Quantum statistical : time dilation due to local information in many-body quantum systems

Dries Sels TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium and Department of Physics, Boston University, Boston, MA 02215, USA

Michiel Wouters TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium (Dated: January 8, 2018) We propose a generic mechanism for the of a gravitational potential that acts on all classical objects in a quantum system. Our conjecture is based on the analysis of mutual information in many-body quantum systems. Since measurements in quantum systems affect the surroundings through entanglement, a measurement at one position reduces the in its neighbourhood. This reduction in entropy can be described by a local temperature, that is directly related to the gravitational potential. A crucial ingredient in our argument is that ideal classical mechanical motion occurs at constant probability. This definition is motivated by the analysis of entropic forces in classical systems, which can be formally rewritten in terms of a gravitational potential.

I. INTRODUCTION verse, it would ideally emerge in any complex quantum theory. In this paper, we will argue that this might be the case. After almost one century of coexistence, the relation between Einstein’s theory of and quan- The first ingredient in our argument is that the pres- tum physics is still not well understood. At the same ence of matter at some place in the universe constitutes time, the precise relation between microscopic , defined as missing entropy [16, 17]. Within physics and macroscopic classical physics has not been the framework of quantum mechanics, when knowledge completely demystified. There are some suggestions that about a particular realisation of the system is available, these two problems are related [1–5]. the incompatible part of the wave function has to be pro- Two fundamentally different strategies are used to re- jected out. When the quantum system has entanglement, late the quantum to the classical. The first one is based a local projection also influences the probability distribu- on the wave-particle duality and is most succinctly ex- tion in its vicinity. We will show that the effect of local pressed in the path integral formulation. The corre- information on its surroundings can be described by a po- spondence between the classical and the quantum is here sition dependent ‘entanglement’ temperature [18, 19], de- mathematically very direct, because it is the same action fined by approximating the local reduced density matrix that appears in both theories. It can therefore be used to by a Gibbs state. We then demonstrate that the inhomo- go from the classical to the quantum and vice versa. A fa- geneity of the entanglement temperature is reflected in a miliar example is electromagnetism: Maxwell’s equations spatial variation of the magnitude of the energy fluctua- are derived from the quantummechanical path integral by tions. means of the stationary phase approximation. This mechanism is most easily illustrated in the EPR The second strategy, statistical physics, is fundamen- setting. When the first qubit of a Bell state |ψi = √1 (|↑↑i + |↓↓i) is not measured, the reduced density ma- tally different. From a quantum theory, thermodynamic 2 relations can be computed, but those thermodynamic re- trix of the second qubit is maximally mixed, equivalent lations cannot be quantised. Still, it is the only method to infinite temperature. On the other hand, when the suited for the description of complex macroscopic sys- first one is measured in the ↑, ↓ basis, the second qubit is arXiv:1602.05707v1 [quant-ph] 18 Feb 2016 tems, about which we only have thermodynamic and hy- in a pure state, which corresponds to a zero temperature drodynamic information [6, 7]. density matrix (see Fig. 1). The main efforts to find a quantum mechanical de- It is actually well known that entanglement allows to scription of gravity have been based on the first method make the connection between pure quantum states and [8, 9] but also the second strategy has been explored [10– statistical density matrices [20, 21]. Tracing over envi- 12]. The latter efforts go under the name of thermody- ronment degrees of freedom leaves the system typically namic or entropic gravity. Conceptually, this strategy in a canonical Gibbs state, even if the composite state is seems preferable, since the physics of gravity deals with pure [22]. We make a natural extension of this work by macroscopic objects, that have nonzero entropy and that considering what happens if the environment is not fully are coupled to environments [13–15]. In thermodynamic traced over. gravity, the gravitational interaction is seen as emergent The main assumption in our quantum statistical ap- rather than as an explicit ingredient in the microscopic proach to gravity is that the magnitude of energy fluc- theory. Gravity being the most universal force in the uni- tuations in a region sets the energy scale for the local 2

