# Clustering of conditional mutual information and quantum Markov structure
at arbitrary temperatures

Tomotaka Kuwahara1,2 1 Analytical quantum complexity RIKEN Hakubi Research
Team, RIKEN Cluster for Pioneering Research (CPR)/ RIKEN Center for Quantum
Computing (RQC), Wako, Saitama 351-0198, Japan  2 PRESTO, Japan Science and
Technology (JST), Kawaguchi, Saitama 332-0012, Japan  1 Analytical quantum
complexity RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research
(CPR)/ RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
2 PRESTO, Japan Science and Technology (JST), Kawaguchi, Saitama 332-0012,
Japan

###### Abstract

Recent investigations have unveiled exotic quantum phases that elude
characterization by simple bipartite correlation functions. In these phases,
long-range entanglement arising from tripartite correlations plays a central
role. Consequently, the study of multipartite correlations has become a focal
point in modern physics. In these, Conditional Mutual Information (CMI) is one
of the most well-established information-theoretic measures, adept at
encapsulating the essence of various exotic phases, including topologically
ordered ones. Within the realm of quantum many-body physics, it has been a
long-sought goal to establish a quantum analog to the Hammersley-Clifford
theorem that bridges the two concepts of the Gibbs state and the Markov
network. This theorem posits that the correlation length of CMI remains short-
range across all thermal equilibrium quantum phases. In this work, we
demonstrate that CMI exhibits exponential decay concerning distance, with its
correlation length increasing polynomially in relation to the inverse
temperature. While this clustering theorem has previously been established for
high temperatures devoid of thermal phase transitions, it has remained elusive
at low temperatures, where genuine long-range entanglement is corroborated to
exist by the quantum topological order. Our findings unveil that, even at low
temperatures, a broad class of tripartite entanglement cannot manifest in the
long-range regime. To achieve the proof, we establish a comprehensive
formalism for analyzing the locality of effective Hamiltonians on subsystems,
commonly known as the ‘entanglement Hamiltonian’ or ‘Hamiltonian of mean
force.’ As one outcome of our analyses, we enhance the prior clustering
theorem concerning bipartite entanglement. In essence, this means that we
investigate genuine bipartite entanglement that extends beyond the limitations
of the Positive Partial Transpose (PPT) class.

##  I Introduction

In quantum many-body physics, a fundamental challenge is uncovering structures
that apply universally, regardless of specific system details. One of the
simplest ways to characterize this is by examining the correlation function
between two observables, which reveals a distinct short-range behavior in non-
critical phases both at finite [1, 2, 3, 4, 5] and zero temperatures [6, 7, 8,
9]. Recent advances in quantum information science have introduced various
methods for comprehending the complexity of quantum phases of matter from an
information-theoretic perspective [10, 11, 12, 13]. Among these, one
particularly renowned and elegant concept is the area law of quantum
entanglement [14]. While the area law has been a conjecture in high dimensions
at absolute zero temperature [15, 16, 17], it has been rigorously confirmed to
hold at non-zero temperatures [18, 19, 20, 21]. The area law has had a
profound impact on various areas of research. It has dramatically influenced
numerical techniques employing the tensor network formalism [22, 23].
Additionally, it has paved the way for developing efficiency-guaranteed
algorithms that compute physical observables [24, 25]. In more recent
developments, the study of bipartite quantum entanglement between distant
subsystems has emerged as a promising avenue for universally revealing short-
range characteristics, even at thermal critical point [26]. So far, quantum
long-range entanglement is believed to manifest primarily in the form of
tripartite (or higher-order multipartite) correlations at non-zero
temperatures.

Over the past two decades, our understanding of quantum phases has evolved,
revealing that they cannot be fully characterized solely through bipartite
correlation measures. A prominent example of this complexity is found in
topologically ordered phases, which exhibit genuinely multipartite
correlations [27, 28]. Efforts in the field have been directed towards
devising comprehensive information-theoretic measures to capture multipartite
correlations in quantum many-body systems. Thus far, one of the most
established measures for quantifying tripartite correlations is Conditional
Mutual Information (CMI) [29, 30, 31]. The CMI has found versatile
applications, including the definition of topological entanglement entropy
[32, 33, 34]. Recent studies have uncovered additional uses for the CMI, such
as characterizing information scrambling [35, 36, 37, 38], identifying
measurement-induced quantum phase transitions [39, 40, 41], and studying
entanglement in conformal field theory [42, 43, 44, 45], among others. In
different contexts, researchers have extensively investigated the operational
significance of the CMI. Remarkably, they have rigorously clarified the
relationship between the CMI and the error of the recovery map [46, 47, 48,
49, 50, 51].

Figure 1: Illustration of the main problem. We consider a finite-dimensional
lattice system, depicted in 2D. The system is partitioned into three
subsystems: $A$, $B$, and $C$, and we examine the conditional mutual
information (CMI) between $A$ and $C$, conditioned on $B$. When the system
Hamiltonian is classical or commuting, the Hammersley-Clifford theorem
indicates that subsystems $A$ and $C$ are conditionally independent. This
independence implies that the CMI is zero when the distance between regions
$A$ and $C$ exceeds the interaction length. The quantum analog suggests a
rapid decay of the CMI with respect to the distance, as conjectured in Eq.
(7). A positive resolution of this conjecture would imply that generic quantum
Gibbs states form approximate Markov networks, which, in turn, implies the
existence of a local recovery map capable of reconstructing the global Gibbs
state from the reduced density matrices [47]. Our main result offers a partial
solution to this conjecture, encapsulated in the primary statements Eqs. (8)
and (9).

Here, our fundamental question is “Can we establish a universal theorem on the
tripartite correlation based on the CMI?” To provide context for this query,
we draw upon the well-known Hammersley-Clifford theorem [52] within classical
many-body systems, which establishes an equivalence between classical Gibbs
states and the concept of a Markov network (or Markov random field). This
network’s characteristics are a finite correlation length of the CMI. Then,
one might envision a similar relationship in quantum systems in the analogy of
classical Gibbs states. Although it breaks down in a strict sense, conjectures
have emerged suggesting that a modified form of the theorem could hold
approximately, with CMI decaying rapidly as the distance [See Ineq. (7)
below]. To date, mathematical proofs have been restricted in three scenarios:
i) commuting Hamiltonians [53, 54, 55] ii) for 1D Gibbs states at arbitrary
temperatures [56], and iii) for high-dimensional Gibbs states above a certain
temperature threshold [57]. Both of these cases correspond to non-critical
thermal phases. Even in high-temperature regimes, the optimal behavior of the
CMI has still been open in all the dimensions [58]. When temperatures are low,
the structure of entanglement becomes exceptionally complex, exhibiting long-
range entanglement even at $\mathcal{O}(1)$ temperatures [59, 60, 61, 62]. Up
to this point, no theoretical efforts have overcome this challenge, and the
clustering of the CMI remains an inaccessible problem.

The resolution of this conjecture holds substantial significance in various
ways. In the realm of quantum many-body physics, it enables us to impose
stricter constraints on the nature of long-range entanglement that persists at
non-zero temperatures. Additionally, it assures us that the recovery map for
the quantum Gibbs state can be constructed using local quantum channels, even
when dealing with low temperatures. From a practical standpoint, the Markov
property is one of the foundational assumptions when dealing with unknown
probability distributions in classical theory [63, 64]. In the quantum world,
the concept of the Markov property plays a fundamental role in various quantum
technologies. Some notable applications include quantum Hamiltonian learning
[65, 66], efficiency-guaranteed quantum Gibbs sampling [67, 68], quantum
marginal problems [69, 70], and more. The establishment of the quantum version
of the Hammersley-Clifford theorem has long been an ambitious goal in quantum
information science.

In the present paper, we report an unconditional proof of the conjecture
regarding the decay of Conditional Mutual Information (CMI) at arbitrary
temperatures and in arbitrary finite dimensions. At this stage, it is
important to note that our result does not provide a complete resolution of
the quantum version of the Hammersley-Clifford theorem, which will be remarked
after the main theorem. Nevertheless, it furnishes strong evidence that the
Markov property generally holds, even at extremely low temperatures.

##  II Results

###  II.1 Setup and notations

We consider a quantum system on a $D$-dimensional lattice, where $\Lambda$
represents the set of all sites. For any subset $L\subset\Lambda$, let $|L|$
denote the number of sites in $L$. For two disjoint subsets $L$ and
$L^{\prime}$, we define the distance $d_{L,L^{\prime}}$ as the length of the
shortest path connecting $L$ and $L^{\prime}$. We also denote the Hilbert
space dimension of $L$ by $\mathcal{D}_{L}$.

We then define the Hamiltonian as follows:

$\displaystyle
H=\sum_{i,i^{\prime}\in\Lambda}h_{i,i^{\prime}}+\sum_{i\in\Lambda}h_{i},$
$\displaystyle\|h_{i,i^{\prime}}\|\leq\bar{J}_{\ell}:=\bar{J}_{0}e^{-\mu\ell},$
(1)

for $d_{i,i^{\prime}}=\ell$, where the operator $h_{i,i^{\prime}}$ represents
an interaction term acting on the two sites $i$ and $i^{\prime}$. The
condition for $\|h_{i,i^{\prime}}\|$ implies that the interaction is short-
range (or exponentially decaying). Here, $\|\cdots\|$ denotes the operator
norm, which is equal to the largest singular value of the given operators. We
can generalize all the results to generic $k$-local Hamiltonians, i.e.,
$H=\sum_{Z\subset\Lambda:|Z|\leq k}h_{Z}$ where $h_{Z}$ is an interaction term
acting on the subset $Z$ ($|Z|\leq k$). For an arbitrary subset
$L\subseteq\Lambda$, we define the subset Hamiltonian as

$\displaystyle H_{L}=\sum_{i,i^{\prime}\in L}h_{i,i^{\prime}}+\sum_{i\in
L}h_{i}.$ (2)

Throughout this paper, we consider the quantum Gibbs state defined as

$\displaystyle\rho_{\beta}:=\frac{e^{\beta H}}{Z_{\beta}},\quad Z_{\beta}=1,$
(3)

where $Z_{\beta}:={\rm tr}(e^{\beta H})$, and we shift the energy origin so
that $Z_{\beta}=1$. For simplicity, we omit the minus sign, so $e^{-\beta H}$
becomes $e^{\beta H}$ without loss of generality. For a given quantum state
$\rho$, we denote the reduced density matrix on the subregion $L$ by
$\rho_{L}$:

$\displaystyle\rho_{L}={\rm tr}_{L^{\rm c}}(\rho),\quad L^{\rm
c}:=\Lambda\setminus L,$ (4)

where ${\rm tr}_{L^{\rm c}}$ means the partial trace for the Hilbert space on
the complementary region $L^{\rm c}$.

For any three sets $A$, $B$, and $C$ within $\Lambda$, we start by defining
the conditional mutual information (CMI) ${\mathcal{I}}_{\rho}(A:C|B)$ between
$A$ and $C$, conditioned on $B$, for a density matrix $\rho$:

$\displaystyle{\mathcal{I}}_{\rho}(A:C|B)$
$\displaystyle:=S_{\rho}(AB)+S_{\rho}(BC)-S_{\rho}(ABC)-S_{\rho}(B),$ (5)

where, for $\forall L\subseteq\Lambda$, we define $S_{\rho}(L)$ as the von
Neumann entropy of the reduced density matrix $\rho_{L}$. Using the mutual
information ${\mathcal{I}}_{\rho}(A:B):=S_{\rho}(A)+S_{\rho}(B)-S_{\rho}(AB)$,
we can also describe the CMI by

$\displaystyle{\mathcal{I}}_{\rho}(A:C|B)={\mathcal{I}}_{\rho}(A:BC)-{\mathcal{I}}_{\rho}(A:B).$
(6)

The quantum Markov conjecture regarding the CMI is now presented as follows:
For an arbitrary quantum Gibbs state $\rho_{\beta}$, the CMI
${\mathcal{I}}_{\rho_{\beta}}(A:C|B)$, where $\Lambda=A\sqcup B\sqcup C$,
rapidly decays with the distance between $A$ and $C$:

[Conjecture]
$\displaystyle{\mathcal{I}}_{\rho_{\beta}}(A:C|B)\leq{\mathcal{G}}_{\mathcal{I}}(R),$
(7)

with $R=d_{A,C}$, where ${\mathcal{G}}_{\mathcal{I}}(R)$ is a super-
polynomially decaying function depending on $\beta$, $\\{A,B,C\\}$, and the
system details. It is important to note that the condition $\Lambda=A\sqcup
B\sqcup C$ is crucial. If $A\sqcup B\sqcup C\subset\Lambda$, even for
commuting Hamiltonians, there exists a counterexample to (7) at low
temperatures [59, 60].

In accordance with the subsystem-size dependence (e.g., $|A|$ or $|C|$), the
Markov property can be classified into three levels [71, 72]:

  1. 1.

Global Markov Property: This is the strongest level, implying that the CMI
decays even when $A$ and $C$ are macroscopic, i.e.,
$|A|,|C|=\mathcal{O}(|\Lambda|)$.

  2. 2.

Local Markov Property: At this level, a small subsystem size for either $A$ or
$C$ is required, i.e., $\min(|A|,|C|)=\mathcal{O}(1)$.

  3. 3.

Pairwise Markov Property: The weakest level, where both subsystems $A$ and $C$
must be small, i.e., $|A|,|C|=\mathcal{O}(1)$.

In classical theory, provided the probability distribution is strictly
positive, these three concepts are equivalent. However, in quantum scenarios,
the conditions for such equivalence remain unclear. To establish the global
Markov property in a quantum system, the decay rate
${\mathcal{G}}_{\mathcal{I}}(R)$ of the CMI should depend on the subsystem
sizes $|A|$ and $|C|$ at most polynomially.

###  II.2 Exponential clustering of the CMI

The central achievement of this work lies in the unconditional proof of the
conjecture (7), which is summarized in the following statement (see [73,
Supplementary Theorems 4 and 5]):

Theorem. For an arbitrary quantum Gibbs state, the conditional mutual
information is upper-bounded as in (7) with the following expression for
${\mathcal{G}}_{\mathcal{I}}(R)$:

$\displaystyle{\mathcal{G}}_{\mathcal{I}}(R)=\mathcal{D}_{AC}e^{-c_{1}R/\left(\beta^{D+1}\log(R)\right)+c_{2}\log(\beta|AC|)},$
(8)

where $\mathcal{D}_{AC}$ is the Hilbert space dimension on the subsystems
$A\sqcup C$. In particular, for one-dimensional systems, the upper bound is
improved to

$\displaystyle{\mathcal{G}}_{\mathcal{I}}(R)=e^{-c_{3}R/\beta+c_{4}\beta\log(\beta
R)}\quad(D=1).$ (9)

Here, the parameters $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ are
$\mathcal{O}(1)$ constants that depend only on the fundamental parameters of
the system (see [73, Supplementary Table I]). These results ensure the
(almost) exponential decay of the conditional mutual information at arbitrary
temperatures.

