Hints of Gravitational Ergodicity: Berry's Ensemble and the Universality of the Semi-Classical Page Curve
CCHEP XXXXX
Hints of Gravitational Ergodicity:
Berry’s Ensemble and the Universality of the Semi-Classical Page Curve
Chethan KRISHNAN a ∗ , Vyshnav MOHAN a † a Center for High Energy Physics,Indian Institute of Science, Bangalore 560012, India
Abstract
Recent developments on black holes have shown that a unitarity-compatible Page curvecan be obtained from an ensemble-averaged semi-classical approximation. In this paper,we emphasize (1) that this peculiar manifestation of unitarity is not specific to black holes,and (2) that it can emerge from a single realization of an underlying unitary theory. Tomake things explicit, we consider a hard sphere gas leaking slowly from a small box intoa bigger box. This is a quantum chaotic system in which we expect to see the Page curvein the full unitary description, while semi-classically, eigenstates are expected to behave asthough they live in Berry’s ensemble. We reproduce the unitarity-compatible Page curve ofthis system, semi-classically. The computation has structural parallels to replica wormholes,relies crucially on ensemble averaging at each epoch, and reveals the interplay between themultiple time-scales in the problem. Working with the ensemble averaged state rather thanthe entanglement entropy, we can also engineer an information “paradox”. Our systemprovides a concrete example in which the ensemble underlying the semi-classical Page curveis an ergodic proxy for a time average, and not an explicit average over many theories. Thequestions we address here are logically independent of the existence of horizons, so we expectthat semi-classical gravity should also be viewed in a similar light. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] F e b Introduction
Recent developments [1, 2] on the information paradox [3, 4, 5, 6] have revealed that onecan reproduce the Page curve for Hawking radiation from semi-classical gravity. This canbe viewed as surprising for a couple of reasons: • Firstly, it reveals that understanding the fine-grained entropy (or at least its qualitativePage evolution) does not require us to know the microstate/density matrix in the fullUV-complete theory; a knowledge of the semi-classical description is enough . Whilethis fact may seem superficially surprising, it should be emphasized that there is nocontradiction here. Entanglement entropy is just one number, and the full densitymatrix is a (possibly infinite dimensional) matrix. So the latter contains a vastlylarger amount of information, which is in principle not required for extracting the fine-grained entropy. It is therefore not implausible, at least in hindsight, that semi-classicalgravity is able to calculate this entropy. • A second and more perplexing feature is that the semi-classical calculation that leadsto the unitarity-compatible Page curve involves the inclusion of replica wormholes intothe Euclidean path integral [7, 8]. When interpreted at face value, this suggests thatwe are in fact dealing with an ensemble average , when we use semi-classical gravityto compute the matrix elements that go into the entropy calculation [7]. Indeed forJT gravity in two dimensions, which is an ensemble average over unitary theories (andtherefore is a non-unitary theory), one can explicitly demonstrate the emergence of thePage curve by evaluating the average over the underlying ensemble [9, 7]. In short, we seem to be finding a unitarity-compatible Page curve from an ensemble-averageddescription .The second bullet point above, raises a puzzle. Our entire premise when looking for atent-shaped (ie., unitarity-compatible) Page curve was that quantum gravity is unitary. Andyet, now we have been dealt a devil’s bargain. We have a unitarity-compatible Page curve,but in the semi-classical (Euclidean) gravity limit where we are working, it seems to bearising in an ensemble average over theories. Even though this is not quite a contradiction– the ensemble average of a quantity that follows the Page curve will also follow the Pagecurve – it does raise a puzzle about how one should think about the relationship betweenthe fundamental description of gravity and its semi-classical description.The Euclidean path integral is believed to be ill-defined as a complete definition of quan-tum gravity in higher dimensions (eg., the wrong sign kinetic term of the conformal modeof the metric). At the conceptual level, an obvious piece that is missing in our present1nderstanding is the connection between semi-classical (bulk-metric based) gravity and theunderlying “true” quantum gravity degrees of freedom, which are presumably holographic.To make matters more confusing, in low dimensions there seem to be non-unitary metrictheories like JT gravity that do have well-defined path integrals. These can be explicitlydemonstrated to be ensemble averages over distinct unitary matrix models [9].The goal of this paper is to make some progress in understanding how to think of semi-classical gravity in more general contexts. More generally, we wish to understand the role(if any) of ensembles in a Page curve calculation in a unitary theory. Does the fact thatsemi-classical gravity is an ensemble average, suggest that the fundamental theory shouldalso necessarily be an ensemble average over distinct theories? This is the case in JT gravity,and it has been suggested that this may be the general paradigm. Such an explicit ensembleaverage however would be disappointing from the point of view of the usual lore of theAdS/CFT correspondence, where individual unitary boundary theories (eg., N = 4 SYM)seem to be dual to individual unitary theories of quantum gravity (eg., type IIB string theoryon AdS × S ) which should each have semi-classical supergravity limits. We do expect blackholes to arise as thermalized states in a single copy of an N = 4 SYM theory.In order to shed some light on this question, we will make two key observations in thispaper regarding the two bulletted points mentioned at the beginning of this Introduction.These observations are – • Neither of the points have a a priori anything to do with gravity, black holes or horizons.By this we mean that both features can be seen in systems that apparently are withoutgravity. • Both features can be seen already at the level of individual unitary theories, withoutexplicit ensemble averages. The ensembles arise much like they do in conventionalstatistical mechanics, where they arise as proxies for time averages when the system isin (approximate) thermal equilibrium.In other words, the first bullet point about the semi-classical accessibility of the unitarity-compatible Page curve is equally valid in non-gravitational unitary theories. Similarly, theredoes not seem to be anything forbidding us from coming up with a non-gravitational theorywhere a Page curve emerges at the semi-classical level via an apparent ensemble-average.Indeed, the bulk of this paper deals with the detailed study of an example that illustratesboth these points. We expect that such examples should be fairly generically constructiblein quantum chaotic systems which can be split into two subsystems. It is an interesting question whether the systems we consider (eg., the hard sphere gas) are secretly dualto some (perhaps exotic) theory of gravity. apparent ensemble average, while thefull quantum gravity indeed remains safely unitary. Closely related ideas have appearedearlier, see [10, 11]. One of the new features in our calculation will be that we are able tofollow the evolution of the system (and the Page curve) explicitly at the semi-classical level.This also enables us to have a clear understanding of the epoch-dependence of the ensemble.Other crucial features of our explicit model will become clear as we proceed.If this picture is correct, low-dimensional examples like JT gravity which come withexplicit ensemble averages and well-defined (but non-unitary) metric path integrals, are tobe viewed as exceptions. The swampland ideas of [12], which suggest that in high enoughdimensions, ensembles for gravity contain only a single theory seem consistent with thispicture. What is nice about JT gravity then, is that it gives us an explicitly doable, well-defined metric path integral unlike more realistic theories of gravity.In what follows, we will work with the concrete example of a hard sphere gas leaking slowly from a small box into a larger one. A hard sphere gas in a box is known to be aquantum chaotic system, whose eigenstates were conjectured by Berry [13, 14] to behavesemi-classically as though they were picked from a Gaussian ensemble. We will call thisconjectural ensemble, Berry’s ensemble. Berry’s conjecture was one of the initial motivationsfor the Eigenstate Thermalization Hypothesis (ETH) [15], see also [16]. The reason for ourinterest in this particular set up involving the hard sphere gas is that based on generalprinciples of unitarity, we expect to see a Page curve in this system if we compute theentanglement entropy of the larger box. Equally importantly, thanks to Berry’s conjecture,we may suspect (and indeed we will demonstrate) that it should be possible to show theemergence of this Page curve via a calculation at the semi-classical level, where an ensembleaverage plays a significant role.Horizons, islands and other geometric objects do not play a role in our calculations, andthere is no genuine information paradox. But note that the questions we are interested inhave only to do with the semi-classical ensemble average aspect, and we will show that oursystem shares that with the black hole system. Therefore, despite the differences, the lessonswe extract from the hard sphere gas have a chance of holding for gravity as well. Indeed,this is our primary motivation behind the present paper.We will find that the semi-classical entanglement entropy of the larger box, follows thePage curve. The assumption of slow leakage , leads to two timescales in the problem and We will make this more precise later.
