# Exponential Decay of Mutual Information for Gibbs states of local
Hamiltonians

Andreas Bluhm<EMAIL_ADDRESS>QMATH, Department of Mathematical Sciences,
University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Ángela Capel<EMAIL_ADDRESS>Fachbereich Mathematik,
Universität Tübingen, 72076 Tübingen, Germany  Zentrum Mathematik, Technische
Universität München, Boltzmannstrasse 3, 85748 Garching, Germany  Munich
Center for Quantum Science and Technology (MCQST), München, Germany  Antonio
Pérez-Hernández<EMAIL_ADDRESS>Departamento de Matemática Aplicada I,
Escuela Técnica Superior de Ingenieros Industriales, Universidad Nacional de
Educación a Distancia, calle Juan del Rosal 12, 28040 Madrid (Ciudad
Universitaria), Spain  Departamento de Análisis Matemático y Matemática
Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

###### Abstract

The thermal equilibrium properties of physical systems can be described using
Gibbs states. It is therefore of great interest to know when such states allow
for an easy description. In particular, this is the case if correlations
between distant regions are small. In this work, we consider 1D quantum spin
systems with local, finite-range, translation-invariant interactions at any
temperature. In this setting, we show that Gibbs states satisfy uniform
exponential decay of correlations and, moreover, the mutual information
between two regions decays exponentially with their distance, irrespective of
the temperature. In order to prove the latter, we show that exponential decay
of correlations of the infinite-chain thermal states, exponential uniform
clustering and exponential decay of the mutual information are equivalent for
1D quantum spin systems with local, finite-range interactions at any
temperature. In particular, Araki’s seminal results yields that the three
conditions hold in the translation-invariant case. The methods we use are
based on the Belavkin-Staszewski relative entropy and on techniques developed
by Araki. Moreover, we find that the Gibbs states of the systems we consider
are superexponentially close to saturating the data-processing inequality for
the Belavkin-Staszewski relative entropy.

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Preliminaries
    1. 3.1 Mutual information and relative entropy
    2. 3.2 Quantum spin chains
    3. 3.3 Locality estimates in 1D
    4. 3.4 Partial trace
  4. 4 Locality of observables
  5. 5 Approximate factorization of the Gibbs state
  6. 6 Correlations in finite intervals vs the infinite chain
  7. 7 Local indistinguishability and clustering
  8. 8 Decay of mutual information for Gibbs states of local Hamiltonians
    1. 8.1 Proof of Step 1
    2. 8.2 Proof of Step 2
    3. 8.3 Proof of Step 3
    4. 8.4 Proof of Step 4
    5. 8.5 Proof of Step 5
  9. 9 Final remarks and Conclusions

##  1 Introduction

The thermal equilibrium properties of quantum systems can be described by
quantum Gibbs states. Such states inherit the locality properties of their
associate Hamiltonian, featuring locality of correlations and allowing for an
efficient description in certain cases. A typical measure of correlations in
many-body systems is the _operator_ or _covariance correlation_ , defined for
a quantum state
$\rho_{ABC}\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C})$:

$\text{Corr}_{\rho}(A:C):=\sup_{O_{A},O_{C}}{\big{|}\text{Tr}[O_{A}\otimes
O_{C}\,(\rho_{AC}-\rho_{A}\otimes\rho_{C})]\big{|}}\,,$

where the supremum is taken over all operator norm-one operators $O_{A}$ and
$O_{C}$ supported on subsystems $A$ and $C$, respectively, and $\rho_{X}$ is
the reduced state of $\rho_{ABC}$ on $X$ for $X\in\\{A,C,AC\\}$. When these
correlations are spatially localized, there are efficient methods to
approximate Gibbs states of local Hamiltonians using tensor network methods
[37, 18].

In his seminal work in 1969, Araki showed that for any infinite 1D quantum
spin system with finite-range and translation-invariant interactions,
bipartite correlation functions decay exponentially with the distance between
$A$ and $C$ [2]. This opened a new field of research that has been
exceptionally active in the last years. Remarkably, Araki’s result was
extended for different correlation functions to higher dimensional systems
above a threshold temperature in a series of papers [22, 39, 31, 44], both for
classical and quantum systems. In a recent work [24], the exponential decay of
correlations property was related to the absence of complex zeroes of the
partition function close to the real axis. Moreover, it is known that systems
with a positive spectral gap exhibit exponential decay of correlations [29].

From an information-theoretic point of view, the most prominent measure of
correlations is however the mutual information, defined for
$\rho_{AC}\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{C})$ by

$I_{\rho}(A:C):=D(\rho_{AC}||\rho_{A}\otimes\rho_{C})\,,$ (1)

where $D(\rho||\sigma):=\operatorname{Tr}[\rho\,(\log\rho-\log\sigma)]$ is the
Umegaki relative entropy between $\rho$ and $\sigma$ [45]. Indeed, it is shown
in [21] that this measure quantifies the total amount of correlation between
subsystems $A$ and $C$ via an operational interpretation. It moreover has
several desirable properties, e.g., it inherits positivity and data processing
from the relative entropy. Note that estimates on the decay of the mutual
information are stronger than those on the decay of the operator correlation
function, since

$I_{\rho}(A:C)\geq\frac{1}{2}\|\rho_{AC}-\rho_{A}\otimes\rho_{C}\|_{1}^{2}\geq\frac{1}{2}\,\text{Corr}_{\rho}(A:C)^{2}\,,$

by Pinsker’s inequality [40] and the duality of Schatten $p$-norms [8].
Furthermore, it is well known that, in the context of data hiding, it is
possible to find a state whose operator correlation is arbitrarily small,
whereas it is still highly correlated in terms of the mutual information [26,
27]. The mutual information has attracted significant attention in the past
years, because it satisfies an area law [46], i.e., the correlations between
adjacent systems scale only as the boundary between them. The connection
between area laws and exponential decay of correlations was subsequently
explored in [11, 12]. See also the recent papers [34, 42] for related results
on the area law in more general settings and for different quantities.
Furthermore, it is known that systems with the so-called rapid mixing property
exhibit exponential decay of mutual information [29].

Another information-theoretic quantity that has been thoroughly studied in the
context of decay of correlations is the conditional mutual information (CMI
for short), given for a state
$\rho_{ABC}\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C})$
by

$I_{\rho}(A:C|B):=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{B})-S(\rho_{ABC})\,,$

where $S(\rho):=-\operatorname{Tr}[\rho\,\log\rho]$ is the von Neumann
entropy. As shown in [20, 43], this quantity is related to the approximate
recoverability of a state, and thus it is widely used as a measure of
independence in quantum systems. Building on Araki’s results, Kato and Brandão
showed in [30] that the CMI of any Gibbs state in the setting of [2] decays
subexponentially. This was subsequently improved in a joint work with Kuwahara
to exponential decay and extended to generic graphs in [33], provided the
system is at high-enough temperature and the interactions are short ranged.
Additionally, in [1], an exponential decay of the CMI for injective matrix
product states (MPS) in finite 1D lattices with finite bond dimension and open
boundary conditions was obtained. See also [17], in which the authors
presented the conjecture that the CMI for any matrix product density operator
(MPDO) should decay exponentially. To support the conjecture, the authors were
able to prove it in some cases of interest. In particular, the conjecture
implies that the parent Hamiltonians of such MPDOs are (quasi-)local.

In this work, we build on Araki’s result to prove an exponential decay of the
(unconditional) mutual information at any temperature for quantum Gibbs states
in 1D. This should be compared to the previous work by Kuwahara et al. [33],
in which an exponential decay of the conditional mutual information was shown
above a threshold temperature for arbitrary lattices.

The main novelty in the approach we follow is the use of another of the
possible extensions of the Kullback-Leibler divergence to the quantum setting,
namely the Belavkin-Staszewski relative entropy (BS-entropy for short) [6].
The interest in this quantity has increased in the past few years, with a
comprehensive study of the fundamental properties of the maximal
$f$-divergences [36, 28], of which the BS-entropy is a prominent example.
Recently, a recovery condition and a strengthened data-processing inequality
for such divergences has been obtained [9], as well as some weak quasi-
factorization results for the BS-entropy [10]. Remarkably, a subclass of
maximal $f$-divergences, namely the geometric Rényi divergences, has found
applications in estimating channel capacities [19]. In the current paper, we
present other important applications of geometric Rényi divergences and the
BS-entropy in the fields of quantum many-body systems and quantum information
theory. In particular, we define a BS-mutual information in the same spirit as
(1) and show that it exhibits an exponential decay under some reasonable
assumptions.

The rest of the paper is organized as follows: In Section 2, we list the main
results of the present article. Some basic notions and preliminary results are
reviewed in Section 3. In Section 4, we provide norm estimates on some local
observables that are necessary for the rest of the article. They are in
particular used in Section 5 to prove one of the main results of the paper,
namely a faster-than-exponential (i.e.  _superexponential_) decay for the
distance of a quantum Gibbs state from being BS-recoverable. In Section 6, we
prove another main result regarding the exponential decay of correlations (on
a finite interval) of a Gibbs state with the distance between two spatially
separated regions in 1D. This condition is further used in Section 7 to show
the local indistinguishability of such states. Subsequently, in Section 8 we
prove our third main result: an exponential decay for the BS-mutual
information of a Gibbs state with the distance between two spatially separated
regions in 1D. Finally, we conclude our manuscript with a discussion of our
results in Section 9.

