Typical entanglement entropy in the presence of a center: Page curve and its variance

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $\Delta S_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

In a quantum system in a pure state, a subsystem A generally has a nonzero entropy because of entanglement with the rest of the system.In this paper we address the question: Is the average entanglement entropy of pure states S A also the typical entropy of the subsystem?To illustrate the significance of this question, let us consider for instance the gas in a room held at fixed temperature.The canonical ensemble allows us to compute the average energy of the gas.However, the configuration of molecules in one room is one realization of this ensemble-we are not averaging over rooms.How close to the average is the energy of this realization?In other words, is the average energy typical?In statistical mechanics, we answer this question by computing the variance (∆E) 2 .In the canonical ensemble, we find ∆E/ E 1 and we conclude that the average energy is typical.Here we investigate the typicality of the entanglement entropy S A of a subsystem by studying its average and variance.
A well-studied special case of subsystem A is described by a subalgebra with a trivial center.In this case the Hilbert space of the system is simply given by a tensor product of the subsystem Hilbert space H A and its complement H B , i.e.H = H A ⊗H B .This case was originally considered by Page [1,[38][39][40][41][42][43] who computed the average in H of the entanglement entropy.The variance was computed recently in [44,45].In this paper we present an FIG. 1. Page curve of a system of N = 10 noninteracting spins.(i) Trivial center.In the absence of an external magnetic field, all states are equiprobable: the entanglement entropy SA of a sample of random pure states is shown as a function of the subsystem size NA (orange circles), together with the band SA ± 3 ∆SA (green dashed line and gray region).The inset shows the data for NA = 4. (ii) nontrivial center.In the presence of a magnetic field, the entanglement entropy of random pure states of given energy En = µ0B n with n = −2 is shown for NA = 4 (red diamonds), together with the band SA E ± 3 (∆SA) E for the Page curve at fixed energy (blue solid line and gray region).
algorithm that reproduces the average and variance, together with the skewness and higher moments of the entanglement entropy probability distribution P (S A )dS A for a trivial center.
In many physically relevant cases, the Hilbert space over which we average the entanglement entropy does not have a tensor product structure, H = H A ⊗ H B .In fact, in general, the subalgebra that identifies the subsystem A has a nontrivial center.As a result, the Hilbert space of the system is a direct sum of tensor products [46,47].In this work we extend the typicality results on the entanglement entropy in the presence of a center.Ex-amples of subsystems with a nontrivial center are subsets of a spin chain with fixed boundary conditions; lattice fermions with fixed boundary conditions [48]; compact scalar fields, Abelian and non-Abelian gauge fields on a lattice [46]; loop quantum gravity [35,36,49,50].
A simple example of subsystems with a center is provided by the energy eigenspace in a noninteracting system with Hamiltonian H = H A + H B .In this case the eigenspace H(E) ⊂ H has the structure of a direct sum of tensor products where H A (ε j ) and H B (ε k ) are eigenspaces of given energy for the subsystems A and B. This structure is relevant for the study of thermal properties of isolated quantum systems.See Fig. 1 for an example.
Building on [1,[38][39][40][41][42][43], we develop methods to compute the exact formula for the average entanglement entropy and the moments m n = (S A − S A ) n of the probability distribution P (S A )dS A of finding a pure state with entanglement entropy S A .Our methods are tailored to the computation of averages over pure states in H and over pure states in the eigenspace H(E).
The exact formula for the average S A over all pure states in H was conjectured by Page in [1], improving earlier works [38,39], and later proved with different methods in [40][41][42][43] and in (7).Here we present the exact formula for the average S A E over all states in the Hilbert H(E) with a nontrivial center.Furthermore, we determine the exact formulas for: the variance m 2 for pure states in H reproducing the result found in [44,45] for a trivial center; the exact formula for third order moment m 3 with a trivial center; the exact formula for m 2 in the presence of a nontrivial center.We compare our results to concentration of measure bounds [24].
