Universal eigenstate entanglement of chaotic local Hamiltonians

This arXiv repository is a bundle of two closely related papers. Abstract of the first paper: In systems governed by"chaotic"local Hamiltonians, we conjecture the universality of eigenstate entanglement (defined as the average entanglement entropy of all eigenstates) by proposing an exact formula for its dependence on the subsystem size. This formula is derived from an analytical argument based on a plausible assumption, and is supported by numerical simulations. Abstract of the second paper: In systems governed by chaotic local Hamiltonians, the first paper conjectured the universality of the average entanglement entropy of all eigenstates by proposing an exact formula for its dependence on the subsystem size. In this note, I extend this result to the average entanglement entropy of a constant fraction of eigenstates in the middle of the energy spectrum. The generalized formula is supported by numerical simulations of various chaotic spin chains.

In this paper we consider "chaotic" quantum many-body systems. We are not able to specify the precise meaning of being chaotic, for there is no clear-cut definition of quantum 1. The entanglement entropy of an eigenstate for a subsystem smaller than half the system size is (to leading order) equal to the thermodynamic entropy of the subsystem at the same energy density.
2. The entanglement entropy of an eigenstate at the mean energy density (of the Hamiltonian) is indistinguishable from that of a random (pure) state.
3. The entanglement entropy of a generic eigenstate is indistinguishable from that of a random state.
We briefly explain the reasoning behind these opinions. The eigenstate thermalization hypothesis (ETH) states that for expectation values of local observables, a single eigenstate resembles a thermal state with the same energy density [42][43][44]. Opinion 1 is a variant of ETH for entropy. Opinion 2 follows from Opinion 1 and the fact that the entanglement entropy of a random state is nearly maximal [45]. Opinion 3 follows from Opinion 2 because a generic eigenstate is at the mean energy density (Lemma 3 in Ref. [46]).
These opinions concern the scaling of the entanglement entropy only to leading order. A more ambitious goal is to find the exact value of eigenstate entanglement. We conjecture that the average entanglement entropy of all eigenstates is universal (model independent), and propose a formula for its dependence on the subsystem size. This formula is derived from an analytical argument based on an assumption that characterizes the chaoticity of the model. It is also supported by numerical simulations of a non-integrable spin chain.
The formula implies that by taking into account sub-leading corrections not captured in Opinion 3, a generic eigenstate is distinguishable from a random state in the sense of being less entangled. Indeed, this implication can be proved rigorously for any (not necessarily chaotic) local Hamiltonian. The proof also solves an open problem of Keating et al. [29].
The paper is organized as follows. Section 2 gives a brief review of random-state entanglement. Section 3 proves that for any (not necessarily chaotic) local Hamiltonian, the average entanglement entropy of all eigenstates is smaller than that of random states. Sections 4 and 5 provide an analytical argument and numerical evidence, respectively, for the universality of eigenstate entanglement in chaotic systems. The main text of this paper should be easy to read, for most of the technical details are deferred to Appendices A and B.

Entanglement of random states
We begin with a brief review of random-state entanglement. We use the natural logarithm throughout this paper.
Definition 1 (entanglement entropy). The entanglement entropy of a bipartite pure state ρ AB = |ψ ψ| is defined as the von Neumann entropy of the reduced density matrix ρ A = tr B ρ AB . It is the Shannon entropy of ρ A 's eigenvalues, which form a probability distribution as ρ A ≥ 0 (positive semidefinite) and tr ρ A = 1 (normalization).
Theorem 1 (conjectured and partially proved by Page [45]; proved in Refs. [47][48][49]). Let ρ AB be a bipartite pure state chosen uniformly at random with respect to the Haar measure. In average, where d A ≤ d B are the local dimensions of subsystems A and B, respectively.
Let γ ≈ 0.5772 be the Euler-Mascheroni constant. The second step of Eq. (2) uses the formula

