SciPost Phys. Core 6, 070 (2023)

Local operator entanglement in spin chains
Eric Mascot1⋆ , Masahiro Nozaki2,3 and Masaki Tezuka4
1 Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA
2 iTHEMS Program, RIKEN, Wako, Saitama 351-0198, Japan
3 Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological
Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China
4 Department of Physics, Kyoto University, Kyoto 606-8502, Japan⋆ eric.mascot@unimelb.edu.au

Abstract
Understanding how and whether local perturbations can affect the entire quantum system is a fundamental step in understanding non-equilibrium phenomena such as thermalization. This knowledge of non-equilibrium phenomena is applicable for quantum
computation, as many quantum computers employ non-equilibrium processes for computations. In this paper, we investigate the evolution of bi- and tripartite operator mutual information of the time-evolution operator and the Pauli spin operators in the onedimensional Ising model with magnetic field and the disordered Heisenberg model tostudy the properties of quantum circuits. In the Ising model, the early-time evolution
qualitatively follows an effective light cone picture, and the late-time value is well described by Page’s value for a random pure state. In the Heisenberg model with strong
disorder, we find that many-body localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian that describes
the time evolution of bi- and tripartite operator mutual information for the Heisenberg
model in the large disorder regime.
Copyright E. Mascot et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 20-01-2021
Accepted 22-09-2023
Check forupdatesPublished 20-10-2023
doi:10.21468/SciPostPhysCore.6.4.070

Contents
1 Introduction and summary2

2 Preliminary
2.1 Information scrambling and operator spreading
2.2 Many-body localization (MBL)
2.3 Operator entanglement and its quantum measures
2.3.1 Definition of operator entanglement
2.3.2 Initial state
2.3.3 Quantum measures

5
5
6
8
8
10
11

3 BOMI and TOMI in a spin chain
3.1 Early-time evolution

13
13
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3.2

Late-time evolution
3.2.1 Factorization of the density matrix

16
18

4 Time evolution of BOMI and TOMI in a disordered spin chain
4.1 Effective theory in the strong disordered phase.

18
20

5 Discussions and future directions

22

A The time evolution of TOMI and BOMI in the one-dimensional Ising model withmagnetic field

23

B The time evolution of BOMI and TOMI in the one-dimensional disordered Heisenberg model

26

References

27

1

Introduction and summary

Introduction
Thermalization is one of the frontier research topics in theoretical and experimental physics,
in which the entanglement of the state plays an important role [1–3]. In closed quantum
systems, the unitarity of the time evolution operator prevents a pure state from becoming athermal state. However, in many cases, after an adequate amount of time, observables in a subsystem are well-approximated by those for a thermal state, known as thermalization. Though
the thermal state does not depend on the entanglement structure of an initial state, thermalization does depend on the initial state and the details of its dynamics. Here, the entanglement
structure refers to the way subsystems are entangled with each other. The reduced density matrix for a subsystem is given by a mixed state. If time evolution washes out the dependence of amixed state on the details of the initial state, the reduced density matrix becomes the reduced
thermal state. An effective temperature of the thermal state is given by the energy scale ofthe initial state. Quantum entanglement has also been found to shed light on the mechanism
behind the anti-de Sitter / conformal field theory (AdS / CFT) correspondence [4–6].
To better understand thermalization, authors in [7–10] studied the entanglement entropy
in a quantum quench of conformal field theories. The entanglement entropy is the von Neumann entropy of the reduced density matrix. A reduced density matrix becomes approximately
equal to the thermal state after enough time passes for the dynamics to wash out the information about the initial state.
The authors in [7] studied the time evolution of entanglement entropy for a subsystem ofwidth l in 1+1 dimensional (1+1D) CFTs. A given initial state depends on an initial energy
scale, 1/β. The time evolution of entanglement entropy after a quench follows the relativistic
propagation of quasiparticles that are created by the quench. An entangled pair of particles
is created on each site. Thereafter, one of the particles moves leftward at the speed of light,
while the other moves rightward. When only one of these two particles remains in the subsystem, the entanglement entropy increases, owing to the entangled pair. For the early-time
window, β ≪ t < 2l , the entanglement entropy is a linear function of t, but when 2l < t, it is
proportional to l. Thus, this volume law of entanglement entropy indicates that the subsystem
has thermalized.
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The authors in [11,12] found that the time evolution of mutual information and entanglement entropy for two disjoint intervals in holographic CFT, a CFT that has a gravity dual, does
not follow the quasiparticle interpretation. The early-time evolution follows the relativistic
propagation of quasiparticles, but the late-time evolution does not. The authors in [13] foundthat holographic tripartite mutual information, a linear combination of holographic mutual
information, must be non-positive. These articles indicate that the multipartite entanglementin CFTs depends on the details of the theories, such as the operator content. We will demonstrate how multipartite entanglement sheds light on the details of thermalization. In [14–24],
it was found that the operator entanglement, the entanglement of state dual to the operator,
elegantly captures the details of dynamics in non-equilibrium processes. The operator entanglement of the time-evolution operator is able to directly show that the quantum informationin the initial subsystem is delocalized in the late-time regime, so that local observers are notable to obtain it.
After thermalization is complete, the entanglement entropy is expected to follow the Page
curve [25, 26]. When the late-time entanglement entropy follows the Page curve, non-local
correlation between the small subsystems vanishes. Quantum computation, a cutting-edge research field, can be implemented as a nonequilibrium process. In the quantum circuit model,
quantum computations are constructed as a sequence of quantum gates and measurements.
The quantum gates are local unitary operators and these local gates and measurements arearranged spatially [27, 28]. If we interpret each step of the computation as each time of the
non-equilibrium process, the quantum circuit may be equivalent to the (non-unitary) nonequilibrium process. It is also important to understand quantum computations or nonequilibrium processes that preserve the quantum nature of the state. In this paper, we examine theproperties of a quantum computational process by studying the operator entanglement in thetwo different dynamical systems. One system has dynamics with the strong scrambling ability
and few degrees of freedom. The other has dynamics that preserve the quantum nature of thestate. This is the spin chain in the many-body localization phase (MBL phase).
In this paper, we will study the properties of a quantum circuit with 1) strong scramblingability, and 2) weak scrambling effect. We will study whether the late-time evolution of the biand tripartite operator mutual information (BOMI and TOMI) of local operators depends on
the operators in the dynamical system. The TOMI measures how much information regarding
the subsystem may be locally hidden, while the BOMI measures how many EPR pairs associatedwith the subsystem considered may be destroyed during the nonequilibrium processes. For 1)
strong scrambling, the fact that the late time evolution of BOMI and TOMI are independentof the local operator suggests that the information of the local operator is washed out by
the dynamics. The authors in [23] found that in the system with strong scrambling ability
and large degrees of freedom, the late-time behavior of BOMI and TOMI of local operators
is independent of the operators and approximated by that of the unitary operator. Dynamics
with a large number of degrees of freedom and strong scrambling has several different featuresfrom dynamics with only a small number of degrees of freedom. For example, the out oftime ordered correlator (OTOC) can measures such properties of dynamics that depend on
the number of degrees of freedom [29, 30]. For this reason, even if the degrees of freedom
are small, we will study whether the late time behavior of BOMI and TOMI depend on local
operators. We will also study how the finiteness of the system affects the scrambling effectof the dynamics. For comparison, we consider the disordered spin chain, which is expected
to be a remarkable example of dynamics with weak scrambling effects. This spin system haschaotic and MBL phases corresponding to weak and strong disorder, respectively. By varying
the strength of the disorder, we will study the difference in the evolution of BOMI and TOMI.

