mation rather than destroy it, and we apply insights from quantum information theory to study the
information content of the Hawking radiation. Our conclusion is that, under plausible dynamical
assumptions, the black hole releases information remarkably quickly, much faster than might have
been naively expected.
Our analysis has two main components.
At first, we assume that a black hole thermalizes
quantum information arbitrarily quickly, so that we may model the internal dynamics of a black
hole by an instantaneous random unitary transformation.
Under this assumption, we show in
Sec. 3 that if a black hole’s internal degrees of freedom are nearly maximally entangled with the

1


previously emitted Hawking radiation (as would be expected for a black hole that has already
radiated away more than half of its initial entropy), then k qubits of quantum information dumped
into the black hole will be revealed after just a few more than k qubits are emitted in the Hawking
radiation. This observation rests on known achievable rates for entanglement-assisted quantum
communication through a quantum erasure channel [5].
Then we reexamine the issue of a black hole’s thermalization time, and we argue in Sec. 4 and
Sec. 5 that a black hole’s internal quantum state becomes thoroughly mixed in a (Schwarzschild)
time of order rS log(rS/lP ), where rS is the black hole’s Schwarzschild radius and lP is the Planck
length (and where the speed of light is c = 1). This argument, based on speculative dynamical
assumptions, relies on a recent construction of efficient quantum circuits that realize approximate
unitary 2-designs [6, 7]. Combining with the preceding result, we infer that, for a black hole whose
evaporation is past the half-way point, k qubits absorbed by the black hole will be reemitted in
Schwarzschild time O(krS) or O(rS log(rS/lP )), whichever is larger.
If we accept that black holes evolve unitarily and encode quantum information in their Hawking
radiation, then we are faced with the challenge of reconciling this phenomenon with the perspective
of an infalling observer who tumbles through the event horizon. We do not attempt to resolve this
mystery here. Rather, we focus on the behavior of the black hole from the perspective of observers
who stay outside. To these observers, a black hole is a seething cauldron of microscopic degrees of
freedom localized close to the horizon, about one qubit per Planck unit of area, undergoing local
unitary dynamics with a characteristic time scale of order the Planck time [8, 9]. We assume that
the observers refrain from attempting to probe these microscopic degrees of freedom directly, which
would be far too dangerous. Rather they are content to infer how the black hole processes infor-
mation indirectly, by investigating the relationship between the infalling matter and the outgoing
radiation.
We will, however, address in Sec. 6 whether our claim that information escapes rapidly from
black holes can be reconciled with the hypothesis of “black hole complementarity,” according to
which no violations of the accepted principles of quantum physics can be detected by any ob-
server, whether outside or inside the black hole. We conclude that rapid escape and black hole
complementarity are compatible, but only just barely so.

2
A classical randomizer

Black holes may not destroy information, but surely they hide it pretty well. How well?
The black hole information puzzle really concerns the processing of quantum information, but
let us to begin our discussion by considering the fate of classical information that enters a black
hole. Suppose that Alice, a citizen of a highly advanced civilization in the distant future who has
recorded her most private thoughts in a very confidential diary, has second thoughts and resolves
to destroy her diary. How should she proceed? Bob, the top forensic scientist of Alice’s era, has
remarkable capabilities — he can recover the contents of an erased hard disk, restore the shredded
pages of a document, even reconstitute burned pages from their ashes and smoke. Presumably,
Alice’s safest option is to toss her diary into a nearby large black hole. Eventually, the black hole
will evaporate completely, encoding Alice’s diary in the outgoing Hawking radiation where it might
be decrypted by Bob. But evaporation of a large black hole is an extremely slow process — Alice’s
secrets will be secure not for all eternity, but at least for many generations to come.
Or will they? Since we are for now discussing only classical information, let us adopt a highly
unrealistic classical model of a black hole. (It will be instructive to contrast this classical model
with a quantum model of a black hole that we will discuss in Sec. 3.) In this classical model, Alice’s

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diary is a bit string of length k and the internal state of the large black hole is regarded as a bit
string of length n − k ≫ k. We assume that Bob, who has been observing the black hole since
its formation and has a thorough understanding of black hole dynamics, knows the black hole’s
internal state, but he does not (yet) know the content of Alice’s diary.
Now Alice tosses in her diary; the black hole’s bit string grows to length n, where Bob knows
n−k of the bits, and the black hole’s internal dynamics processes this length-n string. We model this
dynamics as a permutation (known by Bob) of all of the 2n strings of length n (not a permutation
of the n bits). After the processing, the black hole releases the bits one at a time in the Hawking
radiation, as Bob watches expectantly. How soon will Bob be able to read the diary? We claim
that, for almost any permutation, Bob will only need to receive a few more than k bits before he
will be able to decipher the complete diary, with low probability of error. Alice’s secrets are not
protected for the full lifetime of the black hole; rather they are revealed to Bob almost as quickly
as possible!
The black hole dynamics is deterministic (one particular known permutation is applied to the
n-bit string), but for analyzing the information content of the radiation it is helpful to adopt the
information theorist’s favorite trick — to assume that the permutation has been chosen uniformly
at random from among the (2n)! possible ones. After processing by the black hole, Alice’s k-bit
message has been transformed to one of 2k possible n-bit strings, and if Bob could read all n bits
he would know which of the 2k strings he had and so decode Alice’s message. But even if Bob has
access to just a few more than the first k bits of the string, he is likely to be able to rule out all
messages except for the correct one, so that he can still decode successfully.
For the case of a random encoding, the probability of a decoding failure is easily estimated. If
Bob reads the first s bits of the string, what is the probability that these bits accidently match the
first s of an encoded message other than the correct one? For each message, the probability of an
accidental match is 2−s, and since there are all together 2k encoded messages, the probability Pfail
that any of the wrong messages match the s bits satisfies

Pfail ≤ 2k2−s = 2−c ,
where
s = k + c .
(1)