We summarize several key points. For $D=1$, the result does not depend on the
subsystem sizes $|A|$ and $|C|$, and hence they can be macroscopically large
as $|A|,|C|=\mathcal{O}(|\Lambda|)$. We thus conclude that the global Markov
property holds in one-dimensional quantum Gibbs states at arbitrary
temperatures. This result significantly improves upon the previous one in Ref.
[56], which relies on the exponential clustering for bipartite correlations
and yielded subexponential decay of
${\mathcal{G}}_{\mathcal{I}}(R)=\exp\left(-e^{-\Omega(\beta)}\sqrt{R}\right)$.
Additionally, for some classes of matrix product density operators (MPDO) with
translation invariance, the exponential decay of the CMI has been proven under
the assumption of the gap of the transfer matrix [74, 75]. Our result leads to
the unconditional proof of the CMI decay of MPDOs which have a quasi-local
parent Hamiltonian [76].

On the other hand, for $D\geq 2$, we have achieved the exponential decay of
the CMI at arbitrary temperatures. However, the current bound is insufficient
to prove the global Markov property due to the growth of the coefficient
$\mathcal{D}_{AC}$, which increases as $e^{\Omega(|A|+|C|)}$. Essentially, if
we compare two large subregions (with sizes on the order of $|\Lambda|$), the
influence of $\mathcal{D}_{AC}$ dominates over the decay factor
$e^{-\Omega\left[R/\log(R)\right]}$. In this sense, the statement (8) implies
the pairwise Markov property at this stage, and the complete solution of the
conjecture (7) is still open in high dimensions.

Finally, we mention the temperature dependence of the CMI. The CMI correlation
length increases at most polynomially as $\mathcal{O}(\beta)$ and
$\mathcal{O}(\beta^{D+1})$ for $D=1$ and $D\geq 2$, respectively. The
different scaling for $\beta$ between 1D and higher dimensions arises from the
use of different analytical techniques for high-dimensional cases (see Sec.
V.2). Our results imply that the correlation length of the CMI is much smaller
than that of the bipartite correlation function, which can be infinitely large
at critical points in high dimensions or at least exponentially increases with
$\beta$ as $e^{\mathcal{O}(\beta)}$ in one dimension [1, 77].

##  III Entanglement clustering beyond the PPT class

Regarding the applications of the main inequalities (8) and (9), they allow us
to address the clustering of genuine bipartite entanglement. Previous work
[26] demonstrated that exponential clustering for bipartite entanglement holds
at arbitrary temperatures. However, a severe drawback still remained that it
could not exclude the possibility of the existence of bound entanglement [78,
79], which has been a primary open problem. More precise, the statement was
limited to the positive-partial-transpose (PPT) relative entanglement [80, 81,
82, 83], which has similar properties to entanglement negativity [84].

To capture bound entanglement, we need to analyze a faithful entanglement
measure, meaning that the measure should be zero if and only if the target
quantum state is separable. We call a bipartite quantum state $\rho_{AB}$
separable if it can be decomposed into a mixed state of product states, i.e.,

$\displaystyle\rho_{AB}=\sum_{s}p_{s}\rho_{A,s}\otimes\rho_{B,s}.$ (10)

Conveniently, the CMI serves as an upper bound for bipartite entanglement in
the form of squashed entanglement, which is a faithful measure [85, 86]. For
an arbitrary quantum state $\rho_{AB}$ composed of systems $A$ and $B$,
squashed entanglement is defined as follows:

$\displaystyle E_{\rm sq}(\rho_{AB})$
$\displaystyle:=\inf_{E}\left\\{\frac{1}{2}{\mathcal{I}}_{\rho_{ABE}}(A:B|E)\biggl{|}{\rm
tr}_{E}(\rho_{ABE})=\rho_{AB}\right\\},$ (11)

where $\inf_{E}$ is taken over all extensions of $\rho_{AB}$ such that ${\rm
tr}_{E}(\rho_{ABE})=\rho_{AB}$. Assuming that inequality (7) holds, we can
derive a clustering theorem for squashed entanglement as follows:

$\displaystyle E_{\rm sq}(\rho_{\beta,AB})$
$\displaystyle\leq\frac{1}{2}{\mathcal{I}}_{\rho_{\beta}}(A:B|C)\leq\frac{1}{2}{\mathcal{G}}_{\mathcal{I}}(R),$
(12)

with $R=d_{A,B}$. In the inequality, the ancilla system $E$ is chosen as the
residual system to $AB$, i.e., $E=C=\Lambda\setminus(AB)$. This aspect enables
us to prove entanglement clustering for genuine bipartite entanglement,
including Entanglement of Formation (EoF).

To capture the broadest class of bipartite entanglement measures, we adopt the
EoF since it provides an upper bound for other entanglement measures [87],
such as the relative entanglement [88], the entanglement cost [89], and the
squashed entanglement [29]. The EoF for an arbitrary bipartite quantum state
$\rho_{AB}$ is defined as follows:

$\displaystyle E_{F}(\rho_{AB})$
$\displaystyle:=\inf_{\\{p_{s},\ket{\psi_{s,AB}}\\}}\sum_{s}p_{s}S_{\ket{\psi_{s,AB}}}(A),$
(13)

where $S_{\ket{\psi_{s,AB}}}(A)$ is the von Neumann entropy for the reduced
density matrix onto the subsystem $A$. The convex roof
$\inf_{\\{p_{s},\ket{\psi_{s,AB}}\\}}$ is taken over all decompositions
$\rho_{AB}=\sum_{s}p_{s}\ket{\psi_{s,AB}}\bra{\psi_{s,AB}}$ with $p_{s}>0$.

In general, we can prove the following statement as a consequence of the main
results (8) and (9) (see [73, Supplementary Corollary 42 and Proposition 43]):

Clustering theorem for the EoF. Let $A$ and $B$ be subsystems that are
separated by a distance $R$. Then, for the reduced density matrix
$\rho_{\beta,AB}$, the entanglement of formation obeys the following
clustering theorem:

$\displaystyle
E_{F}(\rho_{\beta,AB})\leq\mathcal{D}_{AB}e^{-c_{1}^{\prime}R/\left(\beta^{D+1}\log(R)\right)+c_{2}^{\prime}\log(R)},$
(14)

which holds in arbitrary dimensions. In particular, for one-dimensional
systems, we can improve the upper bound to

$\displaystyle E_{F}(\rho_{\beta,AB})\leq
e^{c_{3}^{\prime}\beta\log(\beta)-c_{4}^{\prime}R/\beta^{2}}.$ (15)

Here, $c^{\prime}_{1}$, $c^{\prime}_{2}$, $c^{\prime}_{3}$, and
$c^{\prime}_{4}$ are $\mathcal{O}(1)$ constants.

The general upper bound (14) has the same limitation as (8) in that it is
meaningful only when the subsystem sizes of $A$ and $B$ are much smaller than
the length $R$. The dependence on the Hilbert space dimension
$\mathcal{D}_{AB}$ has been removed in the one-dimensional case. We emphasize
that the inequality (15) provides a non-trivial upper bound even for
thermodynamic limit (i.e., $|A|,|B|\to\infty$). This point is a significant
advantage compared to the similar result in Ref. [26, Theorem 12 therein] for
the PPT relative entanglement, which becomes meaningless in the thermodynamic
limit.

##  IV Quasi-locality of the 1D entanglement Hamiltonian

Analyzing the effective Hamiltonian on subsystems poses the most significant
challenge in our work, which is also known as the entanglement Hamiltonian
[90, 91, 92, 93, 94, 95, 96] or Hamiltonian of mean force [97, 98, 99, 100].
The latter terminology is more commonly used in the fields of statistical
mechanics and non-equilibrium physics. When the subsystem is given by
$L\subset\Lambda$, we define the entanglement Hamiltonian by

$\displaystyle\rho_{\beta,L}=e^{\beta H^{\ast}_{L}},\quad
H^{\ast}_{L}=\frac{1}{\beta}\log(\rho_{L}).$ (16)

The quasi-locality of the entanglement Hamiltonian implies the decay of the
CMI [see Eq. (20) below], while the converse does not hold in general [101].
Hence, it is a stronger concept than the conjecture (7). The entanglement
Hamiltonian of the quantum Gibbs state attracts much attention in modern
quantum technologies [66, 99, 102].

As we will show later, we can only prove the existence of a quasi-local
Hamiltonian $\tilde{H}^{\ast}_{L}$ around the subsystem $L$, which
approximates $\rho_{\beta,L}$ as

$\displaystyle e^{\beta\tilde{H}^{\ast}_{L}}\approx\rho_{\beta,L},$ (17)

but this does not ensure that $\tilde{H}^{\ast}_{L}\approx H^{\ast}_{L}$.
Although this approximation is helpful for proving the decay of the CMI (see
Sec. V.4), it does not provide access to the true entanglement Hamiltonian.
This limitation becomes an obstacle to accessing the true entanglement
Hamiltonian, which prevents practical applications such as developing the
polylog sample complexity of Hamiltonian learning [66]. Currently, there
exists a significant gap between the decay of the CMI and the quasi-locality
of the entanglement Hamiltonian, even at high temperatures. In the following
statement, we can partially bridge this gap in the one-dimensional case (see
[73, Supplementary Theorem 6]).

Figure 2: Illustration of 1D entanglement Hamiltonian. For a concatenenated
region $L\subset\Lambda$, we define the entanglement Hamiltonian
$H^{\ast}_{L}$ by $\beta^{-1}\log(\rho_{\beta,L})$ as in Eq. (16). The
effective interaction terms in $H^{\ast}_{L}$ (i.e., $H^{\ast}_{L}-H_{L}$) are
expected to be quasi-local around the boundary of $L$, with the length scale
of the quasi-locality given by $\xi_{\beta}$. The quasi-locality of the
entanglement Hamiltonian imposes a much stronger structural constraint than
the CMI decay (7). In one-dimensional systems with finite-range interactions,
we rigorously prove the error bound (18), which yields
$\xi_{\beta}=e^{e^{\mathcal{O}(\beta)}}$.

Quasi-locality of entanglement Hamiltonian. Let $\tilde{H}^{\ast}_{L,R}$ be
the approximate entanglement Hamiltonian such that the effective interaction
terms are localized around the boundary within a distance $R$ (see Fig. 2).
Then, if the interaction length is finite, i.e., $\bar{J}_{\ell}=0$ for
$\ell>{\rm const.}$ in (1), the approximation error is given by

$\displaystyle\left\|H^{\ast}_{L}-\tilde{H}^{\ast}_{L,R}\right\|\leq
e^{\xi_{\beta}-c_{5}R/\beta},$ (18)

where $\xi_{\beta}=e^{e^{\mathcal{O}(\beta)}}$ and $c_{5}$ is a constant of
order $\mathcal{O}(1)$.

From this theorem, we can ensure that the non-locality of the effective
interaction terms is limited to a distance of at most
$e^{e^{\mathcal{O}(\beta)}}$, which is much larger than the correlation length
of $e^{\mathcal{O}(\beta)}$ [77]. As a reason, the result (18) replies on
analyses based on imaginary time evolution, while the 1D CMI decay (9) is
based on the quantum belief propagation technique [103, 104] (see Sec. V.2).
The assumption of the finite-range interactions also arises from this
approach; for exponentially decaying interactions, the imaginary-time Lieb-
Robinson bound is known to diverge [105]. We conjecture that the quasi-
locality of the entanglement Hamiltonian follows a similar inequality to that
of the CMI (9).

One straightforward application of this inequality (18) is to establish the
logarithmic sample complexity of Hamiltonian learning with respect to the
system size $|\Lambda|$. In Hamiltonian learning [65, 106, 107], given $N$
copies of the same quantum Gibbs state, we reconstruct the Hamiltonian using
$N$ measurement data sets. Using the inequality (18) along with the method
described in Ref. [66]—which is designed for commuting Hamiltonians—we derive
the following expression for sample complexity (see [73, Supplementary
Corollary 48]):

$\displaystyle N=e^{\xi_{\beta}}(1/\epsilon)^{c_{6}\beta^{2}}\log(|\Lambda|)$
(19)

with $c_{6}=\mathcal{O}(1)$ to ensure the estimation error of
$\epsilon=\max_{Z\subset\Lambda}\left\|h_{Z}-\tilde{h}_{Z}\right\|$, where
$\\{\tilde{h}_{Z}\\}_{Z\subset\Lambda}$ are the reconstructed interaction
terms. This is the first result of achieving logarithmic sample complexity for
1D Hamiltonian learning at arbitrary temperatures. For the Hamiltonian
learning problem, the recent flat polynomial approximation techniques [106,
108] might be more effective than using the quasi-locality of the effective
Hamiltonian. Nevertheless, we emphasize that the quasi-local structure of the
entanglement Hamiltonian reveals more fundamental aspects of quantum many-body
physics beyond its application to Hamiltonian learning.

##  V Proof techniques

###  V.1 Overviews

As outlined in Section IV, our main objective is to demonstrate the quasi-
local nature of the entanglement Hamiltonian as suggested in Eq. (17). Using
the notation introduced in Eq. (16), the CMI is represented as:

$\displaystyle{\mathcal{I}}_{\rho_{\beta}}(A:C|B)={\rm
tr}\left[\rho_{\beta}H_{\rho_{\beta}}(A:C|B)\right],$ $\displaystyle
H_{\rho_{\beta}}(A:C|B)=-\beta\left(H^{\ast}_{AB}+H^{\ast}_{BC}-H-H^{\ast}_{B}\right),$
(20)

where we have set $H^{\ast}_{ABC}=H$.

Our significant technical contribution lies in developing a systematic
methodology to analyze the entanglement Hamiltonian, denoted as
$H^{\ast}_{L}$. For simplicity, we focus on the partial trace over a general
subsystem $L$ and examine $H^{\ast}_{L^{\rm c}}$ instead of $H^{\ast}_{L}$.
Our approach is structured into three steps:

  1. 1.