before Alice’s measurement A B

T=∞

after Alice’s measurement A B

T=0

Figure 1. A thermodynamic interpretation of the EPR ex- Figure 2. Illustration of the entropically driven depletion force periment. Before the measurement, Bob’s state is maximally between two particles immersed in a fluid. When the two mixed, corresponding to infinite temperature. Alice’s mea- depletion regions overlap (marked by the dashed circles), the surement reduces Bob’s state to a pure state, which is a zero liquid has more volume and hence a larger entropy. This temperature state. results in the attraction between the two immersed particles. physics. This follows from the fundamental characterisa- are discussed in Sec. V. By considering a global thermal tion of phenomena by their probability. We will demon- equilibrium state, we recover in Sec. VI the Tolman law, strate that it is intimately related to the equivalence prin- that describes how the temperature of a thermal equilib- ciple. ‘Gravitational red shifts’ then appear because the rium state varies in the presence of a gravitational field entanglement temperature is lower closer to a source of [27]. information. We will also show that a position depen- After some considerations on the relation between in- dent entanglement temperature, implies the acceleration formation and entropy (information is missing entropy of semi-classical wave packets. [17]), we speculate on the nature of black holes in Sec. In our scenario, gravitational forces emerge as an inter- VII. In the logic of gravity emerging from information, play between quantum measurement and the statistical we are led to conclude that black holes most naturally meaning of classical mechanics. Usually, we do not think are the objects with maximal information, hence mini- about mechanical systems in terms of probability distri- mal entropy. We relate the origin of the Hawking-Unruh butions, except in systems where ‘entropic’ forces occur temperature [28, 29] to quantum fluctuations in the po- [23, 24]. We therefore start our paper with a short de- sition of particles. scription of entropic forces in Sec. II. It is illustrated Most of the ideas that we use, have appeared in some that the entropic attraction between two objects is due form in other works. The last part of our paper is there- to correlations in their joint probability distribution. We fore devoted to the connections between our viewpoint emphasise that entropic forces are not due to an increase and the related literature. in the entropy, but rather due to the constraint of motion at constant entropy. We point out the close analogy with Born-Oppenheimer forces. In a quantum system, the role of the probability dis- II. CLASSICAL ENTROPIC FORCES tribution of local states is played by the reduced den- sity matrix. Correlations between regions are reflected Theoretical classical mechanics deals with isolated sys- in the quantum mutual information [25, 26]. An entropy tems, that satisfy energy conservation. This excludes sys- and a temperature can be associated to the reduced den- tems that are coupled to baths and therefore it is only an sity matrix. We show that local measurements can af- approximate description of real physical systems, where fect the temperature of the regions that are correlated friction due to coupling with an environment is always with it. Our general arguments about local information present. In addition to friction, environments can also in quantum systems are illustrated by calculations on a exercise forces on mechanical systems. These are known one-dimensional non-interacting fermion model in Sec. as entropic forces [24] and are essential in e.g. deple- III. tion forces between colloidal particles in fluids, osmotic From the requirement of probability conservation of a pressure and the elasticity of polymers. wave packet, we show in Sec. IV that a temperature gra- Let us start with an elementary analysis of entropic dient leads to acceleration. The gravitational redshifts forces. It is the simplest setting in which one can see 3 that probability conservation is a generalisation of en- implicit aspects to the definition of a classical mechan- ergy conservation. We consider the classic example of ical object: it is an unlikely excitation that evolves at two spheres immersed in a fluid (see Fig. 2). Due to constant probability. repulsive interactions between the sphere and the fluid, The above discussion assumed a canonical picture there is a depletion region around each sphere. The pres- where the temperature T is fixed. To further illustrate ence of the spheres therefore lowers the entropy of the the physics behind entropy driven forces, it is instructive fluid (it reduces the available volume for each molecule). to analyse the microcanonical situation, where the total When the depletion regions of the two spheres overlap, energy is fixed. When the spheres are accelerated, their the entropy of the fluid increases. The probability in kinetic energy is provided by reducing the internal energy terms of their positions x1,2 and momenta p1,2 reads of the molecules. We will restrict our discussion to the case where one of the two spheres is fixed and the second 2 2 p1 p2 1 S(x1−x2)− − sphere can move only in one dimension. The discussion p = e 2m1T 2m2T , (1) Z that follows below is generally valid for the motion at constant entropy. One could for example also apply it to where T is the temperature (we set kB = 1) and Z a nor- the adiabatic expansion of a piston filled with a gas. malisation constant. In the absence of the entropic term The condition of no entropy production dS = 0 yields S in the probability (1), energy conservation is equiva- lent to probability conservation and the temperature T ∂S ∂S dS = dx + dE = 0, (4) is irrelevant. ∂x ∂E In the presence of S on the other hand, probability where E refers to the energy of the gas. We reobtain the and energy conservation are no longer the same. Energy from Eq. (3) conservation remains unaltered, but probability conser- vation requires dP dE ∂S = − = T , (5) 2 2 dt dx ∂x p1 p2 − T ln p = + − TS(x1 − x2) (2) where we have used the thermodynamic definition T = 2m1 2m2 dE dS and denote the momentum of the sphere by P . The to be constant. A term −TS is added to the energy, gas loses energy by performing work against the sphere, which gives rise to ‘entropic forces’ in order to keep the entropy constant. In a Born-Oppenheimer language, the energy in the dpi fast degrees of freedom (the gas) decreases under a dis- = T ∇x S. (3) dt i placement of the slow degree of freedom (the sphere). Such forces have been conclusively observed experi- From the requirement of constant entropy for the gas, mentally [30]. We thus conclude that probability con- we can thus derive the Born-Oppenheimer potential, that servation is more powerful than energy conservation in governs the quantum-mechanical dynamics of the sphere. mechanical systems. Note that it is not the increase of the entropy but its We wish to point out that phenomena that involve conservation that is responsible for the acceleration. En- changes of the probability are considered to be outside tropy increase corresponds to dissipation and hence falls of the scope of ideal classical mechanics, as follows from outside of the scope of mechanical motion. Liouville’s theorem. In physical terms, it corresponds to After the displacement, the decrease in internal en- ergy is reflected in a change of the temperature. From the absence of dissipation. In ‘good’ mechanical systems, 3 the entropic term is negligible as compared to the other equipartition, we have that E = 2 NT , where N is the terms in (2), but as we will argue below, this omitted number of molecules in the gas. This gives the relation term could be responsible for the emergence of gravity dE d ln T = E . (6) when analysing classical objects in quantum systems. dx dx The reason why the temperature never appears in classical mechanics is that one implicitly requires that The momentum change can be rewritten in terms of the the mechanical energy E is much larger than the ther- position dependence of the temperature as mal energy. This means that classical mechanics is dP d ln T only concerned with statistically unlikely events, with = −E . (7) dt dx p ∼ e−E/T  1. This may seem contradictory to the standard classical to quantum correspondence, where the We conclude that a temperature gradient leads to a force classical paths are the most likely, but corresponds to that is proportional to the internal energy. classical objects carrying a large amount of information Written in this form, the momentum change under adi- (see Sec. VII). The meaning of the deterministic motion abatic expansion of a gas is formally very close to the is that, conditional on being at position x at time t, there change in momentum due to gravitational forces. It can is unit probability to be at a place x0 at time t0. be written as In quantum physics, where probabilities are the most dP E dφ = − , (8) elementary quantities, it is then natural to elevate these dt c2 dx 4 where φ is the ‘gravitational’ potential defined as In order to make our discussion more concrete, we per-