##  2 Main results

The main results of this paper concern the BS-entropy and some other
quantities derived from it. The BS-entropy for two strictly positive states
$\rho$ and $\sigma$ is given by

$\widehat{D}(\rho\|\sigma):=\text{Tr}[\rho\,\text{log}(\rho^{1/2}\sigma^{-1}\rho^{1/2})]\,.$

It relates to the (Umegaki) relative entropy by means of the following
inequality [38]

$\widehat{D}(\rho\|\sigma)\geq D(\rho\|\sigma)\,,$

which is strict if, and only if, $[\rho,\sigma]\neq 0$ [28]. In particular,
for a bipartite Hilbert space
$\mathcal{H}_{AC}=\mathcal{H}_{A}\otimes\mathcal{H}_{C}$ and a positive state
$\rho_{AC}\in\mathcal{B}(\mathcal{H}_{AC})$, we can define the BS-mutual
information in an analogous way to the usual mutual information, and clearly
the following inequality holds:

$I_{\rho}(A:C)\leq\widehat{I}_{\rho}(A:C):=\widehat{D}(\rho_{AC}\|\rho_{A}\otimes\rho_{C})\,.$

This basic inequality allows us to obtain an estimate for the mutual
information via estimating the BS-mutual information.

The first main result of the paper is the so-called exponential uniform
clustering for Gibbs states.

Theorem I. We consider a quantum spin system on $\mathbb{Z}$ with local,
finite-range, translation-invariant (non-commuting) interactions. In this
setting, there exists a positive function $\ell\mapsto\varepsilon(\ell)$
exhibiting exponential decay such that for every finite interval
$I\subset\mathbb{Z}$ split into three subintervals $I=ABC$, where $B$ shields
$A$ from $C$, the Gibbs state $\rho=e^{-H_{I}}/\operatorname{Tr}(e^{-H_{I}})$
satisfies that the associated covariance correlation decays exponentially with
the size of $B$, namely

$\operatorname{Corr}_{\rho}(A:C)\leq\varepsilon(|B|)\,.$ (2)

We remark that the previous condition, which appears later in the main text as
Theorem 6.2, holds at any inverse temperature $\beta>0$. As far as we know,
there is no proof for this condition in this setting in any previous paper
(although it was shown to hold in any dimension for high-enough temperature in
[31]). It was a necessary condition for some of the main results in [30] and
[13], where the authors claimed that it directly follows from Araki’s result
on infinite-chain exponential decay of correlations. However, it is not
obvious how the former result follows from the latter, and that is the reason
for including a proof of such a result in the current manuscript.

Next, building on the fact that any Gibbs state of a local, finite-range,
translation-invariant Hamiltonian in 1D satisfies exponential uniform
clustering, we prove exponential decay of the mutual information for such
states.

Theorem II. Under the conditions of Theorem I, there is a positive function
$\ell\mapsto\delta_{1}(\ell)$, depending on the local interactions and
$\varepsilon(\ell)$ in (2), that exhibits exponential decay and satisfying

$I_{\rho}(A:C)\,\,\leq\,\,\widehat{I}_{\rho}(A:C)\,\,\leq\,\,\delta_{1}(|B|)\,.$

This result appears in the main text as Theorem 8.2 and builds heavily on the
well-known methods of Araki for complex-time evolutions and finite-range
interactions that constitute a generalization of the classical Lieb-Robinson
bounds [2, 41]. Throughout the paper, we will absorb $\beta$ in the
Hamiltonian so that it is an implicit parameter of the interaction strength
$J$. Given an arbitrary finite-range Hamiltonian $H$, all our results hold for
the Gibbs state $\rho=e^{-H}/\mathrm{Tr}(e^{-H})$ associated to it; thus, they
actually hold for every finite temperature $\beta>0$. The previous theorem
should be compared to [30, Corollary 5], which shows that the CMI of a Gibbs
state in the above setting decays subexponentially in the distance $|B|$.
Considering the (unconditional) mutual information instead, we can improve the
decay to being exponential.

Let us remark, that our results allow to formally complete the equivalence
between several different notions of decay of correlations for Gibbs states of
local, finite-range Hamiltonians in 1D (not necessarily translation-
invariant). More specifically, we show for such states that the following
three conditions are equivalent:

(ii) Exponential Uniform Clustering (i) Exponential Decay of the Mutual
Information (iii) Exponential Decay of Correlations for the Infinite chain KMS
state Figure 1: Equivalence between three different forms of decay of
correlations in 1D.

Dashed lines indicate implications that were previously known or
straightforward consequences of well-known results, while the reverse
implications are formally proved in Theorems 8.2 and 6.2, respectively. The
_translation-invariant_ condition is applied to deduce from Araki’s result [2]
that (i)-(iii) hold. For further detail, we refer the reader to Section 9 and
the diagram in Figure 7 therein.

The use of the BS-entropy in our methods yields another result of independent
interest, namely that the distance of a Gibbs state from being BS-recoverable
(i.e. from saturating the data-processing inequality for the BS-entropy)
decays superexponentially with the size of the middle system. More
specifically, by the data-processing inequality, the following holds for any
quantum channel $\mathcal{T}$ and any positive states $\rho$ and $\sigma$:

$\widehat{D}(\rho||\sigma)\geq\widehat{D}(\mathcal{T}(\rho)||\mathcal{T}(\sigma))\,,$

and by the results of [9], equality in the previous inequality is equivalent
to

$\rho=\sigma\,\mathcal{T}^{*}\left(\mathcal{T}(\sigma)^{-1}\mathcal{T}(\rho)\right)\,,$

where the map
$\mathcal{B}_{\mathcal{T}}^{\sigma}(\cdot):=\sigma\,\mathcal{T}^{*}\left(\mathcal{T}(\sigma)^{-1}(\cdot)\right)$
is called BS-recovery condition. This map represents the analogue to the Petz
recovery map for the saturation in the data-processing inequality for the
Umegaki relative entropy, although the former is not a quantum channel.
Furthermore, another one of the main results of [9] is a strengthened version
of the data-processing inequality for the BS-entropy in terms of the distance
of a state to be BS-recoverable, i.e., to satisfying the latter equality. That
result in particular shows that if there is almost-saturation of the DPI for
the BS-entropy, then the state is close to being BS-recoverable. The converse
implication, though, is as far as we know an open problem. The result below
might shed some light on this question for a particular class of states.

If we consider
$\mathcal{H}_{ABC}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$,
the quantum channel
$\operatorname{Tr}_{A}(\cdot)\mathds{1}_{A}/d_{A}\otimes\operatorname{id}_{BC}$,
a positive state $\rho_{ABC}\in\mathcal{B}(\mathcal{H}_{ABC})$, and if we
define $\sigma_{ABC}=\rho_{AB}\otimes\mathds{1}_{C}/d_{C}$, we say that
$\rho_{ABC}$ is a BS-recoverable state if it is a fixed point of the BS-
recovery condition composed with this quantum channel, i.e., if

$\rho_{ABC}=\rho_{AB}\rho_{B}^{-1}\rho_{BC}\,.$

We are now in the position to informally state our third main result.

Theorem III. Under the conditions of Theorem I, there is a positive function
$\ell\leavevmode\nobreak\ \longmapsto\leavevmode\nobreak\ \delta_{2}(\ell)$,
depending only on the local interactions and exhibiting superexponential
decay, such that for every three adjacent and finite intervals $ABC$, where
$B$ shields $A$ from $C$, the associated Gibbs state $\rho_{ABC}$ satisfies

$\|\rho_{ABC}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\|_{1}\,\leq\,\delta_{2}(|B|)\,.$

This result appears in the main text as Theorem 5.1 and its proof also follows
from the aforementioned Araki’s methods. It should be compared to [30, Theorem
4], which shows that Gibbs states in the above setting are subexponentially
close to being recoverable. Replacing the Petz recovery map by the BS-recovery
condition, we find that the Gibbs states are superexponentially close to
fulfilling the latter. A natural question that arises from this result is,
since Gibbs states in the conditions above are close to being BS-recoverable,
whether there is almost saturation on the DPI of the BS-entropy for such
states. This is left for future work. Additionally to the theorems mentioned
above, we include some other results of independent interest in this text.

We expect that the techniques we developed, based on the BS-entropy, can also
be useful in other contexts, such as for deriving area laws in quantum many-
body systems (see e.g. [42]). Indeed, the main technique in the proof of
exponential decay of the mutual information, namely Proposition 8.1, is to
show that tracing out $B$ in a Gibbs state on an interval $ABC$ leads to the
reduced state being exponentially close in operator norm to a product state,
where the decay is with $|B|$. This finding is one of the key ingredients in
the proof that Davies generators in 1D which converge to a Gibbs state of a
local, finite-range, translation-invariant, commuting Hamiltonian, at any
temperature, satisfy a positive modified logarithmic Sobolev inequality, and
hence exhibit rapid mixing [3, 4].

