Information, information processing and gravity
Information, information processing and gravity
Stephen D. H. Hsu∗
Institute of Theoretical Science
University of Oregon, Eugene, OR 97403
I discuss fundamental limits placed on information and information processing by gravity. Such
limits arise because both information and its processing require energy, while gravitational collapse
(formation of a horizon or black hole) restricts the amount of energy allowed in a finite region.
Specifically, I use a criterion for gravitational collapse called the hoop conjecture. Once the hoop
conjecture is assumed a number of results can be obtained directly: the existence of a fundamental
uncertainty in spatial distance of order the Planck length, bounds on information (entropy) in a
finite region, and a bound on the rate of information processing in a finite region. In the final section
I discuss some cosmological issues, related to the total amount of information in the universe, and
note that almost all detailed aspects of the late universe are determined by the randomness of
quantum outcomes. This paper is based on a talk presented at a 2007 Bellairs Research Institute
(McGill University) workshop on black holes and quantum information.
I. INTRODUCTION
This paper is based on a talk presented at a workshop
on black holes and quantum information (Bellairs Re-
search Institute of McGill University, Barbados). Most
of the participants were quantum information theorists,
so I attempted to keep the technical details concerning
general relativity or particle physics at a minimum. I
tried to summarize, in the most physical and intuitive
way, how gravity enforces some surprising constraints on
information and information processing. From a practi-
cal perspective, due to the feebleness of the gravitational
force, all of the limits deduced are incredibly weak. Our
technologies are nowhere near saturating them, and they
are of much greater interest to theoreticians than exper-
imentalists or engineers. Nevertheless, they are funda-
mental in nature, and, depending as they do both on
quantum mechanics and general relativity, may offer a
view into the properties of quantum gravity.
In the discussion that follows, gravitational collapse
will be our crude but powerful probe of gravitational
physics. Complete gravitational collapse leads to the for-
mation of black holes and causal horizons. Gravity is
a long range force that, as far as we know, cannot be
screened. In this respect, it is fundamentally different
from gauge forces, such as the strong and electroweak in-
teractions, and it is precisely this difference that allows
for dramatic phenomena like complete collapse.
We use Planck units throughout, in which the speed
of light, Planck’s constant and the Planck mass (equiva-
lently, Newton’s constant) are unity. In our expressions,
any energy or mass is therefore measured in units of 1019
GeV (about 10−5 grams), and any length is measured in
units of the Planck length, or about 10−35 meters.
∗Electronic address: hsu@uoregon.edu
II. GRAVITATIONAL COLLAPSE
Ideally, one would like to know precisely what subset
of all possible physical initial data results in gravitational
collapse and the formation of a black hole. This is ob-
viously a difficult problem and it is currently unsolved.
Schoen and Yau [1] proved a celebrated result requiring
the existence of a closed trapped surface if the minimum
density in a region is sufficiently high. However, this
result fails to be useful if the energy of the initial config-
uration is distributed in a very nonuniform manner.
Note that results for black hole or horizon formation
typically require both the assumption of the null or weak
energy condition and of cosmic censorship [2]. Under
those assumptions a closed trapped surface can be shown
to result in a singularity (using the Raychaudhuri equa-
tion and assuming the energy conditions hold), and cos-
mic censorship requires a horizon to conceal the singu-
larity from asymptotic observers.
In our analysis we will use the hoop conjecture, due to
Kip Thorne [3] as a criterion for gravitational collapse.
It states that a system of total energy E, if confined to a
sphere of radius R < ηE (η is a coefficient of order one,
which we neglect below), must inevitably evolve into a
black hole. The condition R < E is readily motivated
by the Schwarzschild radius Rs = 2M . This conjecture
has been confirmed in astrophysically-motivated numer-
ical simulations, and has been placed on even stronger
footing by recent results on black hole formation from rel-
ativistic particle collisions [4]. These results show that,
even in the case when all of the energy E is provided
by the kinetic energy of two highly boosted particles, a
black hole forms whenever the particles pass within a
distance of order E of each other (see Fig. 1). Two par-
ticle collisions had seemed the most likely to provide a
counterexample to the conjecture, since the considerable
kinetic energy of each particle might have allowed escape
from gravitational collapse.