3e find that there is an analogue of a Hawking radiation epoch [1] in the present problemas well. During each epoch, we can compute the entanglement entropy assuming that theeigenstates of the relevant subsystem are taken from Berry’s ensemble . The result, whenplotted against epoch, yields a unitarity-compatible Page curve.Interestingly enough, we also find that despite the absence of horizons, we have a simpleway to obtain an information “paradox” in this system. Instead of computing the ensemble-average of the Renyi entropy from the reduced density matrix, one can consider the Renyientropy of the ensemble-averaged reduced density matrix. By direct calculation through theepochs, we find that the evolution of this object does not have the turnaround and we areleft with Page’s version of the Hawking paradox.A key technical assumption in our calculation is that the leakage is slow so that the gasin each box can come to approximate equilibrium during each epoch. This is what enablesus to take advantage of Berry’s ensemble averaging epoch by epoch. In doing so, we areeffectively assuming that the entanglement entropy during each epoch can be computed viaa suitable time average (thanks to local equilibrium) and that the ensemble average is anergodic stand-in for this, as is often the case in statistical mechanics. The entanglemententropy of the reduced density matrix of the larger system is the thermodynamic entropy ofthe smaller system during that epoch. In the limit when the system has fully thermalizedand both boxes have the same density of particles, this reduces to the result obtained in [17].So our work can be viewed as a type of generalization of the result there. See [18] for somerelated discussions.The structure of these observations strongly suggest that perhaps a similar mechanismis what holds in gravity as well. By analogy with the hard sphere gas, we are thereforetempted to conjecture that semi-classical gravity is providing an ergodic ensemble averageddescription of quantum gravitational dynamics in bulk local equilibrium. Since gravity isholographic, more ideas will be needed to make this into a fully concrete proposal, but let usmake one speculative comment. We suspect that some approximate notion of coarse-grainingwill likely be required in defining the relevant entanglement entropy in flat space gravity.A cut-off has played a role in flat space ever since the work of Gibbons-Hawking [20], andit seems plausible to us that its correct interpretation is in implementing a coarse-graining[21].Our work departs from some of the statements in the literature, which call for gravity tobe viewed as an explicit ensemble average. On the contrary, we view our results as being inline with the ideas of [10, 11]. Our purpose here is to present a concrete non-gravitational In our system, an epoch is characterized by the number of hard spheres in the larger box. This isconceptually parallel to how the clock used in [7] was the string of Hawking photons emitted up to thatpoint.
Let us start by considering a collection of N hard spheres, each with a radius a , enclosedin a cubic box of length L . Assume that there is a larger empty box of length L (cid:48) in contactwith the smaller box. At t = 0, we open a hole in the wall between them so that the gas canleak slowly into the larger box. By tuning the size of the hole, we can take the leakage rateto be slow. Technically, what this means is that the mean free path (cid:96) of the hard sphere ishierarchically larger than the diameter of the hole d . We will also take the size of the holeto be (possibly hierarchically) larger than the sphere radius a . As (cid:96) = ˜ L / √ π ˜ N a (see eg.,[22]), it suffices to have a (cid:28) d (cid:28) ˜ L ˜ N a (2.1)Here ˜ L denotes the fact that we are referring to either of the boxes, and ˜ N is the number ofparticles in it during an epoch (a term we will define below).We will model the system by assuming the hard spheres to be point particles/centerssatisfying the constraint that the distance between any two centers cannot be less than 2 a .If all the particles were enclosed in a single box, this description will reduce to the modeldiscussed in [15]. It is natural to expect our system with the two connected boxes also toexhibit ergodicity and chaos, even though typically the hard sphere gas in a single box isthe one that is studied in the context of chaos and thermalization [15].Let us look at the Hilbert space of the system. We can denote the energy eigenstates by | Ψ α (cid:105) . Let us introduce a position basis | X (cid:105) , where X corresponds to the 3 N dimensionalposition vector of all the particles. In this position basis, we can define the wavefunctionsΨ α ( X ) = (cid:104) X | Ψ α (cid:105) and Ψ ∗ α (cid:48) ( X (cid:48) ) = (cid:104) Ψ α (cid:48) | X (cid:48) (cid:105) . (2.2)To define the domain where the wave function is defined, we first introduce an auxiliary In [15] the box size was taken to be L + 2 a . This makes sure that the centres of the spheres are living ina box of length L . This adds nothing to our discussion, and makes the definition of the hole connecting thetwo boxes slightly unwieldy, so we will let the centers themselves bounce off the box walls. This is purely amathematical convenience. D (cid:48) = (cid:110) X , . . . , X N (cid:12)(cid:12)(cid:12) X i ∈ B S ∪ B L ; | X i − X j |≥ a (cid:111) (2.3)where the three Cartesian coordinates of the individual box domains are B S ≡ [0 , L ] and B L ≡ [ L, L + L (cid:48) ] × [0 , L (cid:48) ] . The crucial extra boundary condition that defines the truedomain of the system is given by the condition that the wavefunction vanishes not on all of ∂B ∪ ∂B , but only on ∂B ∪ ∂B − H where H is the part of the domain which correspondsto the location of the hole. The region within this vanishing condition of the wave function isour true domain, and we denote it by D . We will not need to specify the shape and locationof the hole in detail to do our calculations below, other than the conditions on its size wenoted above. Note that the second box is bigger than the first, ie. L (cid:48) > L , and H is a subsetof ∂B ∩ ∂B .The hierarchy in (2.1) introduces two time-scales into the problem. Since the gas isleaking slowly, the time taken for each of the boxes to reach approximate equilibrium (sep-arately), will be much smaller than the timescale of leakage during which the number ofparticles in the boxes change appreciably. Implicit is also the assumption that the aver-age energies are sufficiently high that each box thermalizes quickly enough compared to theother scales in the problem. In any event, the end result is an epoch where both of theboxes have separately equilibrated and the number of particles in each of the boxes remainsapproximately fixed. Let N S and N L denote the number of particles in the smaller and largerbox at a particular epoch. As the total number of hard spheres in the boxes remain fixedthroughout an epoch, we can use either N S or N L to characterize it. The number of hardspheres are related to each other through the conservation law N S + N L = N. (2.4)To study the evolution of the state of the system, we expand it as a linear combination ofthe eigenstates | Ψ α (cid:105) of the full system (ie., the two boxes connected by the