2 form some calculations on a specific model. The choice φ = c ln T (9) of models is quite limited, since generic interacting quan- For classical systems, these formal manipulations do tum systems have a prohibitively large Hilbert space to not yield any new physics beyond the standard thermo- perform explicit calculations. The simplest systems from dynamic analysis, but we will show below that measure- a theoretical point of view are the ones with quadratic ments in quantum systems lead to a position dependence Hamiltonians, that describe free quasi-particles. We will of the local temperature. Thanks to the analogy between here consider the ground state of the fermionic hopping temperature gradients and gravitational interactions, we Hamiltonian of the form are then led to the appearance of gravity as a consequence X h †  † i Hˆ = − tcˆ cˆ + h.c. + µcˆ cˆ ) , (10) of local information in quantum systems. i+1 i i i i where t is the hopping amplitude and µ is the chemical III. LOCAL INFORMATION AND potential. Apart from a Hamiltonian, we need to specify TEMPERATURE IN QUANTUM SYSTEMS the quantum state of the system. Here, we will consider the ground state of the Hamiltonian, but the analysis After these preliminary remarks on classical mechan- could be extended to excited states as well. ics, we turn our attention to quantum systems. The cen- All expectation values within a subregion A are de- tral object in quantum mechanics of closed systems is its scribed by the reduced density matrix ρA = TrS\Aρ, wave function. We will consider quantum systems that where the trace is over all sites except the ones in A. are defined on a lattice. A typical example is the Hub- Since Hˆ is a quadratic Hamiltonian, its ground state sat- bard model. At first sight, one would have little hope for isfies Wick’s theorem. The reduced density matrix ρA is the emergence of relativity out of such systems, but Lieb therefore also specified by the second order correlation and Robinson showed in their seminal work that causal- functions and is quadratic in the creation and annihila- ity emerges in such systems under very general conditions tion operators on region A as well: [31]. ˆ The time evolution of a quantum system is described ρˆA = exp(−HA). (11) −iHtˆ by a unitary operator U(t) = e (we set ~ = 1). A The ‘modular’ or ‘entanglement’ Hamiltonian can be severe objection for this construction to be related to written as our universe is that it selects preferential coordinates. The point of our work is however that the parametric ˆ X † HA = aˆi hijaˆj. (12) time t does not necessarily correspond to physical time. i,j In the regimes that we will discuss, the physical time depends on parametric time through a space-dependent The matrix hij can be found by a diagonalisation of the time dilation factor, just as in the Newtonian limit of correlation matrix (see Appendix for more information). Einstein’s gravity. An example for NA = 10 sites is shown in Fig. 3. The Before going to the technical analysis of a specific ex- structure of the original Hamiltonian is clearly visible in ample, it is worth spending a few words on the issue the entanglement Hamiltonian. When comparing entan- of locality. While the natural mathematical space of glement Hamiltonians for different subsystem sizes, one quantum mechanics is the Hilbert space of its quantum finds that the main effect is that the magnitude of the states, physical systems are also endowed with a coordi- matrix elements increases. We thus find that the entan- ˜ nate space. This coordinate space is important, because glement Hamiltonian is of the form hA = βAhA. We will physical Hamiltonians are not arbitrary Hermitian oper- call this scale dependent temperature TA the ‘entangle- ators acting on the Hilbert space, but show additional ment’ temperature [18, 19]. Fig. 3b shows the depen- ˜ structure. In particular, the majority of physical Hamil- dence of βA on system size, where hA is normalised by tonians are local, which means that they couple only sites the largest tunneling matrix element. A linear increase that are close to each other. It is for this class of Hamil- of the effective inverse temperature with system size is tonians that Lieb-Robinson causality emerges. apparent, in agreement with analytical calculations ex- A second important property of the local Hamiltonians ploiting the adS/CFT correspondence [33]. concerns the entanglement entropy of their ground states. This behavior can be understood from the fact that, If we only want to know local quantities, it is sufficient unlike the entropy of a Gibbs state, the entropy is not to know the reduced density matrix of the region we are extensive. For 1-D free fermions it is well know that the interested in. Since this involves a loss of information, ground state entanglement entropy is S = 1/3 log(L). the reduced density matrix will have in general nonzero The entanglement temperature must thus decrease such entropy, the entanglement entropy. The entanglement that the entropy of a Gibbs state of the local (smooth) entropy is typically proportional to the number of sites in Hamiltonian equals the entanglement entropy. the local region. For ground states of local Hamiltonians Let us now consider a subregion AB that consists of however, it scales subextensively with system size [32]. two disconnected parts A and B, separated by a distance 5

1 (a) 12 (b) 12 2 (a) (b) 10 10 3 5 4 8 8

5 A

β 6 10

t 6 6 4 7 4 15 8 2 2 9 0 2 4 6 8 0 5 10 20 0 N 5 10 15 20 0 10 20 A

Figure 3. (a) The entanglement Hamiltonian h (left) for the ij Figure 4. (a) The entanglement Hamiltonian h for a subsys- fermionic Hamiltonian (10) on a subregion of 10 sites. The ij tem consisting of two disjoint regions (10 sites each) separated structure of the original Hamiltonian is clearly visible, with by a distance of 15 sites. (b) Tunneling matrix elements (first hopping matrix elements next to the diagonal. (b) The en- off-diagonal) are compared with the case of a single region of tanglement temperature increases as a function of the system 10 sites (dashed lines). size.