Reduction to Connected Exponential Operators: We first approximate the reduced
density matrix $\rho_{\beta,L^{\rm c}}$ as an exponential operator:

$\displaystyle\quad e^{\beta H^{\ast}_{L^{\rm
c}}}\approx\mathcal{T}e^{\int_{0}^{\tau}V_{x}dx}e^{\beta
H_{0}}\left(\mathcal{T}e^{\int_{0}^{\tau}V_{x}dx}\right)^{\dagger},$ (21)

where $V_{x}$ ($0\leq x\leq\tau$) is a quasi-local operator around region $L$,
$\mathcal{T}$ denotes the time-ordering operator, and $H_{0}$ is a $k$-local
Hamiltonian that may differ from the original $H$. The precision of this
approximation is dependent upon $\tau$ and the quasi-local nature of $V_{x}$.

  2. 2.

Logarithm of Exponential Operators: We next consider the logarithm of the
approximated exponential operators as in Eq. (21):

$\displaystyle\quad\quad\hat{H}_{\tau}=\log\left[\mathcal{T}e^{\int_{0}^{\tau}V_{x}dx}e^{\beta
H_{0}}\left(\mathcal{T}e^{\int_{0}^{\tau}V_{x}dx}\right)^{\dagger}\right].$
(22)

Our analysis needs to establish that the effective interaction terms
$\hat{H}_{\tau}-H_{0}$ are localized around region $L$. Additionally, we need
to demonstrate that these effective interactions are predominantly determined
by the interaction terms within the neighboring region of $L$

  3. 3.

Linking Approximations to CMI: To derive the CMI decay, we connect the
approximation in Eq. (21) with the quasi-locality of $\hat{H}_{\tau}$. The
challenge lies in that a close approximation between two operators cannot
translate to the closeness of their logarithms. The qualitatively optimal
bound between $H^{\ast}_{L^{\rm c}}$ and $\hat{H}_{\tau}$ is expressed as:

$\displaystyle\left\|H^{\ast}_{L^{\rm
c}}-\hat{H}_{\tau}\right\|\leq\frac{c\delta_{\tau}}{\lambda_{\min}(\rho_{\beta,L^{\rm
c}})}$ (23)

with $c$ a constant of $\mathcal{O}(1)$, where $\delta_{\tau}$ represents the
approximation error for Eq. (21), and $\lambda_{\min}(\rho_{\beta,L^{\rm c}})$
is the minimum eigenvalue of $\rho_{\beta,L^{\rm c}}$. In general,
$\lambda_{\min}(\rho_{\beta,L^{\rm c}})=e^{-\mathcal{O}(|L^{\rm c}|)}$, and
hence we need to make the error $\delta_{\tau}$ exponentially small with the
system size, i.e., $\delta_{\tau}=e^{-\Omega(|\Lambda|)}$. Achieving an
exponentially small error $\delta_{\tau}$ spoils the quasi-locality of the
effective interactions. Hence, we approach the estimation of the CMI without
directly analyzing the true entanglement Hamiltonian.

In the following, we demonstrate how these steps are taken. To explain the
essence of the analyses, we omit most of the detailed calculations. Further
details and specific calculations are reserved for the supplementary materials
(see [73, Supplementary sections III, IV, V, and VI]).

###  V.2 First step

In the initial phase of our methodology, we adapt two distinct formalisms
based on the dimensional characteristics of the system:

  1. (i)

the Belief-Propagation (BP) formalism for one-dimensional systems, as detailed
in [73, Supplementary section III B],

  2. (ii)

the Partial-Trace-Projection (PTP) formalism for higher-dimensional systems,
described in [73, Supplementary section III C].

We begin with the belief-propagation formalism to derive the exponential form
shown in Eq. (21), which is used to approximate the reduced density matrix. We
briefly review the quantum belief propagation [103, 104]. In essence, it
provides a mechanism to connect two exponential operators,
$e^{\beta{\mathcal{A}}}$ and $e^{\beta({\mathcal{A}}+{\mathcal{B}})}$, for any
operators ${\mathcal{A}}$ and ${\mathcal{B}}$, as follows:

$\displaystyle
e^{\beta({\mathcal{A}}+{\mathcal{B}})}=\Phi_{\mathcal{B}}e^{\beta{\mathcal{A}}}\Phi^{\dagger}_{\mathcal{B}}.$
(24)

The operator $\Phi_{\mathcal{B}}$, known as the quantum-belief-propagation
operator, is defined by:

$\displaystyle\Phi_{\mathcal{B}}=\mathcal{T}e^{\int_{0}^{1}\phi_{\tau}d\tau},\quad\phi_{\tau}=\int_{-\infty}^{\infty}f_{\beta}(t){\mathcal{B}}({\mathcal{A}}_{\tau},t)dt,$
$\displaystyle
f_{\beta}(t):=\frac{2}{\beta\pi}\log\left(\frac{e^{\pi|t|/\beta}+1}{e^{\pi|t|/\beta}-1}\right),$
(25)

where
${\mathcal{B}}({\mathcal{A}}_{\tau},t)=e^{i{\mathcal{A}}_{\tau}t}{\mathcal{B}}e^{-i{\mathcal{A}}_{\tau}t}$
and ${\mathcal{A}}_{\tau}={\mathcal{A}}+\tau{\mathcal{B}}$. The filter
function $f_{\beta}(t)$ exhibits a decay of $e^{-\mathcal{O}(|t|/\beta)}$,
emphasizing the dominance of the time integral within $\mathcal{O}(\beta)$.
Specifically, when ${\mathcal{A}}$ represents a short-range interacting
Hamiltonian, the Lieb-Robinson bound [109, 110] can be applied to the time
evolution ${\mathcal{B}}({\mathcal{A}}_{\tau},t)$. By selecting
${\mathcal{B}}$ as a local operator within region $L\subset\Lambda$, we can
accurately approximate ${\mathcal{B}}({\mathcal{A}}_{\tau},t)$ onto the
neighboring region around $L$.

Figure 3: Belief propagation formalism for the entanglement Hamiltonian. By
tracing out the subsystem $L$, we aim to derive the connected exponential form
given in Eq. (21) to approximate the reduced density matrix $\rho_{L^{\rm
c}}$. For this purpose, we apply the quantum belief propagation method to
segregate the boundary interaction term $\partial h_{Y}$ from the quantum
Gibbs state, which is represented as $e^{\beta
H}=e^{\beta(H_{LX}+H_{Y}+\partial h_{Y})}$. In this setup, the quantum-belief-
propagation operator $\Phi_{\partial h_{Y}}$ serves as a bridge linking
$e^{\beta(H_{LX}+H_{Y})}$ to $e^{\beta H}$. By approximating this operator
within the region $XY$ with $\tilde{\Phi}_{XY}$, we can manage the partial
trace ${\rm tr}_{L}(\cdots)$ independently from the quantum belief propagation
operation. This approximation leads to the desired form of Eq. (28).

In what follows, we partition the one-dimensional system into three parts:
$\Lambda=L\cup X\cup Y$, where $L$ is the target region, and $X$ is a
subsystem of length $\ell$ that connects to $L$ (see Fig. 3). We then
decompose the Hamiltonian as $H=H_{LX}+H_{Y}+\partial h_{Y}$, with $\partial
h_{Y}$ representing the boundary interaction between the subsystems $LX$ and
$Y$. Employing the quantum belief propagation described in Eq. (24) with
${\mathcal{A}}=H_{LX}+H_{Y}$ and ${\mathcal{B}}=\partial h_{Y}$, we derive:

$\displaystyle e^{\beta H}=\Phi_{\partial h_{Y}}e^{\beta H_{LX}}\otimes
e^{\beta H_{Y}}\Phi^{\dagger}_{\partial h_{Y}}.$ (26)

Applying the Lieb-Robinson bound to the time evolution described in Eq. (V.2),
we obtain the following approximation (see [73, Supplementary Lemmas 9 and
10]):

$\displaystyle\left\|\Phi_{\partial h_{Y}}^{-1}\tilde{\Phi}_{XY}-1\right\|\leq
e^{c\beta\left\|\partial h_{Y}\right\|-c^{\prime}\ell/\beta},$
$\displaystyle\tilde{\Phi}_{XY}=\mathcal{T}e^{\int_{0}^{1}\tilde{\phi}_{XY,\tau}d\tau},$
(27)

where $c$ and $c^{\prime}$ are constants of $\mathcal{O}(1)$, and
$\tilde{\phi}_{XY,\tau}$ is supported on $XY$. By applying the approximation
$\Phi_{\partial h_{Y}}\approx\tilde{\Phi}_{XY}$ to Eq. (26), we approximate
the reduced density matrix $\rho_{\beta,L^{\rm c}}$ as

$\displaystyle\rho_{\beta,L^{\rm c}}$
$\displaystyle\approx\tilde{\Phi}_{XY}{\rm tr}_{L}\left(e^{\beta
H_{LX}}\right)\otimes e^{\beta H_{Y}}\tilde{\Phi}^{\dagger}_{XY}$
$\displaystyle=\mathcal{T}e^{\int_{0}^{1}\tilde{\phi}_{XY,\tau}d\tau}e^{\beta(\tilde{H}_{X}^{\ast}+H_{Y})}\left(\mathcal{T}e^{\int_{0}^{1}\tilde{\phi}_{XY,\tau}d\tau}\right)^{\dagger},$
(28)

which is the desired form of Eq. (21), where $\tilde{H}_{X}^{\ast}$ is defined
such that ${\rm tr}_{L}\left(e^{\beta
H_{LX}}\right)=e^{\beta\tilde{H}_{X}^{\ast}}$. The approximation error is
dependent on the length $\ell$ of $X$ and aligns with the error bound in Eq.
(V.2).

In the application of the BP formalism to derive the effective Hamiltonian, it
is necessary to select the length $\ell$ to be larger than
$\beta^{2}\left\|\partial h_{Y}\right\|$ to ensure a small approximation error
from the bound (V.2). In the cases of higher dimensions, the norm
$\left\|\partial h_{Y}\right\|$ scales with the surface size of $X$, which is
$\mathcal{O}(\ell^{D-1})$. However, for $D\geq 2$ and $\beta\gg 1$, it becomes
challenging to satisfy the condition
$\ell\gtrsim\beta^{2}\mathcal{O}(\ell^{D-1})$. To address this limitation, we
develop an alternative approach. We introduce an ancillary system $L_{a}$,
which is a copy of the subsystem $L$, and express the partial trace operation
as follows:

$\displaystyle{\rm
tr}_{L}(\rho_{\beta})={\mathcal{D}}_{L}\bra{{\mathcal{P}}_{L}}\rho_{\beta}\otimes\hat{1}_{L_{a}}\ket{{\mathcal{P}}_{L}},$
$\displaystyle\ket{{\mathcal{P}}_{L}}:=\frac{1}{\sqrt{{\mathcal{D}}_{L}}}\sum_{j=1}^{{\mathcal{D}}_{L}}\ket{j_{L}}\otimes\ket{j_{L_{a}}},$
(29)

where $\ket{{\mathcal{P}}_{L}}$ represents a maximally entangled state between
the subsystems $L$ and $L_{a}$. We define the Partial-Trace-Projection (PTP)
operator ${\mathcal{P}}_{L}$ as the projection onto the state
$\ket{{\mathcal{P}}_{L}}$, which results in:

$\displaystyle\rho_{\beta,L^{\rm
c}}\otimes{\mathcal{P}}_{L}={\mathcal{D}}_{L}{\mathcal{P}}_{L}\rho_{\beta}\otimes\hat{1}_{L_{a}}{\mathcal{P}}_{L}.$
(30)

To approximate Eq. (30) in the exponential form as Eq. (21), we define the
approximate PTP operator ${\mathcal{P}}_{L,\tau}$ as:

$\displaystyle{\mathcal{P}}_{L,\tau}:=e^{-\tau{\mathcal{Q}}_{L}},\quad{\mathcal{Q}}_{L}:=1-{\mathcal{P}}_{L}.$
(31)

Using the approximate PTP operator, we obtain the desired exponential form of
the reduced density matrix:

$\displaystyle\rho_{\beta,L^{\rm c}}\otimes{\mathcal{P}}_{L}$
$\displaystyle\approx{\mathcal{D}}_{L}e^{-\tau{\mathcal{Q}}_{L}}e^{\beta
H}e^{-\tau{\mathcal{Q}}_{L}},$ (32)

where the approximation error is proved to be
${\mathcal{D}}_{L}e^{-\mathcal{O}(\tau)}$. Due to the coefficient
${\mathcal{D}}_{L}$, which scales as $e^{\Omega(|L|)}$, it is required to make
$\tau=\Omega(|L|)$. This requirement results in a problematic dependence on
the Hilbert space dimension for the CMI decay as detailed in Eq. (8).

###  V.3 Second step

The logarithm of connected exponential operators, such as
$\log\left(e^{{\mathcal{A}}}e^{{\mathcal{B}}}\right)$, is commonly analyzed
using the Baker-Campbell-Hausdorff formula or the Magnus expansion [111]. If a
small order truncation of the Magnus expansion is employed, it can ensure the
quasi-locality of the effective Hamiltonian $\hat{H}_{\tau}$ in Eq. (22).
However, a significant challenge arises due to the fact that the Magnus
expansion is not absolutely convergent [112, 113, 114].

To establish the quasi-locality of the effective Hamiltonian, we utilize the
following theoretical framework outlined in [73, Supplementary Lemma 18 and
Corollary 19]:

Effective Hamiltonian Theory. Let ${\mathcal{A}}$ and
${\mathcal{B}}_{\tau_{1}}$ ($0\leq\tau_{1}\leq\tau$) be arbitrary operators.
By defining the effective operators $\hat{{\mathcal{A}}}_{\tau}$ as

$\displaystyle\hat{{\mathcal{A}}}_{\tau}:=\log\left[\mathcal{T}e^{\int_{0}^{\tau}{\mathcal{B}}_{\tau_{1}}d\tau_{1}}e^{\beta{\mathcal{A}}}\left(\mathcal{T}e^{\int_{0}^{\tau}{\mathcal{B}}_{\tau_{1}}d\tau_{1}}\right)^{\dagger}\right],$
(33)

we can simplify it to the form:

$\displaystyle\hat{{\mathcal{A}}}_{\tau}=\beta
U_{\tau}({\mathcal{A}}+\hat{{\mathcal{B}}}_{\tau})U_{\tau}^{\dagger}$ (34)

with

$\displaystyle
U_{\tau}:=\mathcal{T}e^{-i\int_{0}^{\tau}{\mathcal{C}}_{\tau_{1}}d\tau_{1}},$
$\displaystyle\hat{{\mathcal{B}}}_{\tau}:=\int_{0}^{\tau}U_{\tau_{1}}^{\dagger}{\mathcal{B}}_{\tau_{1}}U_{\tau_{1}}d\tau_{1},$
$\displaystyle{\mathcal{C}}_{\tau_{1}}:=\frac{1}{\beta}\int_{-\infty}^{\infty}g_{\beta}(t)e^{i{\mathcal{A}}_{\tau_{1}}t}{\mathcal{B}}_{\tau_{1}}e^{-i{\mathcal{A}}_{\tau_{1}}t}dt,$
(35)

where $g_{\beta}(t):=-{\rm sign}(t)(e^{2\pi|t|/\beta}-1)^{-1}$.

