# How are Entanglement Entropies Related to Entropy Bounds?

Emily Adlam Philosophy Department and Institute for Quantum Studies, Chapman
University, Orange, CA92866, USA<EMAIL_ADDRESS>
##  1 Introduction

As physically embodied observers, we are subject to various kinds of epistemic
limitations, which inevitably have an effect on the way in which we formulate
our physical theories. One particularly important kind of epistemic limitation
results from the locality of interactions, which means that, in a four-
dimensional spacetime, surfaces of two spatial dimensions have a special role
as interfaces through which observers access physical systems and regions of
space. In particular, this means that if there are bounds on the amount of
information that can be stored on a surface, there are also bounds on the rate
at which we can extract information through those surfaces about the systems
we are trying to study, and it is likely that such bounds will have
consequences for the physics that we arrive at on the basis of our
observations.

In ref [1] it was argued that this feature of our situation as physically
embodied observers leads to a possible epistemic interpretation of the
covariant entropy bound: rather than counting the true number of ontological
degrees of freedom inside a region of spacetime, the bound could codify an
epistemic limitation pertaining to the amount of information we can extract
from a region of spacetime. This raises some interesting questions about how
to disentangle the ontological and epistemic features of the bound and related
phenomena.

Ref [1] largely treated entropy in a classical way, as a measure of the number
of degrees of freedom inside a system or the total amount of information that
an observer can get about a system. But of course, our world is really
quantum-mechanical in nature and thus ultimately we should be considering
quantum von Neumann entropies. This makes the issues discussed in ref [1] more
complex, because in addition to measuring classical uncertainty, the von
Neumann entropy can also measure the entanglement that a system shares with
other systems. Thus in this paper, we set out to understand how taking
entanglement entropy into account affects the interpretation of the covariant
bound. In particular, we consider whether interpreting the bound in terms of
entanglement entropies may offer novel ways to explain it.

We begin by discussing how the von Neumann entropy fits into the picture,
arguing that just as in the classical case, a universal bound on the von
Neumann entropy could have either an epistemic or ontological origin. We then
note that if the entropies relevant to the covariant bound are entanglement
entropies, this opens up new possibilities for explaining the bound as a
consequence of features of the entanglement entropy; we discuss several
approaches that one might take to do this. One possibility is to see the bound
as a consequence of some kind of entanglement area law; there are some
obstacles to be overcome before this can be achieved, particularly with
respect to the interpretation of entanglement entropies of null surfaces, but
this may be a fruitful way of proceeding, and in particular may help address
some questions left unresolved by ref [1] about the relationship between
entropy bounds and temporal extension. We also discuss the ‘spacetime from
entanglement’ programme, arguing that entanglement alone may not be able to
full ground spacetime topology, but it could potentially play a role in
determining the spacetime metric, in which case it would certainly be relevant
to explaining the entropy bounds, and this would potentially support a more
ontological reading of the bound.

##  2 Background: Entropy Bounds

Entropy bounds have their origins in an argument due to Bekenstein[2] and
Hawking[3] suggesting that if the second law of thermodynamics is to be obeyed
in the vicinity of black holes, we must generalize it to include an entropy
associated with black holes equal to $\frac{A}{4G\hbar}$, where $A$ is the
surface area of the black hole. Subsequently discussion on this topic
eventually inspired a conjecture known as the ‘strong entropy bound’ which
says that the maximum entropy which can be contained in any region of space is
upper bounded by $\frac{A}{4G\hbar}$, where $A$ is the area of any surface
bounding the region. The strong bound appears to be correct in many physically
relevant situations[4], and it is part of the inspiration for a flourishing
research field aiming to understand quantum gravity in terms of
‘holography’[5, 6, 7].

However, the strong entropy bound is not universally true; it is violated in a
number of special cases, some of which, such as inflation and gravitational
collapse, are believed to occur in our actual universe. Thus an alternative
known as the ‘covariant entropy bound’ (or the ‘Bousso’ bound) has been
proposed[8]: instead of bounding the information inside a volume enclosed by
some surface, this version instead bounds the information on a ‘light-sheet’
associated with the relevant surface. A light-sheet is defined as a set of
null geodesics leaving the surface orthogonally such that the expansion of the
set in the direction going away from the surface is zero or negative, i.e. the
geodesics are remaining parallel or coming closer together as we move away
from the surface. The light-sheet continues up until the geodesics intersect
at a ‘caustic’ (i.e. crossing-point) or encounter a singularity of spacetime.
Bousso’s bound then simply says that the entropy on the light-sheet associated
with a surface of area $A$ is upper bounded by $\frac{A}{4G\hbar}$. The
covariant bound is currently thought to be true everywhere in our universe, at
least at the semi-classical level. There also exists a generalized version of
the bound[9] which suggests that the entropy on a partial light sheet bounded
on one side by a surface of area $A$ and on the other by a surface of area
$A^{\prime}$ is bounded by $\frac{A-A^{\prime}}{4G\hbar}$, but this bound is
not universally true, although it has been proven to hold in many physically
relevant scenarios[10].

Moving to the quantum regime, there have been worries that the covariant bound
would cease to hold in quantum regimes due to the possibility that quantum
fluctuations may violate the null energy condition, meaning that a light-sheet
which starts off with negative expansion may briefly experience some positive
expansion, causing the information on the light-sheet to become larger than
the amount allowed by the covariant bound. However, ref [11] shows that this
will not occur provided that we require that the non-expansion condition holds
everywhere on the light-sheet - then if violations of the null energy
condition occur on the light-sheet, one must begin with a strictly negative
(rather than zero) expansion, such that even when the expansion increases it
still remains less than or equal to zero. This ensures that the von Neumann
entropies of the relevant regions will obey the covariant bound even if the
null energy condition is violated (a more general proof which accounts for
interactions is given in ref [12]). However, it remains unclear whether the
bound can continue to hold in the regime where quantum gravitational effects
become important, for as Smolin notes[13], in that regime surfaces will not
always have a single well-defined light-sheet, so it’s unclear how to even
formulate the bound.

Ref [1] explored the question of how to interpret the covariant bound,
distinguishing between ‘ontological’ interpretations of the bound, where it is
regarded as an upper bound on the true number of degrees of freedom on a
light-sheet, and ‘epistemic’ interpretations of the bound, where it is
regarded as a constraint on the _accessible_ degrees of freedom on a light-
sheet. Much of the discussion around the bound and its connection to
holography seems to implicitly assume an ontological interpretation of the
bound, but there are also good reasons to take the epistemic interpretation
seriously. In particular, the covariant entropy bound is closely related to
the ‘surface postulate,’ which is the hypothesis that the number of degrees of
freedom available on a surface has an upper bound proportional to the area of
the surface. This hypothesis seems independently plausible - certainly, if you
have any reason to believe that the number of degrees of freedom on a surface
may be finite, it seems natural to propose that this number may be
proportional to the surface area.