##  3 Preliminaries

###  3.1 Mutual information and relative entropy

A class of important tools for quantum information theory which we will use in
this work are the generalizations of the classical Kullback-Leiber (KL)
divergence [32] to the quantum setting. The generalization is not unique and
therefore the KL-divergence can be extended to the quantum case in a wide
variety of measures of distinguishability for states. Let
$\mathcal{H}_{AC}:=\mathcal{H}_{A}\otimes\mathcal{H}_{C}$ be a finite-
dimensional Hilbert space and let $\rho_{AC},\sigma_{AC}$ be full-rank states.
Their (Umegaki) relative entropy [38] is given by

$D(\rho_{AC}\,||\,\sigma_{AC})\,:=\,\operatorname{Tr}[\,\rho_{AC}\,(\log{\rho_{AC}}-\log{\sigma_{AC}})\,]\,,$

where by $\operatorname{Tr}$ we are denoting the usual (unnormalized) trace,
and their Belavkin-Staszewski relative entropy (shortened BS-entropy) [6] is

$\widehat{D}(\rho_{AC}\,||\,\sigma_{AC})\,:=\,\operatorname{Tr}\left[\,\rho_{AC}\,\log{\left(\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right)}\,\right]\,.$

An equivalent way to write the BS-entropy, which follows easily from the power
series of the matrix logarithm, is

$\widehat{D}(\rho_{AC}\,||\,\sigma_{AC})\,=\,\operatorname{Tr}\left[\,\rho_{AC}\,\log{\left(\sigma_{AC}^{-1}\,\rho_{AC}\right)}\,\right]\,.$

Let us recall that the two quantum relative entropies are related through

$D(\rho_{AC}\,||\,\sigma_{AC})\,\,\leq\,\,\widehat{D}(\rho_{AC}\,||\,\sigma_{AC}),$
(3)

which is indeed an equality if and only if $\rho_{AC}$ and $\sigma_{AC}$
commute [28, Theorem 4.3]. Additionally, throughout the text we make use of
the following family of maximal $f$-divergences which converge to the BS-
entropy: For $1<q<\infty$, and $\rho_{AC},\,\sigma_{AC}$ as above, their _$q$
-geometric Rényi divergence_ [36] is given by

$\widehat{D}_{q}(\rho_{AC}\|\sigma_{AC}):=\frac{1}{q-1}\log\operatorname{Tr}\left[\sigma_{AC}^{1/2}\left(\sigma_{AC}^{-1/2}\rho_{AC}\sigma_{AC}^{-1/2}\right)^{q}\sigma_{AC}^{1/2}\right]\,.$

In the following result, given $q>1$, we prove upper bounds for the
$q$-geometric Rényi divergence between two full-rank states in terms of an
operator norm, denoted by $\left\|\cdot\right\|$ throughout the text.

######  Lemma 3.1.

For any $q>1$ and any $\rho_{AC},\sigma_{AC}$ full-rank states, we have

$\widehat{D}_{q}(\rho_{AC}\,||\,\sigma_{AC})\,\,\leq\,\,\left\|\sigma_{AC}^{-1}\,\rho_{AC}-\mathds{1}_{AC}\right\|\,.$
(4)

In particular,

$\widehat{D}(\rho_{AC}\,||\,\sigma_{AC})\,\,\leq\,\,\left\|\sigma_{AC}^{-1}\,\rho_{AC}-\mathds{1}_{AC}\right\|\,.$
(5)

Proof. For $q>1$, note that the expression inside the logarithm in the
definition of the $q$-geometric Rényi divergences can be written as

$\displaystyle\operatorname{Tr}\left[\sigma_{AC}^{1/2}\left(\sigma_{AC}^{-1/2}\,\rho_{AC}\,\sigma_{AC}^{-1/2}\right)^{q}\sigma_{AC}^{1/2}\right]$
$\displaystyle=\operatorname{Tr}\left[\rho_{AC}\,\sigma_{AC}^{-1/2}\left(\sigma_{AC}^{-1/2}\,\rho_{AC}\,\sigma_{AC}^{-1/2}\right)^{q-1}\sigma_{AC}^{1/2}\right]$
$\displaystyle=\operatorname{Tr}\left[\rho_{AC}^{1/2}\left(\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right)^{q-1}\rho_{AC}^{1/2}\right]$

where in the last equality we have used that
$f(X^{\dagger}X)=X^{-1}f(XX^{\dagger})X$ whenever $f$ is continuous and $X$ is
invertible. Thus,

$\displaystyle\operatorname{Tr}\left[\sigma_{AC}^{1/2}\left(\sigma_{AC}^{-1/2}\,\rho_{AC}\,\sigma_{AC}^{-1/2}\right)^{q}\sigma_{AC}^{1/2}\right]\leq\left\|\left(\,\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right)^{q-1}\right\|\leq\left\|\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right\|^{q-1}\,.$

By definition of the $q$-geometric Rényi divergence, monotonicity of the
logarithm and the inequality $\log(x+1)\leq x$, we obtain

$\displaystyle\widehat{D}_{q}(\rho_{AC}\|\sigma_{AC})\,\leq\,\frac{1}{q-1}\log\left\|\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right\|^{q-1}\,$
$\displaystyle=\;\log\left\|\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}\right\|$
$\displaystyle\,\leq\;\log\left(\left\|\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}-\mathds{1}\right\|+1\right)$
$\displaystyle\,\leq\;\left\|\rho_{AC}^{1/2}\,\sigma_{AC}^{-1}\,\rho_{AC}^{1/2}-\mathds{1}\right\|$
$\displaystyle\,\leq\;\left\|\sigma_{AC}^{-1}\rho_{AC}-\mathds{1}\right\|\,,$

where we have used triangle inequality and [8, Proposition IX.1.1], which
implies, for $P>0$, that

$\left\|P^{1/2}QP^{1/2}-\mathbbm{1}\right\|\,\,\leq\,\,\left\|QP-\mathbbm{1}\right\|\,.$
(6)

In particular, taking limit when $q\rightarrow 1^{+}$, we deduce that

$\widehat{D}(\rho_{AC}\|\sigma_{AC})=\lim_{q\rightarrow
1^{+}}\widehat{D}_{q}(\rho_{AC}\|\sigma_{AC})\leq\left\|\sigma_{AC}^{-1}\,\rho_{AC}-\mathds{1}_{AC}\right\|\,.$

∎

Let $\rho_{AC}$ be a full-rank state on $\mathcal{H}_{AC}$. Let us recall that
the _mutual information_ of $\rho$ between two regions $A$ and $C$ is defined
as

$I_{\rho}(A:C):=D(\rho_{AC}\,||\,\rho_{A}\otimes\rho_{C})$

The analogous notion for the BS-entropy, which we call BS-mutual information,
can be defined as

$\widehat{I}_{\rho}(A:C):=\widehat{D}(\rho_{AC}\,||\,\rho_{A}\otimes\rho_{C})$

According to (3),

$I_{\rho}(A:C)\,\,\leq\,\,\widehat{I}_{\rho}(A:C)\,.$

Moreover, we can define the Rényi mutual information arising from the
$q$-geometric Rényi divergence by

$\widehat{I}^{q}_{\rho}(A:C):=\widehat{D}_{q}(\rho_{AC}\|\rho_{A}\otimes\rho_{C})\,,$

as done in [42]. Then, we conclude from (4), for any $q>1$,

$\widehat{I}^{q}_{\rho}(A:C)\leq\left\|\rho_{A}^{-1}\otimes\rho_{C}^{-1}\rho_{AC}-\mathds{1}_{AC}\right\|\,.$

Furthermore, as a consequence of (5) it holds that

$\widehat{I}_{\rho}(A:C)\,\,\leq\,\,\left\|\rho_{A}^{-1}\otimes\rho_{C}^{-1}\rho_{AC}-\mathds{1}_{AC}\right\|\,.\\\
$ (7)

To conclude this subsection, let us recap and emphasize that the following
chain of relations, which will appear frequently throughout the text, holds
true for any $q>1$ and any full-rank state $\rho_{AC}$:

$\displaystyle\frac{1}{2}\operatorname{Corr}_{\rho}(A:C)^{2}$
$\displaystyle\leq\frac{1}{2}\|\rho_{AC}-\rho_{A}\otimes\rho_{C}\|_{1}^{2}$
$\displaystyle\leq
I_{\rho}(A:C)\leq\,\widehat{I}_{\rho}(A:C)\,\leq\,\widehat{I}^{q}_{\rho}(A:C)\,\leq\,\|\rho_{A}^{-1}\otimes\rho_{C}^{-1}\rho_{AC}-\mathds{1}_{AC}\|.$