0 R. In terms of the entanglement Hamiltonian, there is 10 (β -β )/β a difference between h and the direct sum h ⊕ h . AB A A AB A B -2 10 I Because more information is present in the compound J AB subsystem AB, larger matrix elements are found in hAB: 10-4 (Θ -Θ )/Θ the joint system is at a lower temperature than the in- A|B A A dividual systems. Physically, this means that the uncer- 10-6 tainty about the state A is reduced when information is obtained about system B. 10-8 100 101 102 It is instructive to look at the limiting cases R = ∞ d and R = 0 when A and B have the same number of sites. When A and B are infinitely far apart they should not Figure 5. The mutual information IAB as a function of the dis- be entangled and the total entanglement Hamiltonian is tance between two subregions of sizes NA = NB = 10 (blue), just the direct sum entanglement Hamiltonian, hence the the effective temperature of the density matrixρ ˆAB (red), the entanglement temperature is the same as for the individ- average conditional entropy J for projection on the Schmidt ual systems. However, when the two systems touch they basis in region A (orange) and the energy average fluctuation simple form one system that is twice as big. It was al- width after projection on the Schmidt basis (green). ready shown above that the entanglement temperature is only half of that of the separate subsystems. This is illustrated in Fig. 4, where we show the en- is gapless [32]. The red line shows the effective temper- tanglement Hamiltonian consisting of two subregions of ature βAB of the joint density matrixρ ˆAB, which shows 10 sites that are separated by 15 sites. The left hand the same behavior at large distances. This shows a clear panel shows the full matrix, where one can clearly iden- connection between the increase in mutual information tify the direct sum of uncoupled entanglement Hamilto- and the decrease in effective temperature when the two nians and some off-diagonal couplings. The right hand subregions are brought closer together. panels show the tunneling matrix elements and compares For classical systems, one can rewrite the mutual in- them to the case of a single subregion (dashed lines). Two formation in terms of conditional as: features stand out. First, the tunneling matrix elements are larger, corresponding to a larger inverse temperature: X βAB > βA. Secondly, there is also an asymmetry in the IAB = S(A) − p(xB)S(A|xB), (14) matrix elements, which could be interpreted as a temper- xB ature gradient. The entropy of the total system AB is also different where the conditional entropy S(A|xB) is defined in from the sum of the entropies of A and B. The difference terms of the conditional probability distribution as between the two is the mutual information S(A|x ) = − P p(x |x ) ln p(x |x ). B xA A B A B For quantum systems unfortunately, the situation is IAB = SA + SB − SAB, (13) more complicated, because conditional probabilities are which is always positive. Fig. 5 shows the mutual infor- no longer simply defined in terms of joint probabilities, mation as a function of the distance for our free fermion but by means of projection operators. It has been found toy system (blue line). The mutual information is seen that for quantum systems, the second definition of the to decay slowly, with a power law behavior at large dis- mutual information is always smaller than the first one. tance. This can be attributed to the fact that the system This has led Zurek to introduce the notion of ‘quantum 6 discord’ as [34] teristic function

