PRX QUANTUM 2, 010344 (2021)

Entanglement Entropies of Equilibrated Pure States in Quantum Many-Body Systems and Gravity

Hong Liu* and Shreya Vardhan † Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 23 December 2020; accepted 24 February 2021; published 17 March 2021)

We develop a universal approximation for the Renyi entropies of a pure state at late times in a noninte- grable many-body system, which macroscopically resembles an equilibrium density matrix. The resulting expressions are fully determined by properties of the associated equilibrium density matrix, and are hence independent of the details of the initial state, while also being manifestly consistent with unitary time evolution. For equilibrated pure states in gravity systems, such as those involving black holes, this approx- imation gives a prescription for calculating entanglement entropies using Euclidean path integrals, which is consistent with unitarity and hence can be used to address the information loss paradox of Hawking. Applied to recent models of evaporating black holes and eternal black holes coupled to baths, it provides a derivation of replica wormholes, and elucidates their mathematical and physical origins. In particular, it shows that replica wormholes can arise in a system with a fixed Hamiltonian, without the need for ensemble averages.

DOI: 10.1103/PRXQuantum.2.010344

I. INTRODUCTION (A) = purpose, we can use the Renyi entropies Sn for n 1, 2, ... Consider a quantum many-body system initially in a [3] far-from-equilibrium pure state |0. If the system is non- (A) 1 n integrable, it should eventually approach a thermal equi- S (t) =− log Tr ρ (t), ρ (t) = Tr ¯ ρ(t), n n − 1 A A A librium, in the following sense. For times t ts, where ρ( ) =|( )( )| ts is a thermalization time scale, |(t)=U(t) |0 can t t t .(1) be associated with macroscopic thermodynamic quantities In a pure state, one must have for any subsystem A and its such as temperature, entropy, and free energy, which obey ¯ the usual thermodynamic relations, and measurements of complement A generic few-body observables in |(t) exhibit the same ( ) ( ¯ ) (eq) A ( ) = A ( ) = ... behavior as in the equilibrium density matrix ρ with Sn t Sn t , n 1, 2, .(2) those macroscopic parameters. Even as the state equilibrates in the above sense, under In an equilibrium density matrix, Eq. (2) is not satisfied. unitary time evolution it must go to a pure state, and Other than brute-force numerical simulations of individ- therefore cannot become equal to the mixed state ρ(eq).A ual cases, we currently do not have an efficient method (A)( ) | natural question then is how we can tell an equilibrated for calculating Sn t for an arbitrary initial state 0 in pure state apart from an equilibrium density matrix. For a general system. For a finite-dimensional Hilbert space instance, this question arises in the context of Hawking’s with no energy constraint, where we expect equilibration to information loss paradox [1,2], in trying to understand the thermal state at infinite temperature, a valuable insight (A) whether the evolution of a black hole formed from the comes from calculating the averages Sn over all pure gravitational collapse of a pure state is unitary. For this states with the Haar measure [4–8]:

(A) = (A¯ ) = ( ) = ... Sn Sn min log dA, log dA¯ , n 1, 2, .(3) *[email protected] †[email protected] Here d and d ¯ are, respectively, the dimensions of the sub- A A ¯ Published by the American Physical Society under the terms of systems A and A, and the above expression is exact only the Creative Commons Attribution 4.0 International license. Fur- in the limit where one of dA, dA¯ is much larger than the ther distribution of this work must maintain attribution to the other. When dA dA¯ , these are equal to the entanglement author(s) and the published article’s title, journal citation, and entropies of a thermal state at infinite temperature. It can DOI. be checked that when the dimension of the Hilbert space is

2691-3399/21/2(1)/010344(33) 010344-1 Published by the American Physical Society HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021) large, the standard deviation about the average, Eq. (3),is (A)( ) 1. The expressions for Sn t in an equilibrated pure small. Thus the right-hand side of Eq. (3) should provide state are time independent, and can be expressed a good approximation for the entanglement entropies of a solely in terms of partition functions and entropies typical pure state. of an equilibrium density operator ρ(eq).Theyare This observation has yielded many important results, thus independent of details of the initial state and such as the prediction of the Page curve for black evap- capture the effects of equilibration. oration [9]. Similarly, a Haar average over pure states has 2. The expressions are manifestly compatible with the been used to predict when information in a black hole can constraint from unitarity in Eq. (2). be transferred to its Hawking radiation [10]. The random (A)( ) 3. While Sn t are defined in terms of Lorentzian path average idea, however, cannot be directly applied in cases (A)( ) integrals, the approximate expression for Sn t for where we expect equilibration to a finite temperature, or in an equilibrated pure state can be expressed in terms field theories and other systems with infinite Hilbert space of a sum of Euclidean path integrals when ρ(eq) dimension, where a canonical and calculable average such has a Euclidean path-integral representation. Each as the Haar average does not seem to exist. Even for a term in the sum has n replicas of the Euclidean path finite-dimensional system that equilibrates to infinite tem- integrals of ρ(eq) connected in a certain way, deter- perature, in order to apply Eq. (3) when the initial state mined by an element of the permutation group Sn. and time-evolution operator is fixed, we need to make the The approximation thus provides a general phys- highly nontrivial assumption that the system can evolve to ical mechanism for how Euclidean path integrals a typical pure state. It would be useful to better understand associated with the equilibrium density operator can the physical basis for this assumption, and have a system- arise as the dominant subset of contributions from atic procedure with which we can improve upon Eq. (3) for intrinsically Lorentzian path integrals. such cases. In this paper, we develop a general approximation Since the only input that goes into our approximation (A) method for calculating Sn for equilibrated pure states method is information about the equilibrium density matrix in systems with a fixed initial state and time-evolution ρ(eq), it can be used to obtain universal results for the operator, in the limit where the effective dimension of the entanglement entropies of a variety of quantum many-body Hilbert space (roughly, the dimension of the accessible part systems when the initial state equilibrates to a given type of of the Hilbert space from the initial state) is large. The ensemble. We explicitly obtain the universal expressions approximation scheme, which we refer to as the equilib- for the microcanonical and canonical ensembles. rium approximation, can be applied to finite temperatures, One important motivation for studying the entangle- systems with infinite Hilbert space dimension, and field ment entropies of equilibrated pure states is to understand theories. the unitarity of black hole evolution. Consider a situation The method we propose builds on an observation in where the initial state |0 describes a star, which under Ref. [11] that the Haar average in a finite-dimensional time evolution collapses to form a black hole. For all prac- Hilbert space can be seen as a projection into a sub- tical purposes, a black hole looks like a thermal state: it space P of the replica Hilbert space used for computing emits thermal radiation at a certain temperature, and has Renyi entropies. This means that for a system with a an entropy that satisfies the standard thermodynamic rela- fixed Hamiltonian, the result Eq. (3) can be considered an tions. Furthermore, correlation functions of a finite number approximation in which at leading order one ignores the of few-body observables in the black hole geometry have contribution from the orthogonal subspace to P. The gen- the same behavior as in a thermal state. The black hole eralization to finite temperatures or systems with infinite- is thus in an equilibrated pure state if the time evolution dimensional Hilbert spaces then boils down to identifying in gravity obeys the usual rule of unitarity in quantum the appropriate subspace of the replica Hilbert space to mechanics. The formalism we develop can thus be applied project into to obtain the leading contribution. Equiva- to a black hole system to obtain its entanglement entropies lently, the method can be thought of as identifying the in a way that respects unitarity. In particular, item 3 above contributions from the most important subset of config- implies that one can get expressions that are compatible urations in the Lorentzian path integrals for the Renyi with unitarity using Euclidean gravity path integrals. entropies. Since we drop certain well-defined contributions Recently, there has been important progress in under- to the time-evolved quantities in making this approxi- standing the unitarity of black hole evolution through mation, it can in principle be systematically improved derivations of the Page curve [12–14](seealso by adding these contributions back. We also develop a Refs. [15–45]) [46]. In particular, in order to obtain Renyi self-consistent criterion to demonstrate the validity of the and von Neumann entropies compatible with unitarity, one approximation. needs to include certain “island” contributions [14] in the Some important general features of the results from our quantum extremal surface prescription [47]. In models for approximation method are as follows: an evaporating black hole in Ref. [17] and for an eternal

010344-2 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) black hole coupled to a bath in Ref. [18], these island structure, and then examine its consequences for a variety contributions were derived by including Euclidean gravity of equilibrated pure states. path integrals with replica wormholes in the calculation. By applying the equilibrium approximation to these mod- A. Equilibrated pure states els, we provide a derivation of the replica wormholes intro- Consider a quantum system in some far-from- duced in these references, explaining how such Euclidean equilibrium pure state |0 at t = 0. At time t, it evolves configurations emerge from Lorentzian time evolution at to late times, and why they lead to answers that are consis- tent with unitarity constraints. Our discussion also clarifies |=U |0 ,(4) an issue raised in Ref. [17], due to which the authors there suggested that an averaging procedure may be necessary to where U is the time-evolution operator for the system. For explain the results from including replica wormholes. We now, we assume the system is compact, so that there exists show that no average over theories is needed, and that the a finite equilibration time scale ts such that for t ts, issue can be resolved within the framework of the equilib- macroscopic properties of | can be well approximated rium approximation. In particular, the results highlight that by some equilibrium density operator ρ(eq) [49]. We refer the success of using Euclidean path integrals to capture the to | as an equilibrated pure state [50,51]. Page curve has to do with the physical nature of entan- We can write the equilibrium density matrix ρeq in the glement entropies, rather than the “magic” of Euclidean form path integrals. Applied to more general holographic sys- tems, our results predict new bulk geometries that must (eq) 1 ρ = Iα, Z(α) = Tr Iα,(5) be summed over in the calculation of Renyi entropies for Z(α) states with black holes. I α We also explore the underlying physical mechanism in where α is an un-normalized density operator, and is a the Heisenberg evolution of operators that underlies the set of equilibrium parameters such as temperature, chem- emergence of the equilibrium behavior of entanglement ical potential, and so on, which can be determined from | entropies. In Ref. [44], we showed that for the second the expectation values of conserved quantities in 0 .We I Renyi entropy, the random void distribution conjectured require that α commute with the evolution operator, i.e., for quantum chaotic systems in Ref. [48] leads to the right- I † = I †I = I hand side of Eq. (3). Using the equilibrium approximation, U αU α, U αU α,(6) we show that the behavior of the Renyi entropies can be ρ(eq) seen as a special example of a general behavior of opera- which can be viewed as a requirement for to be an equilibrium state. tor growth in chaotic systems, and derive a higher-moment I generalization of the random void distribution. Here are some specific examples of α: The plan of the paper is as follows. In Sec. II,we 1. The system has a finite-dimensional Hilbert space, develop the equilibrium approximation, discuss its justifi- and there is no constraint on accessible states from cation and universal consequences, and apply it to a variety | . In this case, the associated equilibrium state of equilibrated pure states. In Sec. III, we apply the approx- 0 ρeq does not need to be labeled with any parameters imation to gravity systems with holographic duals, explain α,andI is the identity operator, how replica wormholes emerge from it, and make com- ments on the need for averaging based on this derivation. I = 1, Z = d,(7) In Sec. IV, we explain how equilibration to infinite tem- perature can be understood in terms of the random void where d is the dimension of the Hilbert space. distribution. In Sec. V, we discuss the applicability of the Below, we refer to this case as the infinite tempera- equilibrium approximation to observables other than the ture case. Renyi entropies, and mention some open questions. 2. The system has a time-independent Hamiltonian H with energy eigenstates |n,and|0 (and hence |) involves only energy eigenstates localized in a = ( −  +  ) II. UNIVERSAL BEHAVIOR OF ENTANGLEMENT narrow energy band I E E, E E . In this ENTROPIES IN EQUILIBRATED PURE STATES case, In this section, we present an approximation for the IE = |nn| , Z(E) = Tr IE = NI ,(8) entanglement entropies of equilibrated pure states, which En∈I can be expressed in a simple, universal form, and applies to a variety of systems. We first explain the physical rea- where NI is the number of energy eigenstates in the soning behind this approximation and its mathematical energy band I.

010344-3 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

3. The system has a time-independent Hamiltonian H, and |0 (and hence |) involves energy eigen- states with a broader range of energies. In this case,

−βH −βH Iβ = e , Z(β) = Tr e ,(9)

where the inverse temperature β is determined by requiring Iβ has the same energy as |0,

1 −βH Tr He = |H| = 0|H|0. Z(β) FIG. 1. The path integrals Eq. (13) for Eq. (11) involve 2n inte- (10) gration contours, with those for U’s going forward in time and † those for U backward in time. ρ0 provides the initial conditions 2 while the contractions implied by the traces in Eq. (11) define the For Eqs. (7) and (8), Iα is a projector, I = Iα, but this α final conditions for the path integrals. is not true for Eq. (9).InEqs.(8)–(9), one can view Iα as an “effective identity operator” defining the accessible n ψ ψ part of the Hilbert space, and the partition function Z(α) i, i × φ ( ) φ( ) D i t D i t as the corresponding “effective dimension.” It is clear that χ χ I i=1 i, i each choice of α in Eqs. (7)–(9) satisfies the requirement, Eq. (6), of invariance under U. n × ( φ − φ ) exp i S[ i] S[ i ] . (13) i=1 B. Renyi entropies as transition amplitudes in a φ φ replicated Hilbert space Here i, i are, respectively, associated with the ith con- We are interested in quantum-informational properties tours going forward and backward in time, as in Fig. 1. {χ χ } {ψ ψ} of | at time scales t t .Thenth Renyi entropy with i, i and i, i denote, respectively, the initial and s ψ respect to a subsystem A is given by final values of the dynamical fields. iA denotes the value of ψi restricted to subsystem A,andψ(n+1)A = ψ1A.The ρ ( ) −( − ) (A) n initial state 0 determines the initial conditions for the inte- A n 1 Sn n †  Z = e = TrAρ = TrA Tr ¯ Uρ0U , ρ χ χ n A A grations through its “wave function” 0[ i, i ] while the ¯ ρ0 =|00| . (11) contractions dictated by TrA and TrA determine the final conditions for the integrals. From Eq. (13), and also intuitively from Fig. 1, we can Recall that in a quantum system with evolution operator U, view Eq. (11) as a transition amplitude in a new “replica” the transition amplitude from an initial state | to a final 0 quantum system consisting of 2n copies of the original sys- state | has the path-integral representation f tem, with an evolution operator given by (U ⊗ U†)n [53]. Z(A) This way of viewing n is particularly convenient for our | | = ψ χ∗ ψ χ discussion below. To write down the explicit form for Z(A) f U 0 D D f [ ] 0[ ] n in the replica system, let us first introduce some notations. φ( )=ψ t  Suppose H is the Hilbert space of the original system.  iS[φ(t )] × Dφ(t )e , (12) The Hilbert space of the replica system can be taken to be φ( )=χ 0 (H ⊗ H)n. If we use an orthonormal basis {|i} for the first H H ⊗ H φ( ) copy of in each , then it is convenient to use a where t collectively denotes the dynamical variables of {|¯} φ( ) basis i for the second copy, defined as the system and S[ t ] is the corresponding action [52]. Equation (11) contains 2nU’s and thus can be written in ¯ ¯ ¯ |i=T |i , i|j = j |i = δij , Uij = i|U|j , terms of path integrals over 2n time integration contours, ¯i|U†|j¯=U∗ , (14) ij n (A)    −1 † Z = ψ ψ δ(ψ − ψ( + ) )δ(ψ ¯ − ψ ) where T is an antiunitary operator such that TUT = U . n D iD i iA i 1 A iA iA¯ i=1 For example, T can be taken to be the time-reversal oper- n ator in systems with time-reversal symmetry, or CPT in   (H ⊗ H)n × DχiDχ ρ0[χi, χ ] more general systems. A basis for can then be i i {| ¯ ¯ ··· ¯ } i=1 written as i1i1i2i2 inin .