Therefore, if Bob wants the probability of failure to be no larger than 2−c for some constant c, he
decodes after receiving k + c bits of Hawking radiation. Here we speak of a probability of failure
because we are averaging over all the possible encodings of k bits in a block of n bits. Our conclusion
is that most encodings work.
In information theory, the capacity C of a noisy communication channel is the maximum achiev-
able asymptotic rate at which coded information can be sent through the channel with a negligible
probability of a decoding error by the receiver. What we have just described is related to two
standard results in the theory of noisy classical channels [10]: (1) The (classical) erasure channel
with erasure probability p has capacity C = 1 − p, and (2) random encodings achieve the capacity.
In our setting, the black hole dynamics transforms Alice’s k-bit message into one of the 2k

codewords of a random code with block length n. We say that the rate of the code is R = k/n, the
number of message bits per bit in the code block. When k + c of the bits in the block have been
revealed to Bob, the remaining n − k − c bits have in effect been erased as far as Bob is concerned.
Yet, despite these erasures, Bob is able to decode Alice’s message with good success probability.
Letting n get large with R and c fixed, we conclude that the message can be decoded even as the
fraction of erased bits approaches p = 1 − R; that is, the rate R = 1 − p is achievable in the limit
of a large code block.
To Alice’s dismay, we conclude that (in this model) a black hole is hardly black at all; it might
more accurately be regarded as a kind of information mirror.
Alice throws her diary into the

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black hole, and it bounces right back! Granted, it may be a strange sort of mirror, since if the
Hawking radiation leaks out slowly and k >> 1, then Alice’s message is obscured for a while;
furthermore, Bob needs to use his knowledge of the black hole’s initial state and its dynamics to
decipher the “reflection.” But once a few more than k bits have been reemitted by the black hole,
the diary comes back into sharp focus, and Alice’s secrets are no longer concealed from Bob. What
is especially ironic about this scenario is that, by modeling the internal black hole dynamics as
a random permutation, we hoped to maximize the black hole’s power to hide the information it
consumes. But at least as a matter of principle we have achieved the opposite of what we intended
— the random permutation encodes the k bits in a form that is optimally protected against the
damaging effects of erasure!
Let’s point out some implicit assumptions underlying this model. We have assumed that the
internal dynamics of the black hole is very fast — the permutation is applied almost instantaneously
after Alice’s message is deposited, before any of the Hawking radiation leaks out. We will return
to the issue of estimating the actual thermalization time scale in Sec. 4 and Sec. 5.
We have
assumed that the permutation is “typical.” This assumption is nontrivial, because if the dynamics
of the black hole is realized by a (reversible) classical computer performing local logic gates, then
most permutations require very long computations, and the computations that can be performed
efficiently may be far from typical. Similarly, we have not worried about the efficiency of Bob’s
decoding operation. We have imagined that Bob consults a huge codebook that lists the 2k valid
message strings, but if k is large then this codebook would be of unmanageable size. We will discuss
the efficiency of the recovery procedure further in Sec. 5. Finally, the most glaring drawback of our
model is that it is classical. Let us now turn to its quantum generalization.

3
A quantum randomizer

Again, we imagine that Alice regrets recording some information and wants to destroy it, but this
time the information is not a bit string — rather it is quantum information stored in a k-qubit
quantum memory. Normally, when we say that a quantum memory stores k qubits, we mean that
the stored quantum state lives in a Hilbert space of dimension 2k, but we also mean something more:
that the Hilbert space has a physically natural decomposition as a tensor product of k two-level
systems. For example, we might envision the memory as a system of k spin-1

2 particles. However,
this tensor product decomposition will not be central to our discussion, so it will for the most part
be adequate to regard Alice’s message system M as a Hilbert space of dimension |M| = 2k without
any special structure (and where k need not be an integer).
Next, we need to reconsider some other features of the classical scenario. For example, what does
it mean to say that Alice’s quantum state can be recovered by Bob from the Hawking radiation?
We don’t necessarily mean that Bob can acquire a complete classical description of the state; that
would be too much to ask. Rather we mean that Bob can do anything with the recovered state
that he would have been able to do with the state of Alice’s memory if he had been able to access
it in the first place, before Alice tried to destroy it.
It is useful to imagine that a third party (Charlie) holds a reference system N with dimension
|N| = |M| that is maximally entangled with Alice’s memory M. That is, the initial joint state of
the memory and the reference system may be chosen to be the pure state

|Φ⟩MN =
1
�

|M|

|M|
�

a=1
|a⟩M ⊗ |a⟩N ;
(2)

we say that the N provides a purification of the state of M. If Charlie holds N, but Alice retains

4


M, then the density operator for N (upon tracing out M) is maximally mixed. Suppose that Alice
tosses M into the black hole. If sometime later Bob is able to extract from the Hawking radiation
a subsystem of dimension |M| that is maximally entangled with N, then we may say that Bob has
recovered the quantum information that had been stored in Alice’s quantum memory. This would
imply in particular that if the initial state of M had been a pure state |ψ⟩ (not entangled with any
reference system), then Bob would be able to recover |ψ⟩ in his chosen subsystem [11].
In the classical scenario, we imagined that Bob knew the initial internal state of the black hole
and the dynamical laws that govern its evolution. We would like to make parallel assumptions
regarding the quantum model, but what should it mean to say that Bob “knows” a quantum state?
When we say that Bob knows the bit string encoded in the classical black hole, we really mean that
Bob holds a record that is perfectly correlated with the state of the black hole (e.g., a perfect copy
of the string). Similarly, in the case of the quantum black hole we may imagine that Bob holds a
quantum memory that is perfectly correlated with the black hole’s internal state, i.e., maximally
entangled with it. This is a strong assumption, but not a crazy one if we grant Bob complete
control over the Hawking radiation.
Indeed, consider how the entanglement of the black hole with the emitted radiation evolves
as the black hole evaporates. We may divide the world into two subsystems — the black hole’s
internal system B and radiated system E. The relative size of these subsystems varies with time; in
particular, we may assume that, at any stage of the evaporation process, log |B| is the black hole’s
Bekenstein-Hawking entropy. Early on, soon after the black hole’s formation, we have |B|/|E| ≫ 1,
and one can plausibly argue [12, 13, 14] that E is very nearly maximally entangled with a subsystem
of B. However, as the evaporation proceeds, log |B| eventually declines to half of its initial value,
and soon after we have |B|/|E| ≪ 1; then we may expect that B is very nearly maximally entangled
with a subsystem of E.
Suppose that Alice, intent on destroying her k-qubit quantum memory, heads for the nearest
large black hole and tosses her qubits in. In her haste, she imprudently fails to investigate the
hole’s history. But this particular black hole actually formed long ago, and Bob has been collecting
its emitted Hawking radiation ever since. By now, the black hole’s internal state is maximally
entangled with a system Bob controls.
How soon will Bob be able to recover Alice’s memory from the Hawking radiation? We assume
that the internal dynamics of the black hole is a deterministic unitary transformation that thor-
oughly mixes the infalling information into the black hole’s preexisting (n−k)-qubit state; then the
black hole’s qubits are released, one by one, in the Hawking radiation. We claim that, for almost
any unitary transformation, Bob needs to wait for only a few more than k qubits to be emitted.
Much as in our classical discussion, the (maximally entangled) black hole is hardly black at all —
it is a quantum information mirror that returns to Bob the information Alice deposited almost as
quickly as possible!
Right after Alice tosses in her qubits, the n-qubit black hole system B is maximally entangled
with the system NE; here B includes Alice’s memory system M, which has now been absorbed
by the black hole, E is the previously emitted Hawking radiation controlled by Bob, and N is
Charlie’s reference system that had been entangled with M. As Bob watches attentively, the black
hole continues to emit Hawking radiation until, after a while, s additional qubits (the subsystem R
of B) have been emitted, with n − s qubits (the subsystem B′) still retained by the black hole. We
suppose for now that the emitted s-qubit subsystem R of B is chosen uniformly at random (we will
revisit this assumption in Sec. 5). That is, we imagine that B is divided into two parts, one with s
qubits and the other with n − s qubits; then a unitary transformation V B chosen uniformly with
respect to the Haar measure on U(2n) is applied to B. After that, the s-qubit system is identified
as R.