−βE (Hˆ −E¯)+βP Pˆ D = IAB − max J A (ρ), (15) φ(βE, βP ) = lnhe i. (17) A Πj Πj For the probability, we have where J is the conditional entropy that depends on A the set of projective measurements Πj . Unfortunately, ln p(E,P ) = min [φ(βE, βP ) + βEE − βP P ]. (18) the optimisation problem over measurements has been βE ,βP proven to be NP complete [35]. We can rewrite this in a more physical way by introduc- If one is interested in the physics of a region A, one ing the velocity v = βP /βE (and we set βE = β). The would want to know the reduced density matrix ρA. probability then reads Imagine now that a measurement in region B is per- formed. When there are correlations (quantified by the ln p(E,P ) = min[φ(β, v) + β(E − vP )]. (19) mutual information) between these two regions, the re- β,v sult of this measurement will affect the density matrix in This gives us the relations region A. In general, the conditional entropy SA|B will be lower than SA, with the mutual information as an up- ∂φ 1 ∂φ per bound on the average entropy reduction for a certain E − vP = − ,P = . (20) ∂β β ∂v type of measurement. In view of the relation between entropy and temperature, this means that the tempera- Let us now consider a situation where the probability ture of the region A depends on information about region distribution depends on the position x of region A. As in B. In other words, a measurement in B results in a posi- the classical case, we can now obtain the entropic force tion dependent temperature in its surroundings. Regions by requiring constant probability for the fluctuation: close to B will be more affected than regions farther away, as indicated by the behavior of the mutual information. d ln p = d[φ(β, v, x) + β(E − vP )] = 0. (21) The simplest class of projections to be performed on this system is a projection on the Schmidt basis in region Using the relations (20), we can simplify this to B. While this is clearly not the best basis in the sense ∂φ of Eq. (15), it gives at least a lower bound on the av- βdE − βvdP + dx = 0. (22) erage amount of information that can be gained with a ∂x projective measurement. From Fig. (5), it is seen that Since there is no external potential acting on the system, conditional entropy J is lower than the mutual informa- the magnitude of the energy fluctuation E should remain tion and decays faster as a function of the distance, but the same. We then find with v d = d that it still shows a power law type decay. dx dt dP 1 ∂φ = . (23) dt β ∂x IV. ENERGY STATISTICS AND MECHANICAL ACCELERATION Let us now consider the situation with small velocity and neglect the momentum in the probability distribu- In order to make contact with fundamental physics, tion. The typical distribution will be Gaussian in energy one should think about the density matrix before mea- fluctuations, which corresponds to a Gaussian character- surement as the ‘vacuum’. An unlikely measurement out- istic function come would then correspond to the detection of matter. Θ2(x) If matter is measured to be present in region B, this φ(β) = β2. (24) measurement then affects the statistics in region A, de- 2 pending on the distance between A and B. The Legendre transform reads The probability to have a fluctuation with energy E and momentum P in region A can be written as the E2 Fourier transform E = −Θ2(x)β, ln p(E) = − . (25) 2Θ2(x) Z dβE dβP ˆ ˆ ¯ ˆ p(E,P ) = eiβE E−iβP P he−iβE (H−EA)+iβP P i, 2π 2π For the force (23), we then get (16) ¯ 1 ∂φ 0 d ln Θ(x) where EA is the average energy in region A. The average = Θ(x)Θ (x)β = −E , (26) β ∂x dx is taken with respect to the density matrix ρA: hOˆi = ˆ Tr(ρAO). so that (for small velocities) For large fluctuations, the saddle point approximation can be used. It allows us to write the probability distri- dP d ln Θ(x) = −E . (27) bution in terms of the Legendre transform of the charac- dt dx 7

The change in momentum is proportional to the energy 3. In order to conserve their probability, excitations in and the gradient of the standard deviation of the en- regions with different temperatures have different ergy. This results in acceleration towards regions with energies, which leads to acceleration. less fluctuations. Note the analogy with the classical en- Looking back at the ingredients that led to the (weak) tropic force, where the internal energy is also reduced by , the most crucial step is the require- the work performed against the spheres. ment that E/Θ(x) is constant, motivated by constant Alternatively, we could have derived the entropic force probability. This is our translation of the physics of ideal (27) by requiring a constant probability for the internal mechanical motion into the language of statistics. energy E int Einstein translated the physics of ideal mechanical mo- 2 tion (free falling objects) into the language of classical Eint ln p(Eint, x) = − . (28) field theory as ‘general covariance’: the form of the equa- 2Θ2(x) tions should be independent of the coordinates. This We then obtain immediately from principle has turned out to be hard to implement in a quantum setting. The reason could be the fact that quan-   Eint tum mechanics is fundamentally a statistical theory, re- d = 0 (29) Θ(x) quiring a direct formulation of physical processes in a probabilistic language. that there is a gradient in the internal energy, correspond- ing to a force V. GRAVITATIONAL REDSHIFTS dE d ln Θ F = − int = −E . (30) dx int dx Gravitational redshifts can also be seen as a direct con- The principal ingredient that is responsible for the en- sequence of the position-dependent temperature. When tropic force is the spatial variation of the energy fluctu- light that is emitted by a certain source (e.g. a burning ations. As we have argued in the previous section, these candle) is detected, we infer its expected energy from the can arise from local information about the system. The observed phenomenology. According to our line of rea- requirement that the internal energy of classical objects soning, it is however more natural to take the dimension- follows the local temperature then implies a spatially de- less ratio of the energy to the local temperature E/Θ(x) pendent speed of time, i.e. a time dilation. In the next to be the fundamental characterisation of a phenomenon. section, we will further discuss how the local temperature This argument is in line with standard thermodynamics is directly related to the red shift. where only relative temperatures can be measured (e.g. In Fig. 5, we plot the energy variance after projection by means of the Carnot efficiency). The element that we on the Schmidt basis (as for the computation of J) with add to this discussion is that al- a green line. The spatial dependence of the energy fluc- lows to define a fundamental temperature of empty space. tuations follows the behavior of the conditional entropy, Under our assumption this temperature sets the scale for establishing a link between entropy and energy fluctua- the local physics, the same phenomenon at a different tions. This suggests that the energy fluctuations Θ are position can have a different energy. We come again to proportional to the entanglement temperature. We will time dilation as a consequence of local information. therefore call Θ the ‘local temperature’ in the following. When the light propagates, it does not change its If we interpret the energy E in Eq. (27) as the rela- frequency, because the propagation is governed by the tivistic rest energy E = mc2 of the excitation in the sense hamiltonian Hˆ and there is no internal energy that can that p = mv, we obtain the acceleration change with position. We then find the frequency of a photon emitted at position x to differ from a photon d ln Θ a = −c2 . (31) emitted by the same phenomenon at position x0 by a dx factor 0 The local temperature can then be identified with the ν(x) Θ(x ) 2 = = e∆φ/c , (33) Newtonian potential as ν(x0) Θ(x) φ = c2 ln Θ. (32) where ∆φ = φ(x) − φ(x0). For the last equality, we have used the definition of the gravitational potential (32). It Let us recapitulate how in our view gravity emerges gives the same relation between the red shift and the from quantum mechanics gravitational potential as in general relativity [36, 37]. 1. Local properties of quantum systems are described by a reduced density matrix, characterised by a lo- VI. THE TOLMAN EFFECT cal temperature. 2. Local information leads to a spatial dependent tem- It is also interesting to consider a thermal equilibrium perature. state of the real Hamiltonian Hˆ at nonzero temperature 8