But if the surface postulate is correct, and if information can only be
transferred along continuous paths in spacetime, as our best theories of
physics seem to suggest, this might be expected to have consequences for the
amount of information that observers can extract about the contents of a
region in an infinitesimal interval. In particular, if there are limitations
on the rate at which information on the surface can be replaced with new
information, then one would expect the surface postulate to induce an upper
bound on the information that can be extracted per unit time, which would
plausibly be proportional to the area of the bounding surface. And the
existence of such rate limitations also seem independently plausible - for
example, if one thinks the surface postulate may be a consequence of the
discretization of spacetime, then one would presumably expect that
discretization to apply to the temporal dimension as well, meaning that
information replacement on the surface could not occur infinitely quickly.

So it would seem quite natural for the surface postulate to give rise to a
bound on the accessible degrees of freedom on a light-sheet, and this bound
would have roughly the form of the covariant bound - thus if we accept the
surface postulate, an epistemic interpretation of the covariant bound seems
quite plausible. Note that this epistemic limitation is epistemic in the
precise sense that it describes limitations on information accessible to an
external observer, but it may nonetheless be entirely objective in the sense
that those limitations could follow from concrete physical facts about the
ontology of the bounding surface. Ref [1] discussed some arguments for and
against both the ontological and epistemic views, and concluded that both
possibilities remain open based on current understanding of physics.

##  3 von Neumann Entopies

Ref [1] largely assumed that the entropies relevant to the covariant bound are
classical - in fact, it was argued that these entropies could plausibly be
understood either as thermodynamic entropies or informational Shannon
entropies, and that either way these entropies could potentially be thought of
as encoding either the total number of degrees of freedom in a region, or the
total number of degrees of freedom accessible to an external observer.

But the covariant bound is also believed to hold in the context of quantum
field theory, and in that context the relevant entropy is the quantum von
Neumann entropy[14], which is perhaps not so naturally understood as counting
degrees of freedom. The von Neumann entropy of a quantum system or spacetime
region is defined by $S=-Tr(\rho\ln(\rho))$, where $\rho$ is the density
matrix characterising the physical state of the system or region. In this
article we will mostly consider cases where $\rho$ is the state of the quantum
fields occupying a given region, although in principle one could imagine a
generalization in which the state $\rho$ might also include degrees of freedom
associated with spacetime itself. It can be shown that the von Neumann entropy
is equal to the Boltzmann entropy and the Shannon entropy in many physically
relevant situations, thus justifying the choice to refer to it as an entropy
in the first place, but there is some controversy about whether these measures
coincide in all possible scenarios[15, 16, 17], and in addition the
interpretational questions surrounding the von Neumann entropy are different
from those associated with the thermodynamical and Shannon entropies, so we
should now consider how the arguments of ref [1] might be affected if we
stipulate that the entropies of interest are von Neumann entropies rather than
classical entropies.

In fact, up to a certain point, moving from classical to quantum entropies
does not significantly change these arguments. This is because, as emphasized
in ref [1], the entropy bound is not the value of any specific entropy; rather
it is a _universal upper bound_ on the entropy, so what matters for
understanding its origin is not necessarily the specific nature of the
entropies that it bounds, but rather the kinds of physical conditions which
could give rise to a universal bound on these entropies. And several of the
physical conditions which might lead to a universal upper bound on classical
entropies could also be expected to give a universal bound on the von Neumann
entropy. For example, one of the main arguments for the ontological view of
the classical entropy bounds is the fact that the total number of degrees of
freedom in a classical system is an upper bound on its classical entropy,
since no physically well-informed observer can assign to a system either a
thermodynamical entropy or a Shannon entropy higher than its total number of
degrees of freedom. And in the same way, the dimension of a system’s Hilbert
space (the quantum analogue of ‘number of degrees of freedom’) is an upper
bound on its von Neumann entropy, since no physically well-informed observer
can assign to a system a von Neumann entropy greater than the dimension of its
Hilbert space. So in the quantum case as in the classical case, one possible
explanation for the covariant bound is that it simply counts the total number
of degrees of freedom, or equivalently the dimension of the full Hilbert
space, in the relevant spacetime region.

Similarly, ref [1] argued that another possible explanation for the classical
entropy bound is that it could follow from a universal restriction on the
amount of information that external observers can obtain about the content of
a spacetime region. This is because external observers will necessarily
describe systems or regions in terms of the number of degrees of freedom that
they believe the system or region to have, and therefore if there exist
degrees of freedom that can’t be accessed from the outside, entropy
assignations made by external observers will likely be based on a state space
which does not include those degrees of freedom, so they will systematically
undercount the real degrees of freedom in the region. And much the same
argument can be made about the von Neumann entropy: observers will assign
quantum states on a Hilbert space defined by the number of degrees of freedom
they associate to the system, so again, if there are degrees of freedom that
can’t be accessed from the outside, their assignations of von Neumann entropy
will likely employ a Hilbert space which does not include those degrees of
freedom, and will thus be upper bounded by the size of the accessible Hilbert
space rather than the true Hilbert space.

This shows that the dilemma raised in ref [1] carries over to the quantum
context: it is not immediately clear whether a universal bound on the von
Neumann entropy has an ontological origin (i.e. it corresponds to the
dimension of the actual Hilbert space) or an epistemic origin (i.e. it
corresponds to the dimension of the _effective_ Hilbert space which is
accessible to external observers).

###  3.1 Entanglement

However, in the quantum case we also have a further issue to consider. For the
von Neumann entropy does not just measure classical uncertainty, in the way
that classical entropy does: it can also measure the degree of entanglement of
a system. That is, the von Neumann entropy is zero for a pure state and non-
zero for mixed states, and there are in principle two kinds of mixed states.
An improper mixed state is produced when we prepare a system by choosing a
pure state according to some probability distribution and then putting the
system into that state, so the resulting system is really in some pure state,
but someone who doesn’t know the outcome of the probabilistic choice cannot
tell which pure state in particular it is in. In this case the von Neumann
entropy can be regarded as a measure of classical uncertainty about the result
of the probabilistic choice, so it is very much like the classical entropy.
But a _proper_ mixed state is produced by taking a system which is entangled
with some other system and then ignoring (i.e. tracing out) the second system.
In this case there are some interpretations of quantum mechanics which would
suggest that there is no longer any classical uncertainty, since they hold
that the proper mixed state is a complete ontological description of the
system, so there is no further fact to be uncertain about. Thus for proper
mixed states the von Neumann entropy may not be so closely analogous to the
classical entropy, depending on one’s chosen interpretation of quantum
mechanics. Moreover since proper and improper mixed states are represented by
the same kind of mathematical object in the density matrix formalism, the way
in which von Neumann entropy is calculated makes no distinction between these
cases, so the entropies relevant to the entropy bound may sometimes measure
the degree of entanglement rather than classical uncertainty, or possibly a
mixture of both.