###  3.2 Quantum spin chains

In this work, we will consider a quantum spin system over $\mathbb{Z}$. For
each finite subinterval $I\subset\mathbb{Z}$ we denote by $|I|$ its cardinal.
At every site $x\in\mathbb{Z}$ we have a finite-dimensional local Hilbert
space $\mathcal{H}_{x}\equiv\mathbb{C}^{d}$ and the corresponding space of
operators $\mathfrak{A}_{x}:=\mathcal{B}(\mathbb{C}^{d})$. For each finite
subset $\Lambda\subset\mathbb{Z}$ let

$\mathcal{H}_{\Lambda}:=\otimes_{x\in\Lambda}{\mathcal{H}_{x}}\,,\quad\mathfrak{A}_{\Lambda}:=\mathcal{B}(\mathcal{H}_{\Lambda})=\otimes_{x\in\Lambda}{\mathfrak{A}_{x}}\,.$

If $\Lambda^{\prime}\subset\Lambda$ we have a canonical linear isometry

$\mathfrak{A}_{\Lambda^{\prime}}\longrightarrow\mathfrak{A}_{\Lambda}=\mathfrak{A}_{\Lambda^{\prime}}\otimes\mathfrak{A}_{\Lambda\setminus\Lambda^{\prime}}\,,\quad
Q\longmapsto Q\otimes\mathbbm{1}_{\Lambda\setminus\Lambda^{\prime}}\,.$

The (normed) algebra of local observables $\mathfrak{A}_{loc}$ is then
formally defined as the inductive limit of the family
$(\mathfrak{A}_{\Lambda})_{\Lambda}$, and its completion with respect to the
operator norm is the algebra of quasi-local observables
$\mathfrak{A}_{\mathbb{Z}}$. In the following, $\left\|A\right\|$ will always
be the operator norm of $A\in\mathfrak{A}_{\mathbb{Z}}$ if we do not specify
the norm. The norms $\left\|\cdot\right\|_{p}$ for $p\in[1,\infty]$ will be
the Schatten $p$-norms. Notably, $p=1$ corresponds to the trace norm.

###  3.3 Locality estimates in 1D

Let us also fix a local interaction $\Phi=(\Phi_{X})_{X}$, where $X$ runs over
all finite subsets of $\mathbb{Z}$, and where $\Phi_{X}\in\mathfrak{A}_{X}$,
$\Phi_{X}=\Phi_{X}^{\ast}$. Recall that the diameter of $X$ is defined as
$\operatorname{diam}(X)=\max_{x,y\in X}|x-y|$. We assume that $\Phi$ has
finite range $r>0$ and strength $J>0$, namely $\|\Phi_{X}\|=0$ if $X$ has
diameter greater than $r$ and $\|\Phi_{X}\|\leq J$ for all $X$. As usual, we
denote for every finite subset $\Lambda\subset\mathbb{Z}$ the corresponding
Hamiltonian by

$H_{\Lambda}:=\sum_{X\subset\Lambda}{\Phi_{X}}\,,$

and the (possibly complex) time evolution of an observable
$Q\in\mathfrak{A}_{\mathbb{Z}}$ by

$\Gamma_{H_{\Lambda}}^{s}(Q):=e^{isH_{\Lambda}}Qe^{-isH_{\Lambda}}\,,\quad
s\in\mathbb{C}\,.$

The following proposition gathers several results by Araki [2] for finite-
range interactions, see also [41] for a more recent result applying to a more
general type of interactions.

######  Proposition 3.2.

Let $\Phi$ be a local interaction with range $r$ and strength $J$ as above.
Then, for every $\beta>0$ there is a constant
$\mathcal{G}=\mathcal{G}(\beta,J,r)>1$ satisfying the following properties:

  1. (i)

For every pair $a,b\in\mathbb{Z}$ with $a\leq b$, every observable $Q$ in
$\mathfrak{A}_{[a,b]}$ , every complex $s\in\mathbb{C}$ with $|s|\leq\beta$
and every pair $n$, $m\in\mathbb{Z}$ with $0\leq n\leq m$

$\displaystyle\|\Gamma_{H_{[a-m,b+m]}}^{s}(Q)-\Gamma_{H_{[a-n,b+n]}}^{s}(Q)\|$
$\displaystyle\leq\mathcal{G}^{b-a+1}\,\frac{\mathcal{G}^{n}}{\left(\lfloor
n/r\rfloor+1\right)!}\,\|Q\|\,,$
$\displaystyle\|\Gamma_{H_{[a-n,b+n]}}^{s}(Q)\|$
$\displaystyle\leq\mathcal{G}^{b-a+1}\,\|Q\|\,.$

  2. (ii)

Denoting

$E_{n}(s):=e^{-sH_{[1-n,n]}}e^{sH_{[1-n,0]}+sH_{[1,n]}}\,,\quad
n\in\mathbb{N}\,,\,s\in\mathbb{C},$

we have for every $1\leq n\leq m$ and every $s\in\mathbb{C}$ with
$|s|\leq\beta$

$\|E_{n}(s)\|\,,\,\|E_{n}^{-1}(s)\|\,\leq\,\mathcal{G}\,\,,$
$\|E_{n}(s)-E_{m}(s)\|\,,\,\|E_{n}(s)^{-1}-E_{m}(s)^{-1}\|\,\leq\,\frac{\mathcal{G}^{n}}{\left(\lfloor
n/r\rfloor+1\right)!}\,.$

The first part of the previous proposition can be interpreted as an extended
version of the celebrated Lieb-Robinson bounds [35] for complex-time evolution
(in the particular setting of 1D and finite-range interactions). These
estimates are one of the main ingredients in Araki’s argument regarding the
absence of (thermal) phase transitions in 1D.

######  Remark 3.3.

To simplify notation, we will absorb $\beta$ in the Hamiltonian so that it
will be an implicit parameter of the interaction strength $J$. Since all our
results hold for Gibbs states $\rho=e^{-H}/\mathrm{Tr}(e^{-H})$ associated to
arbitrary finite-range Hamiltonians, they actually hold for every positive
temperature.

Let $I\subset\mathbb{Z}$ be a finite interval. Let us split $I$ into two
subintervals $X$ and $Y$ so that $I=XY$. Without loss of generality, we can
assume that they correspond to $X=[1-a,0]$ and $Y=[1,b]$ for some
$a,b\in\mathbb{N}$. We present now an alternative formulation of Araki’s
results that will be useful.

######  Corollary 3.4.

Let $\Phi$ be a local interaction with range $r$ and strength $J$ as above.
For a finite interval $I=XY\subset\mathbb{Z}$ split into two subintervals $X$
and $Y$, let us write

$E_{X,Y}(s)\,:=\,e^{-s\,H_{XY}}\,e^{s\,H_{X}+s\,H_{Y}}\quad,\quad
s\in\mathbb{C}\,.$
$E_{X,Y}\,:=\,E_{X,Y}(1)\,=\,e^{-\,H_{XY}}\,e^{\,H_{X}\,+\,H_{Y}}\,.$

Then, there is an absolute constant $\mathcal{G}>1$ depending only on $J$ and
$r$ such that:

  1. (i)

For every $s\in\mathbb{C}$ with $|s|\leq 1$

$\|E_{X,Y}(s)\|\,,\,\|E_{X,Y}(s)^{-1}\|\,\,\leq\,\,\mathcal{G}\,.$

  2. (ii)

If we add two intervals $\widetilde{X}$ and $\widetilde{Y}$ adjacent to $X$
and $Y$, respectively, so that we get a larger interval
$\widetilde{J}:=\widetilde{X}XY\widetilde{Y}$, then

$\left\|E_{X,Y}^{-1}(s)-E^{-1}_{\widetilde{X}X,Y\widetilde{Y}}(s)\right\|,\left\|E_{X,Y}(s)-E_{\widetilde{X}X,Y\widetilde{Y}}(s)\right\|\,\leq\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\quad,\quad|s|\leq
1\,.$

for any $\ell\in\mathbb{N}$ such that $\ell\,\leq\,|X|\,,\,|Y|$.

  3. (iii)

Let us identify $X=[1-a,0]$ and $Y=[1,b]$ for some $a,b\in\mathbb{N}$, and
denote $X_{n}:=X\cap[1-n,0]$ and $Y_{n}:=Y\cap[1,n]$ for each
$n\in\mathbb{N}$. Then, we can decompose

$E_{X,Y}=\sum_{n=1}^{\infty}\widetilde{E}^{(n)}\quad\text{where}\quad\widetilde{E}^{(1)}:=E_{X_{1},Y_{1}}\,\,,\,\,\widetilde{E}^{(n)}:=E_{X_{n},Y_{n}}-E_{X_{n-1},Y_{n-1}}\quad(n\geq
2)$

so that this series is absolutely convergent with

$\widetilde{E}^{(n)}\in\mathfrak{A}_{[1-n,n]},\quad\big{\|}\widetilde{E}^{(1)}\big{\|}\leq\mathcal{G}\quad\text{and}\quad\big{\|}\widetilde{E}^{(n)}\big{\|}\leq\frac{\mathcal{G}^{n-1}}{(\lfloor(n-1)/r\rfloor+1)!}\quad\text{
for every }n\geq 2\,.$

######  Remark 3.5.

One-sided versions of Corollary 3.4.(ii) can be given when adding an adjacent
interval to only one side of $I=XY$:

  1. (ii)′

If we add an interval $\widetilde{Y}$ adjacent to $Y$ so that we get a larger
interval $\widetilde{J}:=XY\widetilde{Y}$, then

$\Big{\|}E_{X,Y}^{-1}(s)-E^{-1}_{X,Y\widetilde{Y}}(s)\Big{\|}\,,\quad\Big{\|}E_{X,Y}(s)-E_{X,Y\widetilde{Y}}(s)\Big{\|}\,\leq\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\quad,\quad|s|\leq
1\,.$

for any $\ell\in\mathbb{N}$ such that $\ell\,\leq\,|Y|$.