T , that is constant in space. According to the discussion how we see possibilities for the understanding of black of gravitational redshifts, this constant temperature will hole physics from our statistical perspective. appear to be different at different positions. The physi- When the amount of matter in a certain region is de- cal temperature measured with a local thermometer then fined by the information in the density matrix, i.e. the corresponds to the ratio of the actual temperature T to missing entropy, the natural upper bound to the amount the local temperature: Tphys(x) = T/Θ(x). of matter a given region can contain is the entanglement Using (33), we recover the same relation between the entropy of the unmeasured quantum state. The remain- spatial dependence of the physical temperature and the ing entropy of the maximum information state is then red shift zero. The objects with maximal information (matter) in a given region of space are most naturally identified with T (x) Θ(x0) ν(x) phys black holes. It seems however unlikely that the entan- 0 = = 0 (34) Tphys(x ) Θ(x) ν(x ) glement entropy can be zero for stable objects, because the Hamiltonian couples all the regions in space, lead- as in the Tolman law in a static metric [27]. ing to decoherence, but it seems reasonable to conjecture that black holes are the stable objects with the minimal VII. SPECULATIONS ON BLACK HOLES AND entropy that can be attained in a given region. THE HAWKING-UNRUH TEMPERATURE Conceptually, this would be very comforting: the ele- mentary objects that appear in the culmination of classi- cal physics, Einstein’s theory of general relativity, would The mysteries concerning black holes have played a carry the minimal quantum uncertainty. The inaccessi- big role in the quest for a quantum mechanical theory of bility of the interior then simply comes from gravity. Specifically, the understanding of their entropy the fact that its state is fixed by the requirement of min- and loss of information constitute important theoretical imal entropy. The explosion of a star at the end of its life challenges. Before speculating on how black holes may fit is the ultimate cooling to a state with the least possible into our picture of quantum statistical gravity, we start entropy. by making some general comments on the relation be- It is impossible to speculate on the quantum nature tween information, probability and entropy. of black holes without mentioning the Hawking temper- When a measurement is performed on a quantum sys- ature, caused by quantum fluctuations [28]. We believe tem with an a priori density matrix ρ , the acquired infor- 0 that the mechanism could be quite simple. In our dis- mation changes the density matrix to ρ. The amount of cussion of mechanical acceleration, we have seen that the information that is obtained by this measurement can be spatially dependent temperature results in a spatially de- quantified by the relative entropy or the Kulback-Leibler pendent internal energy. When there is an uncertainty in distance [25] the position, this results in an uncertainty of the energy ∆E = (∂ E)∆x. When we now take for the minimum I(ρ) ≡ DKL(ρ||ρ0) = −Tr(ρ ln ρ0) + Tr(ρ ln ρ). (35) x uncertainty the ‘Compton’ wave length ∆x = ~c/E, we It expresses how surprising measurements are on a system find ∆E = (~c/E)∂xE, which can be written as with density matrix ρ, when you thought that the density a matrix was ρ0. ∆E = ~ ∼ T . (37) The simplest situation, which is the most relevant for c U typical thermodynamic systems, is when all N states in 0 where we have used that the acceleration is equal to ρ can be assumed to have the same probability (micro- 0 a = (c2/E)(dE/dx). We obtain an uncertainty on the canonical ensemble) and in ρ, a subset of N states in ρ 0 energy that is of the order of the Unruh temperature (defining a ‘macrostate’) are occupied. We then find for T = a/(2πc). For a black hole, the acceleration is re- the information I = S(ρ ) − S(ρ), were S(ρ ) = ln N U ~ 0 i i placed by the surface gravity, resulting in the Hawking is the entropy. In words, the information in a state is temperature T . equal to the missing entropy [17]. It is also immediate to H In our picture, there is no information paradox [38, 39], relate the information to the probability, as first done by since it is the natural evolution of unitary quantum me- Einstein chanics to wash out a local suppression in the entan- p = eS = eS0−I . (36) glement entropy. It is the basic mechanism that leads to thermalisation in quantum many body systems. The Here we see clearly that the most unlikely macrostates evaporation of a black hole is then in principle no dif- contain the largest amount of information. ferent to the mixing of hot and cold tea. The physical In the context of gravity, the macrostates are specified interpretation of this solution is however quite exotic: by a certain distribution of matter. But we should turn it is not a remnant, but the vacuum that is entangled the argument around and try to define matter through with the emitted radiation. In our view, empty space the information in the local density matrix. This prob- has a higher entanglement entropy than space that con- lem is however beyond the scope of the present discus- tains matter excitations (information). This is opposite sion. We will restrict to some speculations, indicating to the traditional assumption that entropy is carried by 9 quasi-particles. in his legendary discussions with Einstein on the Heisen- berg uncertainty principle [43]. Einstein did not want to accept the probabilistic character of quantum mechanics, VIII. LORENTZ TRANSFORMATIONS which Bohr saw to be the very foundation of the theory. When Einstein proposed to violate the energy-time un- So far, we have discussed time dilation of the ‘gravi- certainty principle through a measurement of time with tational’ type. A more elementary type of time dilation a clock and energy with a scale, Bohr saved it by in- already occurs in special relativity under Lorentz boosts. voking the uncertainty in the time measurement induced We want to discuss here how one can understand the by the uncertainty in the position of the clock. As we need for this type of time dilation from an information have discussed in Sec. VII, this could be the underlying analysis. mechanism for the Hawking-Unruh effect. A simple theoretical model in which Lorentz transfor- As discussed in the introduction, the famous EPR pa- mations can be investigated is that of a low temperature per on the incompleteness of quantum mechanics [44] is superfluid Bose gas. It has emergent Lorentz invariance directly connected to our mechanism for gravity. The for its excitations, that have a linear dispersion ωk = c|k|. main conclusion of the paper was phrased as the incom- Let us consider a box of volume V that contains thermal pleteness of quantum theory, because of the instanta- excitations at temperature T and moves at a speed v with neous effect of the projection on distant regions. It is respect to the superfluid. Its entropy equals [40] important to discuss this objection to quantum mechan- ics in the context of our work. An obvious criticism to our T 3V S = C p , (38) mechanism would be that it allows gravitational interac- ( 1 − v2/c2)4 tions to be mediated faster than light. We believe that where C is a numerical constant. This formula shows that this problem should be addressed by a deeper analysis of the entropy depends on the velocity. In order to keep the the ‘measurement problem’. In this paper, we have not entropy constant under a boost, the temperature should addressed measurements in detail, due to a lack of math- change. According to our previous arguments, a change ematical techniques to properly deal with it. But when of the temperature corresponds to a rescaling of time. we say that there is information about a part of the sys- When the lengths are measured with sound waves, the tem, we do not think of suddenly doing a measurement. time dilation will have to be accompanied by a length We are aware of the fact that the earth exists, without us contraction in order to keep the measured distances the being able to decide that we are going to make a measure- same. To a rescaling of the temperature with a factor ment. We rather think of measurement as ‘post-selection’ T → γ−1T then corresponds a rescaling of the volume than an action. For this scenario to work, the informa- with the same factor V → γ−1V . tion that leads to gravity should be stable. The operators The entropy of the moving system is