Rigorous bounds on eigenstate entanglement
This section proves a rigorous upper bound on the average entanglement entropy of all eigenstates. The bound holds for any (not necessarily chaotic) local Hamiltonian, and distinguishes the entanglement entropy of a generic eigenstate from that of a random state. For ease of presentation, consider a chain of n spin-1/2's governed by a local Hamiltonian where H ′ i acts only on spin i, and H ′ i,i+1 represents the nearest-neighbor interaction between spins i and i + 1. We use periodic boundary conditions by identifying the indices i and (i mod n). Suppose H ′ i and H ′ i,i+1 are linear combinations of one-and two-local Pauli operators, respectively, so that tr H ′ i = tr H ′ i,i+1 = 0 (traceless) and tr(H i H i ′ ) = 0 for i = i ′ . We assume translational invariance and H i = 1 (unit operator norm). Let d := 2 n and {|j } d j=1 be a complete set of translationally invariant eigenstates of H with corresponding eigenvalues {E j }.
where ρ j,A := tr B |j j| is the reduced density matrix of |j for A.
Proof. See Appendix A.
We are now ready to prove the main result of this section: In the setting of Lemma 1, where · · · := d −1 tr · · · denotes the expectation value of an operator at infinite temperature.
Proof. Theorem 2 follows from Lemma 1 and the observation that Recall that Theorem 2 assumes H i = 1. Without this assumption, (6) should be modified toS A lower bound can be easily derived from Theorem 1 in Ref. [29] S ≥ m ln 2 − Θ(1/n).
Therefore, both bounds are tight. This answers an open question in Section 6.1 of Ref. [29]. Without translational invariance (e.g., in weakly disordered systems), a similar result is obtained by averaging over all possible ways of "cutting out" a region of length m. Here, H i may be site dependent but should be Θ(1) for all i. Proof. First, we follow the proof of Lemma 1. Without translational invariance, (21) remains valid upon replacing ǫ j,i by ǫ j,i / H i = Θ(ǫ j,i ). By the RMS-AM inequality and Eq. (7), we havē It is straightforward to extend all the results of this section to higher spatial dimensions.

Eigenstate entanglement of "chaotic" Hamiltonians
Suppose the Hamiltonian (4) is chaotic in a sense to be made precise below. This section provides an analytical argument for Conjecture 1 (universal eigenstate entanglement). Consider the spin chain as a bipartite quantum system A ⊗ B. Subsystem A consists of spins 1,2,. . . ,m. For a fixed constant f := m/n ≤ 1/2, the average entanglement entropy of all eigenstates is in the thermodynamic limit n → +∞, where δ is the Kronecker delta.
We split the Hamiltonian (4) into three parts: The locality of H ∂ implies a strong constraint stating that the population of |j A |k B is significant only when ǫ j + ε k is close to E.

Lemma 2.
There exist constants c, ∆ > 0 such that Proof. This is a direct consequence of Theorem 2.3 in Ref. [50].
In chaotic systems, we expect Assumption 1. The expansion (13) of a generic eigenstate |ψ is a random superposition subject to the constraint (14).
This assumption is consistent with, but goes beyond, the semiclassical approximation Eq. (16) of Ref. [33].
We now show that Assumption 1 implies Conjecture 1. Consider the following simplified setting. Let M j be the set of computational basis states with j spins up and n−j spins down, and U j ∈ U(|M j |) = U( n j ) be a Haar-random unitary on span M j . Define M ′ j = {U j |φ : ∀|φ ∈ M j } so that M := n j=0 M ′ j is a complete set of eigenstates of the Hamiltonian The set M captures the essentials of Assumption 1. Every state in M satisfies which is a hard version of the constraint (14). The random unitary U j ensures that Eq. (13) is a random superposition. Thus, we establish Conjecture 1 by Proposition 1. The average entanglement entropy of all states in M is given by Eq. (12).

Numerics
To provide numerical evidence for Conjecture 1, consider the spin-1/2 Hamiltonian [51,52] where σ x i , σ y i , σ z i are the Pauli matrices at site i(≤ n), and σ z n+1 := σ z 1 (periodic boundary condition). For generic values of g, h, this model is non-integrable in the sense of Wigner-Dyson level statistics [51,52]. We compute the average entanglement entropy of all eigenstates by exact diagonalization in every symmetry sector. Figure 1 shows the numerical results, which semiquantitatively support Conjecture 1. Noticeable deviations from Eq. (12) are expected due to significant finite-size effects. However, the trend appears to be that the difference between theory and numerics decreases as the system size increases.

Acknowledgments and notes
We would like to thank Fernando G.S.L. Brandão and Xiao-Liang Qi for interesting discussions and Anatoly Dymarsky for insightful comments. We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907). Additional funding support was provided by NSF DMR-1654340.
After this paper appeared on arXiv, we became aware of a simultaneous work [53] and a slightly later one [54].

A Proof of Lemma 1
Let ǫ j,i := j|H i |j so that |ǫ j,i | ≤ 1. Let ρ j,i be the reduced density matrix of |j for spins i and i + 1. Let I 4 be the identity matrix of order 4. Let X 1 := tr √ X † X be the trace norm. Since H i is traceless, |ǫ j,i | provides a lower bound on the deviation of ρ j,i from the maximally mixed state: Since the Shannon entropy is Schur concave, it suffices to consider For |ǫ j,i | ≪ 1, by Taylor expansion we can prove We have checked numerically that this inequality remains valid for any |ǫ j,i | ≤ 1. Therefore, due to the subadditivity [55] of the von Neumann entropy. We complete the proof using ǫ j,i = E j /n.