One can think of the hoop conjecture as requiring that
the average energy density of an object of size R be less
http://arxiv.org/abs/0704.1154v1
mailto:hsu@uoregon.edu
FIG. 1: The hoop conjecture applied to two relativistic par-
ticles.
than R−2 in order not to collapse to a black hole. Thus,
large objects which are not black holes must be less and
less dense. For example, a sufficiently large object with
only the density of water will eventually form a black
hole!
III. MINIMAL LENGTH
In this section we deduce a fundamental limit on our
ability to measure a distance [5, 6, 7, 8]. The results
suggest that spacetime may ultimately have a discrete
structure. At the end of the section we discuss the impli-
cations for quantum information and the ultimate Hilbert
space of quantum mechanics.
From the hoop conjecture (HC) and the uncertainty
principle, we immediately deduce the existence of a min-
imum ball of size lP . Consider a particle of energy E
which is not already a black hole. Its size r must satisfy
r ∼>max [ 1/E , E ] , (1)
where λC ∼ 1/E is its Compton wavelength and E arises
from the hoop conjecture. Minimization with respect to
E results in r of order unity in Planck units, or r ∼ lP
[9]. If the particle is a black hole, then its radius grows
with mass: r ∼ E ∼ 1/λC . This relationship suggests
that an experiment designed (in the absence of gravity)
to measure a short distance l << lP will (in the presence
of gravity) only be sensitive to distances 1/l. This is the
classical counterpart to T-duality in string theory [10].
It is possible that quantum gravitational corrections
modify the relation between E and R in the HC. How-
ever, if E is much larger than the Planck mass, and R
much larger than lP , we expect semiclassical considera-
tions to be reliable. (Indeed, in two particle collisions
with center of mass energy much larger than the Planck
mass the black holes produced are semiclassical.) This
means that the existence of a minimum ball of size much
smaller than lP does not depend on quantum gravity -
the energy required to confine a particle to a region of
size much smaller than lP would produce a large, semi-
classical black hole.
Before proceeding further, we give a concrete model
of minimum length that will be useful later. Let the
position operator x̂ have discrete eigenvalues {xi}, with
the separation between eigenvalues either of order lP or
smaller. (For regularly distributed eigenvalues with a
constant separation, this would be equivalent to a spa-
tial lattice.) We do not mean to imply that nature im-
plements minimum length in this particular fashion -
most likely, the physical mechanism is more complicated,
and may involve, for example, spacetime foam or strings.
However, our concrete formulation lends itself to detailed
analysis. We show below that this formulation cannot be
excluded by any gedanken experiment, which is strong
evidence for the existence of a minimum length.
Quantization of position does not by itself imply quan-
tization of momentum. Conversely, a continuous spec-
trum of momentum does not imply a continuous spec-
trum of position. In a formulation of quantum mechanics
on a regular spatial lattice, with spacing a and size L, the
momentum operator has eigenvalues which are spaced by
1/L. In the infinite volume limit the momentum operator
can have continuous eigenvalues even if the spatial lattice
spacing is kept fixed. This means that the displacement
operator
x̂(t)− x̂(0) = p̂(0) t
does not necessarily have discrete eigenvalues (the right
hand side of (2) assumes free evolution; we use the
Heisenberg picture throughout). Since the time evolution
operator is unitary the eigenvalues of x̂(t) are the same
as x̂(0). Importantly though, the spectrum of x̂(0) (or
x̂(t)) is completely unrelated to the spectrum of the p̂(0),
even though they are related by (2) [11]. Consequently,
we stress that a measurement of the displacement is a
measurement of the spectrum of p̂(0) (for free evolution)
and does not provide information on the spectrum of x̂.