then equal to the that are measured should therefore be very slow opera- entropy at rest when γ−1 = p1 − v2/c2. We thus recover tors. Mathematically, this corresponds to the operators ˆ the usual Lorentz transformation of the temperature four that almost commute with the Hamiltonian H. Recently, vector [40] (β, 0) → (γβ, γvβ) from the requirement that the question of finding slow observables was addressed for the entropy should be conserved under boosts. one-dimensional quantum spin chains [45]. Here, we used as an example the superfluid Bose gas Further suggestions for a deep connection between with emergent Lorentz invariance, but it could expected gravity and measurements in quantum systems came that the Lorentz invariance of the entropy holds for all from Penrose and Di´osiwho argued that superpositions quantum many body systems that show causality in the of different , and thus gravitational fields, sense of the Lieb-Robinson bound. Unfortunately, we are should be impossible. By this argument, they were led not aware of any proof of this statement. to introduce a fundamental rate of decoherence, deter- mined by gravitational interactions [46, 47]. We propose to solve the same problem in a different way. In our view, IX. RELATION TO OTHER WORKS classical gravity emerges as a consequence of classical in- formation about a quantum system and can therefore The intimate relation between temperature and infor- never be in a superposition. As long as the information mation dates back to Maxwell’s demon thought experi- is not available, it cannot have a gravitational effect and ment [41, 42]. He discussed how a demon that had access not influence the structure of . We agree that to the velocities of individual atoms in a gas can reduce gravity should be restricted to the classical realm, but we the temperature by opening a door when a slow atom disagree with Penrose on his conclusion that the ultimate passes. In other words, when we have information about theory of reality should be classical [1]. We opt for the the system, its effective temperature is lower. We add quantum to be the fundamental level of description. to this that information in a quantum system affects the The idea that gravity is a thermodynamic phenomenon temperature in its vicinity through entanglement. dates back to Jacobson, who presented a derivation of the The connection between measurements in quantum full Einstein equations from the laws of thermodynamics, mechanics and time dilation was already made by Bohr combined with the Unruh effect [29]. We were inspired by 10 the work by Verlinde who, building on the ideas of Jacob- argues that regions that disentangling parts of a system son, made the conjecture of an ‘entropic origin’ of gravity results in the spacetime pulling apart. His arguments are [36]. He attributed an amount of SBH to each volume in based on holography, but are in spirit close to ours. When space, close to our interpretation of the vacuum having two regions are not entangled, the mutual information is high entropy. There is however an important differences zero and they cannot influence each other gravitationally. between our and Verlinde’s works. Where we consider an Van Raamsdonk’s work, as well as Verlinde’s together underlying local temperature Θ(x) as the fundamental the large majority of research activity on quantum grav- object, Verlinde takes Unruh’s relation between temper- ity is nowadays carried out in the context of adS/CFT ature and acceleration as a postulate. Our underlying correspondence [62]. The connection [63] between the analysis allows to justify it. multiscale entanglement renormalisation ansatz (MERA) In spirit, our work is also close to the ‘thermal time hy- [64] for correlated quantum systems and the adS/CFT pothesis’ of Connes and Rovelli [48]. From formal mathe- correspondence may provide some links with our ap- matical considerations, they came to the conclusion that proach. Also the entanglement temperature of subre- the entanglement Hamiltonian should dictate the speed gions was studied in this framework [18, 19, 33, 65, 66]. of time. In subsequent work, they also connected the In particular, in Ref. [33], a force in the direction of lower thermal time hypothesis to the Tolman effect by com- entanglement entropy was found, in agreement with our paring the ‘thermal time flow’ to the ‘mechanical time arguments of a force toward lower effective temperature. flow’ [49, 50]. We would interpret the latter as the time Connections between , black holes and in- flow of the real Hamiltonian Hˆ , where the former is the formation theory have also been made by Lloyd [67, 68]. time we extract from the local temperature. He bases his analyses on the computational interpreta- Gravity as an effective theory for condensed matter tion of entropy (bits), temperature (operations per bit systems is a subject that has attracted significant inter- per unit of time) and energy (total number of operations est, starting with Unruh’s insight that the propagation per unit of time). Similar relations were used by Ng, in of sound waves in a flowing fluid can be described by an the context of space-time foam holographic models [69], effective metric [51]. When going to quantum fluids, this a picture suggested by Wheeler. leads to an effective description in terms of a quantum Wheeler’s disciples also laid the foundation of the mod- field theory on a curved space time [40, 52]. For example, ern consensus on the interpretation of quantum mechan- it in has been predicted that should ics, a hybrid of many worlds and decoherence. Everett be emitted from sonic horizons, boundaries between sub- was the first to make explicit the conclusion that follows sonic and supersonic flow [51]. In these types of emer- inevitably from the linearity of quantum mechanics: in gent gravity, the metric is induced by the superfluid flow, the wave function, all possibilities happen simultaneously which is assumed to be imposed from the outside. [70]. Only when measurements are made, a particular Despite the similarity with the hydrodynamic works on ‘realisation’ of the world appears. Zurek’s decoherence analog gravity in condensed matter systems, the mech- analysis [13] most clearly shows why these different real- anism that we propose is quite different. First, in our isations do not interfere (in the technical and colloquial picture, it is not the hydrodynamic flow that generates sense) with each other. He introduced ‘pointer states’, the metric, but the local information about the quan- the states that are robust under interaction with the en- tum system. Secondly, we use a different principle for vironment. He dubbed the fact that those pointer states our analysis. Where the hydrodynamic gravity is based are selected by the environment ‘einselection’ (environ- on the analysis of waves, we base our analysis on the ment induced selection). These concepts are crucial in statistics of the excitations. our understanding of the emergence of Further in the context of emergence in condensed mat- from quantum mechanics. Our picture of matter as a ter systems, it was shown by d’Alessio and Polkovnikov robust local nonequilibrium implies that all matter ex- how inertia (mass) emerges in adiabatic perturbation the- citations should be pointer states, with as the ultimate ory beyond the Born-Oppenheimer approximation [53]. pointer state a black hole. It will be of great interest to understand the relation of Finally, our work is in a sense also an implementation their work with our view on the equivalence principle. of Wheeler’s ‘it from bit’ [71], since we argued that the The first theoretical works on the entanglement en- most important characteristic of matter is information, a tropy in extended quantum systems [54–57] were moti- local reduction of the entropy. vated by the area law for black holes [28, 58]. This re- search spurred the study of entanglement in condensed matter physics which has led to a great advancement X. CONCLUSIONS AND OUTLOOK of our understanding of correlated quantum systems [32, 59], but to the best of our knowledge, there is not yet In the present work, we have suggested a mechanism a consensus on the connection to the Bekenstein-Hawking for time dilation that emerges out of a quantum system entropy. through a statistical analysis. It is clear that a large effort More recently, it was suggested by Van Raamsdonk will be needed before our suggestion can become a viable that spacetime is built up by entanglement [60, 61]. He candidate for a description of real quantum gravity. 11