  2. (ii)′′

If we add an interval $\widetilde{X}$ adjacent to $X$ so that we get a larger
interval $\widetilde{J}:=\widetilde{X}XY$, then

$\Big{\|}E_{X,Y}^{-1}(s)-E^{-1}_{\widetilde{X}X,Y}(s)\Big{\|}\,,\quad\Big{\|}E_{X,Y}(s)-E_{\widetilde{X}X,Y}(s)\Big{\|}\,\leq\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\quad,\quad|s|\leq
1\,.$

for any $\ell\in\mathbb{N}$ such that $\ell\,\leq\,|X|$.

Both statements $(ii)^{\prime}$ and $(ii)^{\prime\prime}$ are actually a
consequence of Corollary 3.4.(ii). Indeed, to deduce $(ii)^{\prime}$ consider
a new set of interactions $\widehat{\Phi}$ obtained from $\Phi$ by setting
$\widehat{\Phi}_{\Lambda}=\Phi_{\Lambda}$ whenever $\Lambda$ is contained in
$XY\widetilde{Y}$ and $\widehat{\Phi}_{\Lambda}=0$ otherwise. Then, the
corresponding expansionals $\widehat{E}$ for this new interaction satisfy
$\widehat{E}_{X,Y}=E_{X,Y}$ and
$\widehat{E}_{\widetilde{X}X,Y\widetilde{Y}}=E_{X,Y\widetilde{Y}}$. Since
$\widehat{\Phi}$ has the same bounds for interaction range and strength than
$\Phi$, we can apply Corollary 3.4.(ii) to get the desired estimate. A similar
argument shows that $(ii)^{\prime\prime}$ holds.

###  3.4 Partial trace

We will mainly deal with a finite one-dimensional spin system on a finite
interval $I\subset\mathbb{Z}$. Let us split $I$ into two parts, namely
$\Lambda$ and $\Lambda^{c}:=I\setminus\Lambda$. Thus,
$\mathfrak{A}=\mathfrak{A}_{I}=\mathfrak{A}_{\Lambda}\otimes\mathfrak{A}_{\Lambda^{c}}$.
The (unnormalized) trace over $I$ will be written as
$\mathrm{Tr}_{I}:\mathfrak{A}_{I}\to\mathbb{C}$. Moreover, we will consider
the operator (unnormalized) partial trace on $\Lambda$, given for
$\Lambda\subset I$ by the range-extended map

$\operatorname{tr}_{\Lambda}:\mathfrak{A}_{I}\longrightarrow\mathfrak{A}_{I}\quad,\quad
R\otimes
S\longmapsto\operatorname{Tr}_{\Lambda}(R)(\mathbbm{1}_{\Lambda}\otimes
S)\,,\;\;\forall R\in\mathfrak{A}_{\Lambda}\,,\,\,\forall
S\in\mathfrak{A}_{\Lambda^{c}}\,.$ (8)

Note that for every $R\in\mathfrak{A}_{\Lambda}$,
$S\in\mathfrak{A}_{\Lambda^{c}}$ as above

$\|\operatorname{tr}_{\Lambda}(R\otimes
S)\|=|\operatorname{Tr}_{\Lambda}(R)|\,\|S\|\,.$

Sometimes, it will be more convenient to work with the conditional expectation
in the sense of [38, Proposition 1.12], which is a unital completely positive
map [7, Proposition 5.2.2]. We will write
$\mathbb{E}_{\Lambda^{c}}:\mathfrak{A}\to\mathfrak{A}_{\Lambda^{c}}$ for this
conditional expectation. It is defined as in (8), but using the normalized
trace instead of the unnormalized one.

With this notation, if $H_{\Lambda}$ is the Hamiltonian of the system on
$\Lambda$ with local interaction $\Phi$, the local Gibbs state on $\Lambda$ is
given by

$\rho^{\Lambda}:=e^{-H_{\Lambda}}/\operatorname{Tr}_{\Lambda}(e^{-H_{\Lambda}})\,.$

We might omit the superscript when it is clear from the context. The reduced
operator of $\rho=\rho^{I}$ on $\Lambda$ will be denoted by

$\rho_{\Lambda}^{I}=\rho_{\Lambda}:=\operatorname{tr}_{\Lambda^{c}}(\rho)\,.$

Note that $H_{\Lambda}$ has support in $\Lambda$, and so the map

$\mathfrak{A}\longrightarrow\mathfrak{A}\,,\quad
Q\longmapsto\operatorname{tr}_{\Lambda}(e^{-H_{\Lambda}}Q)=\operatorname{tr}_{\Lambda}(e^{-\frac{1}{2}H_{\Lambda}}Qe^{-\frac{1}{2}H_{\Lambda}})$

is positive. The well-known Russo-Dye theorem yields

$\|\operatorname{tr}_{\Lambda}(e^{-H_{\Lambda}}Q)\|\leq\|\operatorname{tr}_{\Lambda}(e^{-H_{\Lambda}})\|\,\|Q\|=\operatorname{Tr}_{\Lambda}(e^{-H_{\Lambda}})\,\|Q\|\,.$

As a consequence, the map

$\mathfrak{A}\longrightarrow\mathfrak{A}\,,\quad
Q\longmapsto\operatorname{tr}_{\Lambda}(\rho^{\Lambda}Q)$

is positive, unital and hence contractive. This observation will be repeatedly
used in the following results. Note that $\operatorname{tr}_{\Lambda}$ is an
operator valued map, whereas $\operatorname{Tr}_{\Lambda}$ maps to scalars.

##  4 Locality of observables

In this section, we will prove some norm estimates on local observables which
will be used repeatedly in the rest of the article.

Let $I\subset\mathbb{Z}$ be a finite interval. For each local observable
$Q\in\mathfrak{A}_{\mathrm{loc}}$, let us define by

$\|Q\|_{I}:=\inf{\\{\|Q-P\|\colon P\in\mathfrak{A}_{I}\\}}$

the distance from $Q$ to the subspace $\mathfrak{A}_{I}$. Beware that this is
clearly not a norm.

######  Lemma 4.1.

Let $Q,Q^{\prime}$ be two local observables. Then

$\|Q\,Q^{\prime}\|_{I}\,\leq\,2\|Q\|_{I}\,\|Q^{\prime}\|\,+\,2\|Q\|\,\|Q^{\prime}\|_{I}\,.$

Moreover, if $Q$ is positive and invertible, then

$\|Q^{-1}\|_{I}\,\leq\,2\,\|Q^{-1}\|^{2}\,\|Q\|_{I}\,.$

Proof. We start with an observation: By compactness, there exists an element
$P_{I}\in\mathfrak{A}_{I}$ satisfying $\|Q\|_{I}=\|Q-P_{I}\|$. But using the
conditional expectation, we have that the element

$Q_{I}:=\mathbb{E}_{I}(Q)\in\mathfrak{A}_{I}$

satisfies

$\|Q\|_{I}\leq\|Q-Q_{I}\|\leq 2\|Q\|_{I}\,.$ (9)

Indeed, the left-hand side is clear, and the right-hand side follows simply
from

$\displaystyle\|Q-Q_{I}\|$ $\displaystyle\leq\|Q-P_{I}\|+\|P_{I}-Q_{I}\|$
$\displaystyle=\|Q-P_{I}\|+\|\mathbb{E}_{I}(Q-P_{I})\|$ $\displaystyle\leq
2\,\|Q-P_{I}\|=2\|Q\|_{I}\,.$

Here, we have used that the conditional expectation is a unital completely
positive map and thus contractive. Hence, we can bound

$\displaystyle\|QQ^{\prime}\|_{I}$
$\displaystyle\,\leq\,\|QQ^{\prime}-Q_{I}Q^{\prime}_{I}\|$
$\displaystyle\,\leq\,\|Q\|\,\|Q^{\prime}-Q^{\prime}_{I}\|\,+\,\|Q-Q_{I}\|\,\|Q^{\prime}_{I}\|$
$\displaystyle\,\leq\,2\|Q\|\,\|Q^{\prime}\|_{I}+2\|Q\|_{I}\,\|Q^{\prime}\|$

If we assume that $Q$ is positive and invertible, that is

$\|Q^{-1}\|^{-1}\mathbbm{1}\leq Q\leq\|Q\|\mathbbm{1}\,$ (10)

we can then apply the conditional expectation to get from (10) that $Q_{I}$ is
also positive and invertible with

$\|Q^{-1}\|^{-1}\mathbbm{1}\leq Q_{I}\leq\|Q\|\mathbbm{1}\,.$ (11)

In particular, it easily follows that $\|Q_{I}^{-1}\|\leq\|Q^{-1}\|$.
Combining this and (9) we conclude

$\displaystyle\|Q^{-1}\|_{I}\leq\|Q^{-1}-Q_{I}^{-1}\|$
$\displaystyle\leq\|Q^{-1}\|\,\|Q-Q_{I}\|\,\|Q_{I}^{-1}\|$
$\displaystyle\leq\|Q^{-1}\|^{2}\,\|Q-Q_{I}\|$ $\displaystyle\leq
2\|Q^{-1}\|^{2}\,\|Q\|_{I}.$

This finishes the proof.∎

The following result appears in the proof of [2, Theorem 4.2], but we will
anyway prove it here for the sake of completeness as an immediate consequence
of Proposition 3.2.