A measurement of arbitrarily small displacement (2) does
not exclude our model of minimum length. To exclude it,
one would have to measure a position eigenvalue x and a
nearby eigenvalue x′, with |x− x′| << lP .
Many minimum length arguments (involving, e.g., a
microscope, scattering experiment or even Wigner’s clock
[6]) are obviated by the simple observation of the mini-
mum ball. However, the existence of a minimum ball does
not by itself preclude the localization of a macroscopic
object to very high precision. Hence, one might attempt
to measure the spectrum of x̂(0) through a time of flight
experiment in which wavepackets of primitive probes are
bounced off of well-localised macroscopic objects. Dis-
regarding gravitational effects, the discrete spectrum of
x̂(0) is in principle obtainable this way. But, detecting
the discreteness of x̂(0) requires wavelengths comparable
to the eigenvalue spacing. For eigenvalue spacing com-
parable or smaller than lP , gravitational effects cannot
be ignored, because the process produces minimal balls
(black holes) of size lP or larger. This suggests a direct
measurement of the position spectrum to accuracy better
than lP is not possible. The failure here is due to the use
of probes with very short wavelength.
A different class of instrument - the interferometer
FIG. 2: An interferometer can be sensitive to path length
differences much smaller than the wavelength of light used.
(Fig. 3) - is capable of measuring distances much smaller
than the size of any of its sub-components [12]. An in-
terferometer can measure a distance
∆x ∼ λ
, (3)
where λ = 1/ν is the wavelength of light used, L is the
length of each arm, τ the time duration of the measure-
ment, and N the number of photons. More precisely, ∆x
is the change over the duration of the measurement in
the relative path lengths of the two arms of the interfer-
ometer. b = τ/L is the number of bounces over which
the phase difference builds, so (3) can also be written as
, (4)
which expresses saturation of the quantum mechanical
uncertainty relationship between the phase and number
operators of a coherent state.
From (3) it appears that ∆x can be made arbitrarily
small relative to λ by, e.g., taking the number of bounces
to infinity. Were this the case, we would have an exper-
iment that, while still using a wavelength λ much larger
than lP , could measure a distance less than lP along one
direction, albeit at the cost of making the measured ob-
ject (e.g., a gravity wave) large in the time direction.
This would contradict the existence of a minimum inter-
val, though not a minimum ball in spacetime. (Another
limit which increases the accuracy of the interferometer
is to take the number of photons N to infinity, but this is
more directly constrained by gravitational collapse. Ei-
ther limit is ultimately bounded by the argument dis-
cussed below.)
A constraint which prevents an arbitrarily accurate
measurement of ∆x by an interferometer arises due to
the Standard Quantum Limit (SQL) and gravitational
collapse. The SQL [13] is derived from the uncertainty
principle (we give the derivation below; it is not specific
to interferometers, although see [14]) and requires that
, (5)
where t is the time over which the measurement occurs
and M the mass of the object whose position is mea-
sured. In order to push ∆x below lP , we take b and t to
be large. But from (5) this requires that M be large as
well. In order to avoid gravitational collapse, the size R
of our measuring device must also grow such thatR > M .
However, by causality R cannot exceed t. Any compo-
nent of the device a distance greater than t away cannot
affect the measurement, hence we should not consider it
part of the device. These considerations can be summa-
rized in the inequalities
t > R > M . (6)
Combined with the SQL (5), they require ∆x > 1 in
Planck units, or
∆x > lP . (7)
(Again, we neglect factors of order one.)
Notice that the considerations leading to (5), (6) and
(7) were in no way specific to an interferometer, and
hence are device independent. We repeat: no device sub-
ject to the SQL, gravity and causality can exclude the
quantization of position on distances less than the Planck
length.