010344-4 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021)

For any operator O acting on H, we can define a set of will generically lead to large oscillations of the integrand n! states |O, σ∈(H ⊗ H)n, where σ is an element of the in the path integral, so that the integral will evaluate to a permutation group Sn of n objects, small value. It is thus natural to expect that configurations satisfying Eq. (19) dominate. An important feature of the    ¯ ¯ ··· ¯ |O σ = O  O  ···O  configurations, Eq. (19), is that they lead to contributions i1i1i2i2 inin , i1iσ( ) i2iσ( ) iniσ( ) , 1 2 n that are independent of t, which also makes it natural for O = |O| ij i j (15) them to describe the behavior of the system after reaching macroscopic equilibrium at t ts. and σ(i) denotes the image of i under σ . The states asso- However, naively setting Eq. (19) in the path inte- ciated with the identity operator 1 are simply denoted as χ = χ  ψ = ψ grals (which also sets i σ(i) and i σ(i)) leads |σ to divergences, for example even in a simple system of two harmonic oscillators. Physically, the divergences come ¯ ¯ ¯ i1i i2i ···ini |σ = δ  δ  ···δ  . (16) 1 2 n i1iσ(1) i2iσ(2) iniσ(n) from the fact that Eq. (19) includes unphysical configura- tions of arbitrarily large energies. Mathematically, such a We note that when H is infinite dimensional, |σ are not procedure corresponds to replacing (U ⊗ U†)n in Eq. (18) normalizable and should be viewed as formal definitions with a projector onto the set of states |σ , σ ∈ Sn asso- rather than genuine states in the replica Hilbert space. ciated with the identity operator 1. In a system of an In the discussion below, we often use the following infinite-dimensional Hilbert space, |σ is not normalizable. properties of the inner products among these states: We now present a systematic procedure, which incorpo- rates the idea of matching the configurations on forward O1, τ|O2, σ evolutions with those on backward evolutions, but avoids such divergences by restricting to configurations that are = O1, λτ|O2, λσ = O1, τλ|O2, σλ, accessible to the evolution. Our discussion does not depend −1 −1 = O1, σ |O2, τ = O1, τ |O2, σ , on whether the system has a finite- or infinite-dimensional n n Hilbert space. Our proposal builds on an important obser- = O†O 1 ··· O†O k ∀σ τ λ ∈ S Tr 1 2 Tr 1 2 , , , n, vation in Ref. [11], that the Haar average in a finite- (17) dimensional Hilbert space can be seen as a projection into the set of states |σ associated with the identity operator. where k is the number of cycles in the permutation στ−1 This observation can be seen as the infinite-temperature and n , s = 1, 2, ..., k, are the lengths of the cycles. case of our discussion. s Let us first introduce some further notation. Consider For H = H ⊗ H ¯ , the associated replica Hilbert space A A the set of states in (H ⊗ H)n associated with the effective inherits a tensor product structure, and we can define the I corresponding states for each tensor factor. identity operator α of Eq. (5), Using the above notation, we can now write Eq. (11) as n ¯ ¯ ··· ¯ |I σ = |I |  σ ∈ S i1i1i2i2 inin α, ia α iσ(a) , n, (20) Z(A) =η ⊗ |( ⊗ †)n|ρ n A eA¯ U U 0, e , (18) a=1 which satisfy [using Eq. (17)] where e is the identity element of Sn and η is the ele- ( − ... ) |η (H ⊗ H )n ment n, n 1, 1 . A is in the space A A and I τ|I σ = ··· I σ |I σ = n α, α, Z2n Z2n , α, α, Z2 , similarly |e ¯ ∈(H ¯ ⊗ H ¯ )n. The equivalence between 1 k A A A ≡ In Eqs. (18) and (11) can be checked by inserting complete Zn Tr α, (21) sets of states of H and (H ⊗ H)n, respectively, in Eqs. (11) στ−1 and (18), and using Eqs. (15)–(16). where k is the number of cycles in the permutation and ns, s = 1, 2, ..., k are the lengths of the cycles. For notational simplicity, we suppress the α dependence in C. Proposal for a general equilibrium approximation Zn,andZ(α) in Eq. (5) is now referred to as Z1.Itis Consider the special set of configurations in the path convenient to define the metric for the normalized states integral Eq. (13) that satisfy corresponding to |Iα, σ ,

¯ Iα τ|Iα σ ··· φi(t) = φσ(i)(t), i = 1, ..., n, σ ∈ Sn. (19) , , Z2n1 Z2nk τσ ≡ = g 1 n . (Iα, σ |Iα, σIα, τ|Iα, τ) 2 Z2 For such configurations, the phase factor in the exponent of (22) Eq. (13) vanishes identically [54]. Heuristically, one may imagine that for sufficiently late times, the contributions Note that gτσ is a symmetric matrix, and from Eq. (17) it is ¯ from configurations of φi, φi, which do not satisfy Eq. (19), invariant under simultaneous multiplication of an element

010344-5 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

λ ∈ Sn on σ and τ from the left or the right, as well as state | after it has reached a macroscopic equilibrium under simultaneously taking inverses of σ and τ.More- [55]. We note that each of the steps in Eqs. (24)–(27) τσ over, we can show that g , the inverse of gτσ,isalso applies to any choice of Iα that is invariant under the (A) invariant under each of these operations. action of U. The approximation that Zn,Q is negligible The projector onto the set of states spanned by {|Iα, σ} should, however, only be valid for Iα chosen such that it then has the form corresponds to the late-time equilibration of |0. In particular, we can obtain a self-consistency condition 1 στ Pα = g |Iα, σ Iα, τ| . (23) from considering the implication of the approximation, Zn 2 σ,τ Eq. (28),forn = 1. In this case, Eq. (11) simply reduces ρ Z(A) = = to the trace of 0 and we should find n 1. For n 1, One key physical input in our approximation is that due the only permutation is the identity e,andEq.(28) then has to the invariance, Eq. (6),ofIα under the action of U, the the form states |Iα, σ are invariant under the action of the time- ( ⊗ †)n Z(A) ≈ 1 ⊗ |I I |ρ = Z1 (ρ I ) evolution operator U U in the replica Hilbert space, 1 eA eA¯ α, e α, e 0, e Tr 0 α , Z2 Z2 † n (29) (U ⊗ U ) |Iα, σ=|Iα, σ , † n (U ⊗ U) |Iα, σ=|Iα, σ , (24) where we use Eqs. (21) and (17). For this result to repro- duce the normalization of ρ0, we thus need which in turn implies that Z2 Tr(ρ0Iα) = . (30) † n † n Z (U ⊗ U ) Pα = Pα(U ⊗ U ) = Pα. (25) 1 We impose Eq. (30) as a self-consistency condition for Then, decomposing the identity on (H ⊗ H)n as 1 |0 to evolve to a state |, which resembles Iα Z1 2 macroscopically [56]. Let us now use the self-consistency 1 = Pα + Q, PαQ = QPα = 0, Q = Q, (26) (A) condition, Eq. (30), in our approximation for Zn .First, using Eq. (17), we find that where Q is the orthogonal projector of Pα, we can rewrite the transition amplitude expression for the nth Renyi Zn I σ |ρ = (ρ I ) n = 2 entropy in Eq. (18) as α, 0, e [Tr 0 α ] n , (31) Z1 (A) † n Z = ηA ⊗ e ¯ |(Pα + Q)(U ⊗ U ) (Pα + Q)|ρ0, e , where we use that since ρ0 is a pure state, n A † n = ηA ⊗ e ¯ |Pα|ρ0, e + ηA ⊗ e ¯ |Q(U ⊗ U ) Q|ρ0, e , k k k A A Tr (Iαρ0) = 0 Iα 0 = [Tr(ρ0Iα)] . (32) ( ) ( ) = Z A + Z A . (27) n,P n,Q Equation (28) can then be written in a form independent of the initial state, Our proposal is that for t ts, for a chaotic system with (A) a large effective dimension Z1, Z is small compared to (A) a τσ n,Q Z ≈ ηA ⊗ e ¯ |Iα, τ , a = g Z(A) n Zn A n,P and can be ignored, so that 1 τ∈Sn σ −1 = ge(στ ) = geσ . (33) (A) (A) 1 τσ Z ≈ Z = g η ⊗ e ¯ |Iα, τ Iα, σ |ρ , e, σ σ n n,P Zn A A 0 2 σ,τ From our usual intuition about statistical systems and = ... n 1, 2, 3, . (28) the comments below Eq. (38) about the standard deviation of the Haar average in the infinite temperature case, we We refer to this approximation as the equilibrium approxi- expect more generally that to suppress potential contribu- mation in the following discussion. In next subsection, we (A) tions from Z in Eq. (27), we should always consider the discuss a justification for the approximation. n,Q regime that the “effective dimension” Z = Tr Iα is large. We thus replace the time-evolution operator (U ⊗ U†)n 1 Since Z contains only a single trace, we should then have of the replica Hilbert space with the projector onto the set n (A) of states spanned by {|Iα, σ } in the expression for Z , Z n n  Z1−n 1, n = 2, 3, .... (34) which can be seen as a realization of the heuristic idea dis- Zn 1 cussed at the beginning of this subsection. Equation (28) 1 is time independent, which is consistent with the proposal For the choices of Iα in Eqs. (7) and (8), the first relation that it captures quantum-informational properties of pure in Eq. (34) is in fact an equality. For Eq. (9), we expect the

010344-6 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) following general behavior: Z(A) As we discuss later, our results for n from the equilib- rium approximation for the microcanonical and canonical = −βH = Ng(β) Z1 Tr e e ensembles agree with previous calculations based on aver- Z ages over special sets of states [58,59], but in these cases → n = eN[g(nβ)−ng(β)] Z1−n 1, (35) Zn 1 there does not seem to be a clear way of interpreting the 1 averages over states as averages over time-evolution oper- where N is proportional to number of degrees of freedom ators from physically relevant Hamiltonians. Moreover, in [57] and is large, and g(β) > 0isanO(1) monotonically the case of an infinite-dimensional Hilbert space, there decreasing function of β.FromEq.(22) we then find that does not exist a canonical averaging procedure over all physical time-evolution operators analogous to the Haar = δ + ( −1) → = + ( −1) average. gτσ τσ O Z1 a 1 O Z1 . (36) Here we propose a self-consistent criterion for decid- We then obtain our final general expression for the approx- ing whether Eq. (28) is a good approximation, which does (A) not depend on whether an average exists. We consider the imation to Zn for an equilibrated pure state: following quantity: (A) (A) (A) 1 Z ≈ Z (τ), Z (τ) = ηA ⊗ e ¯ |Iα, τ . 2 2 n n n n A (A) ( ) (A) Z1 2 ≡ Z = Z A − Z τ∈Sn n,Q n n,P (37) eq app eq app 2 = Z(A) 2 − Z(A) We examine the mathematical structure of Eq. (37) fur- n n,P , (40) ther in Sec. II E. In Sec. II F we show that while Eq. (37) eq app is expressed solely in terms of properties of equilibrium where subscript “eq app” denotes that we apply the equi- density operator Iα, the unitarity constraint Eq. (2) is main- librium approximation, Eq. (28), to the quantity inside tained. The resulting physical properties are discussed in the bracket. We explain more explicitly in Appendix B Sec. II G H. what is meant by applying the equilibrium approxima- The equilibrium approximation for the case where the ( ) tion to Z(A) 2.If Z A , then the approximation is initial state ρ0 is a mixed state is discussed in Appendix A. n n,P self-consistent. The criterion can also be interpreted as the question of whether the equilibrium approximation is com- D. A justification of the equilibrium approximation Z(A) 2  Z(A) I = patible with factorized form of n . When n,P , In the infinite-temperature case, Eq. (7), with 1, the it means that to a good approximation we have equilibrium approximation, Eq. (28), yields results identi- cal with those obtained from the Haar average over unitary 2 matrices U acting on the full system, as it can be checked Z(A) 2 ≈ (Z(A)) n n eq app . (41) that [11] eq app

† n (U ⊗ U ) = PI=1, (38) We can also extend the self-consistency criterion to higher powers: for the approximation for be valid, we need where the overline denotes the Haar average. With this (A) m m interpretation, one can estimate the magnitude of Z by (A) m (A) (A) n,Q m ≡ Z − Z Z , (42) Z(A) n n,P n,P considering the variance of n under the Haar average eq app ( ) 2 ( ) ( ) 2 ( ) ( ) i.e., the equilibrium approximation is compatible with Z A = (Z A )2 − Z A = (Z A )2 − (Z A )2 n,Q n n n n,P . factorization for any power, (39) m m Z(A) ≈ (Z(A)) . (43) 2 n n eq app Z(A) (Z(A))2 eq app If n,Q n,P , then for a randomly chosen time- Z(A) evolution operator U, n,Q has high probability of being In Appendix B, we calculate Eq. (40) explicitly, and (A) − 1 small compared with Z , and the approximation is justi-  2 n,P show that is suppressed by at least a factor Z1 com- fied for most systems, including those where U comes from Z(A) pared with the leading contribution from n,P in the limit a fixed Hamiltonian. ( ) of large Z . However, note that Z A can be comparable to For a general Iα, the projection to Pα in Eq. (27) may 1 n,Q Z(A) not emerge from an over average time-evolution operators. or larger than the next-to-leading term in n,P . We further