By studying the average entanglement entropy S A and the variance we show that, both in the case of a trivial and a nontrivial center, for large systems the average entanglement entropy of a subsystem is also typical.After presenting a detailed derivation of the Page curve and its variance, we discuss the application of our results to a model system where a nontrivial center arises: we determine the Page curve of energy eigenstates of a paramagnetic solid in a magnetic field (Fig. 1), and show how thermal properties of a subsystem arise from entanglement with the rest of the system.A random state |ψ in the Hilbert space H can be generated by choosing an orthonormal basis |n with n = 1, . . ., d and picking a vector |ψ = n ψ n |n at random with respect to the uniform measure dµ(ψ) = Z −1 δ(|ψ| 2 − 1)dψdψ on the unit sphere in C d , with the constant Z defined so that the measure is normalized to unity, dµ(ψ) = 1.The average of a function f over all states |ψ can be computed by integrating uniformly over states, f = f (ψ) dµ(ψ).As the entanglement entropy S A depends on the reduced density matrix ρ A only via its eigenvalues λ a , (the entanglement spectrum, with a = 1, . . ., d A ), to compute the average over |ψ we need only the induced measure over the eigenvalues λ a .This measure dµ(λ 1 , . . ., λ d A ) was computed by Lloyd and Pagels in [39], (see also App.A).
Computing the average entanglement entropy S A and its variance ∆S A is not immediate because they are not polynomial functions of ρ A .Here we follow a strategy that generalizes to the computation of higher moments of S A .We first consider the average of the function Tr A (ρ r A ) with r ≥ 0, To compute the integral over λ a it is useful to introduce the quantity X ij (r), defined as an integral of generalized Laguerre polynomials L (q) with i = 0, . . ., d A −1: Computing the integral (4) is nontrivial.We obtained the result by using the generating function for the Laguerre polynomials, as we describe in more detail in App. A. Formula (4) for X ij (r) provides the main technical tool for our derivation.
The average (3) takes a simple form when expressed in terms of the matrix X ij (r), where we treat X ij (r) as a d A ×d A matrix X(r).We note that ( 5) is a smooth continuous function of r.The average entanglement entropy can be obtained immediately by taking a derivative with respect to r, The result of this computation, expressed in terms of the digamma function Ψ(x) = Γ (x)/Γ(x) (i.e., the logarithmic derivative of the gamma function), is the formula This formula was first conjectured by Page [1] and then proved with different techniques here and in [40][41][42][43].
The technique described above applies directly to the calculation of the average of S A 2 .The strategy is to first consider the average of Tr(ρ r1 A ) Tr(ρ r2 A ) and express it in terms of matrices X(r): × TrX(r Its derivatives with respect to r 1 and r 2 can again be expressed in terms of derivatives of the gamma function, and simplified with the help of Wolfram's Mathematica.The average of S A 2 is obtained as Using the definition (2) and the formula (7), we obtain the exact formula: , discussed also in [44,45].Using the same technique we can determine also the average of higher powers of S A .In particular, we report the exact formula for the third moment m 3 of the entanglement entropy distribution (App.A) We observe that the formulas for the moments m n are exact: they provide the mean (7), the variance (10) and the skewness (11) of the entropy for small systems, as well as for large systems.In the latter case, a Taylor series in 1/d B provides asymptotic expressions for large systems and any subsystem: the average entropy is approximated by the expression As the entropy of A can be at most S max = log d A , this formula shows that for a large system the average entropy of a subsystem is close to maximal.The asymptotic ex-pression for the variance of S A is: We observe that the variance vanishes in the limit of d B → ∞.Therefore we conclude that, in a large system, any subsystem has ∆S A / S A 1. In other words, in a large system, the average entropy of any subsystem is also its typical entropy.We report also the asymptotic expression for the third moment m 3 (11) This formula shows that the skewness m 3 /σ 3 ≈ − 8/(d 2 A − 1) is negative, which results in a right tilt in the distribution that does not vanish as d B → ∞.
One might expect that these asymptotic formulas can be obtained in a simpler way, for instance by first expanding the entropy S A around the maximally mixed state and then taking the average.The expansion in δρ A = ρ A − 1/d A was first proposed by Lubkin and Lubkin in [38] and is commonly found in reviews [7].However, it was shown by Dyer in [43] that the series in δρ A for S A does not converge.This has the consequence that, truncating the expansion in δρ A , only the first order in Page's formula ( 12) is accidentally reproduced.Similarly, the leading order variance ∆S A (13) cannot be obtained by truncating the expansion in δρ A .