This expression significantly simplifies the form of the effective
Hamiltonian. For simplicity, we assume ${\mathcal{B}}_{\tau}={\mathcal{B}}$
without $\tau$ dependence in the following analysis. We then point out that
Eq. (33) originates from the differential equation:

$\displaystyle\frac{d}{d\tau}\hat{{\mathcal{A}}}_{\tau}={\mathcal{B}}-i[{\mathcal{C}}_{\tau},\hat{{\mathcal{A}}}_{\tau}].$
(36)

Assuming the effective operator ${\mathcal{A}}_{\tau}$ satisfies quasi-
locality, the Lieb-Robinson bound can be applied to ensure the quasi-locality
of ${\mathcal{C}}_{\tau}$ in Eq, (V.3), which further implies the quasi-
locality of $d{\mathcal{A}}_{\tau}/d\tau$. Note that $g_{\beta}(t)$
approximately decays as $e^{-\mathcal{O}(|t|/\beta)}$, making the time
integral in the range of $|t|=\mathcal{O}(\beta)$ dominant in the definition
of ${\mathcal{C}}_{\tau}$.

The challenge emerges from the fact that Eq. (36) includes double-bracket
flows, i.e., terms like
$[[{\mathcal{B}},\hat{{\mathcal{A}}}_{\tau}],\hat{{\mathcal{A}}}_{\tau}]$.
According to Ref. [115], the Lieb-Robinson bound for double-bracket flow does
not generally preserve quasi-locality, unlike the standard Heisenberg
equation. To illustrate this, consider:

$\displaystyle-i[{\mathcal{C}}_{\tau},\hat{{\mathcal{A}}}_{\tau}]=\frac{-i}{\beta}\int_{-\infty}^{\infty}g_{\beta}(t)\left[e^{i{\mathcal{A}}_{\tau}t}{\mathcal{B}}e^{-i{\mathcal{A}}_{\tau}t},\hat{{\mathcal{A}}}_{\tau}\right]dt,$
(37)

applying the definition of ${\mathcal{C}}_{\tau}$ from Eq. (V.3). For small
$\delta t\ll 1$, we approximate $g_{\beta}(\delta t)\approx-\beta/(2\pi\delta
t)$, resulting in the double-bracket flow appearing as:

$\displaystyle\frac{-ig_{\beta}(\delta
t)}{\beta}\left[e^{i{\mathcal{A}}_{\tau}\delta
t}{\mathcal{B}}e^{-i{\mathcal{A}}_{\tau}\delta
t},\hat{{\mathcal{A}}}_{\tau}\right]\approx\frac{[[{\mathcal{B}},\hat{{\mathcal{A}}}_{\tau}],\hat{{\mathcal{A}}}_{\tau}]}{2\pi},$
(38)

up to an $\mathcal{O}(\delta t)$ error. This analysis does not necessarily
imply a breakdown in the quasi-locality of the effective Hamiltonian. First,
for sufficiently small $\delta t$, although double-bracket terms appear in the
integral of ${\mathcal{C}}_{\tau}$ in Eq. (V.3), their contribution (i.e., the
integral for $|t|\leq\delta t$) is proportional to $\delta t$. Moreover, it is
observed that the integrand (38) deviates from exact double-bracket flows for
large $t$. By analyzing this deviation more carefully, we can still ensure the
quasi-locality of the effective interaction terms.

In the present analyses, we take ${\mathcal{A}}=H_{0}$ and
${\mathcal{B}}=V_{i_{0}}$ for example, where $H_{0}$ is a short-range
interacting Hamiltonian as specified in Eq. (1), and $V_{i_{0}}$ is a local
operator located at site $i_{0}\in\Lambda$. We focus on estimating the
effective Hamiltonian $\hat{H}_{\tau}=\log(e^{\tau V_{i_{0}}}e^{\beta
H_{0}}e^{\tau V_{i_{0}}})$. Notably, $V_{i_{0}}$ can also be generalized to a
quasi-local operator around any subsystem, as discussed in [73, Supplementary
Sec. V]. A crucial aspect we examine is the quasi-locality of the unitary
operator $U_{\tau}$, particularly through the commutator norm
$\left\|[U_{\tau},u_{i}]\right\|$, where $u_{i}$ is any local unitary operator
on a site $i\in\Lambda$. The dynamic property of this commutator is given by:

$\displaystyle\frac{d}{d\tau}\left\|[U_{\tau},u_{i}]\right\|\leq\left\|[{\mathcal{C}}_{\tau},u_{i}]\right\|.$
(39)

To estimate $\left\|[{\mathcal{C}}_{\tau},u_{i}]\right\|$, we upper-bound the
commutator norm between the time-evolved $V_{i_{0}}$ and $u_{i}$ [73,
Supplementary Lemma 22]:

$\displaystyle\left\|\left[e^{i\hat{H}_{\tau}t}V_{i_{0}}e^{-i\hat{H}_{\tau}t},u_{i}\right]\right\|$
$\displaystyle\leq\left\|[V_{i_{0}},u_{i}(-t)]\right\|$
$\displaystyle\quad+2\left\|V_{i_{0}}\right\|\left(\left\|[U_{\tau},u_{i}]\right\|+\left\|[U_{\tau},u_{i}(-t)]\right\|+\int_{0}^{t}\left\|\left[\hat{V}_{\tau},u_{i}(-t_{1})\right]\right\|dt_{1}\right),$
(40)

where we denote $u_{i}(-t)=e^{-iH_{0}t}u_{i}e^{iH_{0}t}$. In analyzing the
growth of the commutator norm, we employ the upper bound
$\left\|[U_{\tau},u_{i}]\right\|\leq Q(\tau,r)$, where $r=d_{i_{0},i}$, and
$Q(\tau,r)$ is defined as:

$\displaystyle
Q(\tau,r)=e^{\kappa_{0}\tau+\kappa_{1}\tau\log(r+\tau+e)-\kappa_{\beta}r},$
(41)

with $\kappa_{0}$, $\kappa_{1}$, and $\kappa_{\beta}$ being free parameters
chosen afterward.

To ensure consistency between the upper bounds in Eq. (39) and
$\left\|[U_{\tau},u_{i}]\right\|\leq Q(\tau,r)$, we can prove that it is
sufficient to satisfy the condition:

$\displaystyle\left\|[{\mathcal{C}}_{\tau},u_{i}]\right\|\leq\left[\kappa_{0}+\kappa_{1}\log(r+\tau+e)\right]Q(\tau,r).$
(42)

Applying the Lieb-Robinson bound to $u_{i}(-t)$ and utilizing the upper bound
(41) for $\left\|[U_{\tau},u_{i}]\right\|$, we can upper-bound the right-hand
side of the inequality (V.3). After integrating Eq. (V.3) with the filter
function $g_{\beta}(t)$, we further upper-bound
$\left\|[{\mathcal{C}}_{\tau},u_{i}]\right\|$ using the parameters
$\kappa_{0}$, $\kappa_{1}$, and $\kappa_{\beta}$. Under appropriate choices of
parameters such as:

$\displaystyle\kappa_{0}$
$\displaystyle\propto\beta^{D}\left\|V_{i_{0}}\right\|\log(\beta\left\|V_{i_{0}}\right\|),$
$\displaystyle\kappa_{1}$
$\displaystyle\propto\beta^{D}\left\|V_{i_{0}}\right\|,$
$\displaystyle\kappa_{\beta}$ $\displaystyle\propto 1/\beta,$ (43)

we can satisfy the condition (42). This estimation shows that the unitary
operator $U_{\tau}$ maintains quasi-locality around the site $i_{0}$ with
exponentially decaying tails, expressed as
$e^{\mathcal{O}(\beta^{D}\left\|V_{i_{0}}\right\|)-r/\beta}$. The detailed
calculations are complex, and we defer all specifics to the supplementary
materials [73, Supplementary Subtheorem 1]. Particularly for 1D cases, the
dependence of $\kappa_{0}$ and $\kappa_{1}$ on $\beta$ improves to
$\beta^{D-1}$ (i.e., $\beta^{0}$ for $D=1$, as per [73, Supplementary Theorem
3]). We conjecture that this improvement might extend to cases in higher
dimensions.

Figure 4: Quasi-locality of the effective interaction terms. We are examining
the operator logarithm given by $\log\left(e^{\tau V_{i_{0}}}e^{\beta
H}e^{\tau V_{i_{0}}}\right)$ and begin by establishing the quasi-locality of
the unitary operator $U_{\tau}$ in Eq. (V.3), which is instrumental in
defining the effective Hamiltonian presented in Eq. (33). The quasi-locality
of $U_{\tau}$ is evaluated based on the norm
$\left\|\left[U_{\tau},u_{i}\right]\right\|$, where $u_{i}$ represents any
arbitrary unitary operator located on a site $i$ ($d_{i,i_{0}}=r$). However,
confirming the quasi-locality of $U_{\tau}$ alone does not fulfill all our
requirements. We further stipulate that $U_{\tau}$ should primarily be
influenced by the neighboring region around the site $i_{0}$. This requirement
leads us to approximate $U_{\tau}$ by $U_{\tau,i_{0}[r]}$, as expressed in Eq.
(V.3). Here, $U_{\tau,i_{0}[r]}$ is specifically determined by the ball region
centered at the site $i_{0}$ with radius $r$ (inside the dashed circle).

At this stage, we have established only the quasi-locality of the unitary
operator $U_{\tau}$. However, this does not necessarily imply that $U_{\tau}$
is determined solely by the neighboring region around the site $i_{0}$. To
address this mathematically, we consider the subset Hamiltonian
$H_{0,i_{0}[r]}$ on the ball region centered at the site $i_{0}$ with radius
$r$, and define:

$\displaystyle\log(e^{\tau V_{i_{0}}}e^{\beta H_{0,i_{0}[r]}}e^{\tau
V_{i_{0}}})$ $\displaystyle=\beta
U_{\tau,i_{0}[r]}(H_{0,i_{0}[r]}+\hat{V}_{\tau,i_{0}[r]})U_{\tau,i_{0}[r]}^{\dagger},$
(44)

where the unitary $U_{\tau,i_{0}[r]}$ is determined only by the subset
Hamiltonian $H_{0,i_{0}[r]}$. If $U_{\tau}\approx U_{\tau,i_{0}[r]}$, we can
infer that the unitary $U_{\tau}$ is approximately constructed from
$H_{0,i_{0}[r]}$. Utilizing the quasi-locality from Eq. (41), we can also
derive the approximation error between $U_{\tau}$ and $U_{\tau,i_{0}[r]}$,
which qualitatively follows the same upper bound as in Eq. (41) (see [73,
Supplementary Theorem 2]).

###  V.4 Third step

We begin by defining $\tilde{\rho}_{\beta,L^{\rm c}}$ as the approximate
reduced density matrix for $\rho_{\beta,L^{\rm c}}$, following the connected
exponential form presented in Eq. (22). Leveraging the quasi-locality of the
effective Hamiltonian described in Step 2, we can estimate the operator norm
of

$\displaystyle\tilde{H}(A:C|B):=-\beta\bigl{[}$
$\displaystyle\log(\tilde{\rho}_{\beta,AB})+\log(\tilde{\rho}_{\beta,BC})$
$\displaystyle-\log(\tilde{\rho}_{\beta,ABC})-\log(\tilde{\rho}_{\beta,B})\bigr{]}.$
(45)

Employing the quasi-locality function from Eq. (41), we are able to
approximate $\left\|\tilde{H}(A:C|B)\right\|\lesssim Q(\tau,R)$, where
$R=d_{A,C}$ and $\tau$ follows from the approximation in Eq. (21). From the
definition of conditional mutual information (CMI) in Eq. (20), we derive:

$\displaystyle{\mathcal{I}}_{\rho_{\beta}}(A:C|B)-{\rm
tr}\left(\rho_{\beta}\tilde{H}(A:C|B)\right)$
$\displaystyle=-S(\rho_{\beta,AB}\|\tilde{\rho}_{\beta,AB})-S(\rho_{\beta,BC}\|\tilde{\rho}_{\beta,BC})$
$\displaystyle\quad+S(\rho_{\beta,ABC}\|\tilde{\rho}_{\beta,ABC})+S(\rho_{\beta,B}\|\tilde{\rho}_{\beta,B}),$
(46)

where $S(\rho\|\tilde{\rho})$ is the relative entropy, defined as
$S(\rho\|\tilde{\rho}):={\rm
tr}\left[\rho\log(\rho)-\rho\log(\tilde{\rho})\right]$. Consequently, we
establish a simple but loose upper bound:

$\displaystyle{\mathcal{I}}_{\rho_{\beta}}(A:C|B)$
$\displaystyle\leq\left\|\tilde{H}(A:C|B)\right\|+4\max_{L\subseteq\Lambda}\left[S(\rho_{\beta,L^{\rm
c}}\|\tilde{\rho}_{\beta,L^{\rm c}})\right].$ (47)

This approach allows us to demonstrate the decay of CMI without directly
accessing the true entanglement Hamiltonian $H^{\ast}_{L}$, addressing a more
complex challenge described as in Eq. (23).

Despite these advancements, a significant challenge remains. The continuity
inequality does not suffice for estimating $S(\rho_{\beta,L^{\rm
c}}\|\tilde{\rho}_{\beta,L^{\rm c}})$. Utilizing the Alicki-Fannes inequality
[116], we find:

$\displaystyle S(\rho_{\beta,L^{\rm c}}\|\tilde{\rho}_{\beta,L^{\rm
c}})\lesssim|L^{\rm c}|\cdot\left\|\rho_{\beta,L^{\rm
c}}-\tilde{\rho}_{\beta,L^{\rm c}}\right\|_{1}.$ (48)

However, the dependency on the size of $|L^{\rm c}|$ suggests that the simple
inequality (47) is not applicable to the thermodynamic limit as
$|\Lambda|\to\infty$. Notably, our main results, encapsulated in Eqs. (8) and
(9), exclude the $|\Lambda|$ dependency. Since the continuity inequality is
already optimal, we must explore alternative methods to establish the
independence of the relative entropy from $|\Lambda|$. These methods are
elaborated upon in the supplementary materials (see [73, Supplementary Lemma
14 and Corollary 17]).