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Summary of results
We investigate the time evolution of bi- and tripartite operator mutual information of the timeevolution operator and Pauli’s spin operators for three configurations, the fully-overlapping,partially-overlapping, and disjoint configurations, (See Figure 2) in a spin chain with strong
scrambling ability and a disordered spin chain.
We divide the input (original) Hilbert space into A and Ā, and divide the output (scattered)Hilbert space into B and B̄. In the fully- or partially-overlapping configurations, A is initiallyin B. In the disjoint configuration, A is out of B. We summarize the results here.
Spin chain with strong scrambling ability
We consider a 1D Ising model with transverse and longitudinal field for its strong scramblingability. We find that the time-evolution of bi- and tripartite operator mutual information is
described by two effective descriptions, an effective light cone and the Page curve.
Early time evolution: We find the early-time evolution of BOMI and TOMI depend on thevarious operators and boundary conditions, and is described qualitatively by an effective lightcone picture. In this picture, while sections of A, Ā, B and B̄ are partially inside the light cone,
the bipartite mutual information decreases.
Late time evolution: We find the late-time evolution of BOMI and TOMI is independent ofvarious operators and boundary conditions, and the late time value is described by Page’s value
for the average entropy [25,26], which we call the Page curve (See, Section 3.2). This suggeststhat the strong scrambling effect washes out the information about the local operators andboundary conditions, so that the state becomes the typical one, the state which is independentof the initial state. We also find that the late time value of BOMI and TOMI can depend on thesystem size. This suggests that the finiteness of the system can prevent the reduced density
matrix for A ∪ B from factorizing into the reduced density matrices for A and that for B, andthe finiteness of the system can also reduce the amount of information scrambled by the thedynamics.
Disordered spin chain
We consider a 1D Heisenberg model with uniform disorder strength to study the many-body
localization phase. We find parallels to the simpler Ising chain for the weak and strong disorderregimes.
Weak disorder: The spin chain with weak disorder is in the chaotic phase. Therefore, thelate-time values of BOMI and TOMI is well-described by the Page curve. Also, in the weakdisorder region of the disordered spin system, information on local operators and boundary
conditions is lost locally due to dynamics.
Strong disorder: When disorder is large, the late-time value of BOMI in the full overlap
configurations increases with the strength of disorder. We interpret this to be due to the localization effect of the system. When the disorder is large enough, BOMI and TOMI periodically evolve in time. Thus, BOMI and TOMI exhibit quantum revival. This oscillation iswell-described by an effective Hamiltonian. The period is determined by the strength of local
interaction, and the periodic behavior is due to the off-diagonal components of the reduced
density matrix in the σz basis. This suggests this dynamical system is the quantum circuit that
preserves the quantum nature of the initial state.
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The authors in [24] have studied the late-time dynamics of many body localized systems
by using operator mutual information and other quantities such as OTOC. Our results are
consistent with theirs and we extend their results on operator mutual information to earlytime dynamics.

2

Preliminary

In this section, we first describe the concepts of information scrambling and operator spreading. Second, we review many-body localization, which is an interesting dynamical phenomenon far from chaotic dynamics. Finally, we introduce operator entanglement and bipartite and tripartite operator mutual information as quantum measures for operator entanglement.

2.1

Information scrambling and operator spreading

Let us briefly review information scrambling and operator spreading.
Information scrambling
Information scrambling occurs in dynamical processes where the subsystem thermalizes afteran adequate amount of time. To explain information scrambling, we first review thermalization of subsystems. Consider the initial stateX|Ψ〉 =Cα |α〉 ,(1)
Eα <1/εwhere |α〉 is an eigenstate of Hamiltonian, H |α〉 = Eα |α〉 and Eα ≥ 0. his initial state ischaracterized by an energy scale, 1/ε.
Following the time evolution U(t), the reduced density matrix for a subsystem V is givenbyρV (t) = TrV̄ U(t) |Ψ〉 〈Ψ| U † (t) ,(2)where we divide the total Hilbert space into V and V̄ . The reduced density matrix ρV (t)depends not only on the initial state, but also on the details of time-evolution operator U(t).
In this paper, the definition of subsystem thermalization is that ρV (t) is well-approximated bya reduced thermal density with an effective inverse temperature βeff ≈ ε after some time,t→∞ρV (t) −−−→ ρTh.V =TrV̄ e−εH
Tre−εH,(3)

where ρTh. is the thermal state. If the late time behavior of observables in V are approximated by their thermal expectation value, then we say that thermalization of V occurs. For
example, in addition to the mutual information content studied in this paper, the one-point
function and the distance between states [31–33] are also used as indicators of thermalizationof subsystems. If the subsystem thermalizes, these observables behave as follows:〈Ψ| U † (t)V (x)U(t) |Ψ〉≈ 1,t→∞〈V (x)〉Th.

TrV ρV (t)TrV̄ e−εH
lim Ç
≈ 1,
2
t→∞
TrV TrV̄ e−εH TrV (ρV (t))2
lim

5

(4)
(5)

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where, 〈·〉Th. denotes the expectation value of the thermal state. If the approximation in (3)works well, the late-time reduced density matrix depends only on the initial energy 1/ε and noton the entanglement structure of its initial state. Information scrambling is the phenomenon
where the late-time reduced density matrices for any subsystem is independent of the structureof its initial state, and instead depends on 1/ε. If this phenomenon occurs, dynamics has thestrongest scrambling ability.
Operator spreading
One interesting dynamical phenomenon is operator spreading [14, 34–36]. A local operator
in the Heisenberg picture O(x, t) is given byO(x, t) = e iH t O(x)e−iH t= O(x) + (i t) [H, O(x)] +(i t)2[H, [H, O(x)]] + · · · ,2!

(6)where we used the Baker-Campbell-Hausdorff formula. If t is small, O(x, t) is approximately
given by a simple operator O(x). As time proceeds, the larger the contributions of terms withhigher powers of t become. Thus, a local operator in the Heisenberg picuture O(x, t) becomes
more complicated over time.
Consider a probe Q( y), a local operator far from the position of O, to measure how muchO(x, −t) spreads. If t is small, this probe is causally unrelated to O(x, −t),[Q( y), O(x, −t)] = 0 .

(7)

If t is large, this probe is causally related to O(x, −t),[Q( y), O(x, −t)] ̸= 0 .

(8)This is operator spreading. The expectation value of the square of the commutator[Q( y), O(x, −t)]2 measures how much O(x, −t) spreads and how complicated it becomes.
The behavior of this expectation value of the commutator square is related to the out-of-timeordered correlator (OTOC), which has been actively studied [18, 29, 37–39]. The OTOC is
defined as
C(t) = O† (x, −t)Q† ( y)O(x, −t)Q( y) ,(9)where 〈·〉 denotes the thermal expectation value with inverse temperature β. The authorsin [37] suggest that the form of OTOC in the large N theories for the time region t ≫ β is
given byc1
C(t) ≈ c0 − 2 eλt ,
(10)
t≫β
N

where c0 and c1 are positive, order one, quantities that depend on O and Q, and λ is the
Lyapunov exponent, which depends on the scrambling effect of the dynamics. In holographic
CFTs, λ is maximized at 2π
β . In generic systems, including the spin chain considered in thispaper, the behavior of OTOC is discussed in [40].

2.2Many-body localization (MBL)