5


V B

maximal
entanglement

Alice’s
qubits

Bob’s decoder

black
hole

black hole

radiation

radiation

E
R
B'

M

maximal
entanglement

Charlie

N

time

Figure 1: Information retrieval from an evaporating black hole. The black hole, which has been
evaporating for a long time, has become maximally entangled with the previously emitted radiation
system E. At this stage, the black hole swallows Alice’s quantum memory M, which is maximally
entangled with Charlie’s reference system N. The internal dynamics of the black hole applies a
strongly mixing unitary transformation V B, and then the additional radiation system R is emitted,
where the dimension of R is somewhat larger than the dimension of M. Now a subsystem of RE,
controlled by Bob, is nearly maximally entangled with N — the content of Alice’s quantum memory
has escaped from the black hole and is in Bob’s possession.

As the Hawking radiation leaks out, the correlations between the evaporating black hole B′ and
the reference system N gradually weaken. Once R is large enough, the surviving correlation of N
with B′ becomes negligible. At that point, since the overall state of B′RNE is pure, the state of
the reference system N is very nearly purified by the radiation system RE that Bob controls — by
this time, Alice’s quantum information has fallen into Bob’s hands. See Fig. 1.
Let ΨBNE denote the pure density operator of BNE, and let ρBN = TrEΨBNE be the corre-
sponding marginal density operator on BN. The marginal density operator on NB′ is

σNB′(V B) = TrR
�
ρNB(V B)
�
,
where
ρNB(V B) =
�
IN ⊗ V B�
ρNB �
IN ⊗ V B†�
.
(3)

Using standard estimates [15, 16], we find that the L1 distance of σNB′ from a product state,
averaged over V B (and hence over the choice of the subsystem R), can be bounded as
�
dV B ���σNB′(V B) − σN(V B) ⊗ σB′
max
���
2

1 ≤ |NB|

|R|2 Tr
��
ρNB�2�
;
(4)

here σN(V B) = TrB′
�
σNB′(V B)
�
is the marginal density operator on N, and σB′
max = IB′/|B′| is

the maximally mixed density operator on B′. The L1 norm, defined by ∥A∥1 = Tr
√

A†A, is an
appropriate measure because two states that are close in this norm cannot be well distinguished by
any measurement [17].
In the case we are currently considering, in which B is maximally entangled with NE, ρNB is
maximally mixed on a system of dimension |B|/|N|, and therefore

Tr
��
ρNB�2�
= |N|/|B| ;
(5)

6


hence we find

�
dV B ���σNB′(V B) − σN(V B) ⊗ σB′
max
���
2

1 ≤ |N|2

|R|2 = 22k

22s = 2−2c ,
where
s = k + c .
(6)

We see that, for a typical unitary transformation V B and for c sufficiently large, the state of NB′

is nearly maximally mixed after k + c qubits have escaped from the black hole. Alice’s k qubits
have been “forgotten” by the black hole and have been acquired by Bob.
Inconveniently, the information originally encoded in Alice’s memory M has become encoded in
a subsystem M′ of RE that is very diffusely distributed among the emitted radiation quanta. But
in principle Bob could do a quantum computation that maps M′ to a compact system ˆ
M localized
in his laboratory. For any fixed value of the unitary transformation V B, Bob’s decoding map can
be chosen so that, after decoding, the density operator ρ ˆ
MN of
ˆ
MN is close to the maximally
entangled state |Φ⟩ ˆ
MN (see, for example, [16]):

F(V B) ≡ ⟨Φ|ρ
ˆ
MN|Φ⟩ ≥ 1 −
���σNB′(V B) − σN(V B) ⊗ σB′
max
���
1 .
(7)