Indeed, some physicists take the view that _all_ non-zero von-Neumann
entropies are a measure of entanglement. For in fact proper and improper mixed
states are closely related: if we assume that unitary quantum mechanics is
universal and then write down a fully quantum description of the unitary
operation corresponding to the preparation of an improper mixed state (e.g. a
die is rolled and the system is prepared in a pure state depending on the
outcome of the roll) we will find that what it actually produces is an
entangled state of the die and system, of the form
$\sum_{i}\frac{1}{\sqrt{6}}|D_{i}\rangle|\phi_{i}\rangle$ where each $D_{i}$
corresponds to a possible outcome of the die roll and $\phi_{i}$ is the
corresponding state). Tracing over the die yields a proper mixed state for the
system, and this state is mathematically exactly the same as the improper
mixed state we would have obtained directly from the probabilistic preparation
in the non-unitary description. So if one believes that unitary quantum
mechanics is universal, or one subscribes to the church of the larger Hilbert
space (which refers to the view that pure state are fundamental and all mixed
states are derived from pure states[18]), it follows that all classical
uncertainty corresponds to entanglement in this way, so actually _all_ mixed
states are proper mixed states. Thus our analysis of the entropy bounds should
allow for the possibility that all of the entropies relevant to the entropy
bound might ultimately be entanglement entropies.

And indeed, a lot of work on entropy bounds in the context of QFT does assume
that the relevant entropies are entanglement entropies. For example, this is
the approach taken in the quantum proofs of the covariant bound given in refs
[11, 12], in which we define a finite von Neumann entropy on the light-sheet
by means of subtracting the vacuum entropy from the matter entropy. One
exception is the alternative quantum proof of the entropy bound due to
Strominger and Thompson[19], further developed in the form of the quantum
focusing conjecture by refs [20, 21], which proposes to address the problem
posed by violations of the null energy condition by reformulating the bound to
$S\leq\frac{A}{4G\hbar}+S^{\prime}$, where $S$ is the entropy on the light-
sheet and $S^{\prime}$ is the entanglement entropy across the surface. Clearly
in this formulation we should not identify the entropy on the light-sheet $S$
with the entanglement entropy $S^{\prime}$, as if we do we will end up with
the trivially true bound $S^{\prime}\leq\frac{A}{4G\hbar}+S^{\prime}$. Thus
the bounds proved in refs [11, 12] and refs [20, 21] are inequivalent even in
the regimes where they are both well-defined, which is explained by the fact
that they define their associated light-sheets differently[20]. It is likely
that when we move to a strong gravity regime we will ultimately need to use
the quantum focusing conjecture rather than the entanglement entropy vacuum-
subtraction approach of refs [11, 12], for as noted by Bousso et al[20], the
vacuum subtraction approach only works in a weakly gravitating regime: ‘ _When
gravitational backreaction of the state is not negligible, the spacetime
geometry is very different from that of a vacuum state. Then it is unclear
what one would mean by restricting both a general state and the vacuum to the
“same” region or light-sheet. In this case, one cannot define a finite entropy
by vacuum subtraction._ ’ However, since the interpretation in terms of
entanglement entropies appears to give the right result in the weakly
gravitating regime, we will focus on that interpretation in this article.

And in fact, it turns out that taking the entropies to be entanglement
entropies still leads to the same ontic/epistemic dilemma. For on the one
hand, the von Neumann entropy is still upper bounded by the dimension of the
Hilbert space regardless of whether we understand it in terms of classical
uncertainty or entanglement, and therefore a universal bound on _entanglement_
entropies could still be ontological in the sense that it might reflect the
dimension of the actual Hilbert space. Yet on the other hand, entanglement can
also be associated with knowledge: in particular, if we assume that unitary
quantum mechanics is universal and thus the process of gaining knowledge about
a system can be described as a unitary interaction between the system and the
observer, then this interaction will involve the creation of entanglement
between the system and the brain of the observer. So under this assumption,
all of the information that an observer outside a region $R$ has about $R$ can
be represented as entanglement between the observer’s brain and $R$ (although
of course the converse is not true - the brain may be entangled with external
systems in ways that do not amount to conscious knowledge). Here, we take it
that an observer’s brain can reasonably be schematized as wholly outside of
the region $R$ \- this is necessarily an approximation, since the brain itself
is a quantum system which will not be perfectly localised, but nonetheless it
appears that under most circumstances we can think of brains as being fairly
sharply localised in well-defined regions. Under this assumption, it follows
that the entanglement that the brain shares with the region cannot be larger
than the total entanglement of the region with the external world, and thus it
is natural to imagine that the case where the observer has extracted the
maximum possible information from that region corresponds to the case in which
all of the region’s entanglement is with the brain rather than the rest of
exterior - that is, in this case we can think of the brain as simply the
compliment to the region. So if there is some epistemic limitation which
bounds the total amount of information that observers are able to obtain about
the contents of some spacetime region, then it would be natural for that bound
to be manifested in the physics as a bound on the total amount of entanglement
that is able to form between this region and its exterior. Thus a universal
bound on _entanglement_ entropies could still be epistemic in the sense that
it might reflect physical limitations preventing observers from gaining full
information about the content of some region. Note that this interpretation of
the bound is not trivial, because in principle one might imagine that the
observer could have information which does not take the form of entanglement;
but as soon as we accept that all of the information the observer has can be
represented as entanglement between the observer’s brain and $R$, it does seem
to follow quite directly that a bound on entanglement entropy can be
interpreted in an epistemic way.

Indeed, interpreting the entropies in the covariant bound as entanglement
entropies may actually strengthen the epistemic account. For a natural concern
one might have about the argument presented in ref [1] is that the surface
postulate only bounds the information passing out of a region in an
infinitesimal time interval, so one might think that we could eventually
gather information exceeding the number appearing in the covariant bound by
accumulating information as it emerges over time. This would threaten the
epistemic account of the covariant bound: it would entail that if a light-
sheet associated with a surface of area $A$ did in fact typically have more
degrees of freedom than $\frac{A}{4\hbar G}$, then we should eventually be
able to find out about those degrees of freedom, so the fact that our current
theories assign light-sheets associated with surfaces of area $A$ an entropy
no more than $\frac{A}{4\hbar G}$ could not be attributed to our limited
ability to find out about the contents of those regions.

However, if it is assumed that quantum mechanics is universal and hence
information possessed by any observer about the content of the region must
ultimately take the form of entanglement between their brains and the region,
it follows that the total entanglement entropy between a region and its
exterior at a given time represents an upper bound on the _total_ information
that has emerged from that region over time. And the purpose of the light-
sheet construction is that in ordinary cases a light-sheet will traverse the
entire ‘interior’ of a region, as we would normally understand that term. So
if it turns out that the entanglement entropy of a light-sheet associated with
a surface of area $A$ is typically bounded by $\frac{A}{4\hbar G}$, then we
can argue that in fact the total _accumulated_ information available to
external observer about the ‘interior’ as captured by a given light-sheet
projected from the surface at some time is typically bounded by
$\frac{A}{4\hbar G}$, where $A$ is the surface area at that time. Meanwhile,
in cases where the light-sheet fails to traverse the entire ‘interior’ of the
region, that is typically because information from the part of the region
which does not intersect the light-sheet cannot reach the surface at all - for
example, this often occurs in the presence of singularities, in which case it
is known that information content in the surrounding region will remain
trapped behind a horizon and thus will not be accessible to external
observers. So these results concerning entanglement entropy suggest that the
possibility of gathering additional information over time do not in fact
undermine the epistemic interpretation of the entropy bounds: there are strict
limits on how much information we can obtain about the interior of a region,
even if we accumulate information over time.