It is important to emphasize that we are deducing a
minimum length which is parametrically of order lP , but
may be larger or smaller by a numerical factor. This
point is relevant to the question of whether an experi-
menter might be able to transmit the result of the mea-
surement before the formation of a closed trapped sur-
face, which prevents the escape of any signal. If we de-
crease the minimum length by a numerical factor, the
inequality (5) requires M >> R, so we force the experi-
menter to work from deep inside an apparatus which has
far exceeded the criterion for gravitational collapse (i.e.,
it is much denser than a black hole of the same size R
as the apparatus). For such an apparatus a horizon will
already exist before the measurement begins. The ra-
dius of the horizon, which is of order M , is very large
compared to R, so that no signal can escape.
We now give the derivation of the Standard Quantum
Limit. Consider the Heisenberg operators for position
x̂(t) and momentum p̂(t) and recall the standard inequal-
(∆A)2(∆B)2 ≥ −
(〈[Â, B̂]〉)2 . (8)
Suppose that the position of a free test mass is measured
at time t = 0 and again at a later time. The position
operator at a later time t is
x̂(t) = x̂(0) + p̂(0)
. (9)
The commutator between the position operators at t = 0
and t is
[x̂(0), x̂(t)] = i
, (10)
so using (8) we have
|∆x(0)||∆x(t)| ≥ t
. (11)
So, at least one of the uncertainties ∆x(0) or ∆x(t) must
be larger than of order
t/M . As a measurement of the
discreteness of x̂(0) requires two position measurements,
it is limited by the greater of ∆x(0) or ∆x(t), that is, by
t/M .
What are the consequences of a minimum length? In a
discrete spacetime there need not be any continuous de-
grees of freedom, and the number of degrees of freedom in
a fixed volume is finite. Further, one can show that dis-
cretization of spacetime naturally suggests discretization
of Hilbert space itself [15]. Specifically, in a universe with
a minimal length (for example, due to quantum gravity),
no experiment can exclude the possibility that Hilbert
space is discrete.
IV. ENTROPY BOUNDS
In this section we describe two entropy bounds aris-
ing from gravitational collapse. These bounds limit the
number of degrees of freedom in a region of size R, or
equivalently the amount of information in any system of
fixed size.
The first bound uses the area–entropy relation for
black holes. Black holes radiate [16] and have entropy:
S = A/4 [17]. The nature of this entropy is one of the
great mysteries of modern physics, especially due to its
non-extensive nature: it scales as the area of the black
hole (in Planck units), rather than its volume. This pecu-
liar property has led to the holographic conjecture [18, 19]
proposing that the number of degrees of freedom in any
region of our universe grows only as the area of its bound-
ary. (See [20] for a review and discussion of covariant gen-
eralizations of this idea, and [21] for a general discussion
of how area bounds arise in gravitating systems.) The
AdS/CFT correspondence [22] is an explicit realization
of holography.
The entropy of a thermodynamic system is the loga-
rithm of the number of the available microstates of the
system, subject to some macroscopic constraints such as
fixed total energy. In certain string theory black holes,
these states have been counted explicitly [23, 24].
Consider a system of size R and total energy E (e.g.,
the green blob in Fig. 3), which is not a black hole
(E < R). Now imagine a spherical shell of energy R−E
approaching the system at the speed of light. By causal-
ity, the system is unaffected by the shell until the com-
bination of the two already satisfy the hoop conjecture.
The combined system must evolve into a black hole with
entropy A/4. By the second law of thermodynamics,
this final entropy is larger than that of the initial sys-
tem. Since we made no particular assumptions about the
initial system, we deduce that ordinary (non-collapsed)
physical systems have entropy less than their surface area
FIG. 3: A system collapses to form a black hole. By the
second law of thermodynamics, its original entropy is bounded
above by that of the black hole.
in Planck units. This is quite a counterintuitive result,
since in familiar (non-gravitating) systems entropy is typ-
ically extensive.