Eq. (6) implies that, after averaging over V B, the fidelity F(V B) deviates from one by no more
than 2−c; apart from a small error, Bob holds the purification of Charlie’s reference system N.
Thus eq. (6) can well be regarded as the quantum analog of the classical estimate eq. (1); in both
the classical and quantum models, the information that Alice deposits in the black hole escapes
almost as soon as it possibly can.
The conclusion that σNB′ becomes nearly maximally mixed after about k randomly chosen
qubits are discarded can be understood heuristically as follows: The initial density operator ρNB is
uniform on a space of dimension |B|/|N|, and so can be expressed as a uniform ensemble of |B/|N|
mutually orthogonal pure states.
If |R| ≪ |NB′|, then a typical pure state on NB is close to
maximally mixed on R; therefore after tracing out R each pure state in the ensemble representing
ρNB(V B) yields a uniform density operator on a subspace with dimension |R| of NB′. All together
then, ignoring the overlap of these subspaces, the density operator σNB′ is nearly uniform on a
space of dimension |B| · |R|/|N|. This matches the dimension of NB′, |N| · |B′| = |N| · |B|/|R|,
for |R| = |N|. We expect then, that for |R| ≈ |N|, the density operator σNB′ is nearly uniform on
NB′.
As in the classical model, our conclusion can be usefully restated in terms of known results
in the theory of noisy channels: for the quantum erasure channel with erasure probability p, the
entanglement-assisted quantum capacity is QE = 1 − p [5]. In our setting, the n − k − c qubits
retained by the black hole are in effect erased as far as Bob is concerned, yet Bob is able to
extract k qubits of high fidelity quantum information. The quantum communication rate R ≈ k/n
is achieved using a randomly chosen unitary encoder, and by exploiting a supply of pre-existing
quantum entanglement that Bob shares with the black hole.
Suppose on the other hand that Alice is more cautious, and deposits her sensitive quantum
information in a relatively young black hole, such that |E|/|B| ≪ 1. In that event, the previously
emitted Hawking radiation E will be nearly maximally entangled with a subsystem of B. The
radiation will continue to be essentially featureless, revealing none of Alice’s information, until
|B′| = |NRE|. Soon after, the black hole will be nearly maximally entangled with its surroundings
and eq. (6) will begin to apply; at that stage Alice’s information suddenly spills out.
Here too the conclusion is related to a well known property of the quantum erasure channel
with erasure probability p: its quantum capacity (unassisted by entanglement) is Q = 0 for p ≥ 1/2
and Q = 1 − 2p for p < 1/2 [18]. If the black hole initially holds n qubits, then when (n + k)/2

7


qubits have emerged, (n−k)/2 qubits remain inside the black hole. At this point, Bob’s system is k
qubits larger than the black hole, so that Bob has acquired k qubits of quantum information. Thus
the communication rate is R = k/n and the erasure probability is p = 1/2 − k/2n = 1/2 − R/2.
Some years ago, Don Page observed that the information swallowed by a black hole remains
hidden until half of the black hole’s qubits have been radiated away, and then emerges at a constant
rate as the evaporation proceeds [12]. Our conclusion is similar, but our analysis goes further.
Page considered the time dependence of the quantum entanglement of the black hole with its
surroundings, which starts out small, grows until half of the entropy has been radiated away,
and then declines. We have focused instead on when a fixed amount of quantum information of
particular interest can be recovered.
Our simple model of quantum black holes leads us to two main conclusions, under the assump-
tion that Bob has unlimited control over the Hawking radiation. If Alice dumps k qubits into the
black hole after half of the black hole’s initial entropy has been radiated away, the information
bounces right back — Bob recovers Alice’s quantum state after waiting for just a few more than k
qubits to be emitted. On the other hand, if Alice dumps k qubits into the black hole before half
of the black hole’s initial entropy has been radiated away, then Bob needs to wait for a while. But
once the black hole reaches the stage where half of its qubits have been released (and the black
hole has become maximally entangled with its surroundings), Alice’s information pops out almost
immediately!
Both statements seem strange upon first hearing, but especially the latter one, because who is
to say which k qubits are “Alice’s”? In fact, no matter which k qubits of quantum information
swallowed by the black hole are of particular interest, these k qubits are revealed almost right
away when the half-way point of evaporation is finally reached. There is nothing special about the
subsystem M of B that is maximally entangled with N in Fig. 1; for any other k-qubit subsystem
the conclusion would have been the same — that N becomes very nearly maximally entangled with
a k-qubit subsystem of RE.
Therefore, when a black hole that initially held n qubits has evaporated past the half-way point,
so that (n + k + c)/2 qubits have been emitted, Bob gets to decide which k qubits of quantum
information he will retrieve from the Hawking radiation; when he makes up his mind he performs
the decoding operation on RE that maps those k qubits to the quantum memory in his laboratory.
But the catch is that, although Bob can recover almost any k-qubit subsystem at this stage, he
cannot recover more than k qubits.
Returning to the situation depicted in Fig. 1, imagine two different k-qubit subsystems of the
black hole, M1 purified by reference system N1 and M2 purified by reference system N2. When Bob
decodes only M1 his decoding map is permitted to act on N2 (which in that case may be considered
to be part of the radiation system RE), and when he decodes only M2 his decoding map may act
on N1. But if he greedily wanted to decode both M1 and M2, then both N1 and N2 would be out
of his reach; thus trying to decode M2 in addition to M1 only interferes with his ability to decode
M1. If Bob wants to retrieve more than k qubits, he’ll just have to wait for the black hole to emit
more quanta.
We note that if the internal dynamics of the black hole is actually time-reversal invariant, then
it may be more appropriate to model the dynamics by choosing the matrix V B from either the
circular orthogonal or the circular symplectic ensemble, with the choice depending on the total spin
of the black hole [19]. Since time-reversal invariance is not a fundamental symmetry in Nature, we
think it is sensible to consider V B chosen uniformly with respect to the Haar measure on U(2n) as
described above. We have in any case verified that our conclusions would not be much affected were
we to choose V B from the circular orthogonal ensemble or circular symplectic ensemble instead [20].