###  V.5 Analyses of the 1D entanglement Hamiltonian

Finally, we provide a technical overview of accessing the true entanglement
Hamiltonian in a one-dimensional case. The central approach relies on the
continuity of the operator logarithm concerning the relative error, as stated
in [73, Supplementary Theorem 7]:

Continuity of the Operator Logarithm in Terms of the Relative Error: Consider
two density operators $\rho$ and $\sigma$. We define the relative error
$\delta_{\rm R}(\rho,\sigma)$ as:

$\displaystyle\delta_{\rm
R}(\rho,\sigma):=\sup_{\ket{\psi}}\frac{|\bra{\psi}(\rho-\sigma)\ket{\psi}|}{\bra{\psi}\rho\ket{\psi}},$
(49)

where $\sup_{\ket{\psi}}$ is taken over all quantum states. The continuity
inequality for $\log(\rho)-\log(\sigma)$ is then given by:

$\displaystyle\left\|\log(\sigma)-\log(\rho)\right\|\lesssim\delta_{\rm
R}(\rho,\sigma)\log\left[\lambda^{-1}_{\min}(\rho)\right],$ (50)

where $\lambda_{\min}(\rho)$ is the minimum eigenvalue of $\rho$.

With this framework, by analyzing the relative error between $\rho_{\beta,L}$
and its approximation $\tilde{\rho}_{\beta,L}$, we establish the quasi-
locality of the true entanglement Hamiltonian $\log(\rho_{\beta,L})$ compared
to $\log(\tilde{\rho}_{\beta,L})$. Nevertheless, two primary challenges arise:
(i) estimating the minimum eigenvalue of the reduced density matrix, and (ii)
estimating the relative error $\delta_{\rm
R}(\rho_{\beta,L},\tilde{\rho}_{\beta,L})$.

While naively straightforward, the estimation of
$\lambda^{-1}_{\min}(\rho_{\beta,L})$, expected to be proportional to
$\beta|L|$, presents difficulties. This proportionality is established in
commuting cases, as shown in Ref. [66]. Extending this to non-commuting cases
requires strict $k$-locality of the Hamiltonian, utilizing technical methods
from Refs. [117, 65]. Generally, it can be shown (see [73, Supplemental
Proposition 12]):

$\displaystyle\lambda^{-1}_{\min}(\rho_{\beta,L})\leq\beta\bar{J}_{0}|L|+\log(16\bar{J}_{0}|L|\mathcal{D}_{L})$
(51)

for two-body interacting Hamiltonians, where we can choose
$\bar{J}_{0}=3J_{0}+4J_{0}|L^{\rm c}|^{-1}\log(8\mathcal{D}_{L^{\rm c}})+1$
using the Hamiltonian parameter $J_{0}$ in Eq. (1). This approach can be
generalized to any $k$-local Hamiltonians.

To address the second challenge, i.e., the estimation of $\delta_{\rm
R}(\rho_{\beta,L},\tilde{\rho}_{\beta,L})$, we first note that the relative
error for the reduced density matrices is always less than that for the global
ones, i.e., $\delta_{\rm
R}(\rho_{\beta,L},\tilde{\rho}_{\beta,L})\leq\delta_{\rm
R}(\rho_{\beta},\tilde{\rho}_{\beta})$. In one-dimensional cases, this
comparison involves two density matrices derived using the quantum belief
propagation, as specified in Eqs. (26) and (28):

$\displaystyle\rho_{\beta}=\Phi_{\partial h_{Y}}e^{\beta H_{LX}}\otimes
e^{\beta H_{Y}}\Phi^{\dagger}_{\partial h_{Y}},$
$\displaystyle\tilde{\rho}_{\beta}=\Phi_{XY}e^{\beta H_{LX}}\otimes e^{\beta
H_{Y}}\Phi^{\dagger}_{XY}.$ (52)

The upper bound for the relative error can then be derived as:

$\displaystyle\delta_{\rm
R}(\rho_{\beta},\tilde{\rho}_{\beta})\leq\left\|1-\mathcal{W}\mathcal{W}^{\dagger}\right\|,$
(53)

where $H_{0}=H_{LX}+H_{Y}$ and

$\displaystyle\mathcal{W}:=e^{-\beta H_{0}/2}\Phi_{\partial
h_{Y}}^{-1}\Phi_{XY}e^{\beta H_{0}/2}.$ (54)

From the inequality (V.2), we have $\Phi_{\partial h_{Y}}^{-1}\Phi_{XY}\approx
1$. Hence, the primary issue is the amplification by the imaginary time
evolution of $e^{-\beta H_{0}/2}$.

The analysis of this imaginary time evolution reveals that the difficulty
stems from the potential exponential amplification of norms, as expressed by:

$\displaystyle\left\|e^{-\beta H_{0}/2}O_{X}e^{\beta
H_{0}/2}\right\|=e^{\Omega(\beta|X|)+e^{\Omega(\beta)}}\left\|O_{X}\right\|,$
(55)

where $O_{X}$ is an operator supported on $X\subset\Lambda$. On the other
hand, utilizing the Lieb-Robinson bound, we can decompose:

$\displaystyle\Phi_{\partial
h_{Y}}^{-1}\Phi_{XY}=1+\sum_{\ell=1}^{\infty}\Phi_{\ell},$ (56)

where each $\Phi_{\ell}$ is an $\ell$-local operator with a norm that decays
exponentially with $\ell/\beta$. However, the amplification by the imaginary
time evolution for each decomposed term $\Phi_{\ell}$ can be significant,
leading to: $\left\|e^{-\beta H_{0}/2}\Phi_{\ell}e^{\beta
H_{0}/2}\right\|=e^{\Omega(\beta\ell)+e^{\Omega(\beta)}}\left\|\Phi_{\ell}\right\|.$
Such amplification substantially breaks down the approximation
$\mathcal{W}\approx 1$ for large $\beta$. To derive a better bound, we need to
fully utilize the structure of the Hamiltonian in more elaborate ways (see
[73, Supplementary Subtheorem 2]).

##  VI Outlook

In this work, we have addressed the resolution of the conjecture stated in Eq.
(7), proposing that every quantum Gibbs state approximates a Markov network at
any temperature. Our principal findings are encapsulated in the inequalities
presented in Eqs. (8) and (9). Due to the dependency on the Hilbert space
dimension $\mathcal{D}_{AC}$, our analysis only concludes that quantum Gibbs
states constitute approximately pairwise Markov networks, rather than forming
a global Markov network. This study significantly advances the understanding
of the entanglement Hamiltonian by systematically approximating its
properties. As applications of our results, we have identified a clustering
property of genuine bipartite entanglement beyond the PPT class and have
established the logarithmic sample complexity required for learning 1D
Hamiltonians.

Despite these advancements, various open questions remain. One of the most
pressing challenges is improving the subsystem-size dependence (i.e.,
$|A|,|C|$) of the CMI decay to a polynomial form ${\rm poly}(|A|,|C|)$, which
would enable its application to macroscopically large subsystems. The current
analyses are highly intricate, whereas point-by-point calculations have been
optimized. Hence, we believe that further improvement is likely to require a
theoretical leap. This could involve leveraging the quasi-locality of the
Hamiltonians more effectively. One potential approach is to impose additional
constraints on the quantum Gibbs states, such as clustering of bipartite
correlations, as discussed in Ref. [56]. Even under these constraints, the
improvement is still beneficial for applications like quantum Gibbs sampling,
as detailed in Ref. [68].

The second crucial problem we address is identifying the types of long-range
entanglement that robustly persist at finite temperatures. Despite extensive
studies, computing entanglement in quantum Gibbs states remains a significant
challenge, even at the numerical level [118, 59, 119, 120, 121, 122, 123, 124,
125, 126]. Our approaches are twofold. First, we aim to refine the subsystem-
size dependence as described in Eq. (14) for bipartite-entanglement decay in
high dimensions. From the relation (12), this is closely linked to the first
open problem regarding the CMI decay. The second approach considers the
presence of multipartite entanglement. So far, understanding multipartite
entanglement is challenging due to the need for established methods for its
characterization [127, 128, 129, 130, 131, 132]. We still anticipate that
certain types of tripartite entanglement may be absent in generic many-body
systems, as the CMI effectively describes tripartite correlations.

The third open problem focuses on improving the quasi-locality analyses of the
entanglement Hamiltonian. In our studies of the CMI decay, we have only needed
to address the approximate entanglement Hamiltonian in terms of Eq. (17).
However, more than this approximation is required in order to access the true
entanglement Hamiltonian. In the 1D case, this issue is partially resolved, as
demonstrated in Eq. (18), although the temperature dependence remains doubly
exponential. By improving the temperature dependence to the polynomial forms,
we can access the 1D entanglement Hamiltonian at the ground state, where
$\beta=\log(n)$ is taken for approximating the Gibbs state to the ground
state. This may further lead to the solution of the Li-Haldane-Poilblanc
conjecture [90, 133, 134] in a generic setup beyond the CFT class [96, 135].
Our present approach relies on a new continuity inequality for the operator
logarithm, indicated by Eq. (50), suggesting that a small relative error
between two operators implies their logarithms are similarly close. Future
improvements could involve developing a better bound by introducing an
alternative measure of operator distance, distinct from the relative error.

It is also intriguing to explore whether the current analyses may be helpful
in proving the CMI decay or the quasi-locality of the entanglement Hamiltonian
in high-temperature regimes, which has turned out to be still open. By
applying the results to high-temperature cases ($\beta\ll 1$) and finite-range
interactions [$\mu\to\infty$ in Eq. (1)], we can demonstrate the CMI decay in
the form of $e^{-\Omega(R/\beta)}$. Interestingly, this decay rate is
substantially faster than the correlation decay derived from the cluster
expansion, i.e., $(\beta/\beta_{c})^{\Omega(R)}=e^{-\Omega(R)\log(1/\beta)}$
with $\beta_{c}$ as a temperature threshold. This finding indicates that the
cluster expansion technique cannot adequately capture the decay of the CMI. A
similar point has been highlighted in recent results by Gondolf et al. [136],
where the authors demonstrated a super-exponential decay of the conditional
information based on the Belavkin-Staszewski relative entropy [137], an
alternative metric for characterizing conditional independence in quantum
Gibbs states. High-temperature quantum Gibbs states have traditionally been
believed to possess only trivial structure due to the convergence of cluster
expansions [4, 138, 139]. Therefore, it is highly counter-intuitive that
quantum Gibbs states exhibit unexpected behaviors even at high temperatures.

Finally, the primary method used to analyze the behavior of exponential
operators extends beyond studying the quantum Gibbs states. This technique is
particularly valuable for examining the effective Hamiltonians in systems
undergoing open quantum dynamics, which involve interactions with the
environment, as also reviewed in [97]. A thorough analysis of quantum effects
arising from environmental interactions is essential for a better
understanding of quantum thermalization processes as well as for
characterizing mixed quantum phases [140, 141, 142]. Our current work will
pave the way for new analytical approaches to tackle these complex issues.

##  VII Acknowledgment

T. K. acknowledges the Hakubi projects of RIKEN. T. K. was supported by JST
PRESTO (Grant No. JPMJPR2116), ERATO (Grant No. JPMJER2302), and JSPS Grants-
in-Aid for Scientific Research (No. JP23H01099, JP24H00071), Japan. We are
grateful to Angela Capel, Chi-Fang Chen, Shang Cheng, Kim Donghoon, Isaac Kim,
Kohtaro Kato, Quynh The Nguyen, Shengqi Sang, Samuel Scalet, and Chao Yin for
helpful discussions and comments on related topics.

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Supplementary Materials for “Clustering of conditional mutual information and
quantum Markov structure at arbitrary temperatures”
Tomotaka Kuwahara1,2

###### Contents

  1. I Introduction
  2. II Results
    1. II.1 Setup and notations
    2. II.2 Exponential clustering of the CMI
  3. III Entanglement clustering beyond the PPT class
  4. IV Quasi-locality of the 1D entanglement Hamiltonian
  5. V Proof techniques
    1. V.1 Overviews
    2. V.2 First step
    3. V.3 Second step
    4. V.4 Third step
    5. V.5 Analyses of the 1D entanglement Hamiltonian
  6. VI Outlook
  7. VII Acknowledgment
  8. S.I Precise system setup and notations
    1. S.I.1 Convenient lemma for interactions
    2. S.I.2 Conditional mutual information and Quantum Markov structure
      1. S.I.2.1 Relation to entanglement clustering
    3. S.I.3 Upper bound on the norm of commutators
    4. S.I.4 Lieb-Robinson bound and locality analyses
    5. S.I.5 Commutator analysis of different quasi-local operators
    6. S.I.6 1D version
  9. S.II Quantum Belief propagation
    1. S.II.0.1 Proof of Lemma 8
    2. S.II.1 Error bound for approximate quantum Belief propagation
  10. S.III Effective Hamiltonian theory on subsystem
    1. S.III.1 Norm of the entanglement Hamiltonian
      1. S.III.1.1 Proof of Propostion 12
    2. S.III.2 Belief propagation formalism
      1. S.III.2.1 Proof of Lemma 14
    3. S.III.3 Partial-trace-projection formalism
      1. S.III.3.1 Proof of Lemma 16
  11. S.IV Connection of exponential operators
    1. S.IV.0.1 Proof of Lemma 18
    2. S.IV.1 Refined error bound for approximate quantum Belief propagation
      1. S.IV.1.1 Proof of Proposition 20
      2. S.IV.1.2 Proof of Lemma 21
  12. S.V Quasi-locality of the entanglement Hamiltonian
    1. S.V.1 Proof
      1. S.V.1.1 Completing the proof
    2. S.V.2 Proof of Proposition 23
    3. S.V.3 Proof of Proposition 24
      1. S.V.3.1 Preliminary lemmas
      2. S.V.3.2 Proof of Lemma 27
      3. S.V.3.3 Proof of Proposition 24
    4. S.V.4 Proof of Proposition 25
  13. S.VI Refined locality estimation for entanglement Hamiltonian
    1. S.VI.1 Proof of Theorem 2
      1. S.VI.1.1 Completing the proof
    2. S.VI.2 Proof of Proposition 30
    3. S.VI.3 Proof of Proposition 32
      1. S.VI.3.1 Proof of Lemma 35
  14. S.VII Improved bound for 1D cases
    1. S.VII.1 Proof of Theorem 3
    2. S.VII.2 Proof of Proposition 36
  15. S.VIII Clustering theorem for conditional mutual information at arbitrary temperatures
    1. S.VIII.1 high-dimensional systems
      1. S.VIII.1.1 Proof of Theorem 4
    2. S.VIII.2 One-dimensional case: improved bound
      1. S.VIII.2.1 Proof of Theorem 5
  16. S.IX Relation to other information measures
    1. S.IX.1 Clustering of Entanglement of Formation (EoF) at arbitrary temperatures
      1. S.IX.1.1 Proof of Proposition 43
    2. S.IX.2 Clustering of information distribution: strong 1D thermal area law
  17. S.X Quasi-locality of the true entanglement Hamiltonian
    1. S.X.1 1D Hamiltonian learning with log sample complexity
    2. S.X.2 Contiuity inequality for logarithmic operators
    3. S.X.3 Proof of Theorem 6
    4. S.X.4 Proof of Subtheorem 2: estimation of the relative error
    5. S.X.5 Proof of Proposition 51
      1. S.X.5.1 Preliminaries
      2. S.X.5.2 Proof of Propostion 51
  18. A Another proof for Lemma 18
  19. B Exponential $\tau\left\|V\right\|$ dependence
  20. C Continuity bound on the logarithm of operators based on relative error
    1. C.1 Setup and Main result
    2. C.2 Proof of Theorem 7
      1. C.2.1 Proof of Lemma 55