Many-body localization is a counterexample to thermalization, which says the structure ofquantum entanglement is almost preserved under time-evolution. Therefore, we expect quantum information to be preserved locally.
The localization of energy eigenstates up to finite energy density in interacting quantum
systems, termed many-body localization, has received wide attention for more than a decade.
See [41–43] for recent review articles.
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In the non-interacting case the single-particle orbitals can localize due to randomness [44].
It was initially not clear whether localization persists in the presence of interactions. In thetight-binding picture, a particle localizing around a lower energy lattice site would repel otherparticles if the interaction between the particles is repulsive, as would be expected for electrons
with Coulomb interactions. Then the effect of the interaction would be to weaken the randomness in the site energy. In later studies, the possibility of localization was first discussed for the
many-body ground state [45–47], and then to finite-temperature cases [48–50]. Interacting
systems staying insulating at finite temperatures are said to be many-body localized.
An MBL state can escape thermalization when placed in a non-equilibrium initial state.
Integrable systems also do not generally thermalize, but the integrability sensitively depends
on the choice of parameters and thermalization would happen once the integrability is broken,
unless MBL is established. MBL phases are robust against weak perturbations since all theenergy eigenstates are localized up to a finite energy density. In systems with a set of randomparameters (e.g. hoppings, site energies, or on-site interactions) drawn from some probability
distributions, any physical quantities would be probabilistically determined. However, oncethe constants controlling these distributions satisfy certain conditions, the probability that
the MBL behavior is not observed is exponentially small, then the system controlled by these
constants can be said to be in the MBL phase.
Note that systems that exhibit MBL without randomness in the Hamiltonian have been
proposed [51]. Also, note that systems with spatially quasiperiodic modulation, whose nonuniform feature is determined by a few parameters as opposed to lacking long-range correlation, can be many-body localized. Even in these cases, we can generally expect the many-body
localized phase to be robust against time-independent weak perturbations.
Hamiltonians exhibiting the MBL phase provide counterexamples to the most general, or
naive, form of the eigenstate thermalization hypothesis (ETH) [1, 2]. ETH states that for an
isolated quantum mechanical many-body system, for any initial state given as a linear combination of eigenstates close in energy, the expectation value of an operator should “thermalize”,
that is, relax to a value close to the one expected for the microcanonical ensemble determinedby the energy range and will only show small fluctuations at later times.
A kind of emergent integrability exists in the many-body localized systems, as there areexponentially many quasi-local integrables of motion, often abbreviated as LIOMs. Correspondingly, the energy spectrum does not show the random matrix universality as expected ina quantum chaotic system [52] and become uncorrelated.
The time evolution of entanglement entropy from an uncorrelated state, which would be
linear in non-interacting systems or general non-integrable systems without localization [53],becomes logarithmic in time for a many-body localized system [54–57]. In a many-body localized eigenstate the entanglement entropy for a subsystem exhibits area law dependence on
the surface area rather than volume laws that is expected for a thermal reduced density matrix
for the subsystem obeying ETH [58].
For a class of one-dimensional spin chain with random local interactions, the existence of
MBL has been proved. [59]. On the other hand, the best studied model is the one-dimensional
chain of S = 1/2 spins with random fields, eq. (34). While recent studies have suggested
that the MBL transition does not occur at w/J ≈ 3 as was previously believed, but occurs at
w/J ≳ 20 or at a large w/J increasing as the system-size is increased [60,61], for system sizes
that allow exact diagonalization studies, the phenomenology of the many-body eigenstates do
exhibit localizing behavior for w/J ≈ 3.
Experimentally, MBL has been observed in ultracold atoms [62–65] and ions [66] trappedin low dimensions. In [62], the atoms were prepared in a highly non-equilibrium densitywave state, in which most of the particle population is on every second lattice site in a onedimensional lattice. Then, the system was allowed to evolve under the given Hamiltonian7

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and the imbalance between the even and odd sites was measured. In the presence of the onsite interaction, the imbalance remained finite when the quasiperiodic potential was strong
enough.

2.3

Operator entanglement and its quantum measures

In this section, we describe operator entanglement and the quantum measures consideredin this paper, bipartite operator mutual information (BOMI) and tripartite operator mutual
information (TOMI).
2.3.1

Definition of operator entanglement

Operator entanglement is defined as the entanglement structure of the dual state, which is thestate given by a state-channel map:

W=

X
i, j

state-channel map

Wi j |i〉 〈 j| −−−−−−−−−−→ |W 〉 = N

Xi, j

Wi j |i〉 | j ∗ 〉 ,(11)where W is an operator, |·∗ 〉 is a C P T conjugate of |·〉, and N is a normalization factor. The
dual state lives in the Hilbert space whose dimension is the square of the dimension of theoriginal Hilbert space:state-channel map

HW −−−−−−−−−−→ H|W 〉 = HW ⊗ HW ,(12)

where HW is the Hilbert space where the operator acts, and H|W 〉 is the space where the dual
state lives.
In this paper, we consider the operator entanglement of the following operators.
Unitary operator
Time evolution changes the entanglement structure of an initial state. The quantum correlation between a subsystem A of an initial state and subsystem B of the time-evolved state should
show how the time evolution changes the local structure of an initial state. The information
about the local structure propagates (and/or is delocalized) as the structure changes. In thispaper, we define operator entanglement measurements of the unitary time-evolution operator,
and we study the spreading and delocalization of quantum information by using the measuresfor operator entanglement defined in Section 2.3.3. The unitary channel for the time-evolution
operator U(t), represented in the eigenbasis of the Hamiltonian, is
U(t) = e

−iH t

=

N
X

e−i t Ea |a〉 〈a| ,

(13)

a

where the Hamiltonian is a time-independent operator, |a〉 is an eigenstate, and N is the number of states.
The dual state to (13) can be defined by the state-channel map f [U(t)],NN

1 X −i t Ea
1 X −i t Ea
f [U(t)] = |U(t)〉 = pe|a〉 |a〉 = pe|a〉 |a〉N a
N a
N
1 X − i t (Hin +Hout )=pe 2|a〉in |a〉out ,
N a

8

(14)

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A

A

B

B

Figure 1: Input (A) and output (B) subsystems in the doubled Hilbert space
Hin ⊗ Hout . We compute the BOMI and TOMI for the configurations corresponding to
Figure 2.
where we take |a〉 to be a real function because |a〉 is an eigenstate of H. The states, |·〉in
and |·〉out , are in the Hilbert spaces, Hin and Hout . Thus, the state defined in (14) is in
H = Hin ⊗ Hout . Its density matrix is given byN

1 X −i t(Ea −E b )ρ = |U(t)〉 〈U(t)| =e|a〉in 〈b|in ⊗ |a〉out 〈b|out .
N a,b=1(15)

We divide the input Hilbert space Hin into HA and HĀ and the output Hilbert space Hout into
HB and HB̄ as in Figure 1. The input and output states in the spacetime language are givenbyXXb| j〉A |J〉Ā ,C jJ
CiaI |i〉A |I〉Ā , |b〉in =|a〉in =jJiI|a〉out =X
αA

DαaA |α〉B |A〉B̄ , |b〉out =X
βB

DβbB |β〉B |B 〉B̄ ,(16)

where |x〉X are orthogonal vectors in HX and X = A, Ā, B, B̄. Thus, the density matrix in thespacetime language is given by
X
1
b∗ a
ρ=
DαA Dβb∗B |i〉 〈 j|A ⊗ |I〉 〈J|Ā ⊗ |α〉 〈β|B ⊗ |A〉 〈B |B̄ . (17)e−i t(Ea −E b ) CiaI C jJ
N a,b,i, j,I,J,α,β,A,B
Next, we consider a reduced density matrix for A ∪ B as an example. The matrix ρA∪B isgiven by limiting the sum in the above to I = J and A = B ,X
1b∗ aρA∪B = TrA∪B ρ =e−i t(Ea −E b ) CiJa C jJ
DαB Dβb∗B |i〉 〈 j|A ⊗ |α〉 〈β|BN a,b,i, j,J,α,β,BX
ab=e−i t(Ea −E b ) Mab(18)
i j Nαβ |i〉 〈 j|A ⊗ |α〉 〈β|B ,a,b,i, j,J,α,β,BPb∗where Mai jb = J CiJa C jJ
and Niaj b = B DαaB Dβb∗B . Operator entanglement entropy (OEE) forρA∪B is defined as the von Neumann entropy of the reduced density matrix
X
SA∪B = −TrA∪B ρA∪B log2 ρA∪B = −λA∪B log2 λA∪B ,
(19)
P

λA∪B

where λA∪B are the eigenvalues of ρA∪B .
By computing the mutual information (defined in Section 2.3.3) of the unitary timeevolution operator, we study the correlation between a subsystem of the initial state and asubsystem of the output state in which we measure local observables.
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Local operator
A local perturbation, a perturbation due to a local operator, spreads under the time evolution
as in Section 2.1, so that the entanglement structure of the scattered state, the state which isacted with the local operator, should differ from the original state. By measuring the correlation between the subsystem A of the original state and B of the scattered state, we should findwhere information about the entanglement structure in A propagates and how much the operator delocalizes information. Let us define the local operator entanglement as the quantum
entanglement of local operator O(x, t) as follows.
A local operator in Heisenberg picture O(x, t) is defined asO(x, t) = e iH t O(x)e−iH t ,(20)

where we assume that the Hamiltonian is time-independent. We define the state dual to (20)by the channel-state map:
X
X
|O(x, t)〉 = N
e i t Ea 〈a|out O(x) |b〉in e−i t E b |a〉out |b〉in = N〈a|out O(x, t) |b〉in |a〉out |b〉in ,
a,b

a,b

(21)
where the states |·〉in and |·〉out are the eigenstates of the time-independent Hamiltonian, andthe normalization factor N is given by1

N = Z−2 ,Z = Tr O† (x)O(x) .