8


4
The thermalization time

As for our classical model, there are some nontrivial assumptions underlying our model of a quantum
black hole. In particular, when we conclude (in the case where the black hole is maximally entangled
with previously emitted radiation) that Alice’s information “comes out fast,” we are taking it for
granted that the information deposited by Alice is rapidly thermalized by the black hole’s internal
dynamics.
By rapidly, we mean on a time scale comparable to the time interval between the
emission of successive radiation quanta. In terms of the “Schwarzschild time” measured by static
observers who are far from the black hole, this interval is of order the Schwarzschild radius rS for
a nonrotating uncharged black hole (in units where the speed of light is c = 1).
There are several reasons why rS might seem at first to be a reasonable estimate of the black
hole’s thermalization time [21]. Classically, perturbed black holes have damped quasinormal “ring-
ing” modes, where the damping time is comparable to rS. Thus, if one kicks a classical black hole,
rS is the time scale for the black hole to forget about the kick. Furthermore, under AdS/CFT
duality, the dual of a large black hole is a strongly-coupled conformal field theory with the same
temperature T ≈ 1/rS (in units with ¯h = c = 1).
In the field theory the only relevant time
scale governing the approach to thermal equilibrium is the time scale set by the temperature itself:
1/T ≈ rS. Thus, if one kicks the strongly-coupled thermal bath, rS is the time scale for the bath
to forget about the kick.
However, when one says that the strongly-coupled quantum field theory thermalizes in a time of
order 1/T, one ordinarily means that the system relaxes to a state that is locally indistinguishable
from a thermal state on this time scale. Our assertion that information captured by a black hole is
quickly reemitted in the Hawking radiation rests on the validity of a more stringent global criterion
for thermalization. How long does it take for the quantum information deposited by Alice to become
thoroughly mixed with the black hole’s microscopic degrees of freedom?
Since this is really a question about strongly-coupled quantum gravity, we don’t have the tools
to answer it definitively, but we can try to make a plausible guess. For this purpose, we envision
the black hole’s qubits as uniformly distributed over the “stretched horizon” [8, 9]. The stretched
horizon is located about one Planck unit of proper distance above the actual global horizon; here
static observers have a proper acceleration of order one in Planck units, and therefore are surrounded
by a bath of Unruh radiation with temperature of order one. (Quanta from this bath that are
directed nearly radially outward can escape to infinity as Hawking radiation, with a redshifted
temperature of order 1/rS.)
According to the hypothesis of “black hole complementarity” [8, 9], observers who remain
outside the black hole may legitimately regard the stretched horizon as a repository that stores
quantum information absorbed by the black hole, even though the strongly-coupled “Planckian
soup” is invisible to freely falling observers who pass through the horizon. Because of the high
temperature of the Planckian soup at the stretched horizon, we need a thorough grasp of quantum
gravity to understand its detailed dynamics.
The stretched horizon has some dissipative properties that can be analyzed classically [22]. For
example, if electric charge is deposited in the black hole, currents flow on the stretched horizon
to redistribute the charge, which are damped by the stretched horizon’s internal resistance. The
local charge density decays like exp(−t/rS), where t is the Schwarzschild time, and correspondingly
the proper area of a droplet of charge on the stretched horizon grows like exp(t/rS). The droplet
envelops the stretched horizon, and the charge distribution reaches a new steady state, after a
Schwarzschild time of order rS log rS. From the perspective of static observers hovering above the
stretched horizon, the spreading of charge is very fast, reaching a proper distance of order rS in a
time of order log rS as measured by their clocks; yet these observers are unable to manipulate the

9


flow of charge to achieve superluminal communication.
In its classical realization, the superluminal spreading of electric charge at the stretched horizon
is really a kinematic effect — it arises because the static observers in the Schwarzschild geome-
try separate with constant proper acceleration from freely falling charged particles (beneath the
stretched horizon) that are plunging toward the black hole’s global event horizon. One feels hesitant
about drawing any far-reaching dynamic conclusions based on this kinematic spreading. But on the
other hand a complete physical description of physics at the stretched horizon should accommodate
some kind of rapidly mixing dynamics, making it hard to distinguish between effects that arise from
strong local interactions and effects that are merely kinematic. Thus one is tempted to postulate
that the exponential spreading is exhibited not just by the black hole’s classical hair, but also by
its “quantum hair;” i.e., the quantum information that it encodes.
To the observers anchored at the stretched horizon, not just the local charge density but also
other local disturbances of the thermal bath decay like exp(−tstretch), where tstretch is the time in
Planck units as measured by static clocks at the stretched horizon. If we (somewhat fancifully)
envision “information” deposited in the black hole as a locally conserved fluid, the exp(−tstretch)
decay of the local information density suggests that the droplet of information, just like the droplet
of charge, expands exponentially and becomes nearly uniformly distributed over the stretched
horizon in a time tstretch = O(log rS), corresponding to Schwarzschild time t = O(rS log rS). This
picture leads us to suggest that the black hole’s global (Schwarzschild) thermalization time is
O(rS log rS).
In a sense this suggestion (also put forward in [9]) only deepens the information puzzle, as
it poses the challenge of reconciling rapid spreading of quantum information on the stretched
horizon with the black hole’s causal structure. On the other hand, rapid distribution of many-
particle quantum entanglement need not imply superluminal signaling. Furthermore, the black
hole complementarity viewpoint already postulates that the semiclassical causal structure of a
black hole must be highly misleading in some respects; the superluminal spreading is a relatively
gentle extension of a viewpoint we are already taking for granted.
If an observer who is a distance of order rS from the horizon drops a freely falling quantum
memory into the black hole, it takes Schwarzschild time of order rS log rS for the memory to reach
the stretched horizon, and conversely it takes Schwarzschild time of order rS log rS for quanta
emitted from the stretch horizon to reach the observer’s detector. Therefore, a thermalization time
of order rS log rS is compatible with our central metaphor — the black hole is a “mirror” in the
sense that it returns Alice’s information in a time comparable to the time it takes the information
to fall into the black hole.
If one, more conservatively, insists that the spreading “droplet of information” remains confined
to the forward light cone, then the time required for global thermalization is tstretch = Ω(rS),
corresponding to Schwarzschild time t = Ω(r2
S). (Here the “big-Omega” notation indicates a lower
bound on the thermalization time for asymptotically large rS, up to a multiplicative constant.) In
that event, since once global thermalization is achieved k qubits can escape in Schwarzschild time
t = O(krS), one may expect that the Schwarzschild time for a constant number of qubits to escape
from the black hole is t = O(rp
S logq rS), where 1 < p < 2. Here the exponents p and q are sensitive
to the details of the model of thermalization.
Of our two main conclusions in Sec. 3, the conclusion that Alice’s qubits bounce right back (if
the black hole has already radiated away more than half of its initial entropy) really does require
rapid thermalization, i.e., thermalization in Schwarzschild time t = O(rS log rS). But the other
conclusion, that previously absorbed quantum information is released quickly when the evaporation
reaches the half-way point, still applies even if thermalization takes much longer. For the second
conclusion, we only need thermalization to be fast compared to evaporation, which occurs in a

10


Schwarzschild time of order r3
S rather than rS log rS.