hole) as follows: | Ψ (cid:105) = (cid:88) α d α | Ψ α (cid:105) (2.5)The corresponding density matrix of the state is ρ = | Ψ (cid:105)(cid:104) Ψ | = (cid:88) α,α (cid:48) d α d ∗ α (cid:48) | Ψ α (cid:105)(cid:104) Ψ α (cid:48) | (2.6)In the position basis, we have ρ ( X ; X (cid:48) ) ≡ (cid:104) X | ρ | X (cid:48) (cid:105) = (cid:88) α,α (cid:48) d α d ∗ α (cid:48) Ψ α ( X )Ψ ∗ α (cid:48) ( X (cid:48) ) . (2.7)6o analyze the properties of each of the boxes separately with the epochs, it becomesuseful to focus on a subspace of the full Hilbert space. At every epoch, we have (2.4) andthis provides a natural partition of the 3 N components of the vector X into 3 N S and 3 N L components as follows: X = ( x , y ) and | X (cid:105) = | x , y (cid:105) (2.8)where x = ( x , . . . , x N S ) and y = ( y , . . . , y N L ) can loosely be thought of as denoting theposition vectors of the particles in the smaller and larger boxes respectively . In terms ofthese coordinates, we can define the wavefunctions asΨ α ( x , y ) = (cid:104) x , y | Ψ α (cid:105) and Ψ ∗ α (cid:48) ( x , y ) = (cid:104) Ψ α (cid:48) | x , y (cid:105) (2.9)In this notation for the position basis, the density matrix takes the form ρ ( x , y ; x (cid:48) , y (cid:48) ) ≡ (cid:104) x , y | ρ | x (cid:48) , y (cid:48) (cid:105) = (cid:88) α,α (cid:48) d α d ∗ α (cid:48) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x (cid:48) , y (cid:48) ) (2.10) To calculate the entanglement entropy of the larger box, we will compute the n -th Renyientropy of subsystem, and then take the n → n = 2) of the larger box. We start by computing the reduceddensity matrix of the particles “associated to the larger box”, in the notation of the lastparagraphs of the previous section: ρ L ( y ; y (cid:48) ) = (cid:90) D dx ρ ( x , y ; x , y (cid:48) ) = (cid:90) D dx (cid:88) α,α (cid:48) d α d ∗ α (cid:48) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48) ) (3.1)Squaring the matrix, we get ρ L ( y ; y (cid:48) ) = (cid:90) D dy (cid:48)(cid:48) ρ L ( y ; y (cid:48)(cid:48) ) ρ L ( y (cid:48)(cid:48) ; y (cid:48) ) (3.2)= (cid:90) D dy (cid:48)(cid:48) dxdx (cid:48) (cid:88) α ,α (cid:48) ,α ,α (cid:48) d α d ∗ α (cid:48) d α d ∗ α (cid:48) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y (cid:48) )Now let us look at the behavior of this quantity at each epoch. From the discussion inthe previous section, we can see that working in various epochs is equivalent to restrictingourselves to processes occurring at time-scales larger than the equilibrization time of each But note however that at this stage, the ranges of the positions of each of the particles span the fullsystem, and not the left or right box alone. Note that viewing the particles as being localized in one of theboxes is a kind of semi-classical approximation. We will make such an assumption eventually. .It turns out that there is a natural choice for such an ensemble. Consider a quantumchaotic system. Berry’s conjecture [13, 14] says that when the energy of an eigenstate issufficiently high, the state behaves as if it was picked randomly from a fictitious Gaussianensemble. It was shown in [15] that when evaluated in this eigenstate ensemble , the singleparticle momentum distribution function of the hard sphere gas turned out to be equalto the Maxwell-Boltzmann distribution. This is a specific manifestation of the eigenstatethermalization hypothesis (ETH). It is expected that (see eg., [18]) for systems which satisfythe ETH condition, ergodicity is guaranteed. Therefore, we can hope that averaging overBerry’s ensemble acts as an ergodic proxy for the underlying time averaging. A furthercomment worth making, is that Berry’s conjecture is based on semi-classical physics and relieson the connection between classical and quantum chaos [23]. So this further strengthens theparallel with the black hole Page curve calculation, which was done in the setting of semi-classical gravity [7].Adopting this philosophy, we are now ready to compute the purity of the reduced densitymatrix in Berry’s ensemble:Tr (cid:10) ρ L (cid:11) EE = (cid:90) D dy (cid:10) ρ L ( y ; y ) (cid:11) EE = (cid:90) D dy (cid:48)(cid:48) dy (cid:90) D dxdx (cid:48) (cid:88) α ,α (cid:48) ,α ,α (cid:48) d α d ∗ α (cid:48) d α d ∗ α (cid:48) (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE (3.3)where the subscript EE denotes that the quantity is averaged over the eigenstate ensemble.Berry’s conjecture would imply that the four-point function will be given in terms of theWick contractions of the two-point functions, as in [15] (see also appendix A, for related We will assume that the Renyi and entanglement entropies are quantities that can be calculated in thisway. The fact that the results are reasonable (as we will see) will be taken as a posteriori evidence for this. We will refer to this ensemble as Berry’s ensemble in the context of the hard sphere gas. Note that what we are evaluating is actually a product of eigenfunctions, not a correlation function. Butwe will use this slightly distracting terminology because of the obvious parallel in structure, and to not keeprepeating the lengthy phrase “ensemble expectation value of the product of eigenfunctions”. (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE = (cid:104)(cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE + (cid:68) Ψ α ( x , y )Ψ α ( x (cid:48) , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE + (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE (cid:68) Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ α ( x (cid:48) , y (cid:48)(cid:48) ) (cid:69) EE (cid:105) (3.4)Plugging the above expression into the previous one, we getTr (cid:10) ρ L (cid:11) EE = (cid:90) D dy (cid:48)(cid:48) dy (cid:90) D dxdx (cid:48) (cid:88) α ,α (cid:48) ,α ,α (cid:48) d α d ∗ α (cid:48) d α d ∗ α (cid:48) (cid:104)(cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE + (cid:68) Ψ α ( x , y )Ψ α ( x (cid:48) , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE + (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE (cid:68) Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) )Ψ α ( x (cid:48) , y (cid:48)(cid:48) ) (cid:69) EE (cid:105) (3.5)Pulling the sums into the ensemble average, this becomesTr (cid:10) ρ L (cid:11) EE = (cid:90) D dy (cid:48)(cid:48) dy (cid:90) D dxdx (cid:48) (cid:104) (cid:104) Ψ( x , y )Ψ ∗ ( x , y (cid:48)(cid:48) ) (cid:105) EE (cid:104) Ψ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ ( x (cid:48) , y ) (cid:105) EE + (cid:104) Ψ( x , y )Ψ( x (cid:48) , y (cid:48)(cid:48) ) (cid:105) EE (cid:104) Ψ ∗ ( x , y (cid:48)(cid:48) )Ψ ∗ ( x (cid:48) , y ) (cid:105) EE + (cid:104) Ψ( x , y )Ψ ∗ ( x (cid:48) , y ) (cid:105) EE (cid:104) Ψ ∗ ( x , y (cid:48)(cid:48) )Ψ( x (cid:48) , y (cid:48)(cid:48) ) (cid:105) EE (cid:105) (3.6)Now let us evaluate the two-point functions in the above expression. At each epoch, we aremaking a semi-classical approximation that N S particles are in one box and the rest are inthe other. At the level of wave functions, this enables us to assume that the value of thewavefunction Ψ vanishes (at least approximately) at the hole H . Roughly, at each epoch,we choose the boundary condition that Ψ vanishes on the boundary of D S and D L wherethese domains characterize the two separate