Determining the full probability distribution P (S A )dS A of a random pure state is not immediate.The methods introduced here allow us to determine its average m = S A , its variance σ 2 = (∆S A ) 2 and higher order moments m n such as the skewness ( 14), (see also App.A).The normal distribution is the distribution with the largest Shannon entropy at fixed average m and variance σ 2 .In Fig. 2 we compare this distribution to a numerical sample and find that it characterizes well the support of the probability distribution P (S A )dS A of random pure states.In particular, the numerical sample and the analytic formulas ( 12), (13) show that it is unlikely to find a state with maximum entropy S max .The numerical sample shows also a small right tilt with respect to the normal distribution, in accordance with the negative skewness (14).
Previous analysis of typicality have used concentration of measure techniques to provide upper bounds on the probability of finding a state with entropy lower than the average entropy [24].In particular, using Levy's lemma, Theorem III.3 in [24] [51] states that the cumulative distribution P[S A < α ] is bounded from above by a Gaussian function, Probability of finding entanglement entropy SA in a sample of 10 5 random pure states of a spin system (N = 10 and NA = 4, in orange 200 bins).See Fig. 1 (inset) for reference.We compare the sample to the normal probability distribution (15) with mean (7) and variance (10) (solid green line).Inset: Cumulative probability distribution of finding entanglement entropy smaller than α for the same sample (orange circles).We compare the sample to the normal cumulative distribution (solid green line) and to the concentration of measure bound (16) (dashed gray line).(ii) nontrivial center.Probability of finding entanglement entropy SA in a sample of 10 5 random pure states of energy En = µ0B n with n = −2 of a spin system in a magnetic field (N = 10 and NA = 4, in red 200 bins).See Fig. 1 (red diamonds) for reference.We compare the sample to the normal probability distribution with mean (23) and variance (10) We compare this bound to the normal cumulative distribution, and to a numerical sample of random pure states.The inset in Fig. 2 clearly shows that the probability is more concentrated than what the bound ( 16) indicates.
Average entropy and variance in the presence of a nontrivial center.-Considera system with algebra of observables A. We define a subsystem A by choosing a subalgebra of observables A A .The complement of the subsystem A is denoted B and its algebra of observables is the commutant of A A , i.e., A B = {b ∈ A | [b, a] = 0 ∀a ∈ A A }.The intersection of the two subalgebras, In the presence of a center, the Hilbert space of the system decomposes as a direct sum of tensor products [46,47], where the sum is over the spectrum of Z A .We give a concrete example of a center.Let us consider a composite system with Hilbert space H = H A ⊗H B and Hamiltonian H = H A + H B having energy eigenvalues E jk = ε jA + ε kB .On the energy eigenspace H(E) ⊂ H the algebra of the subsystem A has a nontrivial center Z A = H A .Therefore, H(E) has the structure (17), where the sum over j is such that ε kB = E − ε jA .Energy eigenspaces of the subsystem A are denoted H A (ε j ) and have dimension d jA = dimH A (ε j ).Similarly for subsystem B. The energy eigenspaces of the system have then the direct sum structure H(E) = ⊕ j H j (E) where the sector H j (E) = H A (ε j ) ⊗ H B (E − ε j ) has definite energy in each subsystem.We denote d j = dimH j (E) the dimension of each sector, with d j = d jA d jB and d E = j d j the dimension of H(E).
Due to the direct-sum structure (18), Page's formula (7) does not apply.Other instances of systems where the relevant Hilbert space has the form (1) are subsystems in lattice gauge theory [46], in spin chains and in lattice fermions with fixed boundary conditions [48], in loop quantum gravity [35,36,49,50], and in general in presence of an additive constraint.The formulae that we derive below apply equally to all these cases.