##  4 Explanations

It appears that interpreting the entropies relevant to the covariant entropy
bound as von Neumann entropies does not immediately settle the question of
whether the bound is epistemic or ontological in origin. However, it is
possible that we can make progress on this question by appealing to the large
body of research on features of entanglement entropies in various scenarios.

At this point we face a difficult problem, which was also touched on in ref
[1] \- we have available to us many interesting mathematical results
establishing connections between entropy, entanglement entropy, gravity,
spacetime and so on, but in many of these cases it is not straightforward to
determine the correct direction of explanation in a non-question-begging way.
One might be tempted to adopt some kind of quietism in which we simply say
there is no fact of the matter about the ‘correct’ direction of explanation.
However, this approach may be undermined by the fact that the view we take on
these putative explanations seem to suggest different ways of proceeding to
formulate theories of quantum gravity and other new physics - for example, in
ref [22] Jacobson argues that gravity may be an emergent phenomenon which is
explained by entropic considerations, and that if this is true it would likely
have consequences for the correct way to quantize gravity. Thus it is possible
in the long term one explanatory approach or another will actually be
vindicated by future developments in physics, so there is a clear sense in
which there may actually be a fact of the matter about the correct direction
of explanation, and therefore it is worth trying to understand the range of
options available to us.

If the entropies to which the covariant entropy bound applies are indeed
entanglement entropies, we have two main options. First, we could understand
the covariant entropy bound as a fundamental fact about ontology or
epistemology, and then argue that entanglement entropies are constrained to
have certain properties because they have to obey the bound, so these features
of the entanglement entropy can be explained by the bound. In this case, it’s
unlikely that theoretical knowledge about features of the entanglement entropy
can tell us much about the nature of the bound, since these features would all
just be consequences of the bound. Or second, we might imagine that the
covariant bound arises as a consequence of certain independent, pre-existing
features of entanglement entropies in an underlying quantum field theory or
quantum gravity theory, so the bound is explained by the properties of
entanglement entropy, rather than vice versa. In this case, theoretical
knowledge about entanglement entropy may indeed help us understand the origin
of the bound.

Thus, although we think both of these options are potentially interesting, for
the rest of this article we will mostly focus on the latter approach, seeking
to understand what existing theoretical knowledge about entanglement entropies
might tell us about the origins of the covariant bound. We will examine
several different ways in which one might seek to derive the covariant entropy
bound or something similar to it out of features of the entanglement entropy;
in each case we aim to understand whether there are good prospects for
successfully explaining the covariant entropy bound in this way, and to
determine whether a successful explanation of this kind would support an
epistemic or ontological interpretation of the covariant entropy bound.

###  4.1 Area Laws

It has been shown that von Neumann entropies in condensed matter systems often
obey ‘area laws’[23, 24, 25, 26]: given a discrete condensed matter system
which contains mostly short-range interactions (i.e. the Hamiltonian is such
that individual particles in the lattice are coupled only with their immediate
neighbours, or other close-by particles) we often find that in the ground
state most of the entanglement is made up of links between particles at the
surface of the system and particles directly outside the surface, and thus the
entanglement entropy of a subregion scales with the surface area of the
region, rather than the volume. A similar phenomenon occurs in quantum field
theory[27, 28]: although the full von Neumann entropy suffers from an
ultraviolet divergence, the leading divergence scales as the area of the
boundary between the two regions, because the most entangled degrees of
freedom are the high energy ones near the boundary.

Ref [29] offers a heuristic illustration of the origin of such area laws in
QFT. The relevant quantum fields can be decomposed into a set of ‘modes’ of
various wavelengths, and the entanglement entropy counts the modes which cross
the boundary, since those are the modes which correspond to entanglement
between the two regions. So roughly speaking, the entanglement entropy is
proportional to the number of modes which cross the boundary. Now, if we don’t
insist on some minimum wavelength for a mode, then there will be infinite
modes crossing the boundary, so this calculation won’t give a meaningful
result. However, if we impose a minimum wavelength cutoff we will get a finite
answer, and as one might expect, the number of modes crossing the boundary is
roughly proportional to the area of the boundary. In general the constant of
proportionality in this relationship would be expected to depend on the number
of matter fields present, but as shown in ref [30], since matter fields
interact gravitationally, if there are more matter fields we will reach the
Planck scale at a higher cutoff length $L$, and in fact the two effects
exactly cancel out, so we end up with a universal relation between area and
entropy that does not depend on the number of fields.

Now, clearly area laws are similar in form to entropy bounds, so it is
tempting to suppose that the strong or covariant entropy bound may arise as a
consequence of area laws. Indeed, it has often been proposed that the
Bekinstein-Hawking entropy can be explained by an area law[31, 27, 32, 33, 34]
\- in fact this was one of Sorkin’s main motivations for introducing the
concept of ‘entanglement entropy’ in the first place[35] \- and since the
strong and covariant bounds are essentially generalizations of the Bekinstein-
Hawking entropy, it seems reasonable to make a similar proposal here. That
said, as argued in ref [1], this approach requires us to accept that the
cancellation of the number of fields just happens to work out in the right
way, whereas if we take the entropy bound as fundamental we can explain the
cancellation as a consequence of the bound in what is arguably a more
satisfying and unifying approach, so it should not be taken as a foregone
conclusion that the entropy bounds must be explained by area laws and not vice
versa. However, for now let us look further into the possibility of explaining
entropy bounds in terms of area laws.

There are a number of obstacles to be overcome before such an explanation
could be given. First, entanglement area laws do not give _universal_ bounds
on entropy; they simply describe the entropy of some specific state. In
particular, area laws are typically derived for the ground state, whereas it
can be shown that in the context of many-body systems, a quantum state chosen
at random from a Hilbert space will typically obey a volume law rather than an
area law[23]. In a continuum quantum field theory on the other hand there is
some reason to think that excited states will obey the same area law as the
ground state, since it can be proved that at short distances all physical
states of a QFT look alike, so they will all have the same structure as the
ground state[36]. But it is unclear that this result applies generically in
the actual world, since the theory describing the quantum states of spacetime
may have a small-distance cutoff, in which case it is not really a continuum
QFT - and indeed, as described in ref [29], the existence of a cutoff is often
regarded as part of the explanation for the area law scaling111Thanks to an
anonymous referee for raising this point.. In addition, even in the ground
state, area laws apply only when the system is not near its quantum critical
point[37]. So one might question whether it makes sense to explain the
covariant bound, which has been proposed as a _universal_ bound on entropy, by
appeal to area laws which apply only under quite specific circumstances.
However, perhaps it might be argued that most of the quantum fields we
encounter are quite close to their ground state and thus we can expect to find
area-scaling laws applying in a fairly universal way.

Second, even in the case where the system in question _is_ in the ground state
and away from the critical point, the area laws typically apply only to the
leading divergence in the calculation of the entanglement entropy, so there
can still be subleading terms which could lead to the area bound being
exceeded. So if we want to explain the covariant entropy bound in terms of
area laws, we would have to consider how significant these subleading terms
are, and decide whether small violations of the area scaling are compatible
with the existing evidence for the universality of the covariant bound.