(22)

The entanglement structure of the state in (21) strongly depends on the Heisenberg operator. For example, if O(x, t) is the identity operator, then the entanglement structure of thestate in (21) is the same as that of maximally entangled state. In the general case, O(x, t)
can make the quantum entanglement structure of the state in (21) less entangled than thatof the maximally entangled state. By computing the mutual information (defined in Section
2.3.3) of local operators, we study the correlation between subsystem A of the original stateand subsystem B of the scattered state.
2.3.2

Initial state

Here, we consider the dual states at t = 0. At t = 0, the dual states to the unitary and Pauli’s
operators in Heisenberg picture are respectively given by|U(t = 0)〉 =

1 X
2

L
2

a

(i)
|a〉out |a〉in , |σα=x,y,z (t = 0)〉 =

1 X
2

L
2

a

(i)
σα=x,
y,z |a〉out |a〉in ,

(23)

(i)
where σα=x,y,z acts on i-th site of H1 . Thus, |U(t = 0)〉 is equal to the thermofield double state(i)with zero inverse temperature, and |σα=x,y,z (t = 0)〉 is given by acting the Pauli’s operator on
the themofield double state. In terms of the eigenstates of σz on each site, |U(t = 0)〉 and(i)|σα=x,y,z (t = 0)〉 are given by|U(t = 0)〉 =

L
Y

(i)
(i)
|σα=x,y,z (t = 0)〉 = σα=x, y,z|EPR〉i ,i=1|EPR〉 j ,

(24)

j=1

where |EPR〉i is defined by |EPR〉i = 11
the system.

L
Y

22

P

(i)

σz =↑,↓10|σz(i) 〉out |σz(i) 〉in . Here, i means the i-th site of

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2.3.3

Quantum measures

Operator mutual information measures non-local correlation between subsystems. As mentioned in the previous sections, we investigate the time evolution of the operator mutual
information in various configurations in order to find how operator spreading occurs undertime-evolution for a system with strong scrambling ability. We also study the operator mutual
information in a system which has a many-body localized phase.
Bipartite operator mutual information (BOMI)
Bipartite operator mutual information (BOMI) is defined as the linear combination of OEE
IA,B = SA + SB − SA∪B ,(25)

where SA and SB are OEEs for the input (original) subsystem A and the output (scattered)subsystem B. The last term in the right-hand side in (25) is OEE for A∪B. The input and output
subsystems are defined as the subsystems of Hin ⊗Hout where the dual state to U(t) lives, while
the original and scattered subsystems are defined as the subsystems of Horiginal ⊗Hscattered where
the state dual to O(t, x) lives. The BOMI measures the correlation between the input (original)and output (scattered) subsystems. If IA,B vanishes, there are no correlations between A and B,
which means that the reduced density matrices of the subsystems are unrelated to each other.
Tripartite operator mutual information (TOMI)
Tripartite operator mutual information (TOMI) is defined as a linear combination of the BOMI
as follows. We divide the output or scattered space into the subsystems B and B̄, and also
divide the input or original space into A and Ā. The TOMI which we consider is given by
IA,B,B̄ = IA,B + IA,B̄ − IA,B∪B̄ .

(26)

As in [13,14,21,22], the TOMI in (26) measures how much information regarding A is delocalized over time. We will briefly explain why TOMI in (26) can be an indicator of the information
scrambling. Suppose the initial state is given by the product of EPR states in (24). Suppose asubsystem A∪ B is given by the union of a subsystem A on H1 and B on H2 . Let nα denote thenumber of EPR pairs included in the subsystem α. For this initial state, the value of IA,B andIA,B is proportional to nA∪B and nA∪B , respectively, while the value of IA,B∪B is proportional tonA∪B∪B . Since nA∪B + nA∪B = nA∪B∪B , the value of TOMI is zero. In this paper, we use TOMI tostudy how many these EPR pairs are preserved in non-equilibrium processes. Under dynamics
with scrambling ability, the information is locally hidden, so that nA∪B and nA∪B deceases intime. Decrease of nA∪B and nA∪B depends on dynamics. In this paper, we use TOMI to study
how many of these EPR pairs are preserved in a non-equilibrium processes. In the time evolution induced by the time evolution operator with no scrambling ability, the value of TOMI isindependent of time, and it is zero, while in the time evolution induced by the time evolution
operator with stronger scrambling ability, the TOMI will have a larger negative value.
Configurations
As shown in Figure 2, we study the time evolution of the operator mutual information in three
configurations: (a)full overlap, (b)partial overlap, and (c)disjoint configurations. In the full
overlap configuration, the size of A and B are equal, lA = l B , and the edges of each subsystem
are aligned. In the partial overlap configuration, the size of B is larger than that of A by s,l B = lA + s. The left edge of subsystems A and B are aligned. In the disjoint configuration, thesubsystems A and B do not have any overlapping regions and the distance between the right11

SciPost Phys. Core 6, 070 (2023)(a) Full overlap
Scattered space

B

Original spaceAB(b) Partial overlap

B

B

A

(c) Disjoint

B

B

A

Space

Time

Figure 2: Schematic of the three configurations: (a) full overlap, (b) partial overlap,
and (c) disjoint. Full overlap: The size of A and B are equal, lA = l B , and are aligned.
Partial overlap: The size of B is greater than A by s, lA +s = l B , and are aligned on theleft. Disjoint: The distance between the right boundary of A and the left boundary of
B is d.
edge of A and left edge of B is d. These three configurations are useful to consider since in
2D CFTs [21, 23], how the dynamics affects the time evolution of BOMI depends on how thesubsystems are taken. In the full overlap configuration, the behavior in time of BOMI does notdepend as much on the scrambling ability of the time evolution operator. In the partial overlap
and disjoint configurations, the time evolution of BOMI strongly depends on the scramblingability of the dynamics. Therefore, we will study the time evolution of BOMI in these three
configurations.
• Full overlap configuration (Fig. 2 (a)) In this configuration, the initial value of IA,B for|U(t)〉 is proportional to the number of EPR pairs, both edges of which are in A∪ B. The
decreases in time of IA,B will indicates the how many EPR pairs the subsystem A∪ B loses
in the non-equilibrium process. The initial value of IA,B,B will be zero because the initialvalue of BOMI is determined by the number of EPR pairs. In other words, since no EPR
pairs are delocalized, the value of IA,B,B is zero. The negative value of IA,B,B suggests
how many number of EPR pairs are delocalized and locally hidden by the dynamics. For|σα(i) (t)〉, at t = 0, the entanglement structure of a single EPR on i-th site is deformedby σα(i) (t = 0), and we expect the time evolution of IA,B and IA,B,B to be interpreted in
the same manner.
• Partial overlap configuration (Fig. 2 (b)) As explained above, in this configuration of
2D CFTs, the time evolution of IA,B and IA,B,B strongly depends on the scrambling ability.
The 2D holographic CFT is the theory with the large degrees of freedom and strong
scrambling effect. Although the spin systems which we consider in this paper does nothave the large degrees of freedom, we expect the time evolution of IA,B and IA,B,B in thisconfiguration to strongly depend on the scrambling ability of the dynamics. Therefore,
we will consider the partial overlap configuration.
• Disjoint configuration (Fig. 2 (c)) In this configuration, there are no EPR pairs, bothedges of which are in A∪ B. Then, the initial value of IA,B is zero. For |U(t)〉 and σα(i) (t),OEEs for A and B are independent of time. The time evolution of IA,B is determined bythat of SA∪B . The dynamics with strong scrambling makes the entanglement structureof A ∪ B less structure-less, so that the value of SA∪B is expected to increases in the nonequilibrium process. Since the initial value of SA∪B is the one for the maximally entangledstate, we expect that the value of IA,B does not vary in time so much.