To investigate typicality of the entropy in the energy eigenspace H(E), we determine the uniform measure over pure states belonging it.We note that a state |ψ, E in H(E) can be written as a superposition |ψ, E = j √ p j |φ j of normalized states |φ j ∈ H j (E), with weights p j ≥ 0 satisfying the normalization condition j p j = 1.The reduced density matrix for the subsystem A is given by ρ A = j p j ρ jA with ρ jA = Tr B |φ j φ j |.The entanglement entropy of the subsystem, S A (ψ) = −Tr(ρ A log ρ A ), splits into the sum of two terms [52]: where S jA (φ j ) = −Tr(ρ jA log ρ jA ) is the entanglement entropy in the sector H j (E).We note that the entanglement entropy of A is the p j -weighted sum of the entanglement entropy S jA in each sector, plus the Shannon entropy of the weights p j .The uniform measure dµ E (ψ) over pure states in H(E) where dµ(φ j ) is the uniform measure over pure states in each sector H j (E).We derive the measure over the weights p j in App.B and find This measure defines the Dirichlet distribution, also known as the multivariate beta distribution [53].The constant Z is defined so that the measure is normal-ized to unity, dν(p 1 , . . ., p J ) = 1.The average weight is p j = d j /d E and its second moments are Using the technique described in (6), we write the average in H(E) of the entanglement entropy as and find the exact formula where S jA is given by (7).Using the same technique, we compute the exact formula for the average of S A 2 , See App.B for a detailed derivation.In the limit of large dimension of each sector, d j 1, the average entanglement entropy takes the form In the same limit, the variance goes to zero and the ratio of the two quantities scales as Therefore the average entropy of a subsystem is also its typical entropy.Fig. 2 shows that the normal distribution with mean (23) and variance (∆S A ) 2 E characterizes well the support of the distribution of the entanglement entropy of random pure states of energy E.

Discussion and applications.-
We can now answer the question posed in the introduction.As, for large systems, the standard deviation ∆S A is much smaller than the average, we conclude that the average entropy S A is also the typical value of the entanglement entropy of a subsystem.The conclusion holds both in the presence of a trivial and of a nontrivial center.
The result applies to all systems where a subalgebra with a center arises.A notable case is the one of a free isolated quantum system prepared in an energy eigenstate.While interaction between subsystems is necessary for thermalization [10][11][12][13][14][15][16][17][18][19][20][21][22][23], an arbitrarily small interaction is sufficient to select a typical energy eigenstate.We illustrate the relevance of our result with two examples.
A system of N noninteracting spins with magnetic moment µ in a magnetic field B has Hamiltonian We consider a subsystem A consisting of N A spins.The eigenspace H(E) has the structure (1) where the direct sum is over eigenvalues ε of the Hamiltonian H A of the subsystem A.

The average energy in
which shows equipartition of the energy in a typical state.For N 1, the average entanglement entropy of the subsystem (23) evaluates to where ).This is the Page curve of the system and its variance is exponentially small in N .We observe that the entanglement entropy vanishes for the ground state E = −N µB, increases with the energy up to min(N A , N B ) log 2 in the middle of the spectrum, and then decreases indicating that the system behaves as if it had a temperature that is positive for low energy eigenstates and negative for high energy eigenstates [54].In fact for a small subsystem (N A N ) we can define a temperature as the variation of the entanglement entropy with respect to the typical energy with k the Boltzmann constant.This is the familiar relation between the temperature and the energy in a paramagnetic system [55].Fig. (1) shows the Page curve and its variance for a system of N = 10 spins.The temperature of a small subsystem can be read from the slope of the Page curve.