Third, we noted above that in order to obtain a finite value for the
entanglement entropy one must choose a UV cutoff, and as ref [38] points out,
‘ _When we try to compute $S_{entanglement}$ with finite $\ell$ the answer
will depend on the details of how $\ell$ is introduced (sometimes called,
rather misleadingly, “scheme dependence”)._’ Different ways of introducing the
cutoff may may not all give rise to area laws, or they may give different
constants of proportionality. So if area laws in QFT are to explain the
covariant entropy bound, we would need to show that there is some sensible way
of implementing a UV cutoff which typically leads to the right kind of area
law, and we would need to find some way of understanding the physical
significance of this way of taking the cutoff which explains why the specific
scaling behaviour associated with it appears generic at a macroscopic scale.

One possibility is to think of the cutoff scheme as encoding relevant features
of the observer, such as the energy scale at which they are able to probe the
region. This would align entanglement entropy conceptually with the
thermodynamic entropy, whose calculation depends on a choice of coarse-
graining which is usually thought of as encoding relevant features of the
observer, such as the set of exogenous variables which they are able to
manipulate[39, 40]. In this case, to explain the covariant bound based on area
laws it would be necessary to demonstrate that the kind of cutoff scheme
associated with typical observers lead to the right kind of area scaling under
most circumstances. Alternatively one might think that there is some
objectively correct way of taking the cutoff, perhaps corresponding to a real,
physical UV cutoff below which QFT ceases to apply[41, 42, 43, 44]. For
example, it has been shown that in the special case of black holes, if we
choose the Planck length as the cutoff then calculation of the entanglement
entropy across the horizon gives a result proportional to $A$ with a constant
of proportionality of the same order of magnitude as the one in the
Bekinstein-Hawking formula[35]. In this case, to explain the covariant bound
based on area laws it would be necessary to show that the objective physical
cutoff leads to the right kind of area scaling under most circumstances.

What would follow if we could indeed make the case that the covariant entropy
bound is in fact a consequence of entanglement area laws? First, we emphasize
that the word ‘law’ in the term ‘area law’ is not usually understood as
referring to some kind of _fundamental_ law; rather ‘area laws’ are emergent
effects arising in certain physical situations as a result of the structure of
the underlying QFT, and thus if the covariant entropy bounds is ultimately a
consequence of area laws, then it too should probably be thought of as
approximate and emergent rather than fundamental. In particular, we noted in
section 3.1 that the interpretation of the covariant bounds in terms of
entanglement entropies arguably only makes sense in weakly gravitating
regimes, so if the bound is explained by area laws then it may only make sense
in such regimes - and indeed, Smolin points out that it is in any case unclear
how the bound could be properly defined in the strong gravity regime[45].
Second, an account of this kind would clearly favour an epistemic reading of
the covariant bound, since the whole point of the area laws is that in many
cases the entanglement entropy of a region may be significantly smaller than
the real number of degrees of freedom in that region, because the degrees of
freedom away from the boundaries don’t really contribute. And hence, since
unitary quantum mechanics tells us that observers external to a region of
spacetime can only gain information about it by being entangled with it, it
follows that observers are limited in their ability to gain information about
degrees of freedom deeper in the bulk, leading to an entropy bound which
reflects the information accessible to the observer rather than the total
information in the region. For example, ref [46] argues that the entanglement
entropy _‘provides a measure of how the degrees of freedom localized in that
region interact (are “entangled”) with the rest of the theory. In a sense the
entanglement entropy is a measure of the effective operative degrees of
freedom, i.e., those that are active participants in the dynamics, in a given
region of the background geometry.’_

This kind of ‘epistemic’ account of the covariant bound does not entail that
it is in any sense subjective. For facts about the entanglement entropies of
ground states are objective physical facts, and therefore facts about the
information accessible to external observers in the relevant kind of systems
are also objective physical facts. Moreover, although it may be possible to
interpret cutoffs as pertaining to capabilities of observers, the ‘epistemic’
account of the bound is prefectly compatible with the idea that there is a
real, physical UV cutoff. Indeed, the epistemic account would actually make
the relation between a real physical cutoff and the covariant bound more
comprehensible - for as noted in ref [1], we might naturally expect that a
minimum length cutoff would mean the information inside a region scales with
the volume, but the strong and covariant entropy bounds suggest that it scales
with area instead. And the epistemic interpretation in terms of entanglement
entropy can make sense of this fact - we can now see that even if the total
number of degrees of freedom in a region, as defined by the discretization
imposed by the cutoff, does scale with the volume, nonetheless the information
_accessible from the outside_ , also defined by the cutoff, instead scales
with the area, due to the fact that only modes crossing the boundary
contribute. So the entanglement entropy calculations provide a useful
demonstration of the real and concrete ways in which epistemic limitations on
physically embodied observers can come to be reflected in their descriptions
of the physical world.

####  4.1.1 Strong vs Covariant Entropy Bounds

However, accounting for entropy bounds by appeal to entanglement area laws
seems most straightforward if we are trying to explain the _strong_ entropy
bound rather than the covariant entropy bound. This is because entanglement is
most commonly understood as a relation between two systems _at the same time_
, i.e. on the same spacelike surface. So it is quite intuitive to think of the
strong bound as bounding entanglement entropies, since it pertains to the
content of a region defined on a spacelike slice, and less intuitive to think
of the _covariant_ bound as bounding entanglement entropies, since it instead
pertains to the content of a region defined on a null surface.

nullspacelike$B$$B^{\prime}$ Figure 1: The light-sheet construction used in
ref [11]

That said, it is possible to to calculate a von Neumann entropy for a light-
sheet, as done in ref [11], which aims to prove a quantum version of the
generalized entropy bound as discussed in section 2. The calculation proceeds
by extending the light-sheet associated with the surface $B$ to a larger null
surface, as depicted in figure 1, then defining a vacuum state and the actual
state of matter on this null surface, then restricting them to the part of the
null surface occupied by the light-sheet portion bounded by surfaces $B$ and
$B^{\prime}$, thus obtaining reduced density matrices for the vacuum and
actual matter state on the light-sheet. The von Neumann entropies of these
density matrices diverge, but the difference between them is finite, and it is
this value which is shown to be proportional to $\frac{A}{4\hbar G}$. (A more
complex calculation allowing for interacting fields is performed in ref [12]).

However, can the result of this calculation can be interpreted as encoding the
amount of entanglement between the light-sheet and its exterior region? First
of all, it’s not necessarily obvious which exterior region we are talking
about. In the ordinary case when we define entanglement entropies we are
looking at a region defined on a spacelike slice entangled with another region
on the same spacelike slice, so there is no question about the time. But the
light-sheet is not defined on a single spacelike slice, so we can’t simply
take that route. We could perhaps imagine integrating over a stack of
spacelike slices which together foliate the whole light-sheet, but it’s
unclear this would make sense - the resulting calculation of the entropy for
the exterior region would surely end up vastly overcounting the entropy, since
the same entropy-bearing systems would be integrated over many times. But it
would seem arbitrary to select just _one_ spacelike slice. So in fact,
probably the most the natural approach is to extend the null surface on which
the light-sheet is defined beyond the bounding surfaces $B,B^{\prime}$ and
then consider the light-sheet to be entangled with the region outside of the
surfaces _on this null surface_ \- and indeed, this appears to be the
interpretation suggested by the calculation that is actually performed in ref
[11], since as we have noted, it starts by defining a state for an extended
null surface and then restricting it to the light-sheet.