010344-7 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

 −1 k = 1, ..., n, each of which appears twice, are summed show that m is suppressed by at least a factor Z1 com- ( ) pared with the leading contribution from (Z A )m in the over. Equation (46) can be given a diagrammatic represen- n,P tation as in Figs. 2–4. The expression in the parentheses of limit of large Z1. Through analytic continuation, this leads to the equilibrium approximation for the Renyi entropies, the first line corresponds to the “future conditions” indi- cated in Fig. 2(a), where the Kronecker deltas between 1 ∂[(Z(A))m] indices in A are indicated with dashed lines, while those (S(A)) =− lim n eq app , in A¯ are indicated with solid lines. These future condi- n eq app − → ∂ n 1 m 0 m tions are independent of τ, while the τ-dependent factors (A) m 1 ∂(Z ) 1 ( ) in the second line of Eq. (46) correspond to how the indices ≈− lim n,P =− log(Z A ). − → ∂ − n,P should be connected to each other in the interior of the n 1 m 0 m n 1  (44) diagram. We connect each ika to iτ(k) with a dashed line,  a and each ikb to iτ(k) with a solid line, and read off a factor   b Similarly, the approximation for the von Neumann of i i |Iα|i i from each such interior connection ka kb τ(k)a τ(k)b entropy can be obtained by analytic continuation as to obtain Eq. (46). An example of an interior connection is shown in Fig. 2(b), and some examples of diagrams ∂Z(A) associated with different τ are given in Figs. 3–4. ( (A)) =− n,P S1 eq app lim . (45) In particular, each loop of solid lines in a diagram for a n→1 ∂n given permutation corresponds to a trace in A¯ , and each loop of dashed lines to a trace in A. Qualitatively, we E. Diagrammatic structure and path-integral ¯ representation expect that a trace in A (A) should yield a factor that is of the order of the effective dimension of subsystem A (A¯ ). We now examine more closely the mathematical struc- The number of A loops in the diagram associated with τ ture of Eq. (37). Using Eqs. (15)–(16), the inner product in according to our prescription above is given by k(η−1τ), Z(A)(τ) n can be written more explicitly as while the number of A¯ loops is k(τ), where k(σ ) denotes the number of cycles for a permutation σ . Thus we expect η ⊗ |I τ A eA¯ α, that = (δ  ···δ  δ  ···δ  ) i1a iη( ) ina iη( ) i1 i1 in in 1 (η−1τ) (τ) 1 a n a b b b b Z(A)(τ) ∼ dk dk , (47)     n n A A¯ ×i i |Iα|i i ···i i |Iα|i i , Z1 1a 1b τ(1)a τ(1)b na nb τ(n)a τ(n)b (46) where dA, dA¯ denote the effective dimension of subsystems A and A¯ , respectively. Note that one should view Eq. (47) where |i , |i (|i , |i ) denote basis vectors for sub- ka ka kb kb as a heuristic equation, as in general [for example, for ¯ system A (A) in the kth replica. In the above expression, both Eq. (8) and Eq. (9)] there is no precise definition of it should be understood that all indices i , i , i , i with ¯ ka kb ka kb effective dimensions for A and A.

(a) (b)

Z(A)(τ) = FIG. 2. (a) The “future conditions” for each of the n , coming from the factor in the first line of Eq. (46),forn 6. In (b), we show an example of how to connect indices in the interior of the diagram for a permutation τ such that τ(1) = 3, and the factor of   i i |Iα|i i that comes from this interior connection. 1a 1b 3a 3b

010344-8 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021)

(a) (b) (c)

FIG. 3. Examples of planar diagrams corresponding to different choices of planar permutations τ that saturate Eq. (48).

Note that for any τ, of edges is 3n, and hence

k(η−1τ)+ k(τ) ≤ n + 1. (48) k(η−1τ)+ k(τ) = n + 1 − 2h. (50) We can understand this diagrammatically. Let us slightly redraw each of the diagrams in Figs. 3–4, ignoring the The largest value of k(η−1τ)+ k(τ) is thus n + 1, corre- distinction between dotted and dashed loops, and adding sponding to planar diagrams. an extra loop surrounding the diagram. Two examples are Two immediate examples of permutations that corre- shown in Fig. 5. We then get diagrams similar to ’t Hooft’s spond to planar diagrams are τ = e and τ = η.More double-line diagrams for large N matrix-field theories [60]. generally, there is a one-to-one correspondence between With each such diagram, we can associate a polygon by such planar permutations and “noncrossing partitions” replacing the double lines with single lines. If a poly- of n elements [61]. Given a noncrossing partition gon can be drawn without crossing lines on a surface of {{a1, a1, ..., a1 }, {a2, a2, ..., a2 }, ··· , {ak, ak, ..., ak }} of minimum genus h, then the total number of loops in the 1 2 n1 1 2 n2 1 2 nk {1, 2, ...n} into k groups, where the elements of each double-line diagram is equal to the number of faces F of subset in the partition are arranged in descending the polygon on this surface. The total number of loops in order (e.g., a1 > a1 > ···> a1 ), we obtain a planar the original diagrams in Figs. 3–4 is one less than this, so 1 2 n1 permutation with the cycle representation given by 1 1 1 2 2 2 k k k − (a , a , ..., a )(a , a , ..., a ) ···(a , a , ..., a ). k(η 1τ)+ k(τ) = F − 1 = E − V + 2 − 2h − 1, (49) 1 2 n1 1 2 n2 1 2 nk Equation (46) can be further written as where E is the number of edges of the polygon and V is the number of vertices, and we use the theorem relating η ⊗ ¯ |I τ = |I | ··· A eA α, iη(1)a i1b α iτ(1)a iτ(1)b Euler’s characteristic to F, E,andV. But for all diagrams ×η( ) |Iα| τ( ) τ( ) we consider, the number of vertices is 4n and the number i n a inb i n a i n b (51)

(a) (b) FIG. 4. Examples of nonplanar dia- grams corresponding to two choices of τ that do not saturate Eq. (48).

010344-9 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(a) (b)

FIG. 5. Double-line diagrams (and the corresponding polygons) obtained from the diagrams of Figs. 3(a) and 4(a).

so that Eq. (37) has the form n 1   = Dψ Dψ δ(ψ − ψ − ) Zn i i iA τ 1η(i)A (A) 1 1 τ∈S i=1 Z ≈ iη( ) i |Iα|iτ( ) iτ( ) ··· n n n 1 a 1b 1 a 1 b Z1 τ∈S n ψi n  −S [φ ] × δ(ψ ¯ − ψ ) φ E i iA τ −1(i)A¯ D i e , (54) ×iη( ) i |Iα|iτ( ) iτ( ) . (52) ψ n a nb n a n b i=1 i In terms of path integrals, the approximation, Eq. (52), where S [φ] is the Euclidean action for Iα, i.e., corresponds to replacing the second line of Eq. (13) by a E sum of Euclidean path integrals, each of which involves n ψ −S [φ] copies of that for Iα “coupled” together in a certain way Tr Iα = Dψ Dφ e E . (55) specified by permutation τ. More explicitly, we now have ψ n See Fig. 6 for an illustration. Since our approximation (A) 1   Z = ∈ S Dψ Dψ , δ(ψ − ψη( ) ) n Zn n i i iA i A arises from projecting to a subspace of states in the replica 1 τ i=1 theory [with Hilbert space (H ⊗ H)n], it corresponds to n ψi isolating a subset of configurations from the Lorentzian  −S [φ ] × δ(ψ ¯ − ψ ) φ E i iA iA¯ D i e , (53) path integrals Eq. (13), which can further be given a ψ i=1 τ(i) Euclidean formulation. We stress, however, that these

(a) (b) (c)

Z(A)(τ) = I = −βH FIG. 6. Examples of the path-integral representation of n in Eq. (54) for n 3, in the case where α e and the system is (1+1) dimensional. In the ith replica, the final state is labeled by ψi (which we indicate by 1, 2, 3 in the figure) while the initial state ψ     =  = β is labeled by i (which is indicated by 1 ,2,3). The shaded regions represent Euclidean path integrals from t 0tot between  the initial and final states. For each τ, the final states ψ are identified with the initial states ψ , while the final states ψ ¯ are iA τ −1η(i)A iA identified with the initial states ψ , as indicated with the arrows in each case. τ −1(i)A¯

010344-10 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021)

Euclidean path integrals do not arise from analytically con- invariant under A ↔ A¯ so that the full expression satisfies tinuing the Lorentzian path integral. Also note that while the constraint the Lorentzian replica space has 2n copies of the original ¯ system, the Euclidean replica space in Eqs. (53)–(54) has Z(A) = Z(A) (61) only n copies. n n To conclude this subsection, we make some further observations on the structure of Eq. (52), which will be that must be obeyed in a pure state. For this purpose we write Z˜ (A) as useful in the later discussion. Since ika , ikb are dummy n → → μ ν ∈ indices, relabeling ikb iν(k)b , ika iμ(k)a for any , S 1 n leaves Eq. (51) invariant. We thus have Z˜ (A) = Z(A)(τ) + Z(A)(σ −1η−1τσ) n n n , (62) 2 τ= η (A) 1 e, Z (τ) = η ⊗ ¯ |I τ n n A eA α, Z1 −1 −1 1 where σ is a permutation satisfying σησ = η , which = (μη) ⊗ ν |I (μτ) (ντ) μ ν ∈ S −1 A A¯ α, A A¯ , , n, always exists as η and η are in the same conjugacy class Zn 1 (and is in general nonunique). Then note that (56) that is, the inner product is invariant under independent (A) −1 −1 1 −1 −1 Z (σ η τσ) = (σ ησ ) ⊗ e ¯ |Iα, η τ left multiplications for A and A¯ . The inner product is also n n A A Z1 invariant under a simultaneous right multiplication for A ¯ 1 −1 −1 and A by the same permutation, that is, = η ⊗ e ¯ |Iα, η τ , Zn A A 1 ( ) 1 Z A (τ) = η ⊗ |I τ ¯ n A eA¯ α, 1 (A) n = ⊗ η ¯ |Iα τ = Z (τ) Z1 n eA A , n (63) Z1 1 = (ημ) ⊗ μ ¯ |I (τμ) (τμ) ¯ μ ∈ S n A A α, A A , n, Z1 where in the first and third equalities we use Eqs. (56)–(57) (57) repeatedly. Equation (62) can then be written as which simply corresponds to a reordering of the n factors 1 ¯ of the product, Eq. (51). Z˜ (A) = Z(A)(τ) + Z(A)(τ) , (64) n 2 n n τ=e,η F. Unitarity We now examine the physical consequences of Eq. (37) which is manifestly invariant under A ↔ A¯ . further. It is first useful to note that two terms in the sum If the time-evolved state ρ =|| is pure, we should τ over in Eq. (37) have a simple physical interpretation also have Tr ρn = 1. Let us see how this is realized under ( ) n ( ) −( − ) (A,eq) our approximation, Eq. (28). Following arguments exactly Z A ( ) = ρ = Z A,eq = n 1 Sn n e TrA TrA¯ eq n e , parallel to those that lead to Eq. (28) and further to (58) Eq. (37), we obtain the following approximation:

¯ n ¯ (A,eq) 1 1 (A) (A,eq) −(n−1)Sn n n nk Z (η) = Tr ¯ Tr ρ = Z = e , Tr ρ = η|Iα, τ = Tr Iα1 ···Tr Iα n A A eq n Zn Zn (59) 1 τ∈Sn 1 τ∈Sn 1 (A,eq) (A¯ ,eq) = ··· where Sn and n are, respectively, the nth Renyi n Zn1 Znk , (65) ¯ Z1 τ∈S entropy with respect to A and A of the equilibrium den- n sity operator ρeq. These two contributions are represented −1 diagrammatically in Figs. 3(a) and 3(b). We can then write where k = k(τη ),andn1, ...nk, are the lengths of the Eq. (37) as cycles in τη−1. Equation (65) can also be given a diagram- matic representation, as explained in Fig. 7.FromEq.(34), Z(A) = Z(A,eq) + Z(A¯ ,eq) + Z˜ (A) Z˜ (A) = Z(A)(τ) n n n n , n n . the dominant term is given by τ = η shown in Fig. 7(b), τ=e,η leading to (60) Zn The first two terms in Eq. (60) are together manifestly Tr ρn = 1 + O(Z−1) = 1 + O(Z−1). (66) ↔ ¯ Z˜ (A) Zn 1 1 symmetric under A A. We now show that n is also 1

010344-11 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(a) (b) (c)

FIG. 7. (a) The “future conditions” corresponding to η| in the expression, Eq. (65),forTr[ρn]. (b) The diagram corresponding to the dominant contribution from τ = η, while (c) gives an example of a subleading contribution coming from τ = e.

G. Universal behavior of Renyi entropies for 2. In the regime where dA and dA¯ are both large and equilibrated pure states comparable in size, from Eq. (47), the leading terms τ Before applying Eq. (37) to different classes of systems, in Eq. (37) come from those ’s, which saturate here we make some general comments on its implications Eq. (48), that is, from planar diagrams like in Fig. 3. 3. In the infinite temperature case, Iα = 1 = 1 ⊗ 1 ¯ . in various regimes: A A¯ In general, Iα does not factorize between A and A. ¯ 1. Suppose A A (and hence dA dA¯ ). Then, from Nevertheless, as we see more explicitly below and the argument around Eq. (47), the term with the in Sec. III, for various situations of physical interest, maximal number of A¯ loops, which corresponds to one can have an approximate factorization τ = e, should dominate in Eq. (37). In this case, from Eq. (58) we have (A) (A¯ ) Iα ≈ Iα ⊗ Iα . (71) A A¯ ( ) ( ) −( − ) (A,eq) Z A ≈ Z A,eq = n 1 Sn α n n e . (67) Note that here we allow the parameters to be different for A and A¯ , which can happen if A and A¯ interact only for ¯ a finite period of time. Below for notational simplicity, we Similarly, when A A (and hence dA¯ dA), the (A) (A¯ ) term with the maximal number of A loops, which simply write Iα , Iα as IA, I ¯ . Define for any integer m A A¯ A corresponds to τ = η, should dominate. In this case, ( ) ¯ Z A from Eq. (59) (A) = Im (A) = Im ˆ (A) = m Zm TrA A , Zm TrA¯ ¯ , Zm m , A (A) ¯ Z1 ( ) ( ¯ ) −( − ) (A,eq) Z A ≈ Z A,eq = e n 1 Sn . (68) n n ( ¯ ) ¯ Z A Zˆ (A) = m . (72) m ( ¯ ) m Thus, when one of the effective dimensions of A and Z A A¯ is much larger than the other, we have 1 = Iα = ( I ) ¯ I ¯ ≡ ¯ From Eq. (71) we have Z1 Tr TrA A TrA A (A) = (A,eq) (A,eq) = ... ( ) ( ¯ ) Sn min Sn , Sn , n 2, (69) A A Z1 Z1 and Eq. (52) can be expressed as ( ) ( ¯ ) ( ) 1 ( ) ( ) ( ¯ ) ( ¯ ) where S A,eq and S A,eq are, respectively, the equi- Z A ≈ Z A ···Z A Z A ···Z A n n n Zn m1 ml n1 nk librium Renyi entropies with respect to A and A¯ of 1 τ ρ . Analytically continuing Eq. (69) to n = 1, one ( ) ( ) ( ¯ ) ( ¯ ) eq = Zˆ A ···Zˆ A Zˆ A ···Zˆ A , (73) m1 ml n1 nk then finds that for the von Neumann entropy τ (A) ( ) ( ¯ ) where k is the number of cycles of τ with n , ...n the S = min S A,eq , S A,eq , (70) 1 k 1 lengths of the corresponding cycles, and l is the num- ber of cycles of τη−1 with m , ...m the lengths of the ¯ 1 l where S(A,eq) and S(A,eq) are, respectively, the “equi- corresponding cycles. ¯ librium entropy” for subsystems A and A for the We now consider more specifically the examples of Iα system in the equilibrium state ρeq. discussed in Eqs. (7)–(9).