boxes (and are defined precisely below). So wecan decompose the state Ψ as follows:Ψ( x , y ) ≈ (cid:88) i S ,i L c i S i L ψ i S ( x ) φ i L ( y ) (3.7)where ψ i S ( x ) and φ i L ( y ) are the eigenfunctions of the smaller and larger boxes, with N S and N L hard spheres respectively. These wavefunctions are defined in the domains D S and D L where D S = (cid:110) x , . . . , x N S (cid:12)(cid:12)(cid:12) x i ∈ [0 , L ] ; | x i − x j |≥ a (cid:111) (3.8)9nd D L = (cid:110) y , . . . , y N L (cid:12)(cid:12)(cid:12) y i ∈ [ L, L + L (cid:48) ] × [0 , L (cid:48) ] ; | y i − y j |≥ a (cid:111) (3.9)and they vanish on the boundary of their respective domains. The expression for the puritytherefore becomes,Tr (cid:10) ρ L (cid:11) EE = (cid:90) D L dy (cid:48)(cid:48) dy (cid:90) D S dxdx (cid:48) (cid:88) i S ,i (cid:48) S ,i L ,i (cid:48) L ,i S ,i (cid:48) S ,i L ,i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L (cid:104)(cid:68) ψ ∗ i (cid:48) ( x ) φ ∗ i (cid:48) ( y ) ψ i ( x ) φ i ( y (cid:48)(cid:48) ) (cid:69) EE (cid:68) ψ ∗ i (cid:48) ( x (cid:48) ) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) ψ i ( x (cid:48) ) φ i ( y ) (cid:69) EE + (cid:68) ψ ∗ i (cid:48) ( x ) φ ∗ i (cid:48) ( y ) ψ ∗ i (cid:48) ( x (cid:48) ) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) (cid:69) EE (cid:10) ψ i ( x ) φ i ( y (cid:48)(cid:48) ) ψ i ( x (cid:48) ) φ i ( y ) (cid:11) EE + (cid:68) ψ ∗ i (cid:48) ( x ) φ ∗ i (cid:48) ( y ) ψ i ( x (cid:48) ) φ i ( y ) (cid:69) EE (cid:68) ψ ∗ i (cid:48) ( x (cid:48) ) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) ψ i ( x ) φ i ( y (cid:48)(cid:48) ) (cid:69) EE (cid:105) (3.10)As x and y are independent variables, we can again simplify the expression in the squarebrackets using Berry’s conjecture, now for the individual boxes. This gives usTr (cid:10) ρ L (cid:11) EE = (cid:90) D L dy (cid:48)(cid:48) dy (cid:90) D S dxdx (cid:48) (cid:88) i S ,i (cid:48) S ,i L ,i (cid:48) L ,i S ,i (cid:48) S ,i L ,i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L (cid:104)(cid:68) ψ ∗ i (cid:48) ( x ) ψ i ( x ) (cid:69) EE (cid:68) φ ∗ i (cid:48) ( y ) φ i ( y (cid:48)(cid:48) ) (cid:69) EE (cid:68) ψ ∗ i (cid:48) ( x (cid:48) ) ψ i ( x (cid:48) ) (cid:69) EE (cid:68) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) φ i ( y ) (cid:69) EE + (cid:68) ψ ∗ i (cid:48) ( x ) ψ ∗ i (cid:48) ( x (cid:48) ) (cid:69) EE (cid:68) φ ∗ i (cid:48) ( y ) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) (cid:69) EE (cid:10) ψ i ( x ) ψ i ( x (cid:48) ) (cid:11) EE (cid:10) φ i ( y (cid:48)(cid:48) ) φ i ( y ) (cid:11) EE + (cid:68) ψ ∗ i (cid:48) ( x ) ψ i ( x (cid:48) ) (cid:69) EE (cid:68) φ ∗ i (cid:48) ( y ) φ i ( y ) (cid:69) EE (cid:68) ψ ∗ i (cid:48) ( x (cid:48) ) ψ i ( x ) (cid:69) EE (cid:68) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) φ i ( y (cid:48)(cid:48) ) (cid:69) EE (cid:105) (3.11)Using (A.19), we can do the above integrals. This will give usTr (cid:10) ρ L (cid:11) EE = (cid:88) i S ,i (cid:48) S ,i L ,i (cid:48) L ,i S ,i (cid:48) S ,i L ,i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L (cid:104) δ i (cid:48) S ,i S δ i (cid:48) L ,i L δ i (cid:48) S ,i S δ i (cid:48) L ,i L Z i L S (cid:16) U i L , U i (cid:48) L , L (cid:48) (cid:17) + δ i (cid:48) S ,i (cid:48) S δ i (cid:48) L ,i (cid:48) L δ i S ,i S δ i L ,i L Z i S Z i L S (cid:16) U i L , U i (cid:48) L , L (cid:48) (cid:17) S (cid:16) U i S , U i (cid:48) S , L (cid:17) + δ i (cid:48) S ,i S δ i (cid:48) L ,i L δ i (cid:48) S ,i S δ i (cid:48) L ,i L Z i S S (cid:16) U i S , U i (cid:48) S , L (cid:17) (cid:105) (3.12)where we have defined Z i L = (cid:18) L (cid:48) h (cid:19) − N (cid:48) Γ(3 N (cid:48) / mU i L )(2 πmU i L ) N (cid:48) / Z i S = (cid:18) Lh (cid:19) − N Γ(3 N/ mU i S )(2 πmU i S ) N/ (3.13)10nd S ( U i , U j , L ) = exp (cid:34) − m ( U i − U j ) L π (cid:126) U i (cid:35) (3.14)Simplifying the expression, we getTr (cid:10) ρ L (cid:11) EE = (cid:88) i S ,i L ,i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i L S (cid:16) U i L , U i L , L (cid:48) (cid:17) + (cid:88) i S ,i L ,i (cid:48) S ,i (cid:48) L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c i (cid:48) S i (cid:48) L (cid:12)(cid:12)(cid:12) Z i S Z i L S (cid:16) U i L , U i (cid:48) L , L (cid:48) (cid:17) S (cid:16) U i S , U i (cid:48) S , L (cid:17) + (cid:88) i S ,i L ,i (cid:48) S ,i (cid:48) L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c i (cid:48) S i (cid:48) L (cid:12)(cid:12)(cid:12) Z i S S (cid:16) U i S , U i (cid:48) S , L (cid:17) (3.15)We can see that the terms with | U i − U j | /U i smaller than or equal to ( (cid:126) /mU i L ) / will bedominant than the others. In particular, if ¯ U is the average energy, then we can see that thesum will be dominated by the terms with | U i − ¯ U | / ¯ U ≤ (cid:0) (cid:126) /m ¯ U L (cid:1) / ∼ ¯ λ/N / L , where¯ λ = (cid:0) π (cid:126) /mk ¯ T (cid:1) / denotes the thermal wavelength at the temperature ¯ T . For such terms,we can approximate the exponential to be 1 and we get (the sums are now restricted to thenon-vanishing band)Tr (cid:10) ρ L (cid:11) EE = (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i L (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) + (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i L (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i S + (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i S From an analogous calculation, we can also see thatTr (cid:104) ρ L (cid:105) EE = (cid:88) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (3.16)11herefore, we can define the normalized purity of the larger box as follows:Tr (cid:104) ρ L (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) = (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i L (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) + (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i L (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i S (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) + (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) Z i S (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (3.17)This expression can be further simplified toTr (cid:104) ρ L (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) = Tr( ˜ ρ L I L ) + Tr( ˜ ρ L I L )Tr( ˜ ρ S I S ) + Tr( ˜ ρ S I S ) , (3.18)where we have defined I L = (cid:88) i L Z i L | φ i L (cid:105)(cid:104) φ i L | and I S = (cid:88) i S Z i S | ψ i S (cid:105)(cid:104) ψ i S | (3.19)and the normalized reduced density matrices of the boxes as follows:˜ ρ L = 1 (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:88) i L ,i (cid:48) L ,i S c i S i L c ∗ i S i (cid:48) L | φ i L (cid:105)(cid:104) φ i (cid:48) L | (3.20)˜ ρ S = 1 (cid:80) i S ,i L (cid:12)(cid:12)(cid:12) c i S i L (cid:12)(cid:12)(cid:12) (cid:88) i S ,i (cid:48) S ,i L c i S i L c ∗ i (cid:48) S i L | ψ i S (cid:105)(cid:104) ψ i (cid:48) S | (3.21)Here | φ i L (cid:105) and | ψ i S (cid:105) are eigenstates of the larger and smaller box respectively. n -th Renyi Entropy Now let us look at the computation of the n -th Renyi entropy of the larger box. It is straight-forward to calculate Tr ( ρ nL ) by following the steps in section 3. The resulting expression willcontain a product of 2 n eigenfunctions, as in (3.3). However, to evaluate this quantity in theeigenstate ensemble, we will have to perform all the possible pairwise contractions of these2 n eigenfunctions and then do integrals over the resulting expressions. This can turn out tobe quite tedious as the number of possible contractions go as (2 n )! n !2 n for a generic n , see eg.