5
Efficiency

In Sec. 3, our analysis of the quantum model of a black hole was, as a computer scientist might
put it derisively, “merely information theoretic.” We in effect assumed without justification that
the unitary transformation arising from the internal dynamics of an n-qubit black hole is typical
with respect to the Haar measure on U(2n) (and we also ignored the complexity of Bob’s decoding
algorithm). The vague picture offered in Sec. 4, depicting thermalization as the rapid spreading of
a droplet, does not by itself provide much support for describing the black hole’s dynamics as a
random unitary transformation. Can we defend this assumption?
In accord with the black hole complementarity hypothesis, let us suppose that the internal
dynamics of the black hole is governed by a strongly mixing local Hamiltonian acting on n qubits
that are uniformly distributed over the black hole’s stretched horizon [8, 9]. Here n (up to the factor
1
4 log2 e) is the area of the horizon in Planck units, and the Schwarzschild radius is rS = O(√n) in
Planck units. Though we don’t understand quantum gravity well enough to analyze the internal
dynamics in detail, we might nevertheless hope to make a few cogent general observations.
We find it helpful to envision this dynamics as a local quantum circuit, where in each unit of
time two-qubit unitary transformations (“quantum gates”) are applied in parallel to about n/2
pairs of neighboring qubits. It seems natural to choose the unit of time to be the Planck time, but
according to whose clock? This is a subtle question, because, due to the gravitational redshift, the
clocks of static observers hovering above the stretched horizon run much slower (by a factor of order
rS in Planck units) than the clocks of observers residing far from the horizon. As already noted in
Sec. 4, there are physical processes on the stretched horizon (like the distribution of electric charge)
that can act over a proper distance of order rS in a Schwarzschild time of order rS. If we wish to
accommodate such processes in our circuit model, we may take the unit of time to be a Planck unit
of Schwarzschild time. This model of the local dynamics on the stretched horizon is very naive, and
in particular the model does not attempt to address the challenge of reconciling rapid spreading of
information with causality. In any case, no matter how we prefer to map circuit time to physical
time, it is useful to characterize the circuit complexity of global thermalization.
In our model, after a Schwarzschild time t, the unitary transformation acting on the black
hole’s internal state is realized by a circuit that has a number of time steps (the circuit’s “depth”)
of order t (where t is expressed in Planck units) and a total number of (local) quantum gates (the
circuit’s “size”) of order nt. Unfortunately, if t is a polynomial in n, then Haar-random unitary
transformations cannot be generated with circuits of this size. In fact, because the volume of U(2n)
is exponentially large in 2n, reaching a typical unitary transformation requires a circuit whose size
is exponential in n.
However, much smaller circuits can be good encoders for quantum error-correcting codes. In
particular, the capacity of the quantum erasure channel is achieved by typical stabilizer codes,
which have encoding circuits of size O(n2) [23]. Furthermore, good approximate encoding circuits
can be even smaller. If we replace the integral over all unitary transformations by an average over
the elements of the ǫ-approximate unitary 2-design K constructed in [6], then eq. (6) is modified to
become
1
|K|

�

k∈K

���σNB′(V B
k ) − σN(V B
k ) ⊗ σB′
max
���
2

1
≤ 2−2(s−k) + ǫ + O(2−n) .
(8)

Each element of K can be realized by a quantum circuit that has size O(n log(1/ǫ)) and depth
O(log n log(1/ǫ)) [6, 7]. It seems reasonable to expect that random quantum circuits built from

11


two-qubit gates would encode at least as efficiently as the approximate 2-designs that have been
explicitly constructed, and preliminary results from numerical experiments are consistent with this
expectation [24].
This estimate of the circuit size and depth does not take into account any geometric locality
constraints. If the n qubits are uniformly distributed on a two-dimensional sphere with radius of
order √n, then the additional cost of using local gates should be at worst a blowup in size and
depth by a factor of order √n, since the information stored in a qubit can propagate a distance
√n when √n local gates are applied in succession. Therefore, there exist good approximate lo-
cal encoding circuits for the quantum erasure channel that have size O(n3/2 log(1/ǫ)) and depth
O(n1/2 log n log(1/ǫ)). Again, we expect that random local circuits of this size and depth would do
at least as well.
Therefore, if we model the internal dynamics of the stretched horizon by a random local quantum
circuit, we conclude that the global thermalization of the black hole is achieved by a circuit whose
depth is O(rS log rS log(1/ǫ)).
By “global thermalization” we mean such that eq. (8) holds, so
that Bob can approximately decode Alice’s quantum memory with a loss of fidelity of order ǫ.
Modeling the internal dynamics of a physical system by a small random quantum circuit is far
more defensible than modeling it by a unitary chosen uniformly with respect to Haar measure,
because a small circuit can be realized by physically plausible local dynamics. It therefore seems
reasonable to suggest that the dynamics of a black hole could efficiently generate good quantum
error-correcting codes for the quantum erasure channel. Indeed, if each computational step occurs
in one Planck unit of Schwarzschild time, then the thermalization time t = O(rS log rS) scales with
rS as in our crude estimate in Sec. 4.
Decoding, however, can be a much harder computational problem than encoding. Black hole
evaporation maps Alice’s k qubits to a highly nonlocal subsystem M′ of RE.
Bob’s decoding
operation maps M′ back to a compact system ˆ
M localized in his laboratory, returning the qubits
to a usable form. Even if we are willing to grant Bob the power to collect all of the Hawking
radiation emitted by the black hole, Bob might not be able to recover Alice’s k-qubit message by
means of an efficient quantum computation (the resources needed to perform Bob’s computation
might grow faster than any power of k).
Actually, if a stabilizer code is used to protect entanglement-assisted quantum communication
through the erasure channel, then Bob’s decoding computation is easy. Bob can replace each erased
qubit by the standard state |0⟩, and then measure the code’s check operators. With high probability,
there is a unique Pauli operator acting on the erased qubits that restores Bob’s state to the code
space, and the recovery operation can be efficiently computed using linear algebra. However, this
recovery procedure exploits the special structure of stabilizer codes, and would not work for typical
non-stabilizer codes. If thermalization is rapid, then the quantum codes realized by evaporating
black holes have small encoding circuits and therefore they too are rather special, but they are
not stabilizer codes and we do not know whether they are efficiently decodable. Conceivably, the
decoding is hard not only for n large with k/n fixed, but also for n large with k fixed. If Bob’s
decoding problem is intractable, then the physicists of our highly advanced civilization may be hard
pressed to test the hypothesis that black holes reradiate quantum information quickly.
For that matter, one might question the testability of the claim that black holes really preserve
quantum information. We emphasize, though, that even if Bob’s decoding problem is hopelessly
intractable, the physicists of our highly advanced civilization, using only “polynomial resources,”
should be able to test the hypothesis that black holes process quantum information without destroy-
ing it. Once these physicists understand the laws of black hole dynamics well enough, they should
be able, using their quantum computers, to simulate the formation and complete evaporation of a
black hole. Through a “swap test” [25], they could compare the output of their quantum simulation