######  Lemma 4.2.

Let $n\in\mathbb{N}$ and let $I_{n}:=[1-n,n]$. Then, for every observable
$Q\in\mathfrak{A}_{\mathbb{Z}}$ with finite support, every $s\in\mathbb{C}$
with $|s|\leq 1$ and every finite subset $\Lambda\subset\mathbb{Z}$

$\|\Gamma_{H_{\Lambda}}^{s}(Q)\|\leq\mathcal{G}^{2}\|Q\|+(1+\mathcal{G}^{2})\sum_{n\geq
1}\mathcal{G}^{2n}\|Q\|_{I_{n}}\,,\,$

where $\mathcal{G}=\mathcal{G}(1)$ is the constant from Proposition 3.2.

Proof. For each $n\in\mathbb{N}$, compactness allows us to fix
$P_{n}\in\mathfrak{A}_{I_{n}}$ such that $\|Q\|_{I_{n}}=\|Q-P_{n}\|$. We can
then write $Q=\sum_{n\geq 1}{Q_{n}}$, where $Q_{1}:=P_{1}$ and
$Q_{n}:=P_{n}-P_{n-1}$ for each $n>1$. Note that this series is actually a
finite sum, since $Q$ has finite support. On the one hand, we can easily bound

$\|Q_{1}\|\leq\|Q\|_{I_{1}}+\|Q\|\quad,\quad\|Q_{n}\|\leq\|Q\|_{I_{n}}+\|Q\|_{I_{n-1}}\quad(n\geq
2)\,.$

On the other hand, using Proposition 3.2 and noticing that $Q_{n}$ has support
in $I_{n}$, we have

$\|\Gamma_{H_{\Lambda}}^{s}(Q_{n})\|\,\leq\,\mathcal{G}^{2n}\,\|Q_{n}\|\,.$

Combining all these inequalities, we conclude

$\displaystyle\|\Gamma_{H_{\Lambda}}^{s}(Q)\|\leq\sum_{n\geq
1}\|\Gamma_{H_{\Lambda}}^{s}(Q_{n})\|$ $\displaystyle\leq\sum_{n\geq
1}\mathcal{G}^{2n}\|Q_{n}\|$
$\displaystyle\leq\,\mathcal{G}^{2}\|Q\|+(1+\mathcal{G}^{2})\sum_{n\geq
1}\mathcal{G}^{2n}\|Q\|_{I_{n}}\,.$

∎

The next proposition combines the two previous lemmas to give an estimate on
the norm of the complex time evolution of observables of a certain kind.

######  Proposition 4.3.

Let $I=ABC$ be a finite interval split into three adjacent intervals so that
$B$ shields $A$ from $C$, and let $Q\in\mathfrak{A}_{ABC}$ be a positive and
invertible observable. Define

$F(s)\,:=\,e^{sH_{AC}}\,\operatorname{tr}_{B}(\rho^{B}Q)\,e^{-sH_{AC}}\quad,\quad
s\in\mathbb{C}\,.$

Let us identify $B=[1,k]$ for some $k\geq 0$, where $k=0$ corresponds to the
case $B=\emptyset$, and denote $I_{n}=[1-n,k+n]$ for each $n\geq 1$. Then, for
any $s\in\mathbb{C}$ such that $|s|\leq 1$,

$\displaystyle\|F(s)\|$
$\displaystyle\leq(1+\mathcal{G})^{2}\bigg{(}\|Q\|+\sum_{n\geq
1}\mathcal{G}^{2n}\,\|Q\|_{I_{n}}\bigg{)}\,,$ $\displaystyle\|F(s)^{-1}\|$
$\displaystyle\leq(1+\mathcal{G})^{2}\bigg{(}\|Q^{-1}\|+2\|Q^{-1}\|^{2}\sum_{n\geq
1}\mathcal{G}^{2n}\,\|Q\|_{I_{n}}\bigg{)}\,.$

Proof. Let us denote $\widetilde{Q}:=\operatorname{tr}_{B}(\rho^{B}Q)$. Since
$Q$ is positive and invertible,

$\|Q^{-1}\|^{-1}\,\mathbbm{1}\,\leq\,Q\,\leq\,\|Q\|\,\mathbbm{1}\,.$

As a consequence, if we apply the (positive and unital) map
$P\mapsto\operatorname{tr}_{B}(\rho^{B}P)$ we get

$\|Q^{-1}\|^{-1}\,\mathbbm{1}\,\leq\,\widetilde{Q}\,\leq\,\|Q\|\,\mathbbm{1}\,.$
(12)

Since $P\mapsto\operatorname{tr}_{B}(\rho^{B}P)$ is moreover contractive, we
can also get

$\|\widetilde{Q}\|_{I_{n}}\,\leq\,\|Q\|_{I_{n}}\,.$ (13)

At this point, note that all factors

$F(s)=e^{sH_{AC}}\,\widetilde{Q}\,e^{-sH_{AC}}$

are supported in $A\cup C$. Making use of the canonical identification
$\mathfrak{A}_{A}\otimes\mathfrak{A}_{C}\hookrightarrow\mathfrak{A}_{ABC}$
described in Section 3.2, we can “ignore” $B$ and consider $A$ and $C$ as two
adjacent intervals of the form $[1-a,0]$ and $[1,c]$ for some
$a,c\in\mathbb{N}$. In this case, the Hamiltonian $H_{AC}$ is made of
interactions with the same bounds for their range and strength. Moreover, the
interval $I_{n}$ can be identified with $\widetilde{I}_{n}=[1-n,n]$ for each
$n\in\mathbb{N}$, so that applying Lemma 4.2

$\|F(s)\|\,\leq\,\mathcal{G}^{2}\|Q\|+(1+\mathcal{G}^{2})\sum_{n\geq
1}\mathcal{G}^{2n}\,\|Q\|_{I_{n}}\,.$

On the other hand, combining (13) with Lemma 4.1,

$\|\widetilde{Q}^{-1}\|_{\widetilde{I}_{n}}\,\leq\,2\,\|\widetilde{Q}^{-1}\|^{2}\,\|\widetilde{Q}\|_{\widetilde{I}_{n}}\leq
2\|Q^{-1}\|^{2}\,\|Q\|_{I_{n}}\,,$

where in the last inequality we have used that
$\big{\|}\widetilde{Q}^{-1}\big{\|}\leq\left\|Q^{-1}\right\|$ due to (12).
Thus, using that

$F(s)^{-1}=e^{sH_{AC}}\,\widetilde{Q}^{-1}\,e^{-sH_{AC}}\,,$

we can argue as above, applying Lemma 4.2 to estimate

$\|F(s)^{-1}\|\leq\mathcal{G}^{2}\|Q^{-1}\|+2(1+\mathcal{G}^{2})\|Q^{-1}\|^{2}\sum_{n\geq
1}\mathcal{G}^{2n}\,\|Q\|_{I_{n}}\,.$

This finishes the proof.∎

The previous proposition allows us to bound the norm of more complicated local
observables which we will encounter often in later sections.