[24]. Fortunately, we can directly calculate the end result by resorting to a diagrammaticapproach. 12 a) (b) Figure 1: (a) Consider the terms in the computation of the 4th Renyi entropy. We start offby distributing the indices on a circle. The pair of indices belonging to the same copy of thesystem are connected by a thick dashed line through the boundary. (b) Connecting variouspairs of such indices with each other will gives us a particular contraction.We start off by distributing all the 4 n indices present in the higher dimensional analogueof (3.10) on a circle as in fig 1(a). The pair of indices corresponding to each copy of thesystem are connected by a dotted line through the boundary of the circle. Note that thesepairs of indices are placed in such a way that the indices of various copies of the smaller(larger) box are adjacent to each other. Now let us connect one such pair of indices toanother though the interior of the circle using dashed lines. While making the connection,we make sure that an index corresponding to the smaller (larger) box is connected only toanother smaller (larger) box index. Doing this for all the pairs on the circle, we will get adiagram that corresponds to a particular pairwise contraction of all the 2 n eigenfunctions(Refer 1(b)).Now let us compute the value of each of these diagrams. It is useful to introduce someterminology before we proceed. The dashed lines partition the interior of the circle intovarious sub-regions (Refer figure 2). Let us call such a sub-region an m -connected region ifthere are m pairs of indices on the boundary of the region. Depending on the box to whichthe boundary indices belong to, we can attribute each m -connected region to the smaller orlarger box. For example, in fig 2(a), there are two 1-connected and one 2-connected regionsbelonging to the smaller box (These regions are marked in blue).In terms of these regions, we can assign a value to each diagram by using (A.22) and thestructure of the contractions. For every m -connected region, we should introduce a factor of ( Tr ( ˜ ρ S I S )) m − or ( Tr ( ˜ ρ L I L )) m − , depending on which box the region belongs to . Summingover the value of each of these diagrams will give us the n -th Renyi entropy of the larger13 a) (b) Figure 2: The figure highlights the m -connected regions of two diagrams. Let us look atfigure (a). This is the same diagram as figure in 1(b). We can see how the interior dashedlines partition the disk into various sub-regions. In this figure, there are two 1-connectedregions and one 2-connected region corresponding to the indices of the smaller box (marked inblue) and there are one 1-connected and one 3-connected regions corresponding to the largerbox (marked in green). Therefore, this diagram will have a factor of (Tr( ˜ ρ L I L )) Tr( ˜ ρ S I S ).Similarly, the figure (b) will have a factor of (Tr( ˜ ρ L I L )) Tr( ˜ ρ S I S ). We can see that thiscrossing diagram will be sub-leading to figure (a) at every epoch.box.If any interior dashed line of a diagram intersect another, then we will refer to thesediagram as a crossing diagram. As Tr( ˜ ρ L I L ) , Tr( ˜ ρ S I S ) (cid:28) n -th Renyi entropy. This makes the computation easier as the numberof non-crossing partitions, called the Catalan number (see eg., [25]), is much smaller thanthe number of pairwise contractions. We can see that there is a similar contraction structureas well as leading order behavior in [11]. This close resemblance has to do with the fact thatthe “equilibrium approximation” in [11] is equivalent to a time averaging when the systemhas reached an approximate (local) equilibrium.Now let us write an explicit expression for the n -th Renyi entropy by adding the value ofall the leading order diagrams. The structure of the contractions results in a large number ofdegenerate diagrams. Two diagrams can have the same value if one of them can be obtainedby permuting of the m -connected regions of the other diagram. We can also have a degen-eracy when the diagrams have different m -connected regions but the powers of (Tr( ˜ ρ L I L ))and Tr( ˜ ρ S I S ) add up to the same number (Refer figure 3 for an example).14 a) (b) Figure 3: The figure shows two non-crossing diagrams that have the same value. Countingthe number of m -connected regions, we can see that both the diagram will have a value of(Tr( ˜ ρ L I L )) Tr( ˜ ρ S I S ).To take care of these issues, let us first characterize each diagram by the m -connectedregions of the larger box . We can represent the m -connected regions of the larger box bythe notation (1 m m m ...n m n ), where the number m indicates that the diagram contains m m indicates that the diagram contains m m m m ...n m n ) is given by [25] N (1 m m m ...n m n ) = n ( n − . . . ( n − b + 2) m ! . . . m n ! b > b = (cid:80) i m i . When b = 1, N = 1.To account for the second type of degeneracies, let us first look at the partitions of anatural number m , that is, we look at the all the possible ways in which m can be written asa sum over positive integers. We can label each partition by the set { ( j, m j ) } , where j ∈ Z + and m j corresponds to the multiplicity of each j . Therefore, by definition, we have j ≥ (cid:88) j jm j = m (4.2)Let us denote P ( m ) to be the set of all such partitions of m . Using these definitions, wecan can write down the expression for the leading order contribution to the normalized n -th The pairwise contraction structure automatically fixes the m -connected regions of the smaller box in theterms of the m -connected regions of the larger box. Therefore, it suffices to use either one of the regions tocharacterize the diagram. (cid:104) ρ nL (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) n = n − (cid:88) k =0 (cid:88) { ( j,m j ) }∈ P ( k ) N (cid:0) n − r { ( j + 1) m j } (cid:1) (Tr( ˜ ρ L I L )) k (Tr( ˜ ρ S I S )) n − k − (4.3)where r = (cid:80) j ( j + 1) m j . We have used the notation (1 n − r { ( j + 1) m j } ) to represent thenon-crossing diagram consisting of ( n − r ) 1-connected regions and m j ( j + 1)-connectedregions, for all ( j, m j ) ∈ { ( j, m j ) } . When k = 0 and k = ( n − To make explicit statements about the behavior of the entanglement entropy, let us defineTr( ˜ ρ L I L ) ≡ (cid:18) L (cid:48) h (cid:19) − N L Γ(3 N L / m ¯ U L )(2 πm ¯ U L ) N L / (5.1)and Tr( ˜ ρ S I S ) ≡ (cid:18) Lh (cid:19) − N S Γ(3 N S / m ¯ U S )(2 πm ¯ U S ) N S / (5.2)We will call ¯ U S and ¯ U L as average energy of the boxes. To understand the behavior of theentanglement entropy, let us look at early and late times separately.When we make plots, we will assume that the average energy per particle is roughlyconstant. It is possible to relax this assumption somewhat, while retaining the shape ofthe Page curve, but we will not explore it here since it is quite reasonable as a physicalassumption in a closed system of large number of particles [15]. At early times, the larger box will have very small number of particles compared to thesmaller box. Therefore, Tr( ρ L I L ) (cid:29) Tr( ρ S I S ). The n -th Renyi entropy will be dominatedby the k = n − (cid:104) S ( ρ L ) (cid:105) = lim n → − n log (cid:20) Tr (cid:104) ρ nL (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) n (cid:21) = − log Tr( ˜ ρ L I L ) = − log (cid:34)(cid:18) Lh (cid:19) − N L Γ(3 N L / m ¯ U L )(2 πm ¯ U L ) N L / (cid:35) (5.3)16or large N , log Γ(3 N/