12


with the observed product of the black hole evaporation, and so verify that the dynamical black
hole is behaving as their theory predicts. The “feasibility” (in principle) of such a test boosts our
confidence that the assertion that black holes preserve information has a well formulated opera-
tional meaning. Unfortunately, the swap test works for comparing pure states, not mixed states;
it can’t be used to compare the simulation with experiment in the case of a partially evaporated
black hole.

6
Are black holes quantum cloners?

Our analysis has been premised on the assumption that black hole evaporation respects unitarity,
and we have adopted the black hole complementarity viewpoint to describe the process. We have
been led to the conclusion that a black hole (whose evaporation is past the half-way point) is really
a sort of “information mirror” that quickly returns the quantum information it receives. Now we
should reassess whether our assumptions are consistent.
We note that the geometry of the evaporating black hole contains spacelike surfaces that are
crossed both by Alice’s infalling quantum memory (inside the event horizon), and by the outgoing
Hawking radiation that Bob decodes (outside the horizon). Therefore, if Bob can decode success-
fully, then Alice’s quantum information has been “cloned” in the outgoing radiation. But cloning
of arbitrary quantum states is inconsistent with the linearity of quantum mechanics [26, 27]. We
seem to be stuck with a difficult choice: either Alice’s information is not reemitted, or the black
hole is a quantum cloning machine! Either way, the foundations of quantum theory need revision.
This appears to be a powerful argument in favor of information loss [28].
The hypothesis of black hole complementarity was proposed as a way to avoid this quandary
[8, 9]: one chooses not to be bothered by quantum cloning if it occurs where no one can ever find
out. According to this philosophy, we may accept for now that Alice (if she falls into the black hole)
and Bob (if he stays outside) have sharply contrasting descriptions of the same physical process.
Eventually we hope to be able to reconcile Alice’s and Bob’s contrasting viewpoints, but finding
that more complete global description must be postponed until we acquire a deeper understanding
of quantum gravity. In the meantime, we are entitled to insist that Alice’s and Bob’s descriptions
are both compatible with the standard principles of quantum mechanics.
Therefore we should ask: if Alice’s quantum state persists behind the horizon and that state
is also encoded in the outgoing Hawking radiation received by Bob, can Alice or Bob verify the
cloning? Once Alice falls through the horizon, she won’t be able to compare notes with Bob if he
stays outside the black hole. To have any hope of verifying the cloning, Bob needs to remain outside
until he has retrieved Alice’s qubits from the Hawking radiation, and then dive into the black hole
seeking confirmation that both he and Alice have high-fidelity copies of Alice’s original quantum
state. However, as long as the black hole retains Alice’s qubits long enough before reemitting them,
then we are unable to say (based on our current understanding of quantum gravity) whether Bob
will succeed or not [29, 30]. As best we can tell, then, the black hole complementarity hypothesis
seems to be self-consistent, provided the black hole retains quantum information for a sufficiently
long time.
To analyze this thought experiment, it is convenient to describe the Schwarzschild black hole
using the Kruskal null coordinates U, V as depicted in Fig. 2. These coordinates are related to the
Schwarzschild coordinates r, t by

U = −e(r∗−t)/2rS ,
V = e(r∗+t)/2rS ,
where
r∗ = r + rS ln[(r − rS)/rS] .
(9)

In terms of these coordinates, the black hole’s event horizon is at U = 0, and the curvature
singularity occurs at UV = 1. Therefore, if Bob crosses the horizon at V = VBob, then he reaches

13


horizon

singularity

U
V

Alice

Bob

message

radiation

Figure 2: Alice and Bob test whether an evaporating black hole clones quantum information.
Alice, carrying her quantum memory, drops into the black hole. Bob recovers the content of Alice’s
memory from the Hawking radiation, and then enters the black hole, too. Alice sends her qubits
to Bob, and Bob verifies that cloning has occurred.

the singularity at U ≤ V −1
Bob. But if Alice falls freely, the proper time she experiences between
crossing the horizon at V = VAlice and reaching U = V −1
Bob is

τAlice = CrS (VAlice/VBob) ,
(10)

where C is a numerical constant that depends on Alice’s initial data (C = 1/e ≈ .368 if Alice falls
from rest starting at r = ∞). In terms of Schwarzschild time, Bob’s fall into the black hole is
delayed relative to Alice’s by ∆t, where VBob/VAlice = e∆t/2rS, and therefore

τAlice = CrS exp(−∆t/2rS) .
(11)

Thus Alice’s proper time τAlice is of order the Planck time or shorter if Bob’s entry into the black
hole is delayed by ∆t = rS log rS or longer. The conclusion does not change substantially even if
Alice rides a rocket, as long as her proper acceleration is small in Planck units.
If Alice, after crossing the horizon, has less than a Planck time to communicate with Bob about
the status of her qubits, then she is required to send her message to Bob using super-Planckian
frequencies. And with our incomplete understanding of strongly-coupled quantum gravity, we lack
the tools to analyze the emission, transmission, and reception of such signals. On the other hand,
if Alice’s proper time were long compared to the Planck time, then in principle she could send her
qubits to Bob (or send a record of her measurement outcomes), allowing Bob to verify the cloning,
and the verification could be analyzed using controlled semiclassical approximations. We conclude
that, if verifiable cloning does not occur, then the black hole must retain Alice’s information for a
Schwarzschild time interval ∆t = Ω(rS log rS) [29, 30]. (The “big-Omega” notation indicates that
rS log rS is a lower bound on the information retention time for asymptotically large rS, up to a