12

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3

BOMI and TOMI in a spin chain

We study the dynamics of the spin chain,
H=

L
X



(i)
σz(i) σz(i+1) + h x σ(i),x + hz σz

(27)

i=1

where L is the number of site of this system, and σ(i)a are the Pauli spin operators on site
(L+1)
i. For periodic boundary conditions, we define σa= σ(1)
a , whereas for open bound(L+1)
ary conditions, the term involving σa
is ignored. The dynamics of the Hamiltonian with(h x , hz ) = (−1.05, 0.5) is chaotic [33, 37].
The authors in [14] studied the time evolution of the BOMI and TOMI for the unitary
operator U(t), and in particular, studied the scrambling ability of the chaotic chain. In thispaper, we study the scrambling ability of the chain in terms of local operators, and compare
the BOMI and TOMI of local operators to U(t). We consider the time evolution for Pauli’s
iH t (i) −iH t
operators in the Heisenberg picture σ(i)σa efor a spin chain with periodic anda (t) = eopen boundary conditions.
We investigate the time evolution of the BOMI and TOMI in the integrable and chaotic
phases. We briefly summarize the results, and we will give some interpretation on these results
later:
1. Integrable phase: For integrable cases, we study both longitudinal, (h x , hz ) = (0, 2),
and transverse, (h x , hz ) = (0.5, 0), magnetic fields [67]. We focus on the longitudinalcase in the main text. We consider the transverse case in Appendix. A. In the longitudinalcase, H and σz(i) commute. Therefore, the BOMI and TOMI of σz(i) are independent oftime. The BOMI and TOMI of σ x and σ y are either constant or periodic functions intime, depending on the location of the operator and the boundary conditions. The BOMI
and TOMI of U(t) is always a periodic function in time, but the amplitude depends on
the boundary conditions.
2. Chaotic phase: For the parameters (h x , hz ) = (−1.05, 0.5), the system is chaotic.
– Early-time evolution: An effective light cone picture explained in section 3.1 qualitatively describes the early-time evolution of the BOMI and TOMI of the local operators for the full and partial overlap configurations. The value of BOMI in the
disjoint configuration is very small, which implies that little information spreads.
– Late-time evolution: The late-time values for the full and partial overlap configurations are independent of the operators and boundary conditions and become thesame as the unitary operator. The Page curve, explained in 3.2, describes the latetime values of the BOMI for these configurations. The value of the BOMI in the
disjoint configuration is very small since it was initially small.
Although we studied the time evolution of TOMI and BOMI for the full overlap, partial overlap,
and disjoint configurations, we found that even in the full overlap configuration, the time evolution of BOMI and TOMI can capture the dynamical properties of the integrable and chaotic
phases. Therefore, we will explain only the full overlap case in the main text, and the others
will be explain in Appendix. A. In the following subsections, we explain the time evolution of
the BOMI and TOMI in detail, and will give the interpretation on these behaviors in time.

3.1

Early-time evolution

For the three configurations in Figure 2, we divide the input (original) system of size L into
A and Ā of size lA and L − lA, and the output (scattered) system, also of size L, into B and B̄
13

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

Figure 3: Time evolution of the (a,b) BOMI and (c,d) TOMI of the operators U, σ(1)
x ,
(1)
(1)
σ y , and σz in the chaotic chain model for the fully overlapping configuration. The
local operators are located at the left boundary of A and B, and the sizes of A and
B are equal, lA = l B = l. The magnetic field is (h x , hz ) = (−1.05, 0.5) for (a,c) and(h x , hz ) = (0, 2) for (b,d). Dark lines correspond to periodic boundary conditions
and light lines correspond to open boundary conditions. The total number of sites is
L = 10.
of size l B and L − l B . We consider periodic boundary conditions (PBC) and open boundary
conditions (OBC).
In Figure 3, we show the time evolution of the BOMI in panels (a,b) and TOMI in panels(c,d) for the full overlap configuration, as shown in Figure 2(a). The BOMI and TOMI aredefined by (25) and (26). In this configuration, lA = l B = l. The local operators σ(1)a=x, y,z are at
the leftmost site of A and B. In the open spin chain, the local operator site is also the leftmost
site of the system. In panels (a,b), the BOMI always starts at 2l. This is equal to the number of
EPR pairs contained in A ∪ B. In the chaotic phase in panel (a), the BOMI begins to decrease
instantly, but the slope depends on the boundary conditions. This is explained by a light cone
picture. The BOMI then settles on a stable value, described by the Page curve. Both the lightcone and the Page curve will be described in more detail in the next sections. The BOMI for
the integrable phases is shown in panels (b). For longitudinal field, shown in (c), the BOMI
for σz is always constant and the BOMI for U oscillates. For σ x and σ y , the BOMI oscillates
for PBC and is constant for OBC except for l = 1 where it is constant.
The TOMI always starts at zero. In the chaotic phase, shown in (c), the TOMI decreases
and settles on a value depending on l. The late-time value is the same for l and L − l. For thelongitudinal field in (f), the TOMI shows consistent oscillatory behavior below zero or stays
constant at zero.
As in [68], quantum revival occurs in a two-dimensional free field theory on a compact
manifold. Here, quantum revival refers to the BOMI returning to its initial value. The revival
time is proportional to the system size, because the revival in the free field theory follows the14

Random Unitary Dynamics,” Physical Review X, volContents[9] J. Dubail, “Entanglement scaling of operators: a con
a glimpse of simulability of long-time dynamics in 1+
SciPost Phys.
(2023)June 2017.
matical General,
vol. Core50, 6,
p.070
234001,[8] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, “Quan[10] V. Alba, J. Dubail, and M. Medenjak, “Operator Enta
Random Unitary
Dynamics,” Physical Review X, volTime The Case of the Rule 54 Chain,”
Quantum Systems:
Scattered space Scattered interior
250603, “Entanglement
Jun 2019.
[9] p.
J. Dubail,scaling of operators: a con
a glimpse of simulability of long-time dynamics in 1+
matical General, vol. 50, p. 234001, June 2017.
Effective light cone

Original spaceO(x) U † (t)(0.1[10] V. Alba, J. Dubail, and M. Medenjak, “Operator Enta
Quantum Systems: The Case of the Rule 54 Chain,”p. 250603, Jun 2019.

Original interior

Figure 4: Schematic of the effective light cone for a local operator O(x, t). Theinterior of the light cone grows over time. Information in the interior of the lightcone of the original space is communicated through the local operator to the interiorof the light cone of the scattered space. The local operator is approximately identity
in the exterior of the light cone.
relativistic propagation of quasi-particles. The BOMI for PBC in the chaotic phase does not
return to the original one. Therefore, a revival does not occur for PBC in the chaotic phase.

Effective light cone picture
We find the time evolution of the BOMI and TOMI of local operators in the chaotic phasequalitatively follows an effective light cone picture. Let us explain the effective light cone
picture, which well describes the distinctive behaviors of the early-time BOMI and TOMI ofPauli’s spin operators in the chaotic phase of (27). Consider a local operator in the Heisenberg
picture O(x, t). As in Figure 4, we call the original (scattered) subspace in the light coneoriginal (scattered) interior. The interior of effective light cone is the region where O(x, t) has
influence: The local operator makes changes to the entanglement structure of the scatteredsubspace. The exterior of the light cone is approximately equal to a local identity. The sizeof the original and scattered subsystems inside of the light cone, l i , depends on an effective
velocity, veff , determined by the details of the theory. When veff is a constant velocity, l i isdetermined by veff ,l i = 2veff t .
(28)
Here, we divide the original Hilbert space into A and Ā, while we divide the scattered
Hilbert space into B and B̄. In this picture, if Al and Bl (A r and B r ) are in the light cone andA r and B r (Al and Bl ) are not as in (a) of Figure 5((b) of Figure 5), then BOMI and TOMI
decreases. In the early time interval, as in panels (a) and (b) of Figure 5, Al and Bl (A r andB r ) are in the light cone and A r and B r (Al and Bl ) are not, so that IA,B decreases. After enoughtime, as in the panes (c) and (d) of Figure 5, Al , Bl , A r , and Al are in the light cone, so thatthe slope of IA,B flattens. In Figure 6, we show the light cone picture effectively describes thetime evolution of IA,B . For the disjoint configuration, A is almost uncorrelated to B, initially.
Then, IA,B becomes small compared to the other configurations, as seen in Figure 12. In twodimensional holographic CFTs, which are theories that have a gravity dual, the time evolutionof operator BOMI and TOMI is well-described by a very simple explanation, the line tension
picture [23].

3

3
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(a)

(b)

(c)

(d)

Figure 5: Schematic of the effective light cone in the partially overlapping configuration.

3.2

Late-time evolution

Here, we will explain the most important results of the spin system (27) in the chaotic phase.