010344-12 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021)

1. Infinite temperature power series is convergent for |z|≤1, we should use the < Let us first consider the case Eq. (7).FromEq.(37) we first expression in Eq. (78) for dA dA¯ , and the second one > have for dA dA¯ . We then find that ⎧ 1 d 1 1 (τη−1) (τ) ⎪ A (A) k k ⎨log dA − dA < d ¯ Z = ηA ⊗ e ¯ |τA ⊗ τ ¯ = d d ¯ , A n dn A A dn A A (A) = 2 dA¯ τ τ S1 ⎪ . (79) ⎩⎪ − 1 dA¯ < (74) log dA¯ dA¯ dA 2 dA ¯ where dA, dA¯ are, respectively, the dimensions of A and A This agrees with the result from Haar averages [4]. As dis- = with d dAdA¯ . cussed below Eq. (38), subleading corrections to Eq. (76) When one of dA, dA¯ is much greater than the other, we and Eq. (79) beyond the limit of large d will likely not be S(A) = S(A¯ ) = simply find Eqs. (69)–(70) with n log dA and n universal. log dA¯ for all n. Now consider the regime 2. Microcanonical ensemble Now we consider the case of the microcanonical ensem- 1 dA ∼ 2 →∞ = ble, Eq. (8). We expect the result derived below should dA, dA¯ d , finite. (75) dA¯ also apply to a single energy eigenstate of a chaotic system (i.e., to systems satisfying the eigenstate thermalization From Eq. (48) the leading contribution in Eq. (74) comes hypothesis). from permutations τ corresponding to planar diagrams, In this case, IE is a projector, Zn = NI for all n.For|0 which saturate Eq. (48). All permutations with a given lying in the subspace defined by the projector I ,Eq.(31) (τ) = E number of cycles k k give the same contribution, is exactly satisfied. which leads to [62] We can write the Hamiltonian of the system as

n = + ¯ + ¯ (A) 1 n+1−k k H HA HA HAA, (80) Z = N(n, k)d d ¯ , n dn A A k=1 ¯ where HAA¯ denotes the interactions between A and A,and 1 1 n(n − 1) 1 n(n − 1) 1 ¯ = + +···+ + HA, HA involve only, respectively, degrees of freedom of n−1 n−2 n−2 n−1 , ¯ d 2 d d 2 d d ¯ d subsystems A and A. Let us suppose H is local. Then with A¯ A A¯ A A A ¯ (76) sufficiently large subsystems A, A, the contribution of HAA¯ to the energy is much smaller than those of HA, HA¯ in where the coefficients N(n, k) are the number of noncross- macroscopic states whose energies are proportional to the volume of the system. ing partitions of n objects with k blocks, and are known as ¯ ¯ the Narayana numbers When A A or A A, we again have Eqs. (67)–(70). Then using the standard argument of statistical mechanics, (A,eq) (A¯ ,eq) 1 n n we can write Zn and Zn more explicitly as N(n, k) = . (77) − n k k 1 n Z(A) ≈ Z(A,eq) ≈ ρ(A) n n TrA β , In the second line of Eq. (76), we also make explicit −β that N(n,1) = 1 (this comes from τ = η), and N(n, n) = 1 HA ρ(A) = e ¯ (from τ = e). The von Neumann entropy can be obtained β −β , A A, (81) Tr [e HA ] by analytically continuing Eq. (76) to general real values of A n and using Eq. (45). For this purpose, we note that Eq. (76) ¯ n can be written as Z(A) ≈ Z(A¯ ,eq) ≈ ρ(A) n n TrA¯ β ,

(A) 1 Z = ( − − / ¯ ) −βH ¯ n − 2F1 1 n, n;2;dA dA ( ¯ ) e A dn 1 ρ A = , A¯ A, (82) A β −βH ¯ Tr ¯ [e A ] 1 A = F (1 − n, −n;2;d ¯ /d ), (78) n−1 2 1 A A β d ¯ where the inverse temperature is determined from the A β = d log NI density of states as dE . which can be continued to general n. The derivative with Let us now consider the situation in which A and A¯ are respect to n can be found by expanding the hypergeomet- comparable in size. More explicitly, suppose the total sys- ric function 2F1(a, b; c; z) as a power series of z. Since the tem has volume V and VA/V = c < 1, with V →∞. Using

010344-13 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

τ = η = = > / = / again the fact that the contribution of HAA¯ to the energy is (or k1 n, k2 1) when c 1 2. For c 1 2, the small, we can approximate the projector IE as τ = e and τ = η contributions are equal and both are dominant. I ≈ | | | | Note that Eq. (84) can also be obtained by averaging E n A m A¯ n A m A¯ , (83) A A¯ uniformly over all pure states in the subspace I, that is, En +Em∈I by taking a fixed |ψ0∈I and averaging the value of Z(A) |ψ | | n over states U 0 , where U is a Haar-random unitary where n A , m A¯ are, respectively, eigenstates of HA and ¯ matrix acting within the subspace of energy E. Equiva- H ¯ with energies EA and EA. We then have A n m lently, it can be obtained from an average over the “ergodic bipartition” states described in Ref. [59]. (A) 1 Z = η ⊗ ¯ |I τ n n A eA E, NI τ 3. Canonical ensemble 1 −1 ¯ = (dA)k(τη )(dA )k(τ), (84) Let us now consider the situation where the effective n E E−E I NI τ E identity operator α is given by Eq. (9). Note that the result Eq. (37) with this value of Iα can also be obtained by fur- where E runs over the allowed values of energy in A that ther manipulation of the average over random “canonical A can be consistent with total energy E, dE is the dimen- thermal pure quantum states” in Ref. [58]. E A¯ Let us now consider the result in special regimes. When sion of the subspace of A with energy ,anddE−E is the ¯ A A¯ or A¯ A, we again have Eqs. (67)–(70).Letus dimension of the subspace of A with energy E − E.Letus ¯ write now consider the case where A and A are comparable, that is, the total system has volume V and VA/V = c < 1, with ¯ A VADA(A) A V ¯ D ¯ ( ¯ ) V →∞. With the Hamiltonian (80) and assuming local dE = e , d −E = e A A A (85) E interactions, we can approximate, within the traces appear-  = E/  = ( − E)/ ing in various quantities, that Iβ has a factorized form where A VA, A¯ E VA¯ are, respectively, ¯ =   = [64], energy densities for A and A.LetE V, so that A¯ ( − Ac)/(1 − c). Equation (84) can then be written as −β −β −β (A) (A¯ ) −1 I = H ≈ HA ⊗ HA¯ = I ⊗ I [k1 = k(τη ) and k2 = k(τ)] β e e e β β . (89)

( ) (A) 1 Then Z A is given by Eq. (73), and the quantities appear- Z = exp{V k1cDA(A) + (1 − c)k2D ¯ ( ¯ ) }. n n N n A A ing in Eq. (73) have the form I τ A (86) − β −mβH ¯ Tr e m HA ¯ Tr ¯ e A ˆ (A) = A ˆ (A) = A Zm −β , Zm −β . (90)  ( HA )m ( H ¯ )m The sum over A can now be performed by a saddle- TrAe TrA¯ e A point approximation with the saddle point ¯A satisfying the equation Comparing with Eq. (88), we see that while for A A¯ , the Renyi entropies corresponding to the microcanonical and  −¯ c k D (¯ ) = k D A . (87) canonical ensembles have the same form, they differ in the 1 A A 2 A¯ − 1 c regime where VA/V is finite. For a homogeneous system, from Eq. (35), the partition ¯ ¯ Note that A depends on k1, k2. We then find that functions for A and A subsystems can be written as  −β (β) −β (β) (A) 1 HA = NAg HA¯ = NA¯ g Z = (¯ ) TrAe e ,TrA¯ e e , n n exp V k1cDA A NI τ = +  N NA NA¯ . (91)  −¯Ac + (1 − c)k D ¯ . (88) 2 A 1 − c We then have

( ) ( β)− (β) ( ¯ ) ( β)− (β) ˆ A = NA[g m mg ] ˆ A = NA¯ [g m mg ] To proceed further, let us take the system to be homoge- Zm e , Zm e . (92) () = () = () () nous, so DA DA¯ f . We further take f to be described by a power law, i.e., f () = Cα for some To proceed further let us take g(β) to be a power law, exponent α. Conventional statistical systems have α< that is, g(β) = λβ −α with α>0. We then find that the τ = 1 and we restrict to this case [63], where we find that e term is dominant in this expression for c < 1/2, the τ = the dominant contribution to Eq. (88) comes from τ = e η term is dominant for c > 1/2, and for c = 1/2, both τ = (or equivalently k1 = 1, k2 = n) when c < 1/2, and from e and τ = η give equal contributions, which are dominant.

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FIG. 8. Causal constraints and subre- gion thermalization. To find the entan- glement entropies of A in a system with a sharp light-cone structure, it is sufficient to consider the part of the time-evolution operator in J (A),thatis, between the dashed lines.

H. Uncompact systems and subregion equilibration III. GRAVITY SYSTEMS AND REPLICA In our discussion above, we assume that the whole sys- WORMHOLES tem thermalizes after some finite time scale ts.Foran The equilibrium approximation discussed in the last uncompact system, such a time scale does not exist. Nev- section can be applied to gravity systems, with the assump- ertheless, at a finite time t, subregions of certain finite sizes tion that they follow the usual rules of quantum mechanics. can thermalize, and we can apply the approximation of In this context, various quantities in Eq. (37) or Eq. (52) Sec. II C–II E to such subregions. should be seen as amplitudes in an exact theory of quantum As an illustration, we consider an infinite (1 + 1)- gravity. In particular, the Euclidean path integrals Eq. (53) dimensional system, which can be a spin chain or a quan- emerge universally as an approximation to the Lorentzian tum field theory. We assume for simplicity that the system path integral Eq. (13). Furthermore, different Euclidean is governed by a local Hamiltonian, which results in a replica gravity systems have to be “coupled” in specific sharp light cone, with speed c = 1. Suppose we are inter- ways. However, in our current understanding of quantum ested in the entanglement of a finite region A with its gravity, gravity path integrals can only be formulated at a ¯ complement A at time t, as indicated in Fig. 8. Due to the semiclassical level, and hence a direct implementation of causality constraint from the sharp light cone, the region the prescription Eq. (54) may be subtle. For holographic that is relevant for this purpose is J (A), the region at t = 0 systems, the amplitudes in Eqs. (37) and (54) also have a that is causally connected with A. Evolution of the system dual description in terms of the corresponding ones in the ( ) (A) (A¯ ) in J A should not be relevant for finding Sn or Sn ,and boundary system. Here, one has the benefit that the bound- we can replace the time-evolution operator in this part of ary version of path integrals Eq. (54) can be used to provide the system with the identity. See Appendix C for a more boundary conditions for formulating the corresponding explicit argument. bulk ones by using the standard rules of holography. Let us further suppose that the system is sufficiently Intuitively, couplings among different replicas could strongly interacting and chaotic, such that at time t, the lead to replica wormholes, that is, geometries connecting system is locally equilibrated in region J (A), i.e., the equi- different replica manifolds. In this section, we make this libration is maximally efficient as allowed by causality. In idea precise by applying the equilibrium approximation to this case, we can apply Eq. (37), treating J (A) as the full two recently discussed models of black holes [17,18], and system, and the region J (A) − A of length 2t as the com- showing that the prescriptions proposed there for including plement of A. From the discussion of item 1 in Sec. II G certain replica wormholes in the calculation of entangle- we immediately conclude that [65] ment entropies follow from Eq. (37) and Eq. (54).The earlier discussion of Sec. II F then provides an explanation S(A) = seq min(|A|,2t), (93) for why including replica wormholes leads to entangle- n n ment entropies that are consistent with unitarity. We also comment on how the equilibrium approximation provides eq where sn is the nth equilibrium Renyi entropy density an alternative to the need for an averaged description dis- (n = 1 being the equilibrium entropy density). Here the cussed in Ref. [17], and briefly discuss the Renyi entropies entanglement velocity is given by vE = c = 1. Since it is for a big black hole in AdS formed from gravitational expected on general grounds [66,67] that vE ≤ vB, where collapse of a pure state. vB ≤ 1 is the velocity associated with the growth of oper- ators, Eq. (93) implies that we must also have vB = 1, corresponding to the fact that operators grow at the fastest A. A model for black hole evaporation speed allowed by causality. Hence, we can see the maxi- Let us first briefly review the model of an evaporating mally fast growth of operators as a necessary condition for black hole discussed in Ref. [17], where the black hole the assumption of maximally efficient equilibration that we lives in a (1 + 1)-D spacetime with Jackiw-Teitelboim (JT) make above. gravity and has an end-of-the-world (EOW) brane behind

010344-15 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(a) (b) (a) (b)

FIG. 10. (a) Boundary conditions and (b) the bulk path integral (ρ ) for evaluating R ij according to the rules of Ref. [17].