As we have discussed, information should be extracted type of locally deformed quantum states are usually not from the quantum system through slow observables, such considered in calculations on many body quantum sys- that the obtained information is stable. Technically, find- tems, that are concerned with small numbers of exci- ing these observables is a difficult problem: the search tations. It would actually seem implausible that large takes place in an exponentially large Hilbert space. fluctuations in entangled quantum systems do not influ- A further issue concerns the range the gravitational ence their surroundings. Two possibilities then remain potential. In the simple model that we investigated, if quantum mechanics is a fundamental theory. Either we found a power law decay for the mutual information this influence is gravity or it is another, unknown effect. in one-dimensional noninteracting fermions. An obvious Our arguments are encouraging for the gravity scenario question is to find the conditions for which one obtains to be correct. The description of large fluctuations will a 1/r gravitational potential at large distances in three be mathematically challenging, but a thorough under- dimensional systems. standing seems unavoidable if we want to grasp how the So far, we have only provided arguments for one aspect physical world is related to our quantum models. of Einstein gravity to emerge from a statistical analysis of quantum systems, namely time dilation in a static sit- uation. If gravity is really emerging according to our mechanism, one should be able to derive the full set of XI. ACKNOWLEDGEMENTS geodesic and Einstein equations from statistical consid- erations, with far richer phenomenology than Newtonian gravity. It should ultimately include cosmology from the We thank Jacques Tempere and Jozef Devreese for en- perspective of closed quantum systems. couraging discussions. D.S. acknowledges support of the In conclusion, we conjecture that gravity appears as a FWO as post-doctoral fellow of the Research Foundation deformation of statistics due to local information. This - Flanders.

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