14


multiplicative constant.) This finding is just barely compatible with the information retention time
estimated in Sec. 4 and Sec. 5.
We noted in Sec. 4 that it already takes Schwarzschild time O(rS log rS) for the Hawking
radiation to climb from the stretched horizon to Bob’s detectors, if Bob waits at a position that
is a proper distance O(rS) from the horizon. In principle, though, Bob’s “decoding sphere” could
be located at a height where the black hole’s thermal atmosphere has a constant temperature
(independent of rS), At this height, Bob’s clock runs slower than Schwarzschild time by a factor
O(rS), and the outgoing radiation propagates from the stretched horizon to Bob in O(1) time.
From the requirement that cloning is unverifiable, we infer that by Bob’s clock it should take time
Ω(log rS) for Alice’s quantum information to reappear in the thermal atmosphere. This version of
the “no-cloning” argument usefully differentiates between the time delay due to the climb from the
stretched horizon and the time delay due to the thermalization of the stretched horizon.
A serious weakness in the “no-cloning” argument is that we have assumed that, once Alice’s
information is received by Bob, he can decode the information instantaneously. In fact, as empha-
sized in Sec. 5, Bob’s decoding map is a complex quantum operation acting on O(r2
S) qubits. One
could plausibly argue that fundamental laws of physics prevent Bob from performing the decoding
operation fast enough for cloning to be verified behind the horizon, no matter how quickly Alice’s
qubits become encoded in the Hawking radiation. Nevertheless, we find it intriguing that our esti-
mate of the information retention time in Sec. 5 is of the same order as the “no-cloning” limit that
can be inferred if we neglect Bob’s decoding time.

7
Conclusions

The purpose of this article is largely pedagogical. On the one hand, many physicists who are inter-
ested in the quantum behavior of black holes are familiar with Don Page’s observations concerning
the entanglement entropy of an evaporating black hole [12]. On the other hand, most quantum
information theorists know about the quantum capacity of the quantum erasure channel [5, 18].
Here we point out that by invoking the latter we can refine and extend the former. We have tried
to convey our main points without relying too much on information-theoretic jargon, in the hope
of reaching a broad audience among those who are intrigued by these issues.
The essential tool that allows us to go further than Page is the theory of quantum error correc-
tion, which was developed after [12] was written. In fact there is a delicious irony at the core of our
analysis. In Sec. 3, we modeled the internal dynamics of the black hole with a randomly selected
unitary transformation, presuming that black holes, though they do not destroy information, are
nearly optimal thermalizers — they rapidly convert information to a form that is very difficult to
access in practice. However, a random unitary is also a nearly optimal encoder that achieves the
quantum capacity for the quantum erasure channel. The dynamics that conceals quantum infor-
mation effectively in practice, by encoding it in a highly nonlocal manner, also at the same time
protects the information from damage as a matter of principle.
It is not fully realistic to model the internal dynamics with a random unitary transformation,
because typical unitary transformations can be generated only by quantum circuits of exponential
size. However, in Sec. 5 we invoked recent results concerning efficient approximate encoding cir-
cuits for quantum error-correcting codes [6, 7], indicating that plausible local dynamics really can
thermalize quantum information quickly. We argued in Sec. 6 that our estimate of a black hole’s
information retention time is just barely compatible with the principle that black hole evaporation
must not realize verifiable quantum cloning.
Finally, what, if anything, are the broader implications of our observations for the black hole

15


information puzzle? We’re not sure, but we find it intriguing that the encoder for a good quantum
error-correcting code need not necessarily be unitary. Perhaps, then, there could be an interesting
middle ground between unitary black hole dynamics and full-blown information loss — even if some
of the information deposited in a black hole remains inaccessible forever, quantum information
encoded in small subsystems might escape in the Hawking radiation with relatively little damage.
One possible model is to suppose that at any given time the black hole system B′ actually
consists of two parts, the “accessible” system Bacc, for which we might suppose that log |Bacc| is
the Bekenstein-Hawking entropy, and an inaccessible system Bgone which has been lost forever. For
the information to be revealed to Bob, we require that correlations are destroyed not just between
the reference system N and Bacc, but rather between N and B′ = BaccBgone. As the radiation
leaks out, Bacc shrinks, but Bgone may grow, and there is a competition between these two effects.
If R is radiated away, then the effective dimension of the discarded system is not |R| but rather
|R|/|Bgone|; Alice’s quantum information is revealed when this effective dimension becomes larger
than |N|. Quantum information encoded in a sufficiently small subsystem eventually escapes if the
qubits are emitted in the radiation faster than they are destroyed in the black hole’s interior, but
the rate of escape is suppressed to a degree that depends on the rate of destruction.
Since information loss, once allowed, tends to be highly infectious, it is difficult to formulate
deformations of quantum mechanics that incorporate a small amount of information loss, yet are
compatible with low-energy experimental constraints [31]. In the scheme we have just outlined,
when a pure quantum state undergoes gravitational collapse to form a black hole and then evapo-
rates completely, some of the information encoded in the initial quantum state really is destroyed,
but the information encoded in any sufficiently small subsystem survives nearly unscathed. It might
be fruitful to investigate how easily this type of partial information loss can be reconciled with the
experimental successes of quantum mechanics.

Acknowledgments

We are grateful for the hospitality of the Perimeter Institute, where we had the good fortune
to share an office, and JP thanks PH for letting him use the comfortable chair. We also thank
Ashton Anderson, Hilary Carteret, Daniel Gottesman, Dennis Kretschmann, Seth Lloyd, Prakash
Panangaden, David Poulin, Renato Renner, Lenny Susskind, Kip Thorne, Bill Unruh, Andreas
Winter, Jon Yard, and the participants in the 2007 McGill-Bellairs Quantum Information Workshop
for helpful suggestions. This research is supported in part by the Canada Research Chairs program,
the Sloan Foundation, CIFAR, FQRNT, MITACS, NSERC, DoE under Grant No. DE-FG03-92-
ER40701, NSF under Grant No. PHY-0456720, and NSA under ARO Contract No. W911NF-05-
1-0294.

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