the resulting expression gives a derivation of the replica FIG. 9. Lorentzian and Euclidean geometries for a black hole with an end-of-the-world brane (denoted by the green line). wormholes introduced with the ad hoc rules above. The final result matches precisely with that of Ref. [17]. Let us consider a situation where the initial state |0 | in Eq. (4) describes a star, which under time evolution the horizon, see Fig. 9. The state of the full system collapses to form a black hole and subsequently emits resulting from the evaporation process is assumed to be Hawking radiation. The full system at time t, described by such that if |i is an orthonormal basis of N states for the | (ρ ) the state , consists of the black hole and the emitted radiation subsystem R, then the matrix element R ij of radiation subsystem. The radiation subsystem has a Hilbert the reduced density matrix for R can be calculated using space of finite dimension N with no energy constraint, Euclidean path integrals with the following rules, shown in while the black hole can be associated with an inverse tem- Fig. 10. The boundary condition is given by a single open perature β. We assume that the radiation separates from asymptotic boundary segment of JT gravity of length β for β and no longer interacts with the black hole after being some inverse temperature associated with the state. The emitted. We can then write the effective identity operator endpoints of the segment are labeled by i and j , as shown Iα corresponding to | in a factorized form in Fig. 10(a). In the corresponding bulk path integral in Fig. 10(b), the two endpoints are connected with an EOW (B) brane. In addition to a gravity path integral indicated by Iα = 1R ⊗ Iβ ,TrR1R = N, (94) the shaded region, this gives a factor of δij , indicated by the dotted line connecting i and j . Z(R) where R and B denote, respectively, the radiation and black The boundary conditions for the calculation of n involve n open asymptotic boundary segments of length hole subsystems. β as shown in Fig. 11(a) for n = 3, with the dashed lines For an evaporating black hole, the Hamiltonian of the indicating the contraction of indices in the matrix multipli- system cannot be strictly time independent. As a result, cation. The rule for the corresponding bulk path integral Eq. (94) may not strictly satisfy Eq. (24). However, is to sum over all possible ways of connecting the end- Eq. (24) should still be valid to a very good approximation points with EOW branes, like the two examples shown in if the evaporation process happens slowly. Figs. 11(b) and 11(c). Each resulting loop of dashed lines For comparison with the discussion of Ref. [17], we gives a factor of N. Contributions like Fig. 11(c), where assume that the black hole subsystem resembles that in JT multiple asymptotic boundaries are connected by the bulk gravity (or the Sachdev-Ye-Kitaev model model at a suf- geometry, are said to have replica wormholes, and such ficiently low temperature). That is, it has a large number wormhole contributions are important for giving results for of densely spaced states in the energy range accessible at Z(R) inverse temperature β, such that n consistent with unitarity. Z(A) We now describe the evaluation of the quantities n ( ) ( ) according to the equilibrium approximation in a more gen- B B S0 Z = TrBIβ = e z1(β), eral class of quantum-mechanical systems related to the 1 m above model. We then show that applying the standard ( ) (B) Z B = Tr I = eS0 z (β), (95) rules of holography to the Euclidean path integrals in m B β m

(a) (b) (c) FIG. 11. (a) Boundary conditions, and (b),(c) show two contributions to the bulk Z(R) = path integral for evaluating n for n 3 according to the rules of Ref. [17].

010344-16 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) where the “background” density of states eS0 is large, and Eq. (48), can in turn be understood as the number of non- zm(β) are O(1) functions. We are interested in the regime crossing partitions with mi blocks of cardinality i.The explicit expression for this number when the total number − > eS0 , N →∞, Ne S0 = finite. (96) of blocks k 1 is given by [68] ( − ) ···( − + ) Applying Eq. (73) to Eq. (94) we find for the radiation n n 1 n k 2 N({mi}) = . (100) subsystem m1! ···mn! = 1 −1 For k 1, we have only one block consisting of all n Z(R) ≈ k(τη ) (B) ··· (B) n n N Zn Zn , (97) ({ }) = (B) 1 k(τ) elements, and N mi 1. τ NZ1 So far, our discussion is general and applies to any sys- I (B) tem with α as in Eq. (94) and Zm as in Eq. (95).The where n1,...,nk(τ) are the lengths of the cycles in τ. In the specific gravity description enters in the explicit evalua- (B)(β) regime Eq. (96), the leading terms in Eq. (97) are given tion of various partition functions Zm [or equivalently by those τ’s that saturate Eq. (48), that is, by the planar zm(β)], which can be expressed in terms of Euclidean diagrams of Sec. II E. We can write the cycle structure of gravity path integrals and evaluated using a saddle-point a permutation τ as τ = (1m1 2m2 ···nmn ), which indicates approximation. Let us now specify to the gravity system that it has m1 cycles with one element, m2 cycles with two considered in Ref. [17]. This motivates us to write elements, and so on. By definition we have ( ) B −βHB Iβ = f (HB)e , (101) n k(τ) = m1 +···+mn, n = imi. (98) where HB is the Hamiltonian for the black hole subsystem i=1 and we include a factor f (HB), which captures the pres- Since the summand in Eq. (97) depends only on the cycle ence of the EOW brane. The specific form of the function τ f is not important for our discussion. structure of , we can write the leading planar contribution (B)(β) as The calculation of Zm using bulk gravity follows from the standard rules of holography. The partition func- (B) (B) n n tion Z (β) of Iβ can be obtained from the Euclidean ( ) 1 − (B) 1 Z R ≈ 1 k ({ }) ( )mj (B) n ( ) N N mi Zj , black hole geometry of Fig. 9(b). The evaluation of Z (β) ( B )n  m Z = = = 1 k 1 i mi k j 1 for m > 1 is indicated in Fig. 12, and involves replica (B) (99) wormholes. Zm here should thus be identified with the replica wormhole partition function Zm with m boundaries where N({mi}) is the number of planar permutations given in Eq. (2.29) or (2.32) of Ref. [17,69]. The contribu- with cycle structure {mi}, and from our comments below tions from terms corresponding to different τ in Eq. (97)

(a) (b)

( ) −β m B = ( ) HB = FIG. 12. Boundary path integrals for Zm TrB f HB e can be represented as in (a) for n 6. The Euclidean path integral 1   −βHB along each black line from t = 0tot = β represents e , and each green dot represents f 2 (HB). For convenience of presentation we arrange the factor f (HB) symmetrically in each replica. (b) The dual gravity description of the boundary path integral, where one should integrate over all bulk configurations with the specified boundary topology, and we interpret the green dots representing f (HB) as the end-of-the-world branes.

010344-17 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(eq) 1 saturating Eq. (48) can be captured precisely by the pla- ρ = Iα  where Z and ij is the contribution from the Q nar gravity diagrams of Ref. [17] like the ones shown in 1 projector. Using Eq. (94) and dropping ij , we obtain the Figs. 11(b) and 11(c), which correspond, respectively, to equilibrium approximation for these matrix elements, τ = e and τ = η. We can show more explicitly that Eqs. (99)–(100) agree 1 (ρ ) ≈ δ , (106) precisely with the results obtained in the limit Eq. (96) in R ij N ij Z(R) Refs. [17,70]. In Ref. [17], the full expression for n was only given implicitly through a generating functional which would imply called the resolvent, which can be used to obtain a recur- (R) sion relation for Z . More explicitly, the trace of Eq. (R) n 1 n Z = TrRρ ≈ . (107) (2.27) of Ref. [17] can be written in our notation as n R N n−1

 ∞ ∞ (B) ∞ (R) n But Eq. (107) clearly contradicts Eq. (99). For example, 1 1 Z  N Z Z(R) = n + m−1 for n = 2, Eq. (99) gives  ( )  . λn+1 n λ n ( B )n λ λm n=1 n=1 N Z1 m=2 (B) (102) ( ) 1 Z 1 − z2(β) Z R ≈ + 2 = + S0 ( ) e . (108) 2 B 2 2(β) N (Z ) N z1 Equating coefficients of 1/λn+1 on both sides, we find a 1 recursion relation Equations (107) and (108) are compatible only when N eS0 , but the derivation of Eq. (106) uses only Z ∼ NeS0 n−1 (B)  1 ( ) 1 ( ) Z  n ! Z R = Z R + n r1 1 and, in particular, does not need to assume any relative −  ( )  N n n 1 n ( B )n ··· magnitude of eS0 and N. N = N Z  r1! rn! n 2 1 r1+···+ rn=n = In Ref. [17], the same apparent disagreement was i iri n observed, and it was pointed out that Eqs. (106) and (108) n (B) ( ) Z × (Z R )rt + n can be compatible if the Euclidean gravity prescription − ( ) . (103) t 1 ( B )n for computing them is interpreted as an average over an = Z1 t 2 ensemble of theories, so that there is a difference between (ρ ) (ρ )∗ |(ρ ) |2 One can check that Eqs. (99)–(100) indeed satisfy the averages R ij R ij and R ij . We now show that Eq. (103) [71]. the conflict between Eq. (106) and Eq. (108) can be nat- urally resolved using the equilibrium approximation inter- B. Comments on averaging and replica wormholes pretation of the Euclidean gravity prescription, without the need for any averages. In the previous subsection, we demonstrate how the For this purpose, let us estimate the term ij we drop Euclidean gravity prescription for computing the Renyi in reaching Eq. (106) using the analogous procedure to entropies of an evaporating black hole can emerge as an Eq. (40), which is computed in Appendix B. The results approximation to the Lorentzian path integrals Eq. (13). are One important implication of the discussion is that replica ( ) ( ) wormholes can arise in a system with a fixed Hamiltonian, 1 Z B 1 Z B ( )2 = δ 2 | |2 = 2 and it not necessary to have an ensemble-averaged theory. ij ij ( ) , ij ( ) . eq app N 2 ( B )2 eq app N 2 ( B )2 We now further clarify an issue raised in Ref. [17], which Z1 Z1 was used there to interpret replica wormholes as arising (109) from some averaging procedure. Comparing Eq. (109) with Eq. (106), we see that  is Let us first recapitulate the issue. Consider a matrix ele- ij − 1 S ment of the reduced density matrix for the radiation system suppressed by a factor e 2 0 compared with the leading- (here |i denotes a basis for the R subsystem), order contribution, Eq. (106). So the approximation, Eq. (106), appears to be justified in the limit eS0 →∞. † (ρR)ij = TrR(|j i| ρR) = TrR[|j i| TrB(Uρ0U )] This still does not say anything about the relative magni- tude between N and eS0 , so the tension between Eq. (107) = ¯ ⊗ | ⊗ †|ρ iRjR eB U U 0, e . (104) and Eq. (108) remains. But notice that Applying the same procedure as in Eq. (27) to Eq. (104), ( ) 1 Z R = |(ρ ) |2 = |(ρ ) |2 + |(ρ ) |2 = we find 2 R ij R ii R ij = N i,j i i j 1 (eq) (ρ ) = ¯ ⊗ |I +  = ρ +  2 (B) (B) R ij iRjR eB α, e ij R ij , N − N Z 1 Z Z ij + 2 ≈ + 2 1 ( ) ( ) , (110) N 2 ( B )2 N ( B )2 (105) Z1 Z1

010344-18 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) which recovers Eq. (108).InEq.(110), we use Eq. (106) improved within the framework of semiclassical gravity, if for the diagonal terms, and the second equation of we are able to perform a Lorentzian rather than Euclidean Eq. (109) for the off-diagonal terms. calculation of these quantities. We thus see that although the off-diagonal elements For a different perspective on how replica wormholes −S |ij | are higher order in e 0 compared with the diago- can emerge without the need for ensemble averaging, see nal elements, there are many more of them [O(N 2)] than Ref. [31]. Z(R) the number N of the diagonal terms. So 2 can receive significant contributions from these off-diagonal terms out- C. A model for an eternal black hole coupled to a bath S0 side the regime N e , and hence the corrections to We now consider a quantum-mechanical system that the equilibrium approximation for the matrix elements are corresponds to the gravitational system discussed in Z(R) important while estimating 2 . Since the first equation Ref. [18](seealso[15,16,20,72]), which consists of an of Eq. (109) vanishes for off-diagonal elements, we con- eternal black hole in AdS coupled to a flat (1 + 1)-  2 clude that the off-diagonal elements ij must be complex dimensional bath system with speed of light c = 1, see and likely time dependent. Fig. 13(a). In the quantum-mechanical system, shown in Our explanation here is consistent with an idea dis- Fig. 13(b), the eternal black hole is replaced by a boundary cussed in Ref. [17], that an average over time rather than dual. Note that the bath system remains the same in two an average over theories may also explain the disagree- descriptions. The whole system is initially put in a pure ment between Eqs. (106) and (108), as the equilibrium state, which does not have any entanglement between the approximation should agree with an average over time at black hole and bath subsystems [73]. We also assume the late times. interactions between the black hole and the bath are local. We emphasize that the conceptual picture obtained here The full system is uncompact with local interactions, as is different from certain possibilities proposed in Ref. [17]. in the case discussed in Sec. II H. We can therefore apply It was suggested there that bulk geometry may only pro- the discussion of that subsection to the current context, vide “an effective, coarse-grained description” of some replacing the subsystem A there by the subsystem describ- “different, more fundamental degrees of freedom” that ing the eternal black hole. In particular, for the purpose of correspond to the boundary theory, and that such non- studying entanglement between the black hole and the bath geometric degrees of freedom may need to be added to at some time t, it is enough to consider the finite region calculate quantities like Eqs. (106) and (108) to an accu- J (B) around the black hole determined by causality, as racy that allows us to avoid the apparent disagreement. indicated in Fig. 13(b). When the bath is maximally effi- Here, we emphasize that the conflict arises from drop- cient in thermalization as in our discussion of Sec. II H,we  ping ij in Eq. (105), which is equivalent to approximat- can immediately write down the Renyi entropies for either ing these quantities with Euclidean path integrals like in subsystem as a function of time, Eq. (53). It therefore arises even before we use semiclassi- cal gravity to evaluate these Euclidean path integrals, and S(BH) = S(bath) = min(2S(BH), S(bath)(t)], is thus not linked to using semiclassical gravity. In partic- n n n n S(bath)( ) = eq ular, this means that the approximation can in principle be n t 2tsn , (111)

(a) (b)

FIG. 13. (a) The Penrose diagram for an eternal black hole in AdS2 coupled to a bath, the system discussed in Ref. [18]. The shaded region coupled to JT gravity corresponds to the black hole, while the unshaded region with flat Minkowski space corresponds to the bath. (b) A quantum-mechanical system dual to this gravitational system. In this dual theory, the black hole is a (0 + 1)-dimensional system while the bath is (1 + 1) dimensional. At time t, only the part of the time-evolution operator in the region J (B) is relevant for finding the entanglement entropies between the black hole and the bath.