Chaotic chain
Consider the chaotic phase of (27). The late-time evolution of the BOMI and TOMI of the local
operators is independent of operators and boundary conditions, and is approximately equal tothat of the BOMI and TOMI of the unitary operator. This suggests that while the entanglement
structure of the state may depend on the local operators and boundary conditions for theintermediate time region, the entanglement structure of the late time state may not. In otherwords, the strong scrambling effect of the dynamics washes out the information about the local
operators and boundary conditions, so that |σα(i) =x, y,z 〉 with the various boundary conditions
flows to the typical state, the state which is independent of the entanglement structure of theinitial state. To confirm this suggestion, we will compare the late time behavior of BOMI and
TOMI in the chaotic chain to that following the Page curve. The entanglement entropy for atypical state is expected to follow the Page curve. In this case, the Page curve is a function of
the sizes of input and output (original and scattered) subsystems.
The Page curve
The Page curve in this paper will be the effective description of OEEs for a typical state. The
OEE of the local operator state in (21) for the original subsystem A (scattered subsystem B) is
proportional to the subsystem size. It is expected that if the dynamics has strong scramblingability, then the late-time state is structureless. Therefore, the late-time OEE for a subsystem
V with total size L̂ is expected to be the entanglement entropy for the maximally entangledstate:

0 ≤ L̂ < L , L̂ ,SV =L − α,

2L − L̂ ,16

L̂ = L ,L < L̂ ≤ 2L ,(29)

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

Figure 6: Time evolution of (a) BOMI and (b) TOMI for the local operator σz(i) atdifferent sites in the chaotic chain model for the fully overlapping configuration with
open boundary conditions. The y-axis corresponds to the location of the operator.
Inset shows the same information with (a) BOMI and (b) TOMI on the y-axis. The
subsystems, A and B, are centered at site 6 with size lA = l B = 1 and total size L = 11.
The magnetic field is (h x , hz ) = (−1.05, 0.5).
where L̂ is an even integer. The parameter α depends on the parameters h x and hz . If weapply (29) to (25), we get the late time value:
0 ≤ L̂ < L ,0 ,IA,B =α,2 L̂ − L  ,L̂ = L ,(30)

L < L̂ ≤ 2L .

Since IA,B has to be non-negative, α must also be a non-negative number.
We plot the late-time value of SA,B for the full and partial overlap configurations in Figure
7, where s = 0 corresponds to the full overlap configuration. The OEEs for A and B is given bylA = l and l B = l + s, respectively. Then, the behavior of IA,B following the plot of Figure 7 isconsistent with (30) with L̂ = 2l + s. For the full overlap configuration, L̂ = 2l. Thus, when
L is odd, we are not able to take 2l to be L. For the disjoint configuration, the size of A ∪ B isless than or equal to L since A and B are not overlapping. Therefore, IA,B should vanish at latetime according to (29). However, in the numerical computation, IA,B does not vanish, but is asmall value, β. Thus, β is not described by the Page curve.
Applying (30) to (26), we get the late-time value of IA,B,B̄ in Figure 2(a):IA,B,B̄ =α − 2l ,0 ≤ 2l < L ,2α − 2l ,2l = L ,α + 2(l − L) , L < 2l ≤ 2L ,(31)

where L is an even integer, for simplicity.
Thermodynamic limit
Let us consider the late-time value of BOMI in the full overlap, partial overlap, and disjoint
configurations, and TOMI in thermodynamic limit where we take L, the system size, to be
infinite. By using (29) in the Page curve picture, the late-time value of BOMI in the full overlap,
partial overlap, and disjoint configurations is equal to zero. Then, the late-time value of first17

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

Figure 7: Late time entanglement entropy of the operator U(t) in the chaotic chain
model for the partially overlapping configuration (fully overlapping when s = 0). We
show the entanglement entropy at time t = 1012 for an (a) even system size L = 10
and (b) odd system size L = 11. The size of A is lA = l and B is l B = l + s. The magnetic field is (h x , hz ) = (−1.05, 0.5). We show the entanglement entropy for periodic
boundary conditions and operator U(t) but note that the late time entanglement en(i)(i)
tropy is the same for open boundary conditions and operators σ(i)x , σ y , and σz .
and second terms of (26) is zero, whereas that of the last term is constant, so that the late-timevalue of TOMI is
IA,B,B̄ → −2l .
(32)
The late-time value of TOMI, −2l, is consistent with that of the TOMI of a unitary operator in
two-dimensional holographic CFT [21–23]. The late time values in (31) and (32) show that
finite size effects prevent the dynamics from delocalizing quantum information since the latetime value of TOMI in an infinite space is smaller in magnitute compared to a finite space.
3.2.1

Factorization of the density matrix

Finally, we will interpret the late time behavior of BOMI in terms of the reduced density matrix.
For simplicity, we consider the late-time behavior of IA,B in the full overlap configuration. We
assume lA + l B < L. In the chaotic phase, regardless of the states and boundary conditions,the late time value of IA,B is zero. This suggests that ρA∪B in the late time region can beapproximately factored into ρA and ρB :
ρA∪B ≈ ρA ⊗ ρB .
t≫1

(33)

Thus, the correlation between A and B is lost or weak in the time evolution induced by the
chaotic spin chain.

4Time evolution of BOMI and TOMI in a disordered spin chain(i)(i)
In this section, we study the BOMI and TOMI of U(t), σ(i)x , σ y , and σz in a spin system withdisorder:L
L
X
X
HMBL =
Jσ⃗ (i) · σ
⃗ (i+1) +hi σz(i) ,
(34)
i=1

i=1

(i)

where σ⃗ is the vector of Pauli spin operators and hi is a disorder parameter drawn from a
uniform random distribution, hi ∈ [−w, w]. Hereafter we set J = 1 as the unit of energy. The
18

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Time evolution of (a-c) BOMI and (d-f) TOMI of the operators U(t), σ(1)
x ,
(1)
(1)
σ y , and σz in the disordered spin chain model for the fully overlapping configuration. The local operators are located at the left boundary of A and B and the size ofA and B are equal, lA = l B = l. The disorder strength is (a,d) w = 1, (b,e) w = 3, and(c,f) w = 50. Solid lines show the mean over 20 disorder configurations and error
bars show the min and max. Dark lines correspond to periodic boundary conditions
and light lines correspond to open boundary conditions. The total number of sites is
L = 10.
dynamical features of the disordered spin chain are mostly captured by the BOMI and TOMI
in the full overlap configuration, which we will explain in the following section. We consider
the partial overlap and disjoint configurations in Appendix B.
The time evolution of BOMI and TOMI for small, near critical, and large w, are show in
Figure 8.
• Small w region (w = 1): As in [41, 69–72], (34) with small w ≪ 4 is in the chaotic
phase. Therefore, the late-time value of BOMI and TOMI is independent of operatorsand boundary conditions, and is well-described in terms of the Page curve (See, Figure
8.).
• Near critical region (w = 3): For w = 3, near the critical point,1 the time evolution of
BOMI and TOMI depends on the operators and boundary conditions. In the full overlap
configuration, the late time value of BOMI and TOMI is larger than that in (34) withw = 1. When lA + l B < L, the late behavior of BOMI suggests the dynamics of thedisordered spin chain with w = 3 prevents ρA∪B from factorizing to ρA and ρB .
• Large w region (w = 50): In the large w region (w = 50), BOMI and TOMI of σz (t)are almost time-independent. This is because σz(i) (t) is well approximated by
σz(i) (t) ≈ e iHeff t σz(i) e−iHeff t = σz(i) ,(35)

where Heff is described in the next section. For other operators, we see that the BOMI
and TOMI have oscillating behavior. We explain these oscillations using an effective
Hamiltonian in the next section. Moreover, BOMI and TOMI increases with w for all
operators in the large w region.
Recent progress [60] has shown that in the thermodynamic limit, the system may be in the MBL phase if thestrength of the disorder is w > 18. However, for the size of the system studied in this paper, around w = 3, thebehavior of the physical quantities changes from chaotic to that in MBL. Therefore, we will study the time evolutionof BOMI and TOMI at w = 3.
1

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4.1

Effective theory in the strong disordered phase.