So, can we just say the von Neumann entropy of the light-sheet quantifies its
degree of entanglement with the rest of the null surface? This isn’t so
obvious, since the rest of the null surface is in the (lightlike) _future_
and/or _past_ of the light-sheet, so we would not normally say that it is
related to the light-sheet _only_ by entanglement. Of course, there are
certainly ways in which these regions could be connected by what we would
normally think of as entanglement. For example, consider the case where $S$ is
a sphere and the backwards light-sheet of $S$ traverses the entire interior
region of the sphere. Although of course this is a relativistic scenario which
should properly be associated with a relativistic quantum field theory, let us
briefly switch into non-relativistic quantum mechanics, which offers a more
intuitively accessible language for describing entanglement phenomena. So
imagine that there is an entangled pair of particles present, with one
particle on the inside of the sphere and the other particle on the outside. If
neither particle interacts with anything during the relevant time interval,
then there is a sense in which we could say that the interior particle is, at
the time when the light-sheet intersects it, entangled with the exterior
particle not only on the same spacelike slice but also on all future spacelike
slices, until the exterior particle undergoes some interaction which breaks
the entanglement or transfers it elsewhere. After all, the predictions of
quantum mechanics for the outcomes of measurements on these particles are the
same regardless of whether we measure them on the same spacelike slice or the
same null surface, so it makes sense to think of the latter case as
instantiating a kind of temporal entanglement, which is really just the
temporal development of pre-existing spatial entanglement. Thus in this case
the interior particle is ‘entangled’ with the future version of the exterior
particle which intersects with the relevant null surface extended outside of
the sphere. And, if we now switch back into the domain of relativistic quantum
field theory, it seems reasonable to think that something similar could be
said about the quantum fields in these regions, allowing us to say that two
regions on the same null surface can indeed be entangled with each other.

However, unlike in the spacelike case, entanglement is not the only kind of
direct connection we can have between these two regions, because the lightlike
relation between the regions allows for physical systems to cross the boundary
carrying information directly from one to another - in particular, if we are
considering a past-directed light-sheet, so the exterior region outside of the
surface B lies in the future of the light-sheet, we can have physical signals
crossing the boundary to carry information out of the light-sheet into the
exterior, and if we are considering a future-directed light-sheet, so the
exterior region outside of the surface B lies in the past of the light-sheet,
we can have physical signals crossing the boundary to carry information from
the exterior into the lightsheet. So the correlations between the regions may
be due to pre-existing entanglement, but may also be due to physical signals
passing in and out of the light-sheet.

How does this second kind of connection relate to the von Neumann entropy on
the light-sheet calculated in ref [11]? Well, the interpretation of this
entropy is not very clear, but it seems reasonable to think that the
calculated entropy may include both kinds of correlations - i.e. in some sense
it quantifies both connections associated with pre-existing entanglement and
also connections associated with physical signals traveling through the
boundary222Perhaps one could make sense of what is going on here in the
context of an approach akin to the proposal of Aharonov, Popescu and
Tollaksen[47], who argue that time evolution can be thought of as a kind of
‘entanglement’ correlation between Hilbert spaces representing successive
moments of time. This would be an interesting way to unify ‘pre-existing
entanglement’ with ‘physical signals traveling through the boundary’ but it
remains to be seen if it would give the right quantitative results.. After
all, physical signals crossing the boundary are necessary to _produce_
entanglement in the first place, because entanglement cannot be created at a
distance[48]. That is, given an initial state in which a pair of systems are
not originally entangled, in order for them to later become entangled, we must
either perform a joint quantum operation on them before separating them, or we
must entangle a different pair of systems by a joint quantum operation and
then transfer the entanglement from one pair of systems to another by a
process like entanglement swapping[49]. Therefore any entanglement between the
inside and outside of a region must either have existed since the relevant
initial condition - i.e. presumably since the beginning of time - or it must
have originated from some physical system which passed through the boundary at
some point. And since we have no obvious way to determine whether or not
entanglement in the present has its origins in an initial condition or was
created in a later interaction, it’s unclear that one could ever really
separate out entropy coming from entanglement and entropy coming from physical
systems crossing the boundary - and if we were to try, it seems quite likely
that one would find the distinction to be reference-frame dependent. So it
seems reasonable to think that the von Neumann entropy calculated in ref [11]
includes or perhaps even is dominated by contributions from physical signals
crossing the boundary.

This means that the calculation of ref [11] could be interpreted as offering a
quantum version of the epistemic account of the covariant bound previously
suggested in ref [1], i.e. the idea that the bound originates from an
epistemic restriction due to the fact that the amount of information which can
pass through a bounding surface in an infinitesimal interval is upper bounded
by the number of degrees of freedom on the surface. Thus in the case of the
covariant bound, it doesn’t seem to be the case that invoking area laws would
_replace_ or make redundant the kind of epistemic explanations we have
previously considered - the two approaches are quite complementary and the
entropy calculation on a null surface may potentially be regarded as a more
formal expression of the epistemic restrictions motivated in an intuitive way
in ref [1].

Of course, we do still have area laws for areas defined entirely on spacelike
surfaces which presumably include only contributions from entanglement,
without any physical information flow from one region to the other, and one
might worry that if these pure entanglement area laws are not the explanation
of the covariant bound, then it is then a mysterious coincidence that the
entropy featuring in the covariant bound and the entropy featuring in
spacelike area laws both scale with the surface area. But in fact there is no
coincidence here. The entanglement in the area laws is related to the surface
area because entanglement is created by local interactions through the
interfacing surface, and the entropy associated with a light-sheet is related
to the surface area because both the creation of entanglement and information
passing out of the region must be mediated by local interactions through the
interfacing surface. So, as suggested by ref [23], rather than a coincidence
we may have something like a common cause: both phenomena scale with the
surface area due to the locality of interactions through an interfacing
surface.

###  4.2 Spacetime _From_ Entanglement

A more radical approach to connecting entanglement entropy area laws with
entropy bounds is based on the conjecture that the structure of spacetime is
in some sense derived from entanglement. It has been suggested that rather
than explaining area laws in terms of the locality of interactions in
spacetime, we should instead derive the structure of spacetime from
entanglement[50, 51]: for example ref [52] conjectures that ‘ _mutual
information … can be used to associate spatial manifolds with certain kinds of
quantum states_ ’ (in this context the mutual information can be used as a
measure of entanglement entropy). The idea is that the facts about where an
object is located in spacetime are to be derived as a function of its degree
of entanglement: ‘ _degrees of freedom are only “in the interior” in a
geometric sense when they are entangled with their neighbors but not with
distant regions._ ’ It seems plausible that this approach could offer an
explanation of the covariant bound, since clearly if spacetime itself is
derived from entanglement there will necessarily be strong links between
spacetime structure and von Neumann entropy.