B Proof of Proposition 1
Assume without loss of generality that n is even. Let L j (R j ) be the set of computational basis states of subsystem A (B) with j spins up and m − j (n − m − j) spins down so that Thus, any (normalized) state |ψ in M ′ j can be decomposed as where |φ k is a normalized state in span L k ⊗ span R j−k . Let ρ A and σ k,A be the reduced density matrices of |ψ and |φ k for A, respectively. It is easy to see Since |ψ is a random state in span M j , each |φ k is a (Haar-)random state in span L k ⊗ span R j−k . Theorem 1 implies that in average, In average, the population |c k | 2 is proportional to the dimension of span L k ⊗ span R j−k : The deviation of |c k | 2 (from the mean) for a typical state |ψ ∈ span M j is exponentially small. In the thermodynamic limit, j, k can be promoted to continuous real variables so that |M j |, |L k | follow normal distributions with means n/2, f n/2 and variances n/4, f n/4, respectively. Let We have For any fixed constant f < 1/2, it is almost always the case that |L k | ≪ |R j−k |. Hence, Substituting Eqs. (29) and (30) into Eq. (24), Averaging over all states in M, (32) For f = 1/2, we first assume that j ≤ n/2 and k ≤ j/2 (i.e., J ≤ 0 and K ≤ J/2) so that |L k | ≤ |R j−k |. Hence, Let be the complementary error function. Substituting Eqs. (29) and (33) into Eq. (24), This is the average entanglement entropy of a random state in span M j for j ≤ n/2. For j > n/2, Eq. (35) remains valid upon replacing J by −J. Averaging over all states in M, Equation (12) follows from Eqs. (32) and (36).

Introduction
In systems governed by chaotic local Hamiltonians, my previous work [1] conjectured the universality of eigenstate entanglement by proposing an exact formula for its dependence on the subsystem size. This formula was derived from an analytical argument based on an assumption that characterizes the chaoticity of the system, and is supported by numerical simulations. For simplicity, Ref. [1] only considered the average entanglement entropy of all eigenstates explicitly. Due to the recent interest [2], in this note I extend the result to the average entanglement entropy of a constant fraction of eigenstates in the middle of the energy spectrum. The extension is straightforward and does not require any essentially new ideas beyond those in Ref. [1].
For completeness and for the convenience of the reader, definitions and derivations are presented in full so that this note is technically self-contained, although this leads to substantial text overlap with the original paper [1]. It is not necessary to consult Ref. [1] before or during reading this note. However, in this note I do not discuss the conceptual aspects of the work. Such discussions are in Ref. [1].
I recommend related works [3][4][5], which use a similar approach to study other aspects of eigenstate entanglement.
The rest of this note is organized as follows. Section 2 gives basic definitions and a brief review of random-state entanglement. Section 3 presents the main result. Section 4 provides numerical evidence for this analytical result in various chaotic spin chains. The main text of this note should be easy to read, for most of the technical details are deferred to Appendix A.

Preliminaries
Definition 1 (entanglement entropy). The entanglement entropy of a bipartite pure state ρ AB is defined as the von Neumann entropy of the reduced density matrix ρ A = tr B ρ AB .
Theorem 1 (conjectured and partially proved by Page [6]; proved in Refs. [7][8][9]). Let ρ AB be a bipartite pure state chosen uniformly at random with respect to the Haar measure. In average, Let γ ≈ 0.577216 be the Euler-Mascheroni constant. The second step of Eq. (2) uses the formula

Universal eigenstate entanglement
Consider a chain of N spin-1/2's governed by a local Hamiltonian where H i represents the nearest-neighbor interaction between spins at positions i and i + 1. For concreteness, we use open boundary conditions, but our argument also applies to other boundary conditions. Assume without loss of generality that tr H i = 0 (traceless) so that the mean energy of H is 0. We do not assume translational invariance. In particular, H i may be site dependent but should be Θ(1) for all i. Suppose that the Hamiltonian (6) is chaotic in a sense to be made precise below. We provide an analytical argument for Conjecture 1 (universal eigenstate entanglement). Assume without loss of generality that N is even. Consider the spin chain as a bipartite quantum system A ⊗ B. Subsystem A consists of spins at positions 1, 2, . . . , N/2. For a constant 0 < ν ≤ 1, let Λ be such that H has ν2 N eigenvalues in the interval [−Λ, Λ]. The average entanglement entropy of the eigenstates in this energy interval is in the thermodynamic limit N → +∞.
Remark. It is straightforward to extend Eq. (7) to the case where the subsystem size is an arbitrary constant fraction of the system size.
We split the Hamiltonian (6) into three parts: The locality of H ∂ implies a strong constraint stating that the population of |j A |k B is significant only when ǫ j + ε k is close to E. Lemma 1. There exist constants c, ∆ > 0 such that Proof. This is a direct consequence of Theorem 2.3 in Ref. [10].
In chaotic systems, we expect Assumption 1. The expansion (8) of a generic eigenstate |ψ is a random superposition subject to the constraint (9).
This assumption is consistent with, but goes beyond, the semiclassical approximation Eq. (16) of Ref. [11].
We now show that Assumption 1 implies Conjecture 1. Consider the following simplified setting. Let M j be the set of computational basis states with j spins up and N −j spins down, and U j ∈ U(|M j |) = U( N j ) be a Haar-random unitary on span M j . Define M ′ j = {U j |φ : ∀|φ ∈ M j } so that M := N j=0 M ′ j is a complete set of eigenstates of the Hamiltonian The energy of a state in M is defined with respect to this Hamiltonian. The set M captures the essence of Assumption 1. Every state in M satisfies which is a hard version of the constraint (9). The random unitary U j ensures that Eq. (8) is a random superposition. Thus, we establish Conjecture 1 by Proposition 1. The average entanglement entropy of the ν2 N states in M in the middle of the energy spectrum is given by Eq. (7) in the thermodynamic limit N → +∞.