Consider the limit J ≪ w with OBC for simplicity. The energies of (34) to first order in J/w
are
L−1
L
X
X
E≈J
σz(i) σz(i+1) +hi σz(i) ,
(36)
i=1

i=1

which are the same energies as the disordered Ising chain. We consider the effective Hamiltonian
L−1
L
X
X
(i) (i+1)
Heff = Jσz σz+
hi σz(i) .
(37)
i=1

i=1

The effective Hamiltonian is diagonal in the σz(i) basis and the entanglement entropy is exactly
solvable. To demonstrate this, consider the local operator σ(1)x . The operator in the eigenbasis
of Heff is given by
X
(1)
(L)
(1)
(L)
σ(1)
=
|σz(1) . . . σz(L) 〉 e i(Heff [{σz }]−Heff [{τz }])t 〈σz(1) . . . σz(L) |σ(1)xx |τz . . . τz 〉 〈τz . . . τz |{σz ,τz }

=

X




|σz(1) . . . σz(L) 〉 exp 2i t Jσz(1) σz(2) + h1 σz(1) 〈σ̄z(1) . . . σz(L) | ,{σz }where the bar denotes flipping the spin. The density matrix for the dual state to σ(1)x is given
by




1 X
(2)
ρ= L
+ h1 σz(1) − τ(1)exp 2i t J σz(1) σz(2) − τ(1)zz τz
2{σz ,τz }

(L)
(1)
(L)
(1)
(L)
× |σz(1) . . . σz(L) 〉 〈τ(1)z . . . τz |in ⊗ |σ̄z . . . σz 〉 〈τ̄z . . . τz |out .

We see that if both σz(1) and σz(2) are traced out in either the input or output space, then the
exponential becomes unity and the density matrix becomes constant in time. If we trace out
either σz(1) or σz(2) , then we get a cosine dependence on time. As an example, we show themutual information when A and B are the original and scattered spaces that only includes site
1, A = H1in and B = H1out . For subsystems that are not overlapping in the input and output
spaces, we trace out both σz(1) and σz(2) and the exponential becomes unity. Thus, the reduced
density matrix and entanglement entropy is given by
1
1H ,
2 A
1
ρB̄ = L−1 1HB̄ ,2
SA = 1 ,

1
1H ,
2 B
1
ρA∪B̄ = L 1HA ⊗ 1HB̄ ,2
SB = 1 ,SB̄ = (L − 1) ,SA∪B̄ = L ,

ρA =

ρB =

ρĀ =

1
2 L−1

1HĀ ,

1
1HB ⊗ 1HB̄ ,2LSĀ = (L − 1) ,

ρB∪B̄ =

SB∪B̄ = L .

For the overlap, the reduced density matrix and entanglement entropy are given by (1)1 X(1)(1)
ρA∪B =
cos 2t J σz(1) − τ(1)exp 2i th1 σz(1) − τ(1)|σz 〉 〈τ(1)zzz |in ⊗ |σ̄z 〉 〈τ̄z |out ,2 (1) (1)σz ,τzSA∪B = − cos2 (2t J) log2 (cos2 (2t J)) − sin2 (2t J) log2 (sin2 (2t J)) .
Therefore,
IA,B = 2 − SA∪B ,IA,B̄ = 0 ,20

IA,B,B̄ = −SA∪B .

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Time evolution vs. disorder strength of (a-d) BOMI and (e-f) TOMI of theoperator σ(1)x in the disordered spin chain model for the fully overlapping configuration. We use a single disorder configuration to show the evolution over varying
disorder strengths. Dashed lines in (b,d,f) show the BOMI and TOMI using the effective Hamiltonian, corresponding to the J ≪ w regime of the disordered spin chain.
The subsystems, A and B, are centered at site 1 with size lA = l B = 1 and total size
L = 10 using open boundary conditions.
In Figure 9, we show the time evolution of BOMI and TOMI for various disorder strengths.
The BOMI and TOMI are well described by the effective Hamiltonian for w/J > 102 . For
strong disorder, the oscillation of BOMI and TOMI depends only on the local interaction J,
even though the reduced density matrix depends on not only J, but also the disorder, hi . We
can follow a similar procedure to get the density matrix for different operators.

1 X (i) (1)
(L)
(1)
(L)
(1)
(L)
ρ σz(i) = L
σz |σz . . . σz(L) 〉 〈τ(1)z . . . τz |in ⊗ |σz . . . σz 〉 〈τz . . . τz |out ,2{σz ,τz }1 X −i(Heff [{σz }]−Heff [{τz }])t (1)(L)
ρ(U) = Le|σz . . . σz(L) 〉 〈τ(1)z . . . τz |in2{σz ,τz }(L)⊗ |σz(1) . . . σz(L) 〉 〈τ(1)z . . . τz |out .
(1)σ(1)
x and σ y are related by a rotational symmetry of the spins around the ẑ axis. The density

matrix for the σz(i) operators do not have any time dependence and any BOMI or TOMI will
be constant. For the time-evolution operator, we see that if all the σz and τz are the same,then the BOMI and TOMI will be constant. Therefore, we only see oscillations for the fully
overlapping and partially overlapping cases.
Thus, the periodic behavior in time of BOMI and TOMI with large w is due to the offdiagonal part of the density matrix expanded by the eigenstates of σz(i) . This periodic behavior
suggests the state may return back to the initial one and the quantum nature of the state
21

SciPost Phys. Core 6, 070 (2023)

may be preserved by the time evolution induced by the disordered spin chain with strong
disorder. The time evolution of BOMI and TOMI can be periodic in time with the periodπ4J . Thus, the BOMI and TOMI exhibits quantum revival. In 2D non-holographic CFT on
compact space, the time evolution of BOMI will exhibit the quantum revival followed by the
relativistic propagation of quasi-particles. The period of the quantum revival is proportional
to the system size [7, 32, 68, 73, 74]. However, in this spin chain, the period is determined by
the strength of local interaction. Therefore, even if we take L to be infinite, the BOMI and
TOMI may periodically evolve in time. The value of BOMI in Figure 9 is between one and two.
Here, the size of the subsystems, A and B, is one. In the EPR picture, the value of BOMI will
be two or zero. Therefore, we can not explain the time evolution of BOMI in terms of EPR
pairs. This suggests that something beyond the quasi-particle picture is needed to explain theentanglement dynamics in the MBL phase.

5

Discussions and future directions

Discussions
Let us summarize the results, compare with the CFT results [21–23], and comment on a few
future directions.
Summary
We have studied the time evolution of BOMI and TOMI for the unitary time evolution operator
and Pauli’s spin operators in the chaotic chain (27) and MBL chain (34).
We found that:• Chaotic chain: The early-time evolution of TOMI and BOMI is qualitatively described interms of an effective light cone picture. Their late-time evolution is independent of theboundary condition and operators, and their late-time value is quantitatively explainedby the Page curve. For PBC, quantum revival does not occur and the compactness ofspace time prevents the dynamics from delocalizing the quantum information.
• Disordered spin chain: In the weak disordered system (w = 1), the late-time valueof BOMI and TOMI is well-described by the Page curve. When the disorder is large,
BOMI and TOMI in full overlap configuration increases when w increases. In the verystrong disorder system (w = 50), BOMI and TOMI in some of configurations oscillatewith respect to time. This oscillation is well-described by an effective Hamiltonian.
Comparison with CFTs
Here, we compare the late time value of BOMI and TOMI derived by the Page curve with thatof holographic CFTs [21–23], which are theories that have a gravity dual. After taking thethermodynamic limit, the Page curve describes the late time value of BOMI. As in (32), when
we take the thermodynamic limit, the late-time value of TOMI in the chaotic phase of (27) is
I(A, B, B̄) = −2l ,(38)where 2 counts the dimension of the local Hilbert space. In the holographic CFT, the late-timevalue of TOMI is πc 
IA,B,B̄ = −2l log2 e( 6ε ) ,
(39)

22

SciPost Phys. Core 6, 070 (2023)

where c is the central charge, and ε is the scale which characterizes the initial state,
P1
πc
|Ψ〉
∝ ε C e−i t Ea |a〉. Therefore, we expect the chaotic chain with approximately e( 6ε )initialaalocal degrees of freedom to mimic the dynamical properties of holographic CFT.

Future directions
Let us comment on a few future directions:
1 An important future direction is to construct a tractable system with a large number of
local degrees of freedom and strong scrambling ability in order to mimic holographic
CFTs.
2 To compute logarithmic negativity and reflected entropy is also interesting. In holographic CFTs, these values show phase transition-like behavior [20–23]. It would be interesting to see whether logarithmic negativity or reflected entropy in the chaotic chain
also show such a behavior.
3 We numerically found that the late time value of TOMI in the chaotic chain on the compact manifold was non-vanishing. It would be interesting to analytically compute thisin a two-dimensional holographic CFT on a compact manifold to find the finite size correction compared to a non-compact holographic CFT.