2) = (3 N/ −
1) log 3 N/ − N/
2. This gives us (cid:104) S ( ρ L ) (cid:105) = N L log (cid:20) V L (2 m ¯ U L ) / h (cid:21) + N L (cid:34) log (cid:18) π N L (cid:19) / + 32 (cid:35) + O (cid:18) log N L N L (cid:19) (cid:39) N L (cid:40) log (cid:34) V L (cid:18) πm ¯ U L h N L (cid:19) / (cid:35) + 32 (cid:41) (5.4)where V L is the volume of the larger box. This is precisely the thermodynamic entropy ofthe larger box as a function of the number of particles N L at a given epoch. As we discussedabove, if we assume that the average energy per particle is roughly constant, as in [15], weimmediately see that the entanglement entropy will increase with time as N L increases withtime. With the passage of time, more and more particles start moving into the larger box. Thisresults in an increase in the value of Tr( ˜ ρ S I S ) and decrease in Tr( ˜ ρ L I L ). Depending on therelative size of the larger box, there are two types of late time behavior. Let us look at thesecases separately.If the larger box is sufficiently bigger than the smaller box, we will reach an epoch whereTr( ˜ ρ S I S ) = Tr( ˜ ρ L I L ). We call the time taken to reach this epoch the Page time ( t P ) of thesystem. Let us denoted the number of particles in the smaller and larger boxes at the Pagetime by N S P and N L P respectively. For any t > t P , Tr( ˜ ρ L I L ) (cid:28) Tr( ˜ ρ S I S ). Therefore, The k = 0 term in the equation (4.3) will dominate the sum. This gives us (cid:104) S ( ρ L ) (cid:105) = lim n → − n log (cid:20) Tr (cid:104) ρ nL (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) n (cid:21) = − log (cid:34)(cid:18) Lh (cid:19) − N S Γ(3 N S / m ¯ U S )(2 πm ) N S / (cid:35) (cid:39) N S (cid:40) log (cid:34) V S (cid:18) πm ¯ U S h N S (cid:19) / (cid:35) + 32 (cid:41) (5.5)The resulting equation is the thermodynamic entropy of the smaller box . Here N is theequilibrium value of the number of particles in the smaller box.Eventually, both the boxes will completely thermalize w.r.t each other and the net ex-change of particles will drop to zero. This happens when the particle density in each ofthe boxes equalize. Let N S E and N L E denote the final equilibrium values of the number of17igure 4: The figure shows the plot of − n log [Tr (cid:104) ρ nL (cid:105) EE ] computed from (4.3) versus N L when n = 3 for a system with L = 2 meters, L (cid:48) = 2 meters, N = 10 , and particle mass m = 1 amu. We assume that the average energy of the boxes scale linearly with the numberof particles as in [15]. Therefore, at every epoch, we set ¯ U L N (cid:48) = ¯ U S N = k B T and we choose T = 300 K. As more and more particles leak out into the larger box, we can see that theentanglement entropy increases with time initially and then reaches a maximum at the Pagetime (indicated by the dotted line). After the Page time, the entanglement entropy dropsand saturates to the equilibrium value.particles of the boxes. We have the relation N L E L (cid:48) = N S E L (5.6)Using N L E + N S E = N , we can solve the above equation and we get N S E = L NL (cid:48) + L and N L E = L (cid:48) NL (cid:48) + L (5.7)Let us define t E as the time taken for N L E particles to leak into the larger box. For t > t E ,the entanglement entropy will saturate to the value S = N S E (cid:40) log (cid:34) V S (cid:18) πm ¯ U S h N S E (cid:19) / (cid:35) + 32 (cid:41) . (5.8)If we assume that at each epoch we have N L + N S = N , then we can plot the entanglemententropy as a function of N L or N S . We can see from figure 2 that the graph increases at18arly times and the reaches a maximum at the Page time. The graph then decreases andsaturates to (5.8).Now let us look at the case where the size of the larger box is comparable to the size ofthe smaller box. It is possible to have scenario where N L E < N L P . This would mean thatthe system would reach thermal equilibrium before the Page time is reached. Therefore, theentanglement entropy of the system will saturate to S = N L E (cid:40) log (cid:34) V L (cid:18) πm ¯ U L h N L E (cid:19) / (cid:35) + 32 (cid:41) t ≥ t E (5.9)We mention this only for completeness, our primary interest is in the previous scenario. We have managed to reproduce the Page curve by doing an ensemble averaged compu-tation of the entanglement entropy at each epoch. In this section, we will instead computethe entanglement entropy of the ensemble averaged state at each epoch. We will find thatthis leads to an information “paradox”. We put the word in quotes because a genuine infor-mation paradox is tied to the existence of horizons, and also because here we know why theparadox is appearing.Let us start by evaluating the reduced density matrix in the eigenstate ensemble: (cid:104) ρ L ( y ; y (cid:48) ) (cid:105) EE = (cid:90) D dx (cid:88) α,α (cid:48) d α d ∗ α (cid:48) (cid:104) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48) ) (cid:105) EE (6.1)Squaring the matrix, we get (cid:104) ρ L ( y ; y (cid:48) ) (cid:105) = (cid:90) D dy (cid:48)(cid:48) (cid:104) ρ L ( y ; y (cid:48)(cid:48) ) (cid:105) EE (cid:104) ρ L ( y (cid:48)(cid:48) ; y (cid:48) ) (cid:105) EE = (cid:90) D dy (cid:48)(cid:48) (cid:90) D dxdx (cid:48) (cid:88) α ,α (cid:48) ,α ,α (cid:48) d α d ∗ α (cid:48) d α d ∗ α (cid:48) (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y (cid:48) ) (cid:69) EE (6.2)19herefore, the purity of the larger box will be given byTr (cid:104) ρ L (cid:105) = (cid:90) D dy (cid:104) ρ L ( y ; y ) (cid:105) = (cid:90) D dy (cid:48)(cid:48) dy (cid:90) D dxdx (cid:48) (cid:88) α ,α (cid:48) ,α ,α (cid:48) d α d ∗ α (cid:48) d α d ∗ α (cid:48) (cid:68) Ψ α ( x , y )Ψ ∗ α (cid:48) ( x , y (cid:48)(cid:48) ) (cid:69) EE (cid:68) Ψ α ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ α (cid:48) ( x (cid:48) , y ) (cid:69) EE = (cid:90) D dy (cid:48)(cid:48) dy (cid:90) D dxdx (cid:48) (cid:104) Ψ( x , y )Ψ ∗ ( x , y (cid:48)(cid:48) ) (cid:105) EE (cid:104) Ψ( x (cid:48) , y (cid:48)(cid:48) )Ψ ∗ ( x (cid:48) , y ) (cid:105) EE (6.3)Now let us evaluate the two-point functions in the above expression. Using the factorizationin (3.7), we getTr (cid:104) ρ L (cid:105) = (cid:90) D L dy (cid:48)(cid:48) dy (cid:90) D S dxdx (cid:48) (cid:88) i S ,i (cid:48) S ,i L ,i (cid:48) L ,i S ,i (cid:48) S ,i L ,i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L c i S i L c ∗ i (cid:48) S i (cid:48) L (cid:104)(cid:68) ψ ∗ i (cid:48) ( x ) φ ∗ i (cid:48) ( y ) ψ i ( x ) φ i ( y (cid:48)(cid:48) ) (cid:69) EE (cid:68) ψ ∗ i (cid:48) ( x (cid:48) ) φ ∗ i (cid:48) ( y (cid:48)(cid:48) ) ψ i ( x (cid:48) ) φ i ( y ) (cid:69) EE (cid:105) (6.4)This is precisely the first term of the equation (3.10). Therefore, we can immediately carry-over the calculations in section 3 to getTr (cid:104) ρ L (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) = Tr( ˜ ρ L I L )) (6.5)We can also see that Tr (cid:104) ρ nL (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) n = (Tr( ˜ ρ L I L ))) n − (6.6)Therefore, the entanglement entropy of the averaged state will be given by S ( (cid:104) ρ L (cid:105) EE ) = lim n → − n log (cid:20) Tr (cid:104) ρ nL (cid:105) EE (Tr (cid:104) ρ L (cid:105) EE ) n (cid:21) = − log Tr( ˜ ρ L I L ) (cid:39) N L (cid:40) log (cid:34) V L (cid:18) πm ¯ U L h N L (cid:19) / (cid:35) + 32 (cid:41) (6.7)Under the assumptions of the previous section, we can see that the plot of S ( (cid:104) ρ L (cid:105) EE ) v/stime will keep on increasing and then saturate to the value at N L = N L E . Therefore, whenthe size of the larger is box is sufficiently larger than the smaller box, the late time behaviorof S ( (cid:104) ρ L (cid:105) EE ) will be different from that of (cid:104) S ( ρ L ) (cid:105) EE .20e can calculate the purity of the smaller box in the eigenstate ensemble by interchanging ψ ↔ φ , x ↔ y , x (cid:48) ↔ y (cid:48) and x (cid:48)(cid:48) ↔ y (cid:48)(cid:48) in (3.11) and it turns out to be equal to the purity ofthe larger box in the eigenstate ensemble. Therefore, we have (cid:104) S ( ρ L ) (cid:105) EE = (cid:104) S ( ρ S ) (cid:105) EE . Thisbehavior is expected from a unitary theory. However, if we make the same replacements in(6.4), we will be able to see that S ( (cid:104) ρ L (cid:105) EE ) (cid:54) = S ( (cid:104) ρ S (cid:105) EE ). Therefore, there is an apparentloss of unitarity when we work with the averaged state.These results, while simple, are interesting because they provide an explicit mechanismfor understanding how the information paradox may emerge in gravity. It suggests that thevacuum one obtains by quantizing fields in the black hole background has features of an averaged state from the perspective of the fundamental theory. Our calculations in this paper had nothing to do with gravity, horizons or a true in-formation paradox. In fact our primary goal was to illustrate that an ensemble-averagedsemi-classical approximation leading to the Page curve is not limited to gravity. But in do-ing this, we learnt that the ensemble average can arise as a proxy for a time average duringeach epoch, even in single realizations of a unitary theory. This is interesting because theprecise role of the ensemble in the case of gravity has been a bit murky. For one, in 2-d JTgravity there is an explicit ensemble average. But in usual AdS/CFT in higher dimensions,we expect to see black holes in the duals of single copies of the CFT.In our hard sphere gas, we found that an ensemble average can arise in the ergodic senseduring each epoch of local equilibrium. This suggests a similar picture for gravity in higherdimensions. Loosely related ideas have appeared previously in [10, 11], and our goal herewas to find a model that provides a nuts-&-bolts understanding of the origin of the Pagecurve. There are two vastly different timescales during the evaporation of a black hole, andtherefore Hawking temperature is a well-defined approximately constant quantity during anyepoch of evaporation. This makes it possible that the ensemble average in gravity is a proxyfor a time average during each epoch of Hawking radiation. In other words, an explicitaverage over an ensemble of distinct unitary theories may not be necessary.A further observation we made is that the Page version of the Hawking paradox canemerge in our perfectly unitary system, if we did our semi-classical calculation using theensemble-averaged state. We showed that the entropy increases relentlessly until it saturatesat the thermodynamic entropy. This again is a strong suggestion that a similar mechanismmay be at work in the gravity system as well – indeed, a proposal that the Hawking resultis a consequence of an ensemble-averaged state was suggested previously in [26].21n fact, our calculation in Section 6 demonstrates that the “state paradox” formulated in[26] can be resolved without an explicit ensemble average over many theories. Let us take amoment to explain this. We start with the Quantum Extremal Surface [27] formula for thevon Neumann entropy, which we write schematically as S micro = A G + S macro , (7.1)where the S macro on the right hand side is the entropy of the bulk quantum fields as wouldhave been calculated by Hawking. State paradox arises, if we view the latter as a fine-grainedcontribution to the full entropy. It was proposed in [26] that the problem can be solved ifwe view S micro as an ensemble average of the entropy of the state, and S macro as an entropyof the ensemble averaged state. Our calculations in this paper suggest that these ensemblesneed not be explicit collections of distinct theories like in JT gravity — they can be ergodicensembles that stand in for epoch time averages. In particular, the ensembles one thinks ofhere are not fixed, they are implicitly epoch-dependent. In our example of the hard spheregas, it is controlled by the number of particles in either box in a given epoch, which fixesthe appropriate Berry’s ensemble.A key point in the above discussion is that the S macro on the right hand side of (7.1) issupposed to be computed semi-classically, via a state obtained from quantum field theoryin curved space. Even though this object is usually viewed as a fine-grained entropy in theQFT in curved space Hilbert space, it is not clear that it is a fine-grained quantity in the truemicroscopic degrees of freedom in the holographic CFT Hilbert space. This was emphasizedin [28] where the S macro was called a “coarse-grained” entropy. Let us emphasize that this isdistinct from some of the uses of the phrase “coarse-graining” in the literature. What [28]emphasized was that the bulk entropy is calculated in a semi-classical bulk state, and not inthe truly microscopic CFT state. The precise connection between the two descriptions hasnever been very clear in AdS/CFT, but observations in this paper suggest that the states inthe Hilbert space of a quantum field theory in the black hole background have features of anaveraged state. We will elaborate on this preliminary observation, in future work [21]. Moregenerally, if taken at face value, the message of our work is that semi-classical gravity shouldbe viewed as a tool for capturing ergodic averaged gravitational dynamics, for evolutionthat is in bulk local equilibrium. Of course, developing this idea further is something thatwill have to be left for future work. In our hard sphere calculation, the assumption of localequilibrium entered due to the hierarchical timescales that ensured the existence of epochs.We have been quite narrow in our focus in this paper, but let us conclude by emphasizingthat the ideas on the black hole Page curve may have implications even beyond black holes. See [19] for some heterodox statements about the black hole Page curve.
We thank Jude Pereira for discussions and collaborations.
A Hard sphere gas enclosed in a single cubic box