As a second example, we consider a free quantum field: the quantum electromagnetic field in a cubic box of volume L 3 .Previous analysis have focused on the geometric entanglement entropy of a region of space [56][57][58] and its renormalization [8,[59][60][61].Here we identify a different subalgebra of observables that better characterizes what can be measured [62].The Hilbert space of the quantum field H = λ H λ is the tensor product of Hilbert spaces of discrete wavelength λ = (2L/k x , 2L/k y , 2L/k z ) with k x , k y , k z ∈ N. A measuring device, such as an antenna of length , defines a subsystem A corresponding to the dis-crete wavelengths λ A = (2 /k x , 2 /k y , 2 /k z ).If /L ≤ 1 is a rational number, the wavelengths λ A are a subset of the wavelengths λ.The Hilbert space H(E) of eigenstates with energy E inherits the structure (1), where ε is the energy of the antenna A. When the box is large, i.e. when EL c, we can compute the dimension d E of H(E) from the density of states and the occupation numbers of photons.We find . The average energy in the subsystem A is ε = ε ε p(ε) ≈ E 3 /L 3 , which shows that in a typical state the energy of the subsystem is extensive and E/L 3 is the energy per unit volume.The average entanglement entropy of a subsystem with 3 ≤ L 3 /2 is The result follows from (23) together with the fact that p(ε) is sharply peaked at ε and therefore the Shannon entropy contribution is negligible and the dominating term is the average entanglement entropy S εA .As the variance is exponentially small in EL/ c, this is also the typical value of the entropy in an energy eigenstate.For L we can also define a temperature from entanglement: We note that this temperature does not depend on the subsystem size.In terms of this temperature, the entanglement entropy assumes the form of the familiar extensive formula for the canonical entropy of black body radiation.On the other hand, for large subsystems, the typical entanglement entropy is smaller than the thermal entropy and follows a Page curve qualitatively similar to the one in Fig. 1, This entropy is arising from the entanglement between the modes that the antenna can measure, i.e. the wavelengths λ A , and the modes that it cannot measure.The unmeasured modes include both longer wavelengths and wavelengths shorter than that the antenna cannot couple to.
Our results show that, in small noninteracting systems prepared in a typical energy eigenstate, thermal properties can arise from entanglement.Recent experimental developments on measurements of thermalization in small isolated quantum systems, such as ultracold atoms in optical lattices [63][64][65][66], might provide access to the deviations from statistical mechanics predicted by the exact formulas for the average entropy (23) and its variance (24) at fixed energy.
• requires Z = Γ (d A d B ).We then compute the integrals X ij (r) (A12) for any r ≥ 0 which appear in To keep the notation compact we denote X(r) the d A × d A matrix with entries X ij (r).Using the generating function for the generalized Laguerre polynomials we write (A12) as derivatives respect to the parameters x and y of the integral of two generating functions: We compute the derivatives explicitly by applying successively the Leibniz rule, obtaining a closed form for the integrals X ij (r) . (A18) We conclude the computation for S A noticing that Taking the derivative and the limit is a long but straightforward calculation that can be done with the help of Wolfram's Mathematica.The result is the celebrated Page formula, The calculation of S A 2 can be done following a similar strategy.We first compute with r 1 > 0 and r 2 > 0. We perform a similar change of variables q a = ζλ a and eliminate the delta function in the measure by integrating over ζ.We obtain The two sums can be expanded into two terms that can be integrated separately, q r1 a q r2 b . (A23) The integral of the first term is completely analogous to the computation we just performed resulting in Z1 Γ (d The normalization constant Z can be computed using a procedure similar to the one used in (A2), The average and the variance of p j can be easily shown to be given by We illustrate the result with two examples.The first one consists in taking all equal dimensions d j = d E /J.We find: S(p) = Ψ (d E + 1) − Ψ (d E /J + 1) ≈ log(J) for J 1. (B10) As a second example, we consider the case where the dimension d J is much larger than the sum of all the others In this case the exact formulas reduce to The computation of ∆S(p) 2 is straightforward but its expression is convoluted.We report here its leading order for The variance of the Shannon entropy vanishes as 1/d E for d j → ∞.
Average entropy and variance with a trivial center.-Letus consider a quantum system with a bipartite Hilbert space H = H A ⊗ H B .The subsystems A and B have dimension d A = dimH A and d B = dimH B , with A the smaller of the two subsystems, 1 < d A ≤ d B , and the dimension of H given by d = d A d B .The restriction of a pure state |ψ ∈ H to the subsystem A defines the reduced density matrix ρ A = Tr B |ψ ψ|.The en-tanglement entropy S A (ψ) = −Tr(ρ A log ρ A ) is the von Neumann entropy of the reduced density matrix.

j ) 2
= d j (d E − d j ) d 2 E (d E + 1).(B7)The average of the Shannon entropy for the probability distribution (B5) is given by S(p) = − j p j log p j = −

S(p) ≈ 1
d J d R + d R log d J + J−1 i=1 d i Ψ (d i + 1) .(B13)Note in particular that the average Shannon entropy goes to zero as d J → ∞.Furthermore if all the dimensions are large d j 1, we note that the average Shannon entropy equals the Shannon entropy of the average probability, S(p) ≈ J j=1 p j log p j .(B14)