010344-19 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(BH) For a black hole formed from collapse, the HRT surface for where Sn are the thermal entropies for the black hole, eq A¯ is the same as that for A, and thus the unitarity constraint and sn is the equilibrium entropy density for the bath. ¯ = (A) = (A) The entropies increase linearly and then saturate at tn S1 S1 is automatically satisfied [76]. (BH) eq Sn /sn . The expression is valid for t not close to tn, For the Renyi entropies, it is less well understood how and agrees with the results for the entanglement entropy unitarity is maintained. From our discussion of Sec. II G, in Ref. [18]. the Renyi entropies of a subsystem in the boundary field Using the duality between the black hole and its bound- theory can be obtained by the path integrals in Fig. 6, ary description, we can understand the two contributions in and correspondingly in the bulk calculation, the Renyi Eq. (111) more explicitly from the gravity perspective. For entropies can be obtained from certain Euclidean black this purpose it is convenient to start with the path-integral hole geometries, despite the fact that a black hole from representation, Eq. (54), for the boundary description of collapse does not have a Euclidean analytic continua- Fig. 13(b), with A = bath and A¯ = BH. Then the linearly tion. However, the boundary conditions for these gravity increasing contribution to Eq. (111) comes from the con- path integrals include the unconventional ones specified figuration in Fig. 6(a) with τ = e, while the saturation by permutations τ, as indicated for example in Figs. 6(b) value for t > tn comes from the configuration in Fig. 6(b) and 6(c). From Sec. II G, at leading order in the large with τ = η.Forτ = e, one traces over the black hole sub- N expansion, one should consider two types of bulk system within each replica copy. On the gravity side this geometries, one for the boundary conditions with τ = e corresponds to the standard evaluation of the black hole and one for those with τ = η in Figs. 6(a) and 6(b). partition function using the Euclidean black hole geom- Bulk manifolds with more exotic boundary conditions etry. For τ = η, the black hole subsystems for different such as those in Fig. 6(c) provide subdominant correc- replica copies are now connected, and on the gravity side tions, which are exponentially suppressed in the large N this requires the introduction of replica wormholes. The limit. contributions from other values of τ, such as the example in Fig. 6(c), involve other types of replica wormholes, and are only relevant in a relatively short time interval around IV. TYPICALITY AND THE RANDOM VOID the transition time tn between linear growth and saturation. DISTRIBUTION Our assumption that the bath is maximally efficient in In the discussion of the previous sections, we assume thermalization is not important for obtaining the quali- that the time-evolution operator U can take a system from tative features of the results here. The specific physical a far-from-equilibrium state to an equilibrated pure state, nature of the bath system may affect the specific form of (bath) and furthermore that U is such that the contribution from S (t), and the time scale for saturation. However, it (A) n Z in Eq. (27) can be neglected. For a finite-dimensional will not change the fact that entanglement entropies will n,Q system at infinite temperature, the approximation yields S(BH) saturate at 2 n due to the contribution from Fig. 6(b).In the same results as those obtained from the Haar-random particular, the bath can in principle be a free theory instead Z(A) averages of the quantities n over the full Hilbert space. of a chaotic system that leads to rapid thermalization, like (A) Thus, the suppression of Z may be viewed as a dynam- one of the toy models considered in Ref. [44]. n,Q ical criterion for the evolution of a system towards typical states in such systems. However, since we expect on D. Unitarity of Renyi entropies in more general general grounds that evolution to typicality should take holographic systems place in chaotic systems, it would be good to understand In the previous subsections, we consider situations more directly which aspects of chaos are responsible for where only one of the subsystems has a gravity description, it. In this section, we offer some suggestions from the and the path integral involving that subsystem could be perspective of operator growth. turned into a gravity calculation involving replica worm- We conjectured in Ref. [48] that one general feature of holes. However, the unintuitive couplings between replicas operator evolution in chaotic systems at late times is the should appear in more general contexts. Consider an ini- form of the probability of “void formation” in them, which tial pure state, which subsequently undergoes gravitational we referred to as the random void distribution. In Ref. [44], collapse to form a big black hole in AdSd+1, which is sta- based on studies of the second Renyi entropy, we argued ble and does not evaporate. In the boundary language, the that typicality is a direct consequence of the random void system settles into an equilibrated pure state corresponding distribution. to the black hole. The von Neumann and Renyi entropies In this section, we generalize the notion of the random of this final state should satisfy the unitarity constraint, void distribution to higher moments, and show that the Eq. (2). behavior of all higher Renyi entropies of an equilibrated (A) In holography, the von Neumann entropy S1 for a sub- pure state can be seen as special cases of the generalized region A is found from the area of the HRT surface [74,75]. random void distribution. Below, we first review the notion

010344-20 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) of void formation in operator growth, and discuss its rel- purpose, we decompose the initial density matrix as evance for typicality. For simplicity, we consider only a finite-dimensional Hilbert space with no energy constraint 1 ρ =| |= 1 +ˆρ ,Trρˆ = 0. (116) (that is, the infinite temperature case). 0 0 0 d 0 0 Given the initial state is pure, A. Random void distribution and typicality ( ) d − 1 d − 2 Consider the Heisenberg evolution O t of an initial ρ2 = ρ ⇒ˆρ 2 = 1 + ρˆ operator O. Since the identity operator does not evolve, 0 0 0 d2 d 0 in the discussion below we always assume that O does ⇒ Tr ρˆ2 ≈ 1, (117) not have an identity part, i.e., Tr O = 0. With respect to 0 ( ) a subsystem S, we can decompose O t as a sum of two where the final statement is true in the large d limit. To find parts, the reduced density matrix for a subsystem A, we further † decompose ρ(ˆ t) ≡ Uρˆ0U as ( ) = (1)( ) + (2)( ) (1)( ) = ⊗ ˜ O t O t O t , O t 1S OS¯ , (1) (2) (1) (2) ρ(ˆ t) =ˆρ +ˆρ , ρˆ = O ⊗ 1 ¯ ,Tr¯ ρˆ = 0 (2)( ) = A A A TrS[O t ] 0, (112) (118) ˜ where 1 is the identity operator for subsystem S and O¯ is for some OA in subsystem A. Tracelessness of ρˆ implies S ¯ S some operator in S. that TrA OA = 0. It then follows that the reduced density We refer to the presence of O(1)(t) in O(t) as void forma- matrix for A has the form tion in the subsystem S. Using the following inner product 1 between any two operators A and B, ρ ( ) = +ˆρ ρˆ = ρˆ = A t 1A A,TrA A 0, A dA¯ OA (119) dA

1 † ρ ( ) A, B= Tr[A B], (113) and the second Rényi entropy for A t can be written as d − (A) 1 1 ( ) S2 = ρ2( ) = + ρˆ2 = + (ρˆ 1 )2 where d is the dimension of the full Hilbert space, we e TrA A t TrA A dA¯ Tr[ ]. dA dA can define the weight (or “probability”) that an operator (120) O forms a void in the subsystem S at time t as Now assuming the random void distribution, Eq. (115),for (1) (1) =ˆρ = ¯ ( ) O (t), O (t) operator O 0 and the subsystem S A, we have P S (t) = . (114) O,2 O(t), O(t) (ρˆ(1))2 (ρˆ(1))2 (A¯ ) Tr[ ] Tr[ ] 1 Pρˆ = = = 0,2 Tr{[ρ(ˆ t)]2} Tr[(ρˆ )2] d2 Moreover, based on studies in random unitary circuits, 0 A¯ it was conjectured in Ref. [48] that in a chaotic system, 1 1 ⇒ (ρˆ(1))2 = ⇒ ρˆ2 = for a generic traceless initial operator O, at sufficiently late Tr[ ] 2 TrA A , ( ) d ¯ d ¯ times the probability P S (t) has a simple universal form A A O,2 (121)

(S) 1 where we use Eq. (117). We thus find P (t) = , d¯ 1, (115) O,2 d2 S S ( ) −S A 1 1 e 2 = + . (122) which is referred to as the random void distribution. We dA dA¯ note that Eq. (115) should apply essentially to all traceless Equation (3) with n = 2 then follows immediately. The operators, for example, in a spin chain, to local oper- first term in Eq. (122) is the contribution from the iden- ators, basis operators, which cover a finite region, and tity, and the second term comes from processes of void superpositions of nontrivial basis operators. formation in A¯ under the action of U. We now show that if we assume Eq. (115) applies to the nonidentity part of a density matrix for a pure state we obtain Eq. (76) for n = 2. The discussion below can be B. Higher moments of the random void distribution seen as a model-independent version of the argument in Just like the behavior of the second Renyi entropy Ref. [44] for the derivation of the Page curve of black hole may be considered a direct consequence of the random evaporation using the random void distribution. For this void distribution, our discussion of the Renyi entropies in

010344-21 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

Sec. II G can be considered a special case of the random of Eq. (121) to higher n [in the regime of Eq. (75)] void distribution for higher moments of general operators. − More explicitly, consider the generalization of Eq. (114) 1 n 1 1 ρˆn = + ˜ ( ) to higher n: TrA[ A] n−1 N n, p n−p p−1 , (128) d ¯ d d A p=2 A A¯ n ( ) Tr[O (t) ] 1 1 S ( ) = 1 = { ( ) }n ˜ P t − TrS¯ ( TrS[O t ] ). where N(n, p) is the number of noncrossing partitions of n O,n Tr[On] dn 1 Tr[On] S objects into p blocks such that each block has more than (123) one element. N˜ (n, p) are called the Riordan numbers [77]. The consistency of Eq. (128) with Eq. (76) leads to a nice = ρ The nth Renyi entropy for A corresponds to taking O relation between the Riordan and Narayan numbers and S = A¯ . The general approximation scheme we develop in Sec. II G can be used to find Eq. (123), assuming there n 1 − (S) ( ) N(n, p) = CkN˜ (k, p) = Cp Cp 1. (129) exists a time scale such that the quantity PO,n t saturates n n n n and the contribution from projector Q can be neglected. k=p More explicitly, we find in the large d limit that It is tempting to conjecture that Eqs. (126)–(127) apply { ( ) }n to a general chaotic system. In Appendix D we show TrS¯ ( TrS[O t ] ) that they hold in the local random unitary circuits of 1 (η−1τ)− (τ)− Refs. [78,79]. = (Tr O)n + dk ndk n τ|O, e n−1 S¯ S d¯ τ= η S e, V. CONCLUSIONS AND DISCUSSION 1 + n n−1 Tr[O ], (124) In this paper, we develop an approximation to calculate dS the entanglement entropies of an equilibrated pure state. The resulting expressions can be written solely in terms where we separate the contribution from τ = e and τ = η of the partition functions and thermodynamic entropies of explicitly. Recall that the equilibrium density operator ρ(eq), but at the same time are compatible with unitarity. One immediate implication τ|O, e = Tr On1 ···Tr Onk , (125) is that a set of Euclidean path integrals for the equilibrium density operator emerge universally as an approximation where k = k(τ) and ni are the lengths of the cycles of τ.If to the Lorentzian path integrals for Renyi entropies, with O is traceless, a variety of boundary conditions specified by different permutations of the replica systems. We introduce a crite- − n k(η 1τ)−n k(τ)−n rion for checking that the approximation is self-consistent, Tr¯ ({Tr [O(t)]} ) = d d τ|O, e S S S¯ S which at the same time provides an estimate of the con- σ=e,η, Z(A) τhas no cycle with tribution n,Q we neglect. We also extract the universal one element behavior of the entanglement entropies for various classes 1 of equilibrated pure states. + Tr[On]. (126) dn−1 Applied to two recently discussed models of black holes S [17,18], the equilibrium approximation leads to a deriva- In particular, with Tr[On] ∼ O(1) (i.e., independent of tion of the prescriptions proposed in these papers for including replica wormholes in the calculation of entan- d , d¯ , d), then for d d¯ we have S S S S glement entropies, and provides a general explanation for why such a prescription leads to results compatible with ( ) 1 S ( ) = unitarity. Replica wormholes are thus one manifestation of PO,n t 2(n−1) . (127) dS a universal structure, which appears in a large variety of thermalizing systems, including quantum-mechanical sys- Heuristically, Eqs. (115) and (127) suggest that the oper- tems and quantum field theories without holographic duals. ator has become uniformly spread throughout the system, Our derivation can be used to see when and how replica so that the probability that it is trivial in any small subsys- wormholes should be included in more general gravity tem is exponentially suppressed in the number of degrees theories, and in particular it shows that they can arise in of freedom in that subsystem. systems with a fixed Hamiltonian, without any need for an ¯ Setting O in Eq. (124) to be ρ0 and S = A, we then ensemble average. recover Eq. (76). Setting in Eq. (126) O to be ρˆ0 (i.e., the We further discuss a mechanism for equilibration in the traceless part of ρ)andS = A¯ , we obtain a generalization infinite-temperature case from the perspective of operator

010344-22 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021) growth. The underlying property of operator growth, called chaotic systems with conserved energy, both of which are the random void distribution, can be seen as a more direct not captured by the equilibrium approximation. In the JT manifestation of quantum chaos than the assumptions that gravity calculations of Refs. [80,82], the ramp contribution went into the equilibrium approximation. to the spectral form factor is correctly captured by “trum- One important open question for the future is to under- pet” geometries, which involve bulk connections between Z(A) stand better the role of the contribution n,Q, which we disconnected closed boundaries. While these structures neglect, and to develop a further approximation scheme geometrically resemble the replica wormholes of Ref. [17], to capture its effects (see Ref. [11] for a discussion in our observations above imply that these two kinds of the infinite-temperature case). A related question is about connected geometries appearing in two different quanti- the time scales for which our equilibrium approximation ties have distinct physical origins. The replica wormholes should be valid. We expect it to be valid for time scales of Ref. [17], appearing in the calculation of the Renyi much longer than the thermalization scale ts, but perhaps entropies, can be fully explained within the framework not at very long time scales at which large fluctuations of the equilibrium approximation. On the other hand, the could become relevant. ramp contributions to the spectral form factor, and hence It is also interesting to consider for which other observ- the trumpet geometries associated with them, appear to ables the approximation is expected to work well, and for be beyond the scope of the equilibrium approximation. which observables it is expected to fail. Let us discuss two The trumpet geometries lead to a disagreement between examples. For correlation functions of the form the direct evaluation of Tr[U]Tr[U†] and the product of Tr[U]andTr[U†] in the gravity calculation, which is sim- F = ψ0|O1(t1)O2(t2) ···On(tn)|ψ0, (130) ilar to the conflict between Eqs. (106) and (108) discussed in Sec. III B. However, while the equilibrium approxima- if the smallest time tm is much greater than ts, then we tion can resolve the issue discussed in Sec. III B, it does not can apply the equilibrium approximation, which gives the seem to explain the factorization problem of the spectral equilibrium correlation functions associated with ρ(eq) (see form factor. Appendix B for details): In a different future direction, it would also be inter- esting to understand how the mechanism for equilibration (eq) F ≈ Tr[ρ O1(t1 − tm)O2(t2 − tm) ···On(tn − tm)]. based on operator growth in the infinite-temperature case (131) in Sec. IV generalizes to other choices of Iα.