Let us consider N hard spheres, of radii a , enclosed in a single cubic box of length L + 2 a as in [15] . We can denote the energy eigenfunctions of the system by ψ i ( X ), where X =( x , . . . , x N ) is a 3 N -dimensional vector that labels the position of all the particles of thesystem. We can define ψ i ( X ) on the domain D = (cid:26) x , . . . , x N (cid:12)(cid:12)(cid:12)(cid:12) − L ≤ x i , , ≤ + 12 L ; (cid:12)(cid:12)(cid:12)(cid:12) x i − x j |≥ a (cid:27) (A.1)satisfying the boundary condition that ψ i ( X ) vanishes on ∂ D .The eigenfunctions of the box can be chosen to be real and they take the form [15] ψ α ( X ) = N α (cid:90) d N P A α ( P ) δ (cid:0) P − mU α (cid:1) exp( i P · X / (cid:126) ) (A.2)where A ∗ α ( P ) = A α ( − P ) and U α is the energy of the state. Berry’s conjecture says that whenthe energy of ψ α ( X ) is sufficiently high, A α ( P ) acts as if it is a Gaussian random variablewith the two-point function (cid:104) A α ( P ) A β ( P (cid:48) ) (cid:105) EE = δ αβ δ N ( P + P (cid:48) ) /δ (cid:0) P − P (cid:48) (cid:1) (A.3)Moreover, we can compute the four-point functions in terms of the two-point functions asfollows (cid:104) A α ( P ) A β ( P ) A γ ( P ) A δ ( P ) (cid:105) EE = (cid:104) A α ( P ) A β ( P ) (cid:105) EE (cid:104) A γ ( P ) A δ ( P ) (cid:105) EE + (cid:104) A α ( P ) A γ ( P ) (cid:105) EE (cid:104) A δ ( P ) A β ( P ) (cid:105) EE + (cid:104) A α ( P ) A δ ( P ) (cid:105) EE (cid:104) A β ( P ) A γ ( P ) (cid:105) EE (A.4) Note the technical caveat we made in footnote 4. For the single box, we can use either language – normalspheres with box size L + 2 a or spheres whose center can reach the box walls, with box size L . The twoproblems are mathematically identical. (cid:10) ψ ∗ j ( X ) ψ i ( X (cid:48) ) (cid:11) EE = N i N j (cid:90) d N P d N P (cid:48) (cid:104) A j ( − P ) A i ( P (cid:48) ) (cid:105) EE δ (cid:16) P (cid:48) − mU i (cid:17) δ (cid:0) P − mU j (cid:1) exp( i (cid:126) ( P (cid:48) · X (cid:48) − P · X )) (A.5)Using (A.3), we get (cid:10) ψ ∗ j ( X ) ψ i ( X (cid:48) ) (cid:11) EE = δ ij N i N j (cid:90) d N P d N P (cid:48) δ N ( P − P (cid:48) ) δ ( P − P (cid:48) ) δ (cid:16) P (cid:48) − mU i (cid:17) δ (cid:0) P − mU j (cid:1) exp( i (cid:126) ( P (cid:48) · X (cid:48) − P · X )) (A.6)Let us focus on the Dirac delta part of the expression. We have δ ij δ N ( P − P (cid:48) ) δ ( P − P (cid:48) ) δ (cid:16) P (cid:48) − mU i (cid:17) δ (cid:0) P − mU j (cid:1) = δ ij δ N ( P − P (cid:48) ) δ ( P − P (cid:48) ) δ (cid:16) P (cid:48) − mU i (cid:17) δ (cid:0) P − mU i (cid:1) = δ ij δ N ( P − P (cid:48) ) δ ( P − P (cid:48) ) δ (cid:0) P − P (cid:48) (cid:1) δ (cid:16) P (cid:48) − mU i (cid:17) = δ ij δ N ( P − P (cid:48) ) δ (cid:16) P (cid:48) − mU i (cid:17) (A.7)Therefore, (cid:10) ψ ∗ j ( X ) ψ i ( X (cid:48) ) (cid:11) EE = δ ij N i N j (cid:90) d N P d N P (cid:48) δ N ( P − P (cid:48) ) δ (cid:16) P (cid:48) − mU i (cid:17) exp( i (cid:126) ( P (cid:48) · X (cid:48) − P · X ))= δ ij N i (cid:90) d N P exp( i (cid:126) ( P · ( X (cid:48) − X )) δ (cid:0) P − mU j (cid:1) (A.8)In particular, when X (cid:48) = X , we have (cid:10) ψ ∗ j ( X ) ψ i ( X ) (cid:11) EE = δ ij N i (cid:90) d N P δ (cid:0) P − mU j (cid:1) = δ ij N i (2 πmU j ) N/ Γ(3 N/ mU j ) (A.9)Let us normalize this quantity by demanding that (cid:90) D (cid:10) ψ ∗ j ( X ) ψ i ( X ) (cid:11) EE d N X = δ ij (A.10)24his gives us the normalization condition N − j = L N (2 πmU j ) N/ Γ(3 N/ mU j ) (A.11)Now let us look at another important expression: (cid:90) D d N X d N X (cid:48) (cid:104) ψ ∗ i ( X ) ψ j ( X (cid:48) ) (cid:105) EE (cid:104) ψ k ( X ) ψ ∗ l ( X (cid:48) ) (cid:105) EE = δ ij δ kl N i N k (cid:90) D d N X d N X (cid:48) (cid:90) d N P d N P (cid:48) δ (cid:0) P − mU j (cid:1) δ (cid:16) P (cid:48) − mU k (cid:17) exp( i (cid:126) ( P (cid:48) · ( X − X (cid:48) )) exp( i (cid:126) ( P · ( X (cid:48) − X ))= δ ij δ kl N i N k (cid:90) D d N X d N X (cid:48) (cid:90) d N P d N P (cid:48) δ (cid:0) P − mU j (cid:1) δ (cid:16) P (cid:48) − mU k (cid:17) exp( i (cid:126) ( X (cid:48) · ( P − P (cid:48) )) exp( i (cid:126) ( X · ( P (cid:48) − P ))= δ ij δ kl h N N i N k (cid:90) d N P d N P (cid:48) δ (cid:0) P − mU j (cid:1) δ (cid:16) P (cid:48) − mU k (cid:17) (cid:2) δ N D ( P − P (cid:48) ) (cid:3) (A.12)where we have defined δ N D ( K ) ≡ h − N (cid:90) D d N X exp( i K · X / (cid:126) ) (A.13)Let us look at the integral in the last line of (A.12). It is convenient to define this integralas a separate quantity as follows:Φ ij ≡ (cid:90) d N P d N P (cid:48) δ (cid:0) P − mU i (cid:1) δ (cid:16) P (cid:48) − mU j (cid:17) (cid:2) δ N D ( P − P (cid:48) ) (cid:3) (A.14)Now let us first look at the case where i = j . To do the integral explicitly, we will workin the low density limit where N a (cid:28) L . The factor δ N D ( P − P (cid:48) ) will effectively act as aDirac delta function throughout the range of variable P − P (cid:48) . Therefore, it suffices to usethe replacement [15] (cid:2) δ N D ( P ) (cid:3) → ( L/h ) N δ N ( P ) (A.15)this gives usΦ ij = (cid:90) d N P d N P (cid:48) δ (cid:0) P − mU i (cid:1) δ (cid:16) P (cid:48) − mU i (cid:17) δ N ( P − P (cid:48) )= (cid:90) d N P δ (cid:0) P − mU i (cid:1) δ (cid:0) P − mU i (cid:1) = (cid:90) d N P δ (cid:0) P − mU i (cid:1) = (2 πmU i ) N/ Γ(3 N/ mU i ) (A.16)25ow let us look at the case where i (cid:54) = j . If we used the same replacement as in (A.15),then we will see that the integral will be non-zero only when U i = U j . However, we can seefrom the explicit form of δ N D ( P − P (cid:48) ) that the integral (A.12) will be non-zero when U i and U j are sufficiently close to each other. In fact, it turns out that these states are the onesthat dominate the entanglement entropy calculation. Therefore, we will have to use a moreprecise replacement to do the computation. It suffices to use a Gaussian approximation asfollows (see [15] for a closely related but distinct expression) δ N D ( P − P (cid:48) ) (cid:39) ( L/h ) N exp (cid:104) − ( P − P (cid:48) ) L / π (cid:126) (cid:105) (A.17)Using the Gaussian approximation, we can see from [15] thatΦ ij (cid:39) Φ ii exp (cid:2) − m ( U i − U j ) L / π (cid:126) U i (cid:3) (A.18)Substituting this into (A.12) we get (cid:90) D d N X d N X (cid:48) (cid:104) ψ ∗ i ( X ) ψ j ( X (cid:48) ) (cid:105) EE (cid:104) ψ k ( X ) ψ ∗ l ( X (cid:48) ) (cid:105) EE = δ ij δ kl ( Lh ) N N i N k (2 πmU k ) N/ Γ(3 N/ mU k ) exp (cid:2) − m ( U j − U k ) L / π (cid:126) U j (cid:3) = δ ij δ kl (cid:18) Lh (cid:19) − N Γ(3 N/ mU i )(2 πmU i ) N/ exp (cid:2) − m ( U j − U k ) L / π (cid:126) U j (cid:3) = δ ij δ kl Z i exp (cid:2) − m ( U j − U k ) L / π (cid:126) U j (cid:3) (A.19)where we have defined Z i = (cid:18) Lh (cid:19) − N Γ(3 N/ mU i )(2 πmU i ) N/ (A.20)When | U j − U k | /U j smaller than or equal to ( (cid:126) /mU j L ) / , the exponential can effectivelybe replaced by 1 and we will get (cid:90) D d N X d N X (cid:48) (cid:104) ψ ∗ i ( X ) ψ j ( X (cid:48) ) (cid:105) EE (cid:104) ψ k ( X ) ψ ∗ l ( X (cid:48) ) (cid:105) EE = δ ij δ kl Z i (A.21)When the eigenstates are sufficiently close to each other in the energy spectrum, we can alsosee that (cid:90) D (cid:68) ψ ∗ i ( X ) ψ i ( X ) (cid:69) EE (cid:68) ψ i ( X ) ψ ∗ i ( X ) (cid:69) EE · · · (cid:68) ψ i n − ( X n ) ψ ∗ i n ( X ) (cid:69) EE n (cid:89) j =1 d N X j (cid:39) (cid:32) n (cid:89) j =1 δ i j − i j (cid:33) (cid:32) n − (cid:89) j =1 Z i j − (cid:33) (A.22) Note that the integration range that defines (A.13) is non-trivial, which is what makes it different froman ordinary delta function. 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