##  S.I Precise system setup and notations

Consider a quantum system on a $D$-dimensional lattice with $n$ sites with
$\Lambda$ representing the set of all sites. For any arbitrary partial set
$X\subseteq\Lambda$, we denote the cardinality (number of sites in $X$) as
$|X|$. The complementary subset of $X$ is denoted by $X^{\rm
c}:=\Lambda\setminus X$. We define $\mathcal{D}_{X}$ as the dimension of the
Hilbert space on $X$. Also, we often denote $X\cup Y$ by $XY$ for simplicity.
For subsets $X$ and $Y$ of $\Lambda$, the distance $d_{X,Y}$ is defined as the
shortest path length on the graph connecting $X$ and $Y$, with $d_{X,Y}=0$ if
$X\cap Y\neq\emptyset$. When $X$ comprises only one element (i.e.,
$X=\\{i\\}$), we use $d_{i,Y}$ to represent $d_{\\{i\\},Y}$ for simplicity. We
also define ${\rm diam}(X)$ as follows: ${\rm
diam}(X):=1+\max_{i,i^{\prime}\in X}(d_{i,i^{\prime}}).$ The surface subset of
$X$ is denoted by

$\displaystyle\partial X:=\\{i\in X|d_{i,X^{\rm c}}=1\\}.$ (S.1)

Moreover, we define $(\partial X)_{s}$ as follows (see Fig. 5):

$\displaystyle(\partial X)_{s}:=\begin{cases}\\{i\in X|d_{i,\partial
X}=s\\}&\quad\textrm{for}\quad s\leq 0,\\\ \\{i\in X^{\rm c}|d_{i,\partial
X}=s\\}&\quad\textrm{for}\quad s>0.\end{cases}$ (S.2)

where $(\partial X)_{0}=\partial X$ and we have

$\displaystyle X=\bigcup_{s=0}^{\infty}(\partial
X)_{-s},\quad\Lambda=\bigcup_{s=-\infty}^{\infty}(\partial X)_{s}.$ (S.3)

Figure 5: Schematic picture to describe the definition of $(\partial X)_{s}$
for positive and negative $s$.

For a subset $X\subseteq\Lambda$, the extended subset $X[r]$ is defined as

$\displaystyle X[r]:=\\{i\in\Lambda|d_{X,i}\leq r\\},$ (S.4)

where $X[0]=X$, and $r$ is an arbitrary positive number (i.e.,
$r\in\mathbb{R}^{+}$). We introduce a geometric parameter $\gamma$ determined
solely by the lattice structure, satisfying $\gamma\geq 1$ as a constant of
$\mathcal{O}(1)$, which fulfills the inequalities:

$\displaystyle\max_{i\in\Lambda}(|\partial i[r]|\leq\gamma
r^{D-1},\quad\max_{i\in\Lambda}(|i[r]|\leq\gamma r^{D},$ (S.5)

where $r\geq 1$.

We consider a Hamiltonian with short-range (or exponentially decaying)
interactions on an arbitrary finite-dimensional graph:

$\displaystyle H=\sum_{Z}h_{Z},\quad\max_{i\in\Lambda}\sum_{Z:Z\ni
i}\|h_{Z}\|\leq\bar{J}_{0}$ (S.6)

with the interaction decays such that

$\displaystyle
J_{i,i^{\prime}}:=\sum_{Z:Z\ni\\{i,i^{\prime}\\}}\left\|h_{Z}\right\|\leq\bar{J}_{d_{i,i^{\prime}}}:=\bar{J}_{0}e^{-\mu
d_{i,i^{\prime}}},$ (S.7)

where we define $\left\|\cdots\right\|$ as the operator norm, i.e., the
maximum singular value of a target operator. We often utilize the trace norm
$\left\|\cdots\right\|_{1}$ which means $\left\|O\right\|_{1}:={\rm
tr}\left(\sqrt{O^{\dagger}O}\right)$. We note that for $i=i^{\prime}$, the
condition (S.7) reduces to the condition in Eq. (S.6). We denote the subset
Hamiltonian on a region $L$ by $H_{L}$ and all the interaction terms acting on
$L$ by $\widehat{H_{L}}$, i.e.,

$\displaystyle H_{L}:=\sum_{Z:Z\subset
L}h_{Z},\quad\widehat{H_{L}}:=\sum_{Z:Z\cup L\neq\emptyset}h_{Z}=H-H_{L^{\rm
c}}.$ (S.8)

Also, we define the boundary interaction terms on the region $L$ as follows:

$\displaystyle\partial h_{L}\coloneqq H-H_{L}-H_{L^{\rm c}}=\sum_{Z:Z\cap
L\neq\emptyset,\ Z\cap L^{\rm c}\neq\emptyset}h_{Z}\quad{\rm
or}\quad\widehat{H_{L}}=H_{L}+\partial h_{L}$ (S.9)

For an arbitrary operator $O_{1}$, we define the time evolution of $O_{1}$ by
another operator $O_{2}$ as

$\displaystyle O_{1}(O_{2},t):=e^{iO_{2}t}O_{1}e^{-iO_{2}t}.$ (S.10)

For simplicity, we often denote $O_{1}(H,t)$ by $O_{1}(t)$.

We consider the quantum Gibbs state $\rho_{\beta}$ with a fixed inverse
temperature $\beta$:

$\displaystyle\rho_{\beta}:=e^{\beta H}/Z_{\beta},\quad Z_{\beta}={\rm
tr}(e^{\beta H}).$ (S.11)

We also define the reduced density matrix and the corresponding effective
Hamiltonian (a.k.a entanglement Hamilaonian) as

$\displaystyle\rho_{\beta,L}={\rm tr}_{L^{\rm c}}(\rho_{\beta}),\quad
H^{\ast}_{L}=(1/\beta)\log(\rho_{L}),$ (S.12)

where ${\rm tr}_{L^{\rm c}}(\cdots)$ denotes the partial trace operation with
respect to $L^{\rm c}$. We define normalized partial trace $\tilde{{\rm
tr}}_{X}(O)$ as follows:

$\displaystyle\tilde{{\rm tr}}_{X}(O):=\frac{1}{{\rm tr}_{X}(\hat{1})}{\rm
tr}_{X}(O).$ (S.13)

We note that $\tilde{{\rm tr}}_{X}(O)$ is supported on $X^{\rm c}$ and
commutes with arbitrary operators supported on $X$, i.e., $[\tilde{{\rm
tr}}_{X}(O),O_{X}]=0$. Also, the norm $\left\|\tilde{{\rm tr}}_{X}(O)\right\|$
is always smaller than or equal to $\left\|O\right\|$, i.e.,
$\left\|\tilde{{\rm tr}}_{X}(O)\right\|\leq\left\|O\right\|$.

We here introduce the notation $\Theta(x_{1},x_{2},\ldots,x_{s})$ that means a
function with respect to $\\{x_{1},x_{2},\ldots,x_{s}\\}$ as follows:

$\displaystyle\Theta(x_{1},x_{2},\ldots,x_{s})=\sum_{\sigma_{1},\sigma_{2},\ldots,\sigma_{s}=0,1}c_{\sigma_{1},\sigma_{2},\ldots,\sigma_{s}}x_{1}^{\sigma_{1}}x_{2}^{\sigma_{2}}\cdots
x_{s}^{\sigma_{s}}\quad(0<c_{\sigma_{1},\sigma_{2},\ldots,\sigma_{s}}<\infty),$
(S.14)

where the coefficients $\\{c_{\sigma_{1},\sigma_{2},\ldots,\sigma_{s}}\\}$
only depend on the fundamental parameters listed in Table 1. Also, we use the
notation of $\tilde{\mathcal{O}}(x)$ in the following sense:

$\displaystyle\tilde{\mathcal{O}}(x)=\mathcal{O}\left[x\cdot{\rm
polylog}(x)\right].$ (S.15)

Table 1: Fundamental parameters in our statements Definition | Parameters
---|---
Spatial dimension | $D$
Constant for spatial structure [see Ineq. (S.5)] | $\gamma$
Interaction strength and decay rate of short-range interactions [see the definition (S.7)]. |  $\bar{J}_{0}$, $\mu$
Parameters in the Lieb-Robinson bound [see Ineq. (S.49)] |  $v$, $C$

###  S.I.1 Convenient lemma for interactions

######  Lemma 1.

For arbitrary subsets $X$ and $Y$, we obtain the following upper bound:

$\displaystyle\sum_{Z:Z\cap X\neq\emptyset,\ Z\cap
Y\neq\emptyset}\left\|h_{Z}\right\|\leq|X|\cdot|Y|\bar{J}_{d_{X,Y}}.$ (S.16)

In particular, by choosing $Y=X[r]^{\rm c}$, we have

$\displaystyle\sum_{Z:Z\cap X\neq\emptyset,\ Z\cap X[r]^{\rm
c}\neq\emptyset}\left\|h_{Z}\right\|\leq|\partial
X|\mathcal{J}_{r},\quad\mathcal{J}_{r}:=\tilde{J}_{0}e^{-\mu r/2}$ (S.17)

with

$\displaystyle\tilde{J}_{0}:=\frac{\bar{J}_{0}\gamma
e^{\mu}(2/\mu)^{D}D!}{e^{\mu/2}-1},$ (S.18)

where we assume the surface subset $(\partial X)_{-s}$ decreases with $s$.
Note that we adopt the definition of (S.2) for $(\partial X)_{-s}$.

Proof of Lemma 1. We first use

$\displaystyle\sum_{Z:Z\cap X\neq\emptyset,\ Z\cap
Y\neq\emptyset}\left\|h_{Z}\right\|\leq\sum_{i:i\in
X}\sum_{i^{\prime}:i^{\prime}\in
Y}\sum_{Z:Z\ni\\{i,i^{\prime}\\}}\left\|h_{Z}\right\|\leq\sum_{i:i\in
X}\sum_{i^{\prime}:i^{\prime}\in Y}J_{i,i^{\prime}}.$ (S.19)

By applying the condition (S.7) to the above inequality, we obtain the first
main inequality (S.16). We then consider $Y=X[r]^{\rm c}$. We then obtain

$\displaystyle\sum_{i\in X}\sum_{i^{\prime}\in X[r]^{\rm c}}J_{i,i^{\prime}}$
$\displaystyle\leq\bar{J}_{0}\sum_{s=0}^{\infty}\sum_{i\in(\partial
X)_{-s}}\sum_{i^{\prime}\in\Lambda:d_{i,i^{\prime}}>r+s}e^{-\mu
d_{i,i^{\prime}}}.$ (S.20)

using the notation of $(\partial X)_{s}$ in Eq. (S.2). As in Ref. [144, (S.6)
in the supplemental], we can derive

$\displaystyle\max_{i}\sum_{i^{\prime}\in\Lambda:d_{i,i^{\prime}}>r+s}e^{-\mu
d_{i,i^{\prime}}}$
$\displaystyle=\max_{i}\sum_{l=r+s+1}^{\infty}\sum_{i^{\prime}\in\Lambda:d_{i,i^{\prime}}=l}e^{-\mu
l}$ $\displaystyle\leq\gamma\sum_{l=r+s+1}^{\infty}l^{D-1}e^{-\mu
l}\leq\gamma\int_{r+s+1}^{\infty}x^{D-1}e^{-\mu(x-1)}dx$
$\displaystyle\leq\gamma
e^{-\mu(r+s)/2}\int_{0}^{\infty}x^{D-1}e^{-\mu(x-1)/2}dx=\gamma
e^{\mu/2}(2/\mu)^{D}D!e^{-\mu(r+s)/2}$ (S.21)

using the constant $\gamma$, where we use
$\\{i^{\prime}\in\Lambda:d_{i,i^{\prime}}=s\\}=\partial i[s]$ and $|\partial
i[s]|\leq\gamma s^{D-1}$ for $\forall i\in\Lambda$ and $s\geq r+1$. Applying
the inequality (S.I.1) to (S.20), we have

$\displaystyle\sum_{i\in X}\sum_{i^{\prime}\in X[r]^{\rm c}}J_{i,i^{\prime}}$
$\displaystyle\leq\bar{J}_{0}\gamma
e^{\mu/2}(2/\mu)^{D}D!\sum_{s=0}^{\infty}\sum_{i\in(\partial
X)_{-s}}e^{-\mu(r+s)/2}$ $\displaystyle\leq\bar{J}_{0}\gamma
e^{\mu/2}(2/\mu)^{D}D!|\partial X|\frac{1}{1-e^{-\mu/2}}e^{-\mu r/2},$ (S.22)

which gives the main inequality (S.17). This completes the proof of Lemma 1.
$\square$

Remark. We can refine the upper bound (S.17) as

$\displaystyle\sum_{Z:Z\cap X\neq\emptyset,\ Z\cap X[r]^{\rm
c}\neq\emptyset}\left\|h_{Z}\right\|\leq|\partial
X|\bar{J}_{0}\gamma(2D/\mu^{2})^{D}(r+1)^{D-1}e^{-\mu r},$ (S.23)

but we mainly utilize the upper bound (S.17) for simplicity. In the inequality
(S.I.1), we can re-estimate

$\displaystyle\max_{i}\sum_{i^{\prime}\in\Lambda:d_{i,i^{\prime}}>r+s}e^{-\mu
d_{i,i^{\prime}}}\leq\gamma\int_{r+s+1}^{\infty}x^{D-1}e^{-\mu(x-1)}dx\leq\gamma(1/\mu)^{D}(r+s+D)^{D-1}e^{-\mu(r+s)},$
(S.24)

where we use

$\displaystyle\int_{z}^{\infty}x^{m}e^{-\mu
x}dx\leq(z+m)^{m}(1/\mu)^{m+1}e^{-\mu z}.$ (S.25)

Applying the inequality (S.24) to (S.20), we have

$\displaystyle\sum_{i\in X}\sum_{i^{\prime}\in X[r]^{\rm c}}J_{i,i^{\prime}}$
$\displaystyle\leq\bar{J}_{0}\gamma\sum_{s=0}^{\infty}\sum_{i\in(\partial
X)_{-s}}(r+s+D)^{D-1}e^{-\mu(r+s)}\leq\bar{J}_{0}\gamma(1/\mu)^{D}|\partial
X|e^{-\mu r}\sum_{s=0}^{\infty}(r+s+D)^{D-1}e^{-\mu s}$
$\displaystyle\leq\bar{J}_{0}\gamma(1/\mu)^{D}|\partial X|e^{-\mu
r}\int_{r+D}^{\infty}x^{D-1}e^{-\mu(x-1)}dx\leq\bar{J}_{0}\gamma(1/\mu)^{2D}|\partial
X|(r+2D-1)^{D-1}e^{-\mu(r+D-1)}$
$\displaystyle\leq\bar{J}_{0}\gamma(2D/\mu^{2})^{D}|\partial
X|(r+1)^{D-1}e^{-\mu r},$ (S.26)

which gives the improved inequality (S.23).