Now, the ‘spacetime from entanglement’ approach has been criticized in a
number of ways - for example, as noted by Ney[53], if the entangled objects
are not located in a pre-defined spacetime, it’s unclear why the entanglement
they exhibit should have exactly the structure which is compatible with our
ordinary three-dimensional space: to give rise to a 3D space the state must
belong to the space of ‘redundancy-constrained’ states, and ref [52]
emphasizes that these states are not at all generic in the space of possible
states, so it seems we must simply impose this structure in an ad hoc
way333Though of course, proponents of this view might respond that simply
imposing a 3D structure on spacetime itself is equally ad hoc!. But here we
want to focus on a different kind of objection. First, it is important to be
clear about exactly what is meant by the claim that ‘spacetime’ can be derived
from entanglement. Spacetime has both topological features and metrical
features, where the _topology_ of spacetime refers to properties that are
invariant under any continuous deformation of the spacetime, such as facts
about which points are adjacent, while the metric provides a notion of
distance between the points in the space, and thus pertains to properties that
are not invariant under continuous deformations of the spactime. It seems that
ref [52] ultimately aims to extract both the topology and the metric of
spacetime from facts about entanglement - after all, the suggestion that ‘
_degrees of freedom are only “in the interior” in a geometric sense when they
are entangled with their neighbors but not with distant regions,_ ’ seems to
make sense only if we accept that not only the metric but also topology is
determined by entanglement, since whether or not a point is inside of a region
is a topological fact. Thus an appropriate way to assess the success of this
approach is to consider whether or not the structure thus derived is capable
of performing all the characteristic functions of both spacetime topology and
the spacetime metric.

In particular, one key functional role of the topology of spacetime is that it
determines which systems can interact - where by ‘interact’ we mean
specifically processes involving the exchange of information, i.e. the kind of
processes that could be used for signalling. According to our best current
understanding of physics, such processes always occur locally in spacetime,
meaning that information transfer can only occur between systems which are
adjacent: this feature of physics is sometimes known as ‘no-signalling,’ or
‘locality’ or ‘causality’ and it features prominently in quantum mechanics,
QFT and special relativity. Moreover, ‘causality’ is not just an arbitrary
feature that these theories happen to have: for if superluminal signalling
were possible then in principle we would be able to use it to construct closed
causal loops and causal paradoxes and thus to create logical
inconsistencies[54], so there are reasons that go beyond any of these
individual theories to think that the locality of interactions must be a deep
and significant feature of our reality. And to enforce local interactions one
needs at least adjacency relations encoded in a topological manifold, and
possibly also one also needs conformal structure, depending on whether one
expects the lightcone structure of spacetime to be determined as part of the
locality constraint or whether it is expected to emerge in some more dynamical
way. Thus the role played by the topology of spacetime in defining adjacency
relations, and/or the role of conformal structure in defining possible
signalling relations, is one of its most important functions. Moreover, we
have already noted that covariant bound appears to be closely related to the
locality of interactions across surfaces, so the role of spacetime topology
and/or conformal structure in defining adjacency relations is particularly
relevant if we are seeking to explain the bound.

However, it seems unlikely that spacetime could play this role if its topology
and/or conformal structure were in fact derived entirely from entanglement and
nothing else. It is of course possible to get adjacency relations from the
approach of ref [52] \- specifically, the authors first use the proposed
relation between mutual information and distance to assign distances to pairs
of regions, and then use multidimensional scaling to embed them in a manifold,
and thus, because they have arrived at a manifold at the end of this process,
they have necessarily also arrived at a notion of adjacency. However, it is a
little unclear how adjacency could come to have the dynamical significance
with respect to interactions that it in fact has if it were solely grounded on
entanglement: for in this picture ‘adjacency’ essentially just corresponds to
being highly entangled, but being highly entangled doesn’t usually have
anything to do with being able to exchange information. Indeed, the ‘no-
signalling’ theorem in quantum mechanics refers precisely to the fact that
information _cannot_ be exchanged using entanglement alone, so it would be
surprising if being highly entangled suddenly made systems capable of
exchanging information444Recall that the ‘no-signalling theorem’ simply refers
to the fact that measurements performed on distinct quantum systems commute,
and therefore classical information cannot be sent from one system to another
using entanglement, regardless of when and where in spacetime they are
located. Thus this use of the no-signalling theorem does not presuppose the
existence of conformal structure - it still makes sense in a pure topological
manifold.. Furthermore, ‘degree of entanglement’ is a continuous measure,
whereas ‘adjacency’ is not - either two systems are adjacent and can interact,
or they are not adjacent and cannot interact - so although the procedure of
ref [52] does arrive at a notion of adjacency by creating an embedding based
on entanglement, from a dynamical point of view it seems hard to see how a
sharp distinction which is not present in the original entanglement structure
but which is merely a byproduct of the mathematical procedure of creating an
embedding could suddenly come to have so much dynamical significance in
determining how the system can interact. And finally, there seems to be a
potentially problematic circularity in this vicinity, because one of the
distinctive features of entanglement is that local interactions are required
to _create_ entanglement, and it seems difficult to make sense of this
constraint if we derive spacetime structure entirely from entanglement: surely
there needs to be some pre-existing spacetime structure to determine how that
entanglement can come into being in the first place? This point is related to
Ney’s worry in ref [53], for the usual way of explaining why entanglement in
our world has the structure that it does is to suppose that it is produced in
a dynamical process involving local interactions in some pre-existing
spacetime, but we will not be able to explain it this way if we take it that
entanglement is ontologically prior to facts about locality in spacetime,
which leads to a version of the dilemma posed by Ney: if spacetime structure
emerges from entanglement and not vice versa, it seems mysterious that the
entanglement in our world should have the specific kind of structure that
_would_ have been produced by dynamically local interactions in a four-
dimensional spacetime, even though it cannot in fact have been produced that
way.

So in our view, it is unlikely that _all_ important facts about spacetime can
be derived entirely from entanglement. That said, there may be a viable
intermediate position: although it seems unlikely that entanglement alone can
fully define the topology of spacetime, entanglement could potentially be used
to define the _metric_ of spacetime. That is, rather than first obtaining
distance relations from entanglement and then using these to obtain an
embedding on a manifold as suggested in ref [52], we could instead _start_
from a bare topological manifold or conformal manifold with no metric, with
quantum systems living at each point of the manifold; and then, using the
facts about adjacency defined by the underlying manifold, allow entanglement
to form by means of local interactions between adjacent systems. (Or
alternatively, we could start from kind of non-spatiotemporal substratum which
is not to be identified with spacetime but which does define some standard of
adjacency - for example the edges in loop quantum gravity[55]). And we could
then postulate that the _metric_ of the spacetime is subsequently defined by
facts about entanglement, for example by specifying that the distance between
two regions (a metrical feature) is given by a function of the quantum mutual
information between the regions.