The second bound, obtained by ’t Hooft [25], shows
that if one excludes states from the Hilbert space whose
energies are so large that they would have already caused
gravitational collapse, one obtains S = lnN < A3/4,
where N is the number of degrees of freedom and A the
surface area. To deduce this result, ’t Hooft replaces the
system under study with a thermal one. This is justified
because, in the large volume limit, the entropy of a sys-
tem with constant total energy E (i.e., the logarithm of
the phase space volume of a microcanonical ensemble) is
given to high accuracy by that of a canonical ensemble
whose temperature has been adjusted so that the average
energies of the two ensembles are the same. (This is a
standard, and central, result in statistical mechanics.)
Given a thermal region of radius R and temperature
T , we have S ∼ T 3R3 and E ∼ T 4R3. Requiring E < R
then implies T ∼ R−1/2 and S < R3/2 ∼ A3/4. We stress
that the thermal replacement is just a calculational trick:
temperature plays no role in the results, which can also
be obtained by direct counting.
In [26], it was shown that imposing the condition
Tr[ ρH ] < R on a density matrix ρ implies a simi-
lar bound SvN < A
3/4 on the von Neumann entropy
SvN = Tr ρ ln ρ. For ρ a pure state the result reduces
to the previous Hilbert space counting.
We note that these bounds are more restrictive than
the bound obtained from black hole entropy: S < A/4.
One can interpret this discrepancy as a consequence of
higher entropy density of gravitational degrees of freedom
relative to ordinary matter [27].
The consequences of these bounds are rather striking:
they suggest that gravitating systems in d dimensions
contain only as much information as analogous, but non-
gravitating, systems in d− 1 dimensions. A concrete re-
alization of this is the AdS/CFT duality in string theory
[22].
V. BOUND ON RATE OF INFORMATION
PROCESSING
In this section we derive an upper bound on the rate
at which a device can process information [28]. We de-
fine this rate as the number of logical operations per unit
time, denoted as the ops rate R. The operations in ques-
tion can be those of either classical or quantum comput-
ers. The basis of our result can be stated very simply:
information processing requires energy, and general rel-
ativity limits the energy density of any object that does
not collapse to a black hole. Replacing information pro-
cessing by information in the previous sentence leads to
holography or black hole entropy bounds, a connection
we will explore further below. For related work on funda-
mental physical limits to computation, see [29] and [30].
Our result is easily deduced using the Margolus–
Levitin (ML) theorem [31] from quantum mechanics, and
the hoop conjecture.
The Margolus–Levitin theorem states that a quantum
system with average energy ǫ requires at least ∆t > ǫ−1
to evolve into an orthogonal (distinguishable) state. It
is easy to provide a heuristic justification of this result.
For an energy eigenstate of energy E, E−1 is the time
required for its phase to change by order one. In a two
state system the energy level splitting E is at most of or-
der the average energy of the two levels. Then, the usual
energy-time uncertainty principle suggests that the sys-
tem cannot be made to undergo a controlled quantum
jump on timescales much less than E−1, as this would
introduce energy larger than the splitting into the sys-
Consider a device of size R and volume V ∼ R3, com-
prised of n individual components [32] of average energy
ǫ. Then, the ML theorem gives an upper bound on the
total number of operations per unit time
R < nǫ , (12)
while the hoop conjecture requires E ∼ nǫ < R. Com-
bined, we obtain
R < R ∼ V 1/3 . (13)
It is interesting to compare this bound to the rate of
information processing performed by nature in the evo-
lution of physical systems. At first glance, there appears
to be a problem since one typically assumes the num-
ber of degrees of freedom in a region is proportional to
V (is extensive). Then, the amount of information pro-
cessing necessary to evolve such a system in time grows
much faster than our bound (13) as V increases. Re-
call that for n degrees of freedom (for simplicity, qubits),
the dimension of Hilbert space H is N = dimH = 2n
and the entropy is S = lnN ∼ n. In the extensive case,
n ∼ S ∼ V .
However, as noted in the previous section, gravity also
constrains the maximum information content (entropy S)
of a region of space. ’t Hooft [25] showed that if one ex-
cludes states from the Hilbert space whose energies are so
large that they would have already caused gravitational
collapse, one obtains S = lnN < A3/4, where N is the
number of degrees of freedom and A the surface area.