Comparison with numerics
In this section, we compare Eq. (7) with the numerical results in the literature [2,12,13]. All these numerical results are obtained by exact diagonalization. They are limited to relatively small system sizes N ≤ 20 and suffer from non-negligible finite-size effects. Although they cannot confirm Conjecture 1 conclusively, they are quite suggestive: Eq. (7) is supported by numerical simulations of various (not necessarily translation-invariant) chaotic spin chains for various values of ν.
Sometimes an incorrect analytical formula with one or more fitting parameters can fit the numerical data well when the number of data points is limited. We do not need to worry about such false positives here, for Eq. (7) does not contain any fitting parameters.

ν = 0 +
In fact, the original paper [1] also presented the result of the case ν = 0 + or the entanglement entropy of the eigenstates at the mean energy of the Hamiltonian. In this case, Eq.
Let {h i } be a set of independent random variables uniformly distributed on the interval [−1, 1]. In the spin-1/2 chain the entanglement entropy of an eigenstate with energy close to 0 was calculated for the system size N = 16 [12]. The numerical result, averaged over 10 samples of {h i }, is 8 ln 2 − 0.5733 ± 0.0015, which is closer to Eq. (14) than to Eq. (3).

0 < ν < 1
Let ∆S be the difference between the right-hand sides of Eqs. (3) and (7). Its values as a function of ν are listed in Table 1.
In the spin-1/2 chain 1 If the size of the smaller subsystem is a constant f < 1/2 fraction of the system size, in the thermodynamic limit N → +∞. This is the special case J = 0 of Eq. (31) in Ref. [1]. 2 We thank the authors of Ref. [13] for sharing the exact value of the data point at β = 0.0 and L A = 10 in their Fig. 3.  (17). The dots are numerical results from Fig. 7 of Ref. [2]. The dashed lines are our model-independent theoretical results in the thermodynamic limit (Table 1). Although one cannot conclude whether the dots approach the dashed lines of the same color as N → +∞, the trend looks promising.
the average entanglement entropyS of the ν = 1/4, 1/8, 1/16 fraction of eigenstates in the middle of the energy spectrum was calculated up to the system size N = 16 [2]. As shown in Fig. 1, the numerical results semi-quantitatively support Eq. (7).

Declaration of competing interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments
where |φ k is a normalized state in span L k ⊗ span R j−k . Let ρ A and σ k,A be the reduced density matrices of |ψ and |φ k for A, respectively. It is easy to see Since |ψ is a random state in span M j , each |φ k is a (Haar-)random state in span L k ⊗ span R j−k . Theorem 1 implies that in average, S(σ k,A ) = ln min{|L k |, |R j−k |} − min{|L k |, |R j−k |} 2 max{|L k |, |R j−k |} .
In average, the population |c k | 2 is proportional to the dimension of span L k ⊗ span R j−k : The deviation of |c k | 2 (from the mean) for a typical state |ψ ∈ span M j is exponentially small. In the thermodynamic limit, j, k can be promoted to continuous real variables so that |M j |, |L k | follow normal distributions with means N/2, N/4 and variances N/4, N/8, respectively. Let J := j/ √ N − √ N /2, K := k/ √ N − √ N /4.
Consider the case that j ≤ N/2 and k ≤ j/2 (i.e., J ≤ 0 and K ≤ J/2) so that |L k | ≤ |R j−k |. Hence, This is the average entanglement entropy of a random state in span M j for j ≤ N/2. For j > N/2, Eq. (27) remains valid upon replacing J by −J. We determine the energy cutoff Λ such that ν2 N states in M have energies in the interval [−Λ, Λ]: Averaging over these ν2 N states in M,