The idea that entanglement may define only the metric and not the topology of
spacetime is not necessarily a problem for the ‘spacetime from entanglement’
program, for it would still be very interesting if we could understand the
spacetime metric as a function of entanglement relations. It would however
presumably entail that the covariant bound cannot be defined entirely by
reference to entanglement entropies, for the bound clearly has something to do
with topology - the whole point of the light-sheet construction is to give a
covariant way of identifying the _interior_ of a region, and the notion of an
interior is at least in part a topological notion. But nonetheless, if
entanglement did play a role in defining the metric of a spacetime, then it
would certainly have a close connection to the covariant entropy bound, since
area is determined by the metric and hence it is the metric which gives us the
area $A$ appearing in the bound. For example, if one were to say that the area
of a surface is, by definition, equal to some constant times the entanglement
entropy passing through it, then it would be true by definition that the
entanglement entropy of some region must have a bound proportional to its
surface area.

What would such an account mean for the epistemic/ontic distinction? On the
one hand, it is still true in this picture that the entanglement entropy of a
region quantifies the total amount of information that external observers can
have about that region; so if we then understand the covariant bound as a
consequence of a bound on the entanglement entropy, it would not be wrong to
say that the covariant bound reflects some kind of epistemic limitation.
However, in the spacetime-from-entanglement approach the entanglement entropy
is also understood as defining the ontology of spacetime, so it’s not clear
that it still makes sense in this picture to say that there there can be more
degrees of freedom inside a region than external observers are able to access
- in a sense, the spacetime-from-entanglement approach blurs or erases the
distinction between epistemic and ontic accounts of the ontology of spacetime.
Thus the approach might be most compatible with some kind of relational or
perspectival picture in which we say that the ‘number of degrees of freedom’
of a region can only be defined relative to an external observer, in which
case the maximum entanglement entropy of a region is by definition equal to
the total number of degrees of freedom in that region.

Moreover, as noted in ref [1], one might think that the ontology of spacetime
_should_ be defined in such a way that the total information content closely
matches the accessible information. After all, our intuitive picture of
spacetime as a continuum completely filled up with degrees of freedom has the
consequence that the ontology of spacetime far outstrips the evidence
available to us, which seems problematic if we value ontological parsimony,
and in combination with the surface postulate apparently leads to the
troubling possibility that there could exist whole realms of physics which are
permanently inaccessible to us. So if it can be shown that a ‘spacetime-from-
entanglement’ approach does indeed help close the gap between the ‘true’
information content of spacetime and the accessible information, that might be
an argument in its favour.

Before closing our discussion of ‘spacetime from entanglement’ let us mention
one obvious objection to the conjecture that the metric of spacetime arises
out of entanglement structure. The argument of ref [52] show only how to use
entanglement structure to define spatial distances, not temporal intervals;
and yet in Special and General Relativity the spacetime metric defines both
spatial and temporal distances. The authors of ref [52] suggest that their
approach to defining the geometry of space at a single time could be combined
with time evolution to describe the geometry of spacetime as a whole, but this
seems somewhat in tension with the underlying principles of special
relativity: for in a relativistic context local Lorentz transformations can be
used to transform a spatial interval into a combination of spatial and
temporal intervals, so one might naturally wonder how deriving spacetime from
entanglement could possibly be compatible with relativity if this approach
requires us to treat spatial and temporal distance as two entirely different
categories with different underlying ontologies. Of course, enacting a split
between space and time is compatible with a Hamiltonian formulation of general
relativity, so if there were a natural way to derive the constraint equations
of the Hamiltonian formulation this approach might be made at least formally
compatible with relativity555Thanks to an anonymous referee for raising this
point. Still, since this approach would presumably require us to choose some
privileged foliation of spacetime, it would arguably still be somewhat
unappealing from a relativistic standpoint. In addition, note that every
entangled state is a quantum superposition of unentangled states, so the
linearity of quantum mechanics would suggest that the dynamics of an entangled
state should simply be a linear combination of the dynamics for the
unentangled states. Thus if entanglement in and of itself is capable of
producing dynamical effects, this would potentially violate the unitary of
quantum mechanics, since dynamics due specifically entanglement would
presumably not be expressible as a linear combination of unentangled
dynamics666Thanks to an anonymous referee for raising this point..

However, it is possible that rather than defining spatial geometry at a point
and then performing time-evolution, one could use methodology similar to ref
[11] to define entropies on null surfaces, or methodology similar to refs [38,
56] to define entropies in 4D regions, thus giving something like
‘entanglement entropy’ which could be used to define timelike and lightlike
distances as well as spacelike ones. But as noted in section 4.1.1, it’s
unclear that these kinds of entropies measure only ‘entanglement’ in the usual
sense of the word - they would seem to include also physical signals passing
between regions, so this might require us to change the interpretation of the
construction in some way. Thus as in section 4.1.1, it seems that improving
our understanding of the correct interpretation of von Neumann entropies
defined for null surfaces or 4D regions would be useful to help elucidate the
connection between entanglement and entropy bounds.

##  5 Conclusion

In this article we set out to understand how to interpret the covariant
entropy bound in a context where the entropies bounded may be be quantum von
Neumann entropies rather than classical entropies. Our initial observation is
that both the ontological and epistemic interpretations of the bound remain
possible, and we also noted that this conclusion holds regardless of whether
the von Neumann entropy is understood as encoding classical uncertainty,
entanglement entropy, or any combination of the two.

Now, if the entropies relevant to the covariant bound are entanglement
entropies, one possible option would be to regard the covariant bound as a
fundamental fact about ontology or epistemology, which would allow us to use
the bound to explain various features of entanglement entropies in the
underlying quantum field theory or gravitational theory. But alternatively,
one might also suppose that the covariant bound should be viewed as emerging
from deeper features of entanglement entropy; in this article we have
discussed several ways in which one might try to make that case. First we saw
that entanglement area laws could potentially offer what appears to be an
epistemic explanation of the covariant bound, although this depends on some
open questions about how to interpret entropies defined on null surfaces. An
explanation by appeal to properties of the entanglement entropy might end up
being closely related to the intuitive ‘epistemic’ approach suggested in ref
[1], since both seem closely tied to facts about the locality of interactions
through boundaries.

Next, we discussed the idea that spacetime structure is derived from
entanglement. We argued that although entanglement might potentially be the
origin of the metric of spacetime, it is less plausible that it is also the
origin of the topology. But nonetheless, we concluded that the spacetime-from-
entanglement approach might point to a more ontological way of thinking about
the bound, since in this picture entanglement entropy defines the ontological
content of spacetime.

Overall, our conclusion is that even if the covariant bound is ultimately a
consequence of properties of the entanglement entropy, both ontological and
epistemic interpretations of it remain possible. Current established knowledge
about entanglement entropies seems to point towards an epistemic
interpretation of the bound, but this could change if the ‘spacetime from
entanglement’ approach were to gain more mainstream acceptance. In any case,
if theoretical knowledge about entanglement entropies is indeed pointing us
towards an epistemic interpretation of the bounds, that seems interesting and
significant, due to the close connection between the covariant bound and
gravity; the role of entanglement entropies here may perhaps point to a novel
way of understanding gravity in the context of quantum mechanics.

##  6 Acknowledgements

Thanks to Yasaman Yazdi for very helpful comments on a draft of this paper.

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