Given this result we can calculate the maximum rate of
information processing necessary to simulate any physical
system of volume V which is not a black hole. The rateR
FIG. 4: The visible universe is exponentially larger than the
initial region from which it evolved.
is given by the number of degrees of freedom S ∼ R3/2
times the maximal ML rate T ∼ R−1/2. This yields
R ∼ R as in our bound (13).
Finally, we note that black holes themselves appear to
saturate our bound. If we take the black hole entropy to
be S ∼ A ∼ R2, and the typical energy of its modes to
be the Hawking temperature TH ∼ R−1, we again obtain
R ∼ R.
VI. HOW MUCH INFORMATION IN THE
UNIVERSE?
In this final section we ask how much information is
necessary to specify the current state of the universe,
and where did it come from?
There is convincing observational evidence for the
big bang model of cosmology, and specifically for the
fact that the universe is and has been expanding. In
a radiation-dominated universe, the FRW scale factor
grows as R(t) ∼ t1/2, where t is the comoving cosmolog-
ical time. From this, it is clear that our universe evolved
from a much smaller volume at early times. Indeed, in
inflationary cosmology (Fig. 4) the visible universe re-
sults from an initial patch which is exponentially smaller
than our current horizon volume. The corresponding ra-
tio of entropies is similarly gigantic, meaning that there is
much more information in the universe today than in the
small primordial patch from which it originated. There-
fore, the set of possible early universe initial conditions
is much, much smaller than the set of possible late time
universes. A mapping between all the detailed rearrange-
ments or modifications of the universe today and the set
of possible initial data is many to one, not one to one
[33].
Thus, the richness and variability of the universe we
inhabit cannot be attributed to the range of initial con-
ditions. The fact that I am typing this on a sunny day,
or that our planet has a single moon, or that the books
on my office shelves have their current arrangement, was
not determined by big bang initial data.
How, then, do the richness and variability of our world
arise? The answer is quantum randomness – the random-
ness inherent in measurements of quantum outcomes.
Imagine an ensemble Ψ of n qubits, each prepared in
an identical state ψ. Now imagine that each qubit is
measured, with a resulting spin up (+) or spin down
(−) result. There are 2n possible records, or histories,
of this measurement. This is an exponentially large set
of outcomes; among them are all possible n-bit strings,
including every n-bit work of literature it is possible to
write! Although the initial state Ψ contained very little
information (essentially, only a single qubit of informa-
tion, since each spin is in an identical state), n bits of
classical information are required to specify which of the
2n outcomes is observed in a particular universe. For
n → ∞ the set of possible records is arbitrarily rich and
varied despite the simplicity of initial state Ψ.
In the same way, given an initial quantum state Ψ de-
scribing the primordial patch of the big bang from which
our horizon volume evolved, one must still know the out-
comes of a large number of quantum measurements in
order to specify the particulars of the universe today.
From a many worlds perspective, one must specify all
the decoherent outcomes to indicate a particular branch
of the wavefunction – a staggering amount of information.
Equivalently, from the traditional Copenhagen perspec-
tive, each quantum measurement injects a bit (or more)
of truly random information into our universe, and this
randomness accounts for its variability.
The most familiar cosmological quantum randomness
comes from fluctuations of the inflaton field, which deter-
mine the spectrum of primordial energy density fluctua-
tions. It is these density fluctuations that determine the
locations of galaxies, stars and planets today. However,
from entropic or information theoretic considerations we
readily deduce that essentially every detailed aspect of
our universe (beyond the fundamental Lagrangian and
some general features of our spacetime and its contents)
is a consequence of quantum fluctuations!
Acknowledgements— The author thanks the partici-
pants of the Bellairs workshop, and especially the or-
ganizer Patrick Hayden, for a stimulating and pleasant
environment. A. Zee provided some important comments
on an earlier draft of the manuscript. The author is sup-
ported by the Department of Energy under DE-FG02-
96ER40969.
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