######  Corollary 4.4.

Let $I=ABC$ be three adjacent finite intervals (we admit the possibility of
some being empty). Then, there is an absolute constant $\mathcal{C}$ depending
only on the strength $J$ and range $r$ of the local interactions, such that

$\displaystyle\big{\|}\operatorname{tr}_{B}(\rho^{B}Q)\big{\|},\big{\|}\operatorname{tr}_{B}(\rho^{B}Q)^{-1}\big{\|}\,\leq\,\mathcal{C}\,\,,\quad\quad
Q\in\\{E_{B,C}^{\dagger}\,,\,E_{B,C}\,,\,E_{A,B}^{\dagger}\,,\,E_{A,B}\\}\,,$
(14)
$\displaystyle\big{\|}\operatorname{tr}_{AB}(\rho^{AB}Q)\big{\|}\,,\,\big{\|}\operatorname{tr}_{AB}(\rho^{AB}Q)^{-1}\big{\|}\,\leq\,\mathcal{C}\,\,,\quad\quad
Q\in\\{E_{A,B}^{\dagger\,-1}\,,\,E_{A,B}^{\,-1}\\}\,,$ (15)
$\displaystyle\big{\|}\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}E_{AB,C}^{\dagger}\big{)}\big{\|}\,,\,\big{\|}\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}E_{AB,C}^{\dagger}\big{)}^{-1}\big{\|}\,\leq\,\mathcal{C}\,.$
(16)

Proof. Let us identify $B$ with the interval $[1,k]$ for some $k\geq 0$, where
$k=0$ corresponds to $B$ being empty, and define $I_{n}=[1-n,k+n]$ for each
$n\in\mathbb{N}$. We prove (14) for the case $Q=E_{A,B}$, since the argument
for the other three cases is analogous. Let

$F_{A,B}:=\operatorname{tr}_{B}(\rho^{B}E_{A,B})=e^{-\frac{1}{2}H_{A}}\,\operatorname{tr}_{B}(\rho^{B}Q_{A,B})\,e^{\frac{1}{2}H_{A}}\,,$

where

$Q_{A,B}:=E_{A,B}^{\dagger}(\tfrac{1}{2})\,E_{A,B}(\tfrac{1}{2})\,.$

Using Lemma 4.1 and Corollary 3.4,

$\|Q_{A,B}\|_{I_{n}}\,\leq\,2\,\|E_{A,B}^{\dagger}(\tfrac{1}{2})\|_{I_{n}}\mathcal{G}+2\,\mathcal{G}\|E_{A,B}(\tfrac{1}{2})\|_{I_{n}}\,\leq\,\frac{4\,\mathcal{G}^{n+1}}{(\lfloor
n/r\rfloor+1)!}\,.$

The last inequality can be seen by choosing $E_{A\cap I_{n},B\cap I_{n}}(1/2)$
and its adjoint as local approximations. Thus, by Proposition 4.3

$\|F_{A,B}\|\leq(1+\mathcal{G})^{2}\bigg{(}\mathcal{G}^{2}+\sum_{n\geq
1}\frac{4\,\mathcal{G}^{3n+1}}{(\lfloor n/r\rfloor+1)!}\,\bigg{)}$

and

$\|F_{A,B}^{-1}\|\leq(1+\mathcal{G})^{2}\bigg{(}\mathcal{G}^{2}+\sum_{n\geq
1}\frac{8\,\mathcal{G}^{3n+5}}{(\lfloor n/r\rfloor+1)!}\,\bigg{)}\,.$

Let us next prove (15). We just argue in the case where $Q=E_{A,B}^{-1}$,
since the other cases are completely analogous. Let

$\widetilde{F}_{A,B}:=\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{-1}\big{)}=\operatorname{tr}_{AB}\big{(}\rho^{AB}\,\widetilde{Q}_{A,B}\big{)}\,,$

where

$\widetilde{Q}_{A,B}:=E_{A,B}^{\dagger\,-1}\big{(}\tfrac{1}{2}\big{)}E_{A,B}^{\,-1}\big{(}\tfrac{1}{2}\big{)}\,.$

Arguing as before we get the desired inequalities. Finally, let us prove (16).
Let us first rewrite

$\displaystyle
F_{A,B,C}:=\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,$
$\displaystyle=\,\operatorname{tr}_{B}\big{(}\rho^{B}\,e^{H_{A}+H_{B}+H_{C}}e^{-H_{ABC}}\big{)}$
$\displaystyle=e^{\frac{1}{2}H_{A}+\frac{1}{2}H_{C}}\,\operatorname{tr}_{B}\big{(}\rho^{B}\,Q_{A,B,C}\big{)}\,e^{-\frac{1}{2}H_{A}-\frac{1}{2}H_{C}}\,,$

where

$Q_{A,B,C}:=E_{A,B}^{\dagger}\big{(}\tfrac{1}{2}\big{)}E_{AB,C}^{\dagger}\big{(}\tfrac{1}{2}\big{)}E_{AB,C}\big{(}\tfrac{1}{2}\big{)}E_{A,B}\big{(}\tfrac{1}{2}\big{)}\,.$

According to Corollary 3.4, we can bound

$\big{\|}Q_{A,B,C}\big{\|},\big{\|}Q_{A,B,C}^{-1}\big{\|}\leq\mathcal{G}^{4}\,.$

Moreover, using Lemma 4.1 and Corollary 3.4

$\displaystyle\big{\|}Q_{A,B,C}\big{\|}_{I_{n}}\,$
$\displaystyle\leq\,2\,\big{\|}E_{A,B}^{\dagger}\big{(}\tfrac{1}{2}\big{)}\,E_{AB,C}^{\dagger}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}\,\mathcal{G}^{2}+2\,\mathcal{G}^{2}\,\big{\|}E_{AB,C}\big{(}\tfrac{1}{2}\big{)}\,E_{A,B}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}$
$\displaystyle\leq\,4\,\mathcal{G}^{3}\Big{(}\,\big{\|}E_{A,B}^{\dagger}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}\,+\,\big{\|}E_{AB,C}^{\dagger}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}\,+\,\big{\|}E_{AB,C}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}\,+\,\big{\|}E_{A,B}\big{(}\tfrac{1}{2}\big{)}\big{\|}_{I_{n}}\,\Big{)}$
$\displaystyle\leq\,\frac{16\,\mathcal{G}^{n+3}}{\big{(}\lfloor
n/r\rfloor+1\big{)}!}\,.$

Thus, applying Proposition 4.3,

$\big{\|}F_{A,B,C}\big{\|}\leq(1+\mathcal{G})^{2}\,\bigg{(}\,\mathcal{G}^{4}+\sum_{n\geq
1}\frac{16\,\mathcal{G}^{3n+3}}{\big{(}\lfloor
n/r\rfloor+1\big{)}!}\,\bigg{)}$

and

$\big{\|}F_{A,B,C}^{-1}\big{\|}\leq(1+\mathcal{G})^{2}\,\bigg{(}\,\mathcal{G}^{4}+\mathcal{G}^{8}\,\sum_{n\geq
1}\frac{32\,\mathcal{G}^{3n+3}}{\big{(}\lfloor
n/r\rfloor+1\big{)}!}\,\bigg{)}\,.$

∎

##  5 Approximate factorization of the Gibbs state

The following Theorem is our first main result. It shows that the Gibbs state
of a local Hamiltonian on $ABC$ can be approximated by a certain product of
its marginals. The error made in the approximation decays superexponentially
in the length scale of the system under consideration. Note that by the
results of [9], the two following conditions are equivalent for any quantum
channel $\mathcal{T}$ and any strictly positive states $\rho$ and $\sigma$:

$\rho=\sigma\,\mathcal{T}^{*}\left(\mathcal{T}(\sigma)^{-1}\mathcal{T}(\rho)\right)\qquad\iff\qquad\widehat{D}(\rho||\sigma)=\widehat{D}(\mathcal{T}(\rho)||\mathcal{T}(\sigma))\,,$

where the map
$\mathcal{B}_{\mathcal{T}}^{\sigma}(\cdot):=\sigma\,\mathcal{T}^{*}\left(\mathcal{T}(\sigma)^{-1}(\cdot)\right)$
is called BS-recovery condition, in analogy to the Petz recovery map for the
analogous equality in the data-processing inequality for the relative entropy.
In the particular case of a tripartite space
$\mathcal{H}_{ABC}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$,
two strictly positive states $\rho_{ABC}$, $\sigma_{ABC}\in\mathfrak{A}_{ABC}$
such that $\sigma_{ABC}=\rho_{AB}\otimes\mathds{1}_{C}/d_{C}$ and a quantum
channel $\mathcal{T}:=\mathbb{E}_{A}$, we say that $\rho_{ABC}$ is a BS-
recoverable state if it is a fixed point of the BS-recovery condition composed
with the partial trace, i.e., if

$\rho_{ABC}=\rho_{AB}\rho_{B}^{-1}\rho_{BC}\,.$

Therefore, any quantum Markov chain is a BS-recoverable state but the converse
is not true [28]. Moreover, note that, by the equivalence stated above, the
following clearly holds

$\rho_{ABC}=\rho_{AB}\rho_{B}^{-1}\rho_{BC}\qquad\iff\qquad\widehat{D}(\rho_{ABC}||\rho_{AB})=\widehat{D}(\rho_{BC}||\rho_{B})$

We emphasize that the BS-entropy on the very right is the one on
$\mathcal{H}_{BC}$. The main result of this section thus states that the
distance of a quantum Gibbs state from being BS-recoverable decays
superexponentially with the size of the middle system. This result should be
compared to the findings in [30], since the equality
$D(\rho_{ABC}||\rho_{AB})=D(\rho_{BC}||\rho_{B})$ is equivalent to
$\rho_{ABC}$ being a quantum Markov chain.

$A$$B$$C$$I$ Figure 2: Representation of an interval $I$ split into three
subintervals $I=ABC$, where $B$ shields $A$ from $C$. Here we are taking
$\ell=5$.