Using an analogous criterion to the one discussed in ACKNOWLEDGMENTS Sec. II D, we show in Appendix B that Eq. (131) is We thank Ping Gao and Sam Leutheusser for helpful valid provided that the minimal subsystem A in which discussions. This work is supported by the Office of High O ≡ O (t − t )O (t − t ) ···O (t − t ) is contained 1 1 m 2 2 m n n m Energy Physics of U.S. Department of Energy under Grant is much smaller than its complement A¯ . This is also consis- Contracts No. DE-SC0012567 and No. DE-SC0019127. tent with what we expect based on the behavior of Renyi entropies in this regime [recall Eqs. (69)–(70)]. As a final example, let us consider the quantity APPENDIX A: EQUILIBRIUM APPROXIMATION FOR A GENERAL INITIAL DENSITY MATRIX † n ˆ † n Z ≡ (Tr U Tr U ) = Tr(U ⊗ U ) , (132) Here we discuss the generalization of the equilibrium approximation to a general initial density operator rather where in the second equality we express the quantity in the than a pure state, which has been the focus of the main ˆ n replica system, with Tr denoting the trace in (H ⊗ H) . text. This is the nth power of the spectral form factor stud- Consider the quantities iedinRef.[80]. This quantity does not correspond to the = ρn = ρn = ( ρ †)n = η|( ⊗ †)n|ρ equilibration of a far-from-equilibrium initial state or have zn Tr 0 Tr Tr U 0U U U 0, e . an equilibrium value. We therefore intuitively expect that (A1) the equilibrium approximation should not make sense for Eq. (132). Indeed, for any choice of Iα, on applying the Applying the equilibrium approximation to the above equilibrium approximation to the above expression, we expression, we have Z ≈ get n!, which does not satisfy the self-consistency criterion of Sec. II D [81]. = 1 η|I τI τ|ρ zn n α, α, 0, e . (A2) Based on studies in random matrix theory and the SYK Z2 τ model, the spectral form factor is expected to have a lin- S ear “ramp” and eventually a constant “plateau” of order e In the case where ρ0 is a pure state, recall that An = (where S is the thermodynamic entropy) at late times in Iα, τ|ρ0, e is independent of τ, and by imposing Eq. (30)

010344-23 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021) we obtain zn ≈ 1asdiscussedinEq.(66).Forρ0, which is expressions for Eq. (A8) are somewhat complicated. a mixed state, requiring the n = 1 equation to be satisfied For example, for n = 2andn = 3, we find again fixes ( ) Z ¯ Z A ≈ Z(A,eq) + − 2 Z(A,eq) 2 n z2 2 n (A10) Z2 Z1 Tr Iαρ0 = . (A3) Z1 and For n = 2, (A) ( ) Z2 (eq) (eq) Z ≈ Z A,eq + 3 z − Tr[(ρ ⊗ ρ )ρ(eq) ] 3 n 2 2 A A¯ Z1 1 z = (η|Iα, eIα, e|ρ , e + η|Iα, ηIα, η|ρ , e) Z3 Z2 Z2 2 2 0 0 + z − − 3 z − Z2 3 3 2 2 2 Z1 Z1 Z1 2 1 Z2 2 2 ( ¯ ) − = Z + Z Tr(Iαρ ) , (A4) × Z A,eq +η ⊗ |ρ(eq) η 1 2 2 2 1 0 n A eA¯ , . (A11) Z2 Z1 Z(A) which requires APPENDIX B: ESTIMATE OF n,Q IN VARIOUS CASES 2 2 Z2 Z2 Here we discuss the calculation of Eq. (40) to show (Iαρ ) = − Tr 0 z2 2 2 . (A5) Z1 Z1 the self-consistency of the equilibrium approximation, and also the analogous quantities for the equilibrium approxi- 2 −1 mation of other observables. Since Z2/Z ∼ Z 1, for z2 ∼ O(1) we then have 1 1 For any quantity T that can be written as a transition T 2 amplitude in a replica Hilbert space, we can separate as 2 Z2 Tr(Iαρ0) ≈ z2 . (A6) a sum of two parts, Z2 1 † n T = b|(U ⊗ U ) |a = TP + TQ, TP = b|Pα|a,

Continuing this further, the equation for zn can be used (B1) n to determine Tr[(Iαρ ) ]intermsz , ...z and Z , ...Z . 0 1 n 1 n for some states |a , |b∈(H ⊗ H)n. The definition of Each of these equations should be imposed as a self- T T I ∼ ( ) P, Q are in complete analog with Eq. (27) for the Renyi consistency condition on α. In particular, for zn O 1 , T we have entropies. Note that in general may not be real. To dis- cuss the self-consistency of the equilibrium approximation n for the quantity T , similar to Eq. (40), we consider n Z2 Tr(Iαρ0) ≈ zn . (A7) n 2 2 2 Z1 |T | = (|T | ) −|T | Q eq app eq app P = ¯ ⊗ | ˜ |¯ ⊗ −|T |2 Now for a general density operator ρ0 we have in the b b Pα a a P ,(B2) large Z1 limit T 2 = (T 2) − T 2 = ⊗ | ˜ | ⊗ − T 2 Q eq app P b b Pα a a P , (A) (A) 1 Z ≈ Z = η ⊗ ¯ |I τ I τ|ρ eq app n n,P n A eA α, α, 0, e , Z2 τ (B3) n = 2, 3, .... (A8) where |¯a , |b¯∈(H ⊗ H)n are defined from |a , |b by the following procedure: Now using Eq. (A5) and it higher n counterparts we can ¯ ¯ ¯ ... i1i i2i ···ini |a = a   ···  express Eq. (A8) in terms of z1, zn and various partition 1 2 n i1i1i2i2 inin I ∼ ( ) functions of α. When zn O 1 for all n, we have → ¯ ¯ ··· ¯ |¯ = ∗ i1i1i2i2 inin a a   ···  (B4) i1i1i2i2 inin (A) (A) 1 Z ≈ Z = bτ η ⊗ e ¯ |Iα, τ , and n n,P Zn A A 1 τ 1 ˜ α = |Iα τIα τ| = ··· = ... P 2n , , (B5) bτ zn1 znk , n 2, 3, , (A9) Z 2 τ∈S2n

n where k is the number of cycles of τ with n1, ...nk is the projector associated with Iα in (H ⊗ H) ⊗ (H ⊗ n the lengths of the cycles. For general zn, the explicit H) . We again assume Z1 is large and Eq. (36).

010344-24 ENTANGLEMENT ENTROPIES OF EQUILIBRATED PURE STATES. . . PRX QUANTUM 2, 010344 (2021)

We then have 2 2 ≡ Z(A) n,Q eq app |T |2 Q eq app = 1 η ⊗ ⊗ η ⊗ |I τ A eA¯ A eA¯ α, .(B9) 1 ¯ Z2n = b ⊗ b|Iα, τ Iα, τ|¯a ⊗ a 1 τ=σ ⊗τ  Z2n 2 τ∈S2n 1 ¯     − b|Iα, τ Iα, τ |¯a b|Iα, σ Iα, σ |a . As in Eq. (47), we can estimate the magnitude of the above Z2n ¯ 2 τ ,σ ∈S expression by counting the number of traces for A and A, n ∝ ∝ (B6) with TrA dA,TrA¯ dA¯

When τ ∈ S can be written in a factorized form τ = 2n η ⊗ ⊗ η ⊗ |I τ ∼ k1 k2 = (ν−1τ)     A eA¯ A eA¯ α, d d ¯ , k1 k , σ ⊗ τ for σ , τ ∈ Sn (meaning that it can be seen as a A A σ  composition of a permutation involving only the first k2 = k(τ), ν = η ⊗ η. (B10) n elements with a permutation τ  involving only the last n elements), the contribution from τ in the first term can- σ  τ  cels with the contribution from the associated , in the k1 and k2 can be interpreted, respectively, as the number of second term. One can write down a similar expression for dashed and solid loops in the diagrams shown in Fig. 14. (T 2) Q eq app. Hence, These diagrams are obtained by following exactly the same rules as those for Fig. 2–4. The contractions correspond- η ⊗ ⊗ η ⊗ | 1 ing to the final condition A eA¯ A eA¯ are given in |T |2 = ¯ ⊗ |I τ I τ|¯ ⊗ Q b b α, α, a a , Fig. 14(a). Notice that the contractions for the first group eq app Z2n 2 τ=σ ⊗τ  of n elements and those for the second group of n ele- (B7) ments, respectively, form separate subdiagrams in such a τ = σ  ⊗ τ  1 way that for any , we always get nonplanar dia- T 2 = ⊗ |I τI τ| ⊗ Q b b α, α, a a , grams, as such a τ necessarily connects some elements of eq app Z2n 2 τ=σ ⊗τ  the first group to those of the second group. An example is (B8) Fig. 14(c), while τ for Fig. 14(b) is factorizable and does not contribute to Eq. (B9).   Like in the discussion around Fig. 5 in Sec. II E, we can where τ = σ ⊗ τ indicates that τ ∈ S2n cannot be written in a factorized form. obtain a double-line diagram from each of these diagrams We now consider Eq. (B7) or Eq. (B8) for a few different by adding an extra surrounding loop. The total number of observables. loops in the double-line diagram is again equal to the num- ber of faces F of the polygon associated with it, when it is placed on a manifold where it does not have crossing lines. 1. Renyi entropies In this case, the total number of edges of the polygon is For the nth Renyi entropy we take |a=|ρ0, e and E = 6n − 1 and the total number of vertices is V = 4n − 2, | =|η ⊗ τ b A eA¯ , for which we have so if the diagram associated with can be drawn without

(a) (b) (c)

FIG. 14. Examples of the diagrammatic representation for Eq. (B9) for n = 3. (a) The contractions of indices from “future η ⊗ ⊗ η ⊗ | τ conditions” A eA¯ A eA¯ . (b),(c) Examples of interior connections for two different choices of .

010344-25 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021) crossings on a surface of minimum genus h, then We see from the above discussion that the suppression Z(A) + = − = − + − − = + − of the correction n,Q has to do with the trace structure in k1 k2 F 1 E V 2 2h 1 2n 2 2h. the definition of the Renyi entropies. (B11) To see the factorization condition for higher m in Since we always get nonplanar diagrams with h ≥ 1for Eq. (43), let us now consider the quantity   τ = σ ⊗ τ ,  = Z(A) m − (Z(A))m k1 + k2 ≤ 2n. (B12) m n n,P eq app Also, since any τ = σ  ⊗ τ  is not equal to either e or ν, 1 m τ = (τ) ≤ − = (η ⊗ e ¯ ) |Iα, τ . (B16) and for any e, k 2n 1, Zmn A A 1 τ=σ  ⊗σ  ···⊗σ  1 2 m k1, k2 ≤ 2n − 1. (B13)    1 ( ) Here τ = σ ⊗ σ ···⊗σ indicates a permutation in Smn ∼ ∼ 2 Z A 1 2 m Recall that for dA dA¯ Z1 , the leading term of n,P that is not factorized among each of the m consecutive sets 1 (1−n) − 1 n 2  ∼ 2 of n elements. The discussion completely parallels the m = scales as Z1 , while from the above Z1 . When ∼ Z(A) 1, 2 cases discussed earlier, with dA dA¯ Z1, the leading contribution of n,P scales as / − − 1 1−n 1 2 n 2 k k − d while from the above  ∼ d d ¯ . In both cases (η ⊗ )m|I τ ∼ 1 2 = (ν 1τ) A A A A eA¯ α, dA d ¯ , k1 k m , we have A k2 = k(τ), νm = η ⊗ η ⊗···⊗η, (B17)  − 1    ∼ Z 2 1. (B14) Z(A) 1 m times n,P

1 k1 and k2 can again be seen as the number of dashed and ∼ ∼ 2 ∼ Also note that in both cases dA dA¯ Z1 and dA dA¯ solid loops in the diagrams of Fig. 15 (shown for the m = 3 Z1, the next-to-leading order correction to Zn,P is sup- case). Now for a diagram we have E = 3mn − m + 1, V = −1 − + pressed by Z1 relative to the leading contribution, so that 2mn 2m 2. Hence, the contribution from ZQ is larger than this contribution. = η = ( ) For n 1, we have [with 12 below] k1 + k2 = F − 1 = E − V + 2 − 2h − 1 = mn + m − 2h. (B18) 2 1 Z2  = e|Iα, η = 1, (B15) Z2 Z2 1 1 Examples of τ corresponding to h = 0andh = 1are which is consistent with the other self-consistency condi- shown, respectively, in Figs. 15(b) and 15(c). For any τ = σ  ⊗ σ  ···⊗σ  tion, Eq. (30), that we introduce earlier. 1 2 m, we get a nonplanar diagram, so for

(a)

(b) (c)

FIG. 15. Examples of the diagrammatic representation for Eq. (B16) for n = 3andm = 3. (a) The “future conditions” common to Z(A) m = τ all diagrams for n with m 3. (b),(c) Examples of interior connections for two different choices of . eq app