###  S.I.2 Conditional mutual information and Quantum Markov structure

In this subsection, we introduce the quantum conditional mutual information
${\mathcal{I}}_{\rho_{ABC}}(A:C|B)$ and the related conjecture on it. First of
all, the definition of the conditional mutual information between the subsets
$A$ and $C$ conditioned on the subset $B$ is defined as follows:

$\displaystyle{\mathcal{I}}_{\rho}(A:B|C):=$ $\displaystyle
S_{\rho}(AC)+S_{\rho}(BC)-S_{\rho}(ABC)-S_{\rho}(B),$ (S.27)

where we define $S_{\rho}(L)$ for $\forall L\subseteq\Lambda$ as the Von-
Neumann entropy for the reduced density matrix on the subset $L$:

$\displaystyle S_{\rho}(L):=-{\rm tr}\left[\rho_{L}\log(\rho_{L})\right].$
(S.28)

Note that we have referred to the definition (S.12).

The conditional mutual information is deeply connected to the quantum Markov
property. For an arbitrary density matrix $\rho$, the quantum Markov property
implies the following equation for an arbitrary tripartition of the total
systems ($\Lambda=A\sqcup B\sqcup C$):

$\displaystyle{\mathcal{I}}_{\rho}(A:C|B)=0\quad\textrm{for}\quad d_{A,B}\geq
r_{0},$ (S.29)

where $r_{0}$ is a constant of $\mathcal{O}(1)$. That is, when the subset $A$
(or $C$) is shielded by the subset $B$, the two subsets are conditionally
independent. The quantum Markov property is known to have a proper operational
meaning in terms of the recovery map [31]. The existence of the recovery map
is utilized in improving the entanglement clustering in one-dimensional
systems (see Proposition 43).

An interesting relation between the quantum Markov structure and the quantum
Gibbs state is given by Hammersley-Clifford theorem. It says that for the
classical probability distribution, the quantum Markov structure and the
classical Ising model are equivalent; that is, any probability distribution
with the Markov structure (S.29) is expressed in the form of the Gibbs state
by the classical Ising Hamiltonian. Conversely, any Gibbs state by the
classical Ising Hamiltonian has the Markov structure. On the other hand, it
has been a long-standing open question whether there exists a similar
relationship between the quantum Markov structure and the quantum Gibbs
states.

As a partial solution, when the Hamiltonian is short-range and commuting, the
above Markov property strictly holds for quantum Gibbs states at arbitrary
temperatures [54, 55]. On the other hand, when the Hamiltonian is non-
commuting, the quantum Markov property (S.29) breaks down in the exact sense.
Still, even for non-commuting Hamiltonians, it has been conjectured [68, 56]
that the quantum Markov property generally holds in an approximate sense:

######  Conjecture 1.

For arbitrary quantum Gibbs states, the conditional mutual information
${\mathcal{I}}_{\rho_{\beta}}(A:C|B)$ ($\Lambda=A\sqcup B\sqcup C$) rapidly
decays with the distance between $A$ and $C$:

$\displaystyle{\mathcal{I}}_{\rho_{\beta}}(A:C|B)\leq{\mathcal{G}}_{\mathcal{I}}(R),\quad
R=d_{A,C},$ (S.30)

where ${\mathcal{G}}_{\mathcal{I}}(R)$ is a super-polynomially decaying
function which depend on $\beta$, $\\{A,B,C\\}$ and fundamental parameters as
in Table 1.

This approximate version of the quantum Markov property also possesses a
similar operational meaning to the exact Markov property [47], and it plays a
critical role in preparing the quantum Gibbs states on a quantum computer
[145, 146, 68]. In one-dimensional cases, the conjecture has been proved at
arbitrary $\mathcal{O}(1)$ temperatures [56], where the decay rate is given by
a subexponential form of $e^{e^{-\mathcal{O}(\beta)}\mathcal{O}(R^{1/2})}$. In
dimensions greater than 1, Conjecture 1 is known to be true only in high-
temperature regimes, i.e., above a threshold temperature [57]. The primary
goal of this paper is to prove the conjecture in arbitrary temperature
regimes.

####  S.I.2.1 Relation to entanglement clustering

We also mention a connection between the quantum Markov structure and the bi-
partite entanglement. As has been mentioned in Ref. [26], the solution of the
quantum Markov conjecture immediately includes the exponential clustering of
the quantum squashed entanglement. For an arbitrary quantum state $\rho_{AB}$
that is composed of the systems $A$ and $B$, the squashed entanglement is
defined as follows:

$\displaystyle E_{\rm
sq}(\rho_{AB}):=\inf_{E}\left\\{\frac{1}{2}{\mathcal{I}}_{\rho_{ABE}}(A:B|E)\biggl{|}{\rm
tr}_{E}(\rho_{ABE})=\rho_{AB}\right\\},$ (S.31)

where $\inf_{E}$ is taken over all extensions of $\rho_{AB}$ such that ${\rm
tr}_{E}(\rho_{ABE})=\rho_{AB}$. As one of the convenient properties, the
squashed entanglement satisfies the faithfulness; that is, it is equal to zero
if and only if the quantum state is not entangled [85, 86].

Under the assumption that the inequality (S.30) holds, we can derive the
exponential clustering for the squashed entanglement in the following way:

$\displaystyle E_{\rm sq}(\rho_{\beta,AB})$
$\displaystyle\leq\frac{1}{2}{\mathcal{I}}_{\rho_{\beta}}(A:B|C)$
$\displaystyle\leq\frac{1}{2}{\mathcal{G}}_{\mathcal{I}}(R).$ (S.32)

Now, the ancilla system $E$ is chosen as the residual system to $AB$, i.e.,
$E\to C=\Lambda\setminus(AB)$.

To utilize the quantum squashed entanglement to upper-bound other entanglement
measures, we use the quantity $\delta_{\rho_{AB}}$, which characterizes the
minimum distance between the target state and separable (i.e., non-entangled)
states:

$\displaystyle\delta_{\rho_{AB}}:=\inf_{\sigma_{AB}\in{\rm
SEP}(A:B)}\left\|\rho_{AB}-\sigma_{AB}\right\|_{1},$ (S.33)

where ${\rm SEP}(A:B)$ is a set of quantum states defined on the subsets $A$
and $B$ that have no entanglement. One can derive the following upper bound
for $\delta_{\rho_{AB}}$ in terms of the squashed entanglement as shown in
Ref. [147, Corollary 3.13] (see also Ref. [148]):

$\displaystyle\delta_{\rho_{AB}}\leq
4\min(\mathcal{D}_{A},\mathcal{D}_{B})\sqrt{E_{\rm sq}(\rho_{AB})}\leq
2\min(\mathcal{D}_{A},\mathcal{D}_{B})\sqrt{2{\mathcal{I}}_{\rho_{\beta}}(A:B|E)},$
(S.34)

where $\mathcal{D}_{A}$ and $\mathcal{D}_{B}$ are the dimensions of the
Hilbert spaces of $A$ and $B$, respectively. Therefore, if $E_{\rm
sq}(\rho_{AB})\ll 1/\mathcal{D}_{AB}$, we can conclude that
$\delta_{\rho_{AB}}$ is as small as ${\mathcal{G}}_{\mathcal{I}}(R)$ from
(S.I.2.1). In the case where $|AB|$ is macroscopically large as
$|AB|=\mathcal{O}(|\Lambda|)$, $\mathcal{D}_{AB}$ is exponentially large with
$|\Lambda|$ and hence the inequality (S.I.2.1) cannot be utilized to upper-
bound other entanglement measures such as the relative entanglement entropy
and the EoF. In the case of one-dimensional systems, we can overcome the
problem by employing the prescription introduced in Ref. [26, Theorem 12] (see
also Sec. S.IX).

###  S.I.3 Upper bound on the norm of commutators

In this section, we show several convenient lemmas on the norm of commutators
that are often utilized in the analyses.

######  Lemma 2.

Let $O$ be an arbitrary operator. Also, for an arbitrary unitary operator
$u_{i}$ acting on a site $i\in\Lambda$, we assume the following inequality:

$\displaystyle\sup_{u_{i}}\left\|[O,u_{i}]\right\|\leq\delta_{i},$ (S.35)

where $\sup_{u_{i}}$ is taken for the set of all the unitary operators acting
on the site $i$. Then, we obtain

$\displaystyle\left\|O-\tilde{{\rm
tr}}(O)\right\|\leq\sum_{i\in\Lambda}\delta_{i}.$ (S.36)

Also, for an arbitrary operator $O_{X}$ which is supported on subset
$X\subset\Lambda$, we obtain

$\displaystyle\left\|[O,O_{X}]\right\|\leq 2\left\|O_{X}\right\|\sum_{i\in
X}\delta_{i}.$ (S.37)

Remark. The inequality (S.36) gives upper bound for the norm of
$\left\|O\right\|$ when $O$ is traceless (i.e., ${\rm tr}(O)=0$):

$\displaystyle\left\|O\right\|\leq\sum_{i\in\Lambda}\delta_{i}\quad\textrm{for}\quad{\rm
tr}(O)=0.$ (S.38)

Proof of Lemma 2. For the proof, we consider

$\displaystyle\tilde{{\rm tr}}_{X}(O)=\frac{\hat{1}_{X}\otimes{\rm
tr}_{X}(O)}{{\rm tr}_{X}(\hat{1})},$ (S.39)

where we use the definition (S.13). Note that $\tilde{{\rm tr}}_{X}(O)$ is
supported on $X^{\rm c}$, and hence $[\tilde{{\rm tr}}_{X}(O),O_{X}]=0$. We
now label the sites in $X$ by $X=\\{1,2,\ldots,|X|\\}$ without loss of
generality. Using the discussion in Ref. [110], we have

$\displaystyle\tilde{{\rm tr}}_{X}(O):=\int d\mu(u_{1})d\mu(u_{2})\cdots
d\mu(u_{|X|})(u_{1}u_{2}\cdots u_{|X|})^{\dagger}O(u_{1}u_{2}\cdots u_{|X|}),$
(S.40)

and hence we obtain

$\displaystyle\left\|O-\tilde{{\rm tr}}_{X}(O)\right\|$ $\displaystyle\leq\int
d\mu(u_{1})d\mu(u_{2})\cdots d\mu(u_{|X|})\sum_{i\in
X}\left\|[O,u_{i}]\right\|\leq\sum_{i\in
X}\sup_{u_{i}}\left\|[O,u_{i}]\right\|\leq\sum_{i\in X}\delta_{i},$ (S.41)

where $\mu(u_{i})$ is the Haar measure for the unitary operator on the site
$i\in\Lambda$. Therefore, by choosing $X=\Lambda$, we obtain the first main
inequality (S.36). Also, by using the inequality of

$\displaystyle\left\|[O,O_{X}]\right\|=\left\|[O-\tilde{{\rm
tr}}_{X}(O),O_{X}]\right\|\leq 2\left\|O_{X}\right\|\cdot\left\|O-\tilde{{\rm
tr}}_{X}(O)\right\|,$ (S.42)

we obtain the second main inequality (S.37). This completes the proof
$\square$

######  Lemma 3.

Let $O_{X}$ be an arbitrary operator supported on a subset $X\subset\Lambda$.
If $O_{X}(t)$ satisfies the Lieb-Robinson bound as

$\displaystyle\left\|[O_{X}(t),O_{Y}]\right\|\leq\mathcal{G}(X,Y,t,d_{X,Y})$
(S.43)

with $\mathcal{G}(X,Y,t,d_{X,Y})$ an appropriate function, we have

$\displaystyle\left\|O_{X}(t)-O_{X[r]}^{(t)}\right\|\leq\mathcal{G}(X,X[r]^{\rm
c},t,r+1),\quad O_{X[r]}^{(t)}:=\tilde{{\rm tr}}_{X[r]^{\rm
c}}\left[O_{X}(t)\right].$ (S.44)

Similarly, we also obtain

$\displaystyle\left\|O_{X[r+\delta
r]}^{(t)}-O_{X[r]}^{(t)}\right\|\leq\mathcal{G}(X,X[r+\delta r]\setminus
X[r],t,r+1),$ (S.45)

where the integers $r$ and $\delta r$ can be arbitrarily chosen.

Proof of Lemma 3. The inequality (S.44) is immediately proved by using

$\displaystyle\tilde{{\rm tr}}_{X[r]^{\rm c}}\left[O_{X}(t)\right]=\int
d\mu(U_{X[r]^{\rm c}})U_{X[r]^{\rm c}}^{\dagger}O_{X}(t)U_{X[r]^{\rm c}},$
(S.46)

which yields

$\displaystyle\left\|O_{X}(t)-\tilde{{\rm tr}}_{X[r]^{\rm
c}}\left[O_{X}(t)\right]\right\|\leq\left\|\left[O_{X}(t),U_{X[r]^{\rm
c}}\right]\right\|\leq\mathcal{G}(X,X[r]^{\rm c},t,r)$ (S.47)

from the inequality (S.43) and $d_{X,X[r]^{\rm c}}=r+1$.