Acknowledgments
We would like to thank Shinsei Ryu, Jonah Kundler-Flam, and Mao Tian Tan for useful discussions. M.N. and M.T. thank the Yukawa Institute for Theoretical Physics at Kyoto University
for hospitality during the workshops YITP-T-18-04 “New Frontiers in String Theory 2018” andYITP-T-19-03 “Quantum Information and String Theory 2019”.
Funding information The work of M.T. was partially supported by Grant-in-Aid No.
JP17K17822, No. JP20H05270, No. JP20K03787, and JP21H05185 from JSPS of Japan. The
work of M.N. was partially supported by Grant-in-Aid No. JP19K14724 from JSPS of Japan.

A

The time evolution of TOMI and BOMI in the one-dimensional
Ising model with magnetic field

Here, we will explain the time evolution of BOMI and TOMI in the partial overlap and disjoint
configurations of the one-dimensional Ising model with magnetic field.
In Figure 10, we show the time evolution of TOMI and BOMI in the full overlap configuration in the one-dimensional Ising model (0.5, 0). The parameters (h x , hz ) = (0.5, 0) are also
in the integrable phase. However, the BOMI and TOMI of all operators are neither constantnor periodic functions. Even the late-time evolution of the BOMI and TOMI depends on theoperators and boundary conditions. For transverse field, shown in (b), the dynamics has somescrambling ability and shows characteristics similar to (a), but also has some oscillatory behavior similar to (c). For transverse field in (e), the TOMI shows unpredictable behavior, but
is in general, more negative for PBC.
In Figure 11, we show the time evolution of the BOMI and TOMI for the partial overlap
configuration. In this configuration, the number of sites in B is larger than A by s. The sizeof B is equal to l B = lA + s In Figure 11, we show the time evolution of IA,B and IA,B,B̄ for23

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

Figure 10: Time evolution of the (a) BOMI and (b) TOMI of the operators U, σ(1)
x ,
(1)
σ(1)
,
and
σ
in
the
chaotic
chain
model
for
the
fully
overlapping
configuration.
yz
The local operators are located at the left boundary of A and B, and the sizes of
A and B are equal, lA = l B = l. The magnetic field is (h x , hz ) = (0.5, 0). Dark
lines correspond to periodic boundary conditions and light lines correspond to open
boundary conditions. The total number of sites is L = 10.
lA = 3. The BOMI in the chaotic phase, shown in (a), decreases instantly for PBC but has adelay for OBC and this delay is later for larger s. The BOMI then settles on a value close tozero for l B < L/2. In the integrable phase, the BOMI for transvere field, shown in (b), shows
unpredictable behaviour, but is negative and decreases in general. The BOMI for longitunal
field in (c) shows the same behavior as the full overlap configuration in Figure 3(c). The TOMI
in the partial overlap configuration is similar to the full overlap configuration. For the chaotic
phase, however, we see that the decrease of the TOMI is delayed for OBC with increasing delay
for larger s.
In Figure 12, the BOMI and TOMI for the disjoint configuration is shown. In this figure,lA = l B = l = 3. the distance between the rightmost of A and the leftmost of B is d. The BOMI
and TOMI for both chaotic and integrable phases start at zero. The BOMI in the chaotic phasein (a) shows a small bump at an early time and then returns close to zero. The BOMI in the
integrable phase increases slightly for transverse field (b) and remains zero for longitudinal
field (c). The TOMI for the disjoint configuration is similar to the full and partial overlap
configurations.

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SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Time evolution of the (a-c) BOMI and (d-f) TOMI of the operators U, σ(1)
x ,
(1)
(1)
σ y , and σz in the chaotic chain model for the partially overlapping configuration.
The local operators are located at the left boundary of A and B and the size of B isgreater than the size of A by s, lA +s = l B . The magnetic field is (h x , hz ) = (−1.05, 0.5)
for (a,d), (h x , hz ) = (0.5, 0) for (b,e), and (h x , hz ) = (0, 2) for (c,f). Dark lines correspond to periodic boundary conditions and light lines correspond to open boundary
conditions. The total number of sites is L = 10 and the number of sites in A is lA = 3.
(a)

(b)

(c)

(d)

(e)

(f)

Figure 12: Time evolution of the (a-c) BOMI and (d-f) TOMI of the operators U,
(1)
(1)
σ(1)
x , σ y , and σz in the chaotic chain model for the disjoint configuration. The
local operators are located at the left boundary of A and the left boundary of B is adistance d from the right boundary of A. The magnetic field is (h x , hz ) = (−1.05, 0.5)
for (a,d), (h x , hz ) = (0.5, 0) for (b,e), and (h x , hz ) = (0, 2) for (c,f). Dark lines correspond to periodic boundary conditions and light lines correspond to open boundary
conditions. The total number of sites is L = 10 and the number of sites in A and B
are lA = l B = 3.

25

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 13: Time evolution of (a-c) BOMI and (d-f) TOMI of the operators U, σ(1)
x ,
(1)
σ(1)
,
and
σ
in
the
disordered
spin
chain
model
for
the
partially
overlapping
conyz
figuration. The local operators are located at the left boundary of A and B and thesize of B is greater than the size of A by s, lA + s = l B . The disorder strength is (a,d)
w = 1, (b,e) w = 3, and (c,f) w = 50. Solid lines show the mean over 20 disorderconfigurations and error bars show the min and max. Dark lines correspond to periodic boundary conditions and light lines correspond to open boundary conditions.
The total number of sites is L = 10 and the number of sites in A is lA = 3.

B

The time evolution of BOMI and TOMI in the one-dimensional
disordered Heisenberg model

Here, we will explain the time evolution of BOMI and TOMI in the partial overlap and disjoint
configurations of the one-dimensional disordered Heisenberg model.
• Small w region (w = 1): As in the full overlap configuration, the late time value of
BOMI and TOMI in the partial overlap and disjoint configurations follows the Page’s
curve (See, Figures 13, and 14).
• Near critical region (w = 3): The time evolution of BOMI and TOMI depends on theoperators and boundary conditions. In the partial overlap configurations, the late timevalue of BOMI and TOMI is larger than that in (34) with w = 1. In the disjoint configuration, the late time value of BOMI and TOMI in the chaotic phase (w = 1) is independentof the distance between two intervals, d, but that of BOMI and TOMI near the criticalpoint (w = 3) depends on d. In (34) with OBC, the late time value of BOMI is a monotonically decreasing function of d, but that of TOMI is a monotonically increasing functionof d. The d-dependence of BOMI shows MBL prevents the information from spreading,and that of TOMI shows MBL prevents the information from being delocalized.
• Large w region (w = 50): As in Figure 13, for the partial overlap configuration, thevalue of BOMI and TOMI with the enough strong disorder increases, when w increases.
As in Figure 14, when w increases, the value of the BOMI with the enough strong disorder in the disjoint configuration decreases, but the value of the TOMI decreases. In
the disjoint configuration, the late time value of BOMI and TOMI in the chaotic phase(w = 1) is independent of the distance between two intervals, d, but that of BOMI and
TOMI near the critical point (w = 3) depends on d. In (34) with OBC, the late time valueof BOMI is a monotonically decreasing function of d, but that of TOMI is a monotonically26

SciPost Phys. Core 6, 070 (2023)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14: Time evolution of (a-c) BOMI and (d-f) TOMI of the operators U, σ(1)
x ,
(1)
(1)
σ y , and σz in the disordered spin chain model for the disjoint configuration. The
local operators are located at the left boundary of A and the left boundary of B is adistance d from the right boundary of A. The disorder strength is (a,d) w = 1, (b,e)
w = 3, and (c,f) w = 50. Solid lines show the mean over 20 disorder configurations
and error bars show the min and max. Dark lines correspond to periodic boundary
conditions and light lines correspond to open boundary conditions. The total numberof sites is L = 10 and the number of sites in A and B are lA = l B = 3.
increasing function of d. The d-dependence of BOMI shows MBL prevents the information from spreading, and that of TOMI shows MBL prevents the information from being
delocalized. As in Figure 14, when w increases, the value of the BOMI with w > wcri inthe disjoint configuration decreases, but the value of the TOMI decreases.

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