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¯ all terms appearing in Eq. (B16), where a, b denote indices for a basis of A. Now suppose Iα can be factorized Iα = I ⊗ I ¯ , we then find that k + k ≤ mn + m − 2. (B19) A A 1 2 τ = σ  ⊗ σ  ···⊗σ  2 ={(ρ(eq)) }2 (ρ(eq))2 Also, again since any is not equal to ij A ij TrA¯ [ ¯ ], 1 2 m eq app A e or νm, 2 (eq) (eq) (eq) 2 | | = (ρ ) (ρ ) Tr ¯ [(ρ )] . (B27) ij eq app A ii A jj A A¯ k1, k2 ≤ mn − 1. (B20) Now as an illustration let us consider the example of ∼ ∼ 1/2 We then find that in both limits, dA dA¯ Z1 and dA Sec. III A with Eq. (94) and A = R. We have ∼  −1 dA¯ Z1, m is suppressed by at least a factor of Z1 (Z )m (B) relative to the leading term in n,P . ( ) 1 1 Z (ρ eq ) ≈ δ 2 = δ 2 R ij ij , ij ij ( ) N N 2 (Z B )2 2. Matrix elements and correlation functions 1 ( ) 1 Z B Let us now consider | |2 = 2 ij ( ) . (B28) N 2 ( B )2 Z1 T = ψ0|O1(t1)O2(t2) ···On(tn)|ψ0 † = Tr[OU(tm)ρ0U(tm) ], (B21) b. Equal-time correlation functions where O ≡ O (t − t )O (t − t ) ···O (t − t ),andt Let us now consider Eqs. (B22)–(B24) for some generic 1 1 m 2 2 m n n m m operator O (which can be a product of observables), with is the smallest time among t1, ..., tn.Iftm ts, then we ( ) A the smallest subsystem containing O. can apply the equilibrium approximation for U tm , with ¯ |a=|ρ , e, |b=|O†, e and n = 1. We find In the case A A, from our earlier discussion in 0 Sec. II G, in particular, Eqs. (69)–(70), the reduced den- (eq) ρ ρ = ρ † ψ0|O1(t1)O2(t2) ···On(tn)|ψ0 ≈ Tr(Oρ ) sity operator A of U 0U is well approximated by ρ(eq) O( ) = A , and indeed we expect Eq. (B22) to hold as t = 1 (I O) Tr α , (B22) TrA(OρA) ≈ TrA[O(ρe)A] = Tr(Oρe). But going away Z1 from the regime A A¯ , we expect the approximation, which implies that the expectation value of O should be Eq. (B22), to break down. Let us see how this comes equal to that in the equilibrium density operator ρ(eq).For about from examining Eqs. (B23)–(B24). As an illustration Eq. (B22),Eqs.(B7) and (B8) become [below η = (12)] we consider a finite-dimensional Hilbert space at infinite temperature with I = 1. 1 1 T 2 = O† |I η = (OI OI ) For this purpose, consider Q , e α, Tr α α , eq app 2 2 Z1 Z1 ⎛ ⎞ (B23) O = ⎝ | |⎠ ⊗ cij i j 1A¯ , (B29) i,j 2 1 † |T | = O ⊗ O |Iα η Iα η|ρ Q eq app 2 , e , e , , 0, e Z2 where |i again denotes a basis for A. Suppose cij are ran- domly picked numbers with comparable magnitude (which 1 † = (O IαOIα) 2 Tr . (B24) we can take to be 1) but random phases. From Eq. (B22) Z1 1 1 a. Matrix elements of reduced density matrix O(t) ≈ c ∼ , (B30) d ii 1 O =| |⊗ | A i d 2 A special example is j i 1A¯ , where i is a A basis for a subsystem A, in which case we have while (ρ ) = (ρ(eq)) +  A ij ij ij , (B25) 1 1 1 1 1 A 2 ≡ |T |2 ≈ | |2 ∼ 2 ∼ Q cij dA . eq app d ¯ d2 d ¯ d2 d ¯ where ij is dropped under the equilibrium approximation. A A ij A A A Furthermore, (B31) 1 2 = (I ) (I ) | |2 We thus find the approximation, Eq. (B22), is good if ij α ia,jb α ib,ja, ij eq app 2 eq app Z1 a,b 1  2 1 dA = (I ) (I ) = 1 ⇐⇒ d d ¯ . (B32) α ja,jb α ib,ia, (B26) O( ) 1 A A 2 t 2 Z1 d a,b A¯

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FIG. 16. Time evolution of the initial state |ψ0 to the state |ψ(t)=U |ψ0 at time t by a circuit of local two-site unitary operators in an uncompact system.

APPENDIX C: CAUSALITY ARGUMENT any state ρ, Here we illustrate the causality argument of Sec. II H by (A) † † (A) S (U ⊗ U ¯ )ρ(U ⊗ U ) = S (ρ),(C1) modeling the time evolution using discrete steps, that is, by n A A A A¯ n a unitary circuit. More explicitly, we consider an infinite (A¯ ) † † (A¯ ) S (U ⊗ U ¯ )ρ(U ⊗ U ) = S (ρ) (C2) spin chain in 1+1-D with local Hilbert space dimension q, n A A A A¯ n and its local time evolution U is modeled by a circuit of two-site unitary operators, as shown in Fig. 16. We want for all n. Let us act on the state |ψ(t) created by the circuit ( −) ⊗ − to show that the entanglement entropies of the region A in Fig. 16 with U A¯ 1A, where the operator U acting shown in Fig. 8 are independent of the part of the time- on A¯ is constructed from local unitary operators as shown evolution operator outside the region J (A), which is in in Fig. 17. U− has t layers of local unitary operators like the causal contact with A. original time-evolution operator U, and each local unitary Recall that for any subsystem A, for any unitary opera- in U− is equal to the inverse of the operator in U located ¯ tors UA and UA¯ acting only on A and A, respectively, and at its mirror image with respect to the line at time t.By

FIG. 17. Blue rectangles indicate the local unitaries of the original circuit, red rectangles indicate their inverses used to construct U−, and white rectangles indicate the identity operator. We show the action of U− ⊗ 1 on |ψ(t), and some of the local random A¯ A unitaries in the definition of U−.

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FIG. 18. Due to the cancellation of inverses between local operators in U and U−, the final state in Fig. 17 can also be produced by A¯ ˜ ⊗ ˜ the circuit shown above, where the time-evolution operator is of the form UJ (A) 1J (A), with U given by the part of the circuit between the green dashed lines. construction, the state prepared by the circuit in Fig. 17 83], by using the Haar average of (V ⊗ V†)n, each V in is equal to the state prepared by that of Fig. 18, where the Fig. 19(a) can be associated with a “spin” σ taking val- ˜ ⊗ overall time-evolution operator is of the form UJ (A) 1J (A) ues in the permutation group S ,andEq.(D1) becomes ˜ n with UJ (A) nontrivial. the partition function for a classical spin system, which has the same lattice structure as the circuit in Fig. 19(a). P(S) See Fig. 19(b). The final state and initial state in Eq. (D1), APPENDIX D: O,n IN RANDOM UNITARY CIRCUITS respectively, determine the spin configurations at the top and bottom layers of the systems. In the top layer, the A In this section, we show that the result, Eq. (124), subsystem has e spins while A¯ has η spins. The bottom obtained from the equilibrium approximation holds for layer has a superposition of spin states determined from sufficiently late times in local random unitary circuits. the matrix elements of operator O. The spin system has We use the setup and methods of Refs. [78,79,83], where interactions among the three spins on each shaded triangle the system is a spin chain in (1 + 1) dimensions with local in Fig. 19(b), characterized by a factor J (σb, σc; σa).The Hilbert space dimension q. The time-evolution operator contribution to Eq. (D1) from a given spin configuration is is constructed from local unitary operators as shown in given by the product of J (σb, σc; σa) from all shaded tri- Fig. 19(a), where each V appearing in the circuit is an inde- angles in Fig. 19(b), and a factor involving O that comes pendent random unitary matrix acting on two sites, drawn from the bottom layer. from the Haar measure of U(q2). We denote the number As in the usual Ising model, the partition function for of sites in the full system and in the subsystem A, respec- | | | |−| | this spin system can be obtained by summing over different tively, as |L| and |A|.Thend = q A and d ¯ = q L A , with ¯ A A domain-wall configurations. A domain wall between spins the subsystems A and A = B ∪ B as shown in Fig. 19(a). − 1 2 σ and τ is labeled as σ 1τ (see Fig. 20), and thus there We take both |A| and |L|−|A| to be large, and consider | | | ¯ | are altogether n! different types of domain walls, one for times t A , A . S Now consider each element of n. We refer to domain walls associated with transpositions of two elements as elementary domain { ( ) }n =⊗ η| ⊗ | ( ⊗ †)n | S TrA¯ ( TrA[O t ] ) m∈A¯ m m∈A e m U U O, e , walls. Just like an element of n can be decomposed into a (D1) product of transpositions, a domain wall can decomposed into compositions of elementary domain walls. For exam- where we can express (U ⊗ U†)n as a product of (V ⊗ V†)n ple, given that η = (n, n − 1, ...1) can be decomposed as for the local unitaries V. As explained in Refs. [78,79, η = (1, 2)(1, 3) ···(1, n − 1)(1, n),anη-domain wall may

010344-29 HONG LIU and SHREYA VARDHAN PRX QUANTUM 2, 010344 (2021)

(a) (b)

¯ FIG. 19. (a) The tensor network for the time-evolution operator in random unitary circuits and the regions A and A = B1 ∪ B2.(b) The corresponding triangular lattice for the evaluation of Eq. (D1), which can have a spin labeled by elements of Sn at each vertex, like σa,b,c shown explicitly in the figure.

− −( − )E(l)(v) ( ) be considered as a composite of n 1 elementary domain q l 1 t. With finite q, for general v, E l (v) is dif- walls associated with these transpositions. ferent from E (2(v) due to interactions among domain walls When the difference in position x and the difference (they become equal in the limit q →∞). It was argued in in time t between the initial and final points are both Ref. [84] based on the dynamics of entanglement growth large for all domain walls relevant for a quantity of interest, in chaotic systems that the conditions there exists a coarse-grained description where we can characterize domain walls by their velocities x/t,and  E (l)(v ) = v E (l)(v) ≥ v E (l) (v ) = collectively take into account the contributions from all B B, , B 1 (D2) detailed configurations that correspond to these velocities v [83]. Then a domain wall with velocity contributes a should be satisfied for all l in any chaotic system. Here vB (2) factor q−E (v)t to the partition function, where E (2)(v) is the butterfly velocity of the system. The explicit form of is independent of the type of the elementary domain wall. E (l)(v) in random unitary circuits with finite q is not known (3) Furthermore, if there are l − 1 elementary domain walls for l > 2, but the condition E (vB) = vB was checked up traveling together at velocity v for time t, we get a factor to next-to-leading order in 1/q in Ref. [83].

(a) (b)

FIG. 20. A domain wall that separates σ spins on the left and σμ spins on the right is labeled by μ, as shown in (a). For the configuration shown in (b) (where we do not show the details of the lattice), the combined contribution from all domain walls is −E(2)(v ) − E(3)(v ) − E(4)(v ) q 1 tq 2 2 tq 3 3 t.

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(a) (b)

FIG. 21. (a) A general domain-wall configuration that can contribute in the scaling limit and at late times. (b) The leading contribution in the scaling limit for a particular choice of σ , where each of the domain walls has velocity vB.

From the spin configuration indicated at the top bound- In the above derivation, we only make use of the fact that ary of Fig. 19(b), we have an η−1 domain wall at the left the average over local random unitaries could be expressed edge of A,andanη domain wall at the right edge. As in terms of “spins” associated with permutations, and that discussed above, each of these domain walls can be seen as there is a membrane tension associated with the domain composites of n − 1 elementary walls, which can in prin- walls between such spins in the scaling limit, which satis- ciple travel independently through the lattice. The lower fies the conditions, Eq. (D2). The discussion of Ref. [11] endpoints of each of these elementary domain walls in the implies that the above features are also present in the lattice can be either on the left or right edges, or at the scaling limit in a variety of chaotic systems involving no bottom [85]. One example of a possible configuration is random averaging, such as the Floquet spin chains studied Fig. 20(b). there. We therefore expect that it may be possible to show Now suppose t |A|, |A¯ |. Since the E (l)(v) are all O(1), using the methods of Refs. [11] that the result, Eq. (124), in all configurations where any domain walls reach the holds in the systems considered there. lower boundary, we get factors exponentially suppressed in t relative to configurations where all domain walls meet in the middle or end at the edges of the system. So at such times, it is sufficient to consider configurations that do not [1] S. W. Hawking, Particle creation by black holes, Commun. reach the lower boundary. A general example of such a Math. Phys. 43, 199 (1975). [2] S. W. Hawking, Breakdown of predictability in gravita- configuration is indicated in Fig. 21(a).FromEq.(D2),we tional collapse, Phys.Rev.D14, 2460 (1976). see that for a pair of domain-wall configurations, which [3] Here n = 1 corresponds to the von Neumann entropy and is meet in the middle, the maximal contribution comes from the n → 1 limit of Eq. (1). the case where both have velocity vB. Similarly with those [4] D. N. Page, Average Entropy of a Subsystem, Phys. Rev. ending on the edges. Thus any domain wall, which travels Lett. 71, 1291 (1993). to the left or right, must have velocity vB at leading order [5] E. Lubkin, Entropy of an nsystem from its correlation with in our limit. For a decomposition of η a kreservoir, J. Math. Phys. 19, 1028 (1978). [6] S. Lloyd and H. Pagels, Complexity as thermodynamic η = σ × ν, (D3) depth, Ann. Phys. (N.Y.) 188, 186 (1988). [7] C. Nadal, S. N. Majumdar, and M. Vergassola, Statisti- σ cal distribution of quantum entanglement for a random where corresponds to the composite domain walls, bipartite state, Journal of Statistical Physics 14, 403 (2011). which meet in the middle and ν to those ending on the [8] S. Leutheusser and M. Van Raamsdonk, Tensor network edge, the corresponding domain-wall configuration con- models of unitary black hole evaporation, JHEP 8, 141 tributes (2017). [9] D. N. Page, Information in Black Hole Radiation, Phys. ¯ −1 ¯ q−|σ||A|q−|ν||A| = q−[n−k(σ)]|A|q−[n−k(η σ)]|A|. (D4) Rev. Lett. 71, 3743 (1993). [10] P. Hayden and J. Preskill, Black holes as mirrors: Quantum In the regime we are interested in, the above expression information in random subsystems, JHEP 709, 120 (2007). is maximized when k(σ ) + k(η−1σ)is maximized, that is, [11] T. Zhou and A. Nahum, The entanglement membrane in chaotic many-body systems, arXiv:1912.12311 [cond- they should saturate Eq. (48). mat.str-el]. | σ −1 On the lower boundary, O, e is attached to a spin [12] G. Penington, Entanglement Wedge Reconstruction and the −1 at each site, so that we get a factor of σ |O, e. Putting Information Paradox, arXiv:1905.08255 [hep-th]. together the contributions from such leading domain- [13] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, wall contributions for different choices of σ that saturate The entropy of bulk quantum fields and the entanglement Eq. (48), we then obtain the result, Eq. (124). wedge of an evaporating black hole, JHEP 12, 63 (2019).

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