We then let $X^{\prime}=X[r+\delta r]\setminus X[r]$. The second inequality
(S.45) is derived in the same way as follows:

$\displaystyle\left\|O_{X[r+\delta r]}^{(t)}-O_{X[r]}^{(t)}\right\|$
$\displaystyle=\left\|\tilde{{\rm tr}}_{X[r+\delta r]^{\rm
c}}\left[O_{X}(t)\right]-\tilde{{\rm tr}}_{X[r]^{\rm
c}}\left[O_{X}(t)\right]\right\|$ $\displaystyle=\left\|\int
d\mu(U_{X[r+\delta r]^{\rm c}})U_{X[r+\delta r]^{\rm
c}}^{\dagger}\left(O_{X}(t)-\int
d\mu(U_{X^{\prime}})U_{X^{\prime}}^{\dagger}O_{X}(t)U_{X^{\prime}}\right)U_{X[r+\delta
r]^{\rm c}}\right\|$ $\displaystyle\leq\left\|O_{X}(t)-\int
d\mu(U_{X^{\prime}})U_{X^{\prime}}^{\dagger}O_{X}(t)U_{X^{\prime}}\right\|$
$\displaystyle\leq\int
d\mu(U_{X^{\prime}})\left\|\left[O_{X}(t),U_{X^{\prime}}\right]\right\|\leq\mathcal{G}(X,X^{\prime},t,r+1).$
(S.48)

This completes the proof. $\square$

###  S.I.4 Lieb-Robinson bound and locality analyses

We here introduce the Lieb-Robinson bound which characterizes quasi-locality
by a time evolution. It plays a central role in our analyses alongside the
quantum belief propagation. One can prove the following statement in general
[109, 8, 149, 9]:

######  Lemma 4 (Lieb-Robinson bound [9]).

For arbitrary operators $O_{X}$ and $O_{Y}$ with unit norm and $d_{X,Y}=R$,
the norm of the commutator $[O_{X}(t),O_{Y}]$ satisfies the following
inequality:

$\displaystyle\left\|[O_{X}(t),O_{Y}]\right\|\leq C\min(|\partial X|,|\partial
Y|)\left(e^{v|t|}-1\right)e^{-\mu R},$ (S.49)

where we have adopted $C$ and $v$ as fundamental parameters in Table 1, but
they can be expressed using the parameters, i.e., $D$, $\gamma$, $\bar{J}_{0}$
and $\mu$.

Remark. Originally derived inequality in Ref. [9] is slightly weaker than
(S.49):

$\displaystyle\left\|[O_{X}(t),O_{Y}]\right\|\leq
C_{0}\min(|X|,|Y|)\left(e^{v|t|}-1\right)e^{-\mu R}.$ (S.50)

Using the inequality (S.37) in Lemma 2, we improve the bound (S.50) with $O\to
O_{X}(t)$, $O_{X}\to O_{Y}$ and $\delta_{i}\to C\left(e^{v|t|}-1\right)e^{-\mu
d_{i,X}}$, we have

$\displaystyle\left\|[O_{X}(t),O_{Y}]\right\|\leq
C\left(e^{v|t|}-1\right)\sum_{i\in Y}e^{-\mu d_{i,X}},$ (S.51)

and

$\displaystyle\sum_{i\in Y}e^{-\mu
d_{i,X}}\leq\sum_{s=0}^{\infty}\sum_{i\in(\partial
Y)_{-s}}e^{-\mu(d_{X,Y}+s)}\leq|\partial Y|e^{-\mu
R}\sum_{s=0}^{\infty}e^{-\mu s}=\frac{|\partial Y|e^{-\mu R}}{1-e^{-\mu}}.$
(S.52)

Combining (S.51) and (S.52) yields the inequality (S.49) by letting
$C=C_{0}/(1-e^{-\mu})$.

We define $O_{X}(H,t,\tilde{X})$ as an approximation of $O_{X}(H,t)$ on the
subset $\tilde{X}$,

$\displaystyle O_{X}(H,t,\tilde{X}):=\frac{1}{{\rm tr}_{\tilde{X}^{\rm
c}}(\hat{1})}{\rm tr}_{\tilde{X}^{\rm
c}}\left[O_{X}(H,t)\right]\otimes\hat{1}_{\tilde{X}^{\rm c}},$ (S.53)

where ${\rm tr}_{\tilde{X}^{\rm c}}(\cdots)$ is the partial trace with respect
to the subset $\tilde{X}^{\rm c}$. We then obtain from Lemma 3

$\displaystyle\left\|O_{X}(H,t)-O_{X}(H,t,X[r])\right\|\leq
C\|O_{X}\|\cdot|\partial X|\left(e^{v|t|}-1\right)e^{-\mu r},$ (S.54)

where the notation $X[r]$ has been defined in Eq. (S.4).

We here consider the Lieb-Robinson bound for short-range interacting (or with
exponentially decaying interactions) systems, but the Lieb-Robinson bound
holds for more general classes of Hamiltonians with power-law decaying (i.e.,
long-range) interactions [150, 151, 152, 153]. As long as the Lieb-Robinson
bound holds, we expect that our main results will be generalized to the long-
range interacting systems.

###  S.I.5 Commutator analysis of different quasi-local operators

We here consider the following problem. Let us consider two quasi-local
operators $O_{1}$ and $O_{2}$. Then, how can we estimate the quasi-locality of
the commutator $\left[O_{1},O_{2}\right]$? The locality analysis for different
quasi-local operators plays a crucial role in our analyses. Similar techniques
have been considered in the context of the Lieb-Robinson bound [152, 144]. The
following lemma characterizes the commutator norm of
$\int_{-\infty}^{-\infty}f(t)\left\|\left[O,u_{i}(t)\right]\right\|dt$ (see
also Fig. 6), where $u_{i}(t)$ is characterized by the time evolution for a
unitary operator at the site $i\in\Lambda$ and $f(t)$ is a filter function.
This also characterizes the quasi-locality of the operator
$\int_{-\infty}^{-\infty}f(t)O(-t)dt$.

Figure 6: Schematic picture of the setup of Lemma 5. We consider two quasi-
local operators; the former one is $O$ that satisfies the quasi-locality
described by (S.55), while the latter one is given by $u_{i}(t)$ [or
$u_{i}(\lambda t)$ with $0\leq\lambda\leq 1$]. The commutator between $O$ and
$\int_{-\infty}^{\infty}f(t)u_{i}(\lambda t)dt$ is upper-bounded as in (S.56).
In the proof, we decompose the operator $u_{i}(\lambda t)$ as in Eq. (S.59)
and take the commutator with each of the decomposed terms as in (S.I.5).

######  Lemma 5.

Let $O$ be a quasi-local operator around a subset $\mathfrak{L}$ in the sense
that

$\displaystyle\left\|O,u_{i}\right\|\leq{\mathcal{F}}(\ell),\quad\ell=d_{i,\mathfrak{L}}$
(S.55)

for an arbitrary unitary operator on a site $i$, where ${\mathcal{F}}(\ell)$
is a monotonically decaying function. Then, under the Lieb-Robinson bound in
short-range interacting systems, we obtain

$\displaystyle\int_{-\infty}^{\infty}f(t)\left\|[O,u_{i}(\lambda
t)]\right\|dt\leq
2f_{1}\mathcal{F}(\ell)+2\gamma(\Delta\ell)^{D}\sum_{s=1}^{\infty}s^{D}\mathcal{F}(\ell-s\Delta\ell)\left[Cf_{2}ve^{-\mu
s\Delta\ell/2}+f_{t_{0}}(s)\right]$ (S.56)

for $0\leq\lambda\leq 1$, where we define

$\displaystyle f_{t_{0}}(s):=\int_{|t|\geq t_{0}}f(t)dt,\quad
f_{1}:=\int_{-\infty}^{\infty}f(t)dt,\quad f_{2}:=\int_{|t|\leq
t_{0}}|t|f(t)dt\leq\int_{-\infty}^{\infty}|t|f(t)dt,$ (S.57)

where $t_{0}=\max(0,\mu s\Delta\ell/(2v)-\mu(\Delta\ell-1)/v)$, and $f(t)$ is
an arbitrary positive function. In particular, by taking $\Delta\ell=1$, the
inequality (S.56) reduces to

$\displaystyle\int_{-\infty}^{\infty}f(t)\left\|[O,u_{i}(\lambda
t)]\right\|dt\leq
2f_{1}\mathcal{F}(\ell)+2\gamma\sum_{s=1}^{\infty}s^{D}\mathcal{F}(\ell-s)\left[Cf_{2}ve^{-\mu
s/2}+f_{t_{0}}(s)\right],$ (S.58)

where $t_{0}=\mu s/(2v)$.

Remark. The lemma is utilized in Propositions 23 and 25 for example, where we
let the filter function as the function $g_{\beta}(t)$ in the quantum belief
propagation (S.89), which plays an essential role in Lemma 18 for connection
of exponential operators.

Proof of Lemma 5. We first decompose

$\displaystyle u_{i}(\lambda
t)=u_{i}^{(t)}+\sum_{s=1}^{\infty}\left(u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right),\quad
u_{i[\ell_{s}]}^{(t)}:=\tilde{{\rm tr}}_{i[\ell_{s}]^{\rm
c}}\left[u_{i}(\lambda t)\right],$ (S.59)

where we use $u_{i[\infty]}^{(t)}=u_{i}(\lambda t)$ and define $\ell_{s}$ as

$\displaystyle\ell_{s}:=s\Delta\ell.$ (S.60)

Using the inequality (S.37) in Lemma 2, we obtain

$\displaystyle\left\|\left[O,u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right]\right\|$
$\displaystyle\leq
2\left\|u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right\|\sum_{i^{\prime}\in
i[\ell_{s}]}\sup_{u_{i^{\prime}}}\left\|[O,u_{i^{\prime}}]\right\|$
$\displaystyle\leq
2\left\|u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right\|\cdot|i[\ell_{s}]|\mathcal{F}(\ell-\ell_{s})$
$\displaystyle\leq
2\gamma\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})\left\|u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right\|,$
(S.61)

where we use $d_{i^{\prime},\mathfrak{L}}\geq
d_{i,\mathfrak{L}}-\ell_{s}=\ell-\ell_{s}$ for $i^{\prime}\in i[\ell_{s}]$ and
$d_{i,\mathfrak{L}}=\ell$. In the same way, we have

$\displaystyle\left\|\left[O,u_{i}^{(t)}\right]\right\|$ $\displaystyle\leq
2\left\|u_{i}^{(t)}\right\|\mathcal{F}(\ell)=2\mathcal{F}(\ell),$ (S.62)

where we use $\left\|\tilde{{\rm tr}}_{i^{\rm c}}\left[u_{i}(\lambda
t)\right]\right\|\leq\left\|u_{i}(\lambda t)\right\|=1$.

Moreover, by combining the Lieb-Robinson bound and the inequality (S.45) in
Lemma 3, we have

$\displaystyle\left\|u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right\|\leq\min\left[1,C\left(e^{v|\lambda
t|}-1\right)e^{-\mu(\ell_{s-1}+1)}\right]\leq\min\left[1,C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}\right],$
(S.63)

where we use $0\leq\lambda\leq 1$. Combining the inequalities (S.I.5) and
(S.63), we obtain

$\displaystyle\left\|\left[O,u_{i[\ell_{s}]}^{(t)}-u_{i[\ell_{s-1}]}^{(t)}\right]\right\|$
$\displaystyle\leq
2\gamma\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})\min\left[1,C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}\right].$
(S.64)

Therefore, we can derive for $\left\|\left[O,u_{i}(t)\right]\right\|$

$\displaystyle\left\|\left[O,u_{i}(\lambda t)\right]\right\|\leq
2\mathcal{F}(\ell)+2\gamma\sum_{s=1}^{\infty}\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})\min\left[1,C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}\right],$
(S.65)

whose integral then reads

$\displaystyle\int_{-\infty}^{\infty}f(t)\left\|\left[O,u_{i}(\lambda
t)\right]\right\|dt$ $\displaystyle\leq
2\mathcal{F}(\ell)\int_{-\infty}^{\infty}f(t)dt+2\gamma\sum_{s=1}^{\infty}\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})\int_{-\infty}^{\infty}f(t)\min\left[1,C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}\right]dt.$
(S.66)

Let us set $t_{0}=\mu\ell_{s}/(2v)-\mu(\Delta\ell-1)/v$ and define
$f_{t_{0}}(s)$ as in Eq. (S.57), where we let $t_{0}=0$ if
$\ell_{s}/2-\Delta\ell\leq 0$ (or $s\leq 2$). We then obtain

$\displaystyle\int_{-\infty}^{\infty}f(t)\min\left[1,C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}\right]dt$
$\displaystyle\leq\int_{|t|\leq
t_{0}}f(t)C\left(e^{v|t|}-1\right)e^{-\mu(\ell_{s}-\Delta\ell+1)}dt+\int_{|t|\geq
t_{0}}f(t)dt$ $\displaystyle\leq Cve^{-\mu\ell_{s}/2}\int_{|t|\leq
t_{0}}|t|f(t)dt+f_{t_{0}}(s)$ $\displaystyle\leq
Cve^{-\mu\ell_{s}/2}\int_{|t|\leq t_{0}}|t|f(t)dt+f_{t_{0}}(s),$ (S.67)

where we use $e^{v|t|}-1\leq v|t|e^{v|t|}\leq
v|t|e^{\mu\ell_{s}/2+\mu(-\Delta\ell+1)}$ for $|t|\leq t_{0}$.

By applying the inequality (S.I.5) to Eq. (S.I.5), we obtain

$\displaystyle\int_{-\infty}^{\infty}f(t)\left\|\left[O,u_{i}(t)\right]\right\|dt$
$\displaystyle\leq
2f_{1}\mathcal{F}(\ell)+2Cf_{2}v\gamma\sum_{s=1}^{\infty}\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})e^{-\mu\ell_{s}/2}+2\gamma\sum_{s=1}^{\infty}\ell_{s}^{D}\mathcal{F}(\ell-\ell_{s})f_{t_{0}}(s),$
(S.68)

which reduces to the desired inequality (S.57) by letting
$\ell_{s}=s\Delta\ell$. This completes the proof. $\square$

[ End of Proof of Lemma 5]

We can also prove a similar statement using the same proof technique:

######  Corollary 6.

Under the same setup as in Lemma 5, we consider

$\displaystyle\int_{-\infty}^{\infty}f(t)\int_{0}^{t}\left\|[O,u_{i}(t_{1})]\right\|dt_{1}dt.$
(S.69)

Then, the same inequality as (S.56) holds for (S.69) by replacing $f(t)$ by
$|t|f(t)$, where the definitios of $f_{t_{0}}(s)$, $f_{1}$ and $f_{2}$ are
replaced correspondigly.

Proof of Corollary 6. We start from the inequality (S.65). By integrating it
from $t_{1}=0$ to $t_{1}=t$ in (S.65), we have