######  Theorem 5.1.

Let $\Phi$ be a finite-range interaction over $\mathbb{Z}$. Then, there exists
a positive and decreasing function $\ell\longmapsto\epsilon(\ell)$ with
superexponential decay and depending only on the strength $J$ and the range
$r$ of the interaction with the following property: for every
$I\subset\mathbb{Z}$ finite interval split into three subintervals $I=ABC$,
where $B$ shields $A$ from $C$ and $|B|\geq\ell$ $($see Figure 2$)$, and for
$\rho=\rho^{I}$ the Gibbs state on $I$,

$\big{\|}\rho_{ABC}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\big{\|}_{1}\,\leq\,\epsilon(\ell)\,.$

Proof. Using Hölder’s inequality

$\displaystyle\big{\|}\rho_{ABC}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\big{\|}_{1}$
$\displaystyle\leq\big{\|}\mathbbm{1}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\rho_{ABC}^{-1}\big{\|}\,\|\rho_{ABC}\|_{1}$
$\displaystyle=\big{\|}\mathbbm{1}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\rho_{ABC}^{-1}\big{\|}\,.$

We are actually going to bound

$\big{\|}\mathbbm{1}-\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}\big{\|}\,\leq\,\tilde{\epsilon}(\ell)\,.$
(17)

The above result will follow by simply recalling that if an operator $Q$
satisfies
$\|\mathbbm{1}-Q\|\leq\tilde{\epsilon}<1$, then

$\big{\|}\mathbbm{1}-Q^{-1}\big{\|}=\big{\|}\mbox{$\sum_{k\geq
1}(\mathbbm{1}-Q)^{k}$}\big{\|}\leq\tilde{\epsilon}/(1-\tilde{\epsilon})\,.$
(18)

For large $\ell$, the assertion follows from (18) by taking e.g.
$\epsilon(\ell)=2\tilde{\epsilon}$. For small $\ell$, we can directly bound
the LHS of (17) by a constant, as we show below. To estimate (17), let us
first rewrite

$\displaystyle\rho_{ABC}\,\rho_{BC}^{-1}$
$\displaystyle=e^{-H_{ABC}}\,\operatorname{tr}_{A}\big{(}e^{-H_{ABC}}\big{)}^{-1}\,$
$\displaystyle=e^{-H_{ABC}}\,e^{H_{A}+H_{BC}}\,e^{-H_{A}}\,\operatorname{tr}_{A}\big{(}\,e^{-H_{ABC}}e^{H_{A}+H_{BC}}e^{-H_{A}}\big{)}^{-1}\,$
$\displaystyle=E_{A,BC}\,\rho^{A}\,\operatorname{tr}_{A}\big{(}E_{A,BC}\,\rho^{A}\big{)}^{-1}\,,\,$

where we are using the notation $\rho^{\Lambda}$ and $E_{\Lambda,\Lambda^{c}}$
introduced in Section 3.4. Analogously

$\displaystyle\rho_{B}\,\rho_{AB}^{-1}$
$\displaystyle=\operatorname{tr}_{AC}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}\big{)}^{-1}$
$\displaystyle=\operatorname{tr}_{AC}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{tr}_{C}\big{(}e^{-H_{BC}}\big{)}^{-1}\,\left(\,\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{tr}_{C}\big{(}e^{-H_{BC}}\big{)}^{-1}\,\right)^{-1}$
$\displaystyle=\operatorname{tr}_{A}\left(\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}\big{)}\operatorname{tr}_{C}\big{(}e^{-H_{BC}}\big{)}^{-1}\right)\,\left(\,\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{tr}_{C}\big{(}e^{-H_{BC}}\big{)}^{-1}\,\right)^{-1}$
$\displaystyle=\,\operatorname{tr}_{A}\big{(}\widetilde{E}_{A,BC}\,\rho^{A}\big{)}\,(\rho^{A})^{-1}\,\widetilde{E}_{A,BC}^{-1}\,,$

where

$\widetilde{E}_{A,BC}:=\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{tr}_{C}\big{(}e^{-H_{BC}}\big{)}^{-1}\,e^{H_{A}}\,.$

Therefore

$\rho_{ABC}\,\rho_{BC}^{-1}\,\rho_{B}\,\rho_{AB}^{-1}=E_{A,BC}\,\operatorname{tr}_{A}\big{(}E_{A,BC}\,\rho^{A}\big{)}^{-1}\,\operatorname{tr}_{A}\big{(}\widetilde{E}_{A,BC}\,\rho^{A}\big{)}\,\,\widetilde{E}_{A,BC}^{\,-1}\,,$
(19)

where the outer $\rho^{A}$’s have cancelled each other. We include now two
statements whose proof is postponed by now.

Statement 1: The factors on the right-hand side of (19) and their inverses are
uniformly bounded by a constant $\widehat{\mathcal{C}}>0$ depending on $J$ and
$r$ of the local interaction $\Phi$ $($but independent of $A,B,C$$)$.

Statement 2: There is a superexponentially decaying positive function
$\ell\longmapsto\delta(\ell)$ depending only on $J$ and $r$ of the local
interaction $($but independent of $A,B,C$$)$ such that

$\big{\|}\widetilde{E}_{A,BC}-E_{A,BC}\big{\|}\leq\delta(\ell)\,.\\\ $

Using that $Q\longmapsto\operatorname{tr}_{A}(Q\,\rho^{A})$ is contractive, we
immediately deduce from the second statement that

$\|\operatorname{tr}_{A}(\widetilde{E}_{A,BC}\,\rho^{A})-\operatorname{tr}_{A}(E_{A,BC}\,\rho^{A})\|\leq\delta(\ell)\,.\\\
$

Therefore, replacing in the right-hand side of (19)

$E_{A,BC}\leadsto\widetilde{E}_{A,BC}\quad\mbox{ and
}\quad\operatorname{tr}_{A}\big{(}\widetilde{E}_{A,BC}\,\rho^{A}\big{)}\leadsto\operatorname{tr}_{A}\big{(}E_{A,BC}\,\rho^{A}\big{)}\,,$

we can estimate

$\displaystyle\big{\|}\rho_{ABC}\,\rho_{BC}^{-1}\,\rho_{B}\rho_{AB}^{-1}-\mathbbm{1}\big{\|}$
$\displaystyle\leq\widehat{\mathcal{C}}^{\,3}\,\big{\|}E_{A,BC}-\widetilde{E}_{A,BC}\big{\|}+\widehat{\mathcal{C}}^{\,3}\,\big{\|}\operatorname{tr}_{A}\big{(}\widetilde{E}_{A,BC}\,\rho^{A}\big{)}-\operatorname{tr}_{A}\big{(}E_{A,BC}\,\rho^{A}\big{)}\big{\|}$
$\displaystyle\leq
2\,\widehat{\mathcal{C}}^{\,3}\,\delta(\ell)=:\epsilon(\ell)\,.$

For large $\ell$, the assertion follows from (18). For small $\ell$, we can
use statement 1 to upper bound
$\big{\|}\mathbbm{1}-\rho_{AB}\rho_{B}^{-1}\rho_{BC}\rho_{ABC}^{-1}\big{\|}$
by a constant. This concludes the proof. ∎

It remains to argue that both statements hold.

Proof of Statement 1. By Corollaries 3.4 and 4.4, we can bound

$\big{\|}E_{A,BC}\big{\|}\leq\mathcal{G}\quad\quad\text{and}\quad\quad\big{\|}\operatorname{tr}_{A}(\rho^{A}E_{A,BC})^{-1}\big{\|}\leq\mathcal{C}\,.$

To bound the third and fourth factors, let us first rewrite

$\displaystyle\widetilde{E}_{A,BC}$
$\displaystyle=\operatorname{tr}_{C}\big{(}e^{-H_{ABC}}e^{H_{AB}}\big{)}\,e^{-H_{AB}}\,e^{H_{A}+H_{B}}\,\operatorname{tr}_{C}\big{(}e^{-H_{BC}}e^{H_{B}}\big{)}^{-1}$
$\displaystyle=\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}\,E_{A,B}\,\operatorname{tr}_{C}\big{(}E_{B,C}\,\rho^{C}\big{)}^{-1}\,.$
(20)

We can apply once again Corollaries 3.4 and 4.4 to bound

$\big{\|}\widetilde{E}_{A,BC}\big{\|}\,\leq\,\mathcal{G}\,\mathcal{C}^{2}\quad\mbox{
and
}\quad\big{\|}\widetilde{E}_{A,BC}^{\,-1}\big{\|}\,\leq\,\mathcal{G}\,\mathcal{C}^{2}\,.$

Finally, since $Q\longmapsto\operatorname{tr}_{A}(Q\,\rho^{A})$ contractive,

$\big{\|}\operatorname{tr}_{A}\big{(}\widetilde{E}_{A,BC}\,\rho^{A}\big{)}\big{\|}\leq\big{\|}\widetilde{E}_{A,BC}\big{\|}\leq\mathcal{G}\,\mathcal{C}^{2}\,.$

The inverse of this term can be bounded by the same quantity using Jensen’s
operator inequality [23]. This finishes the proof of the first statement.∎

$A$$B_{\text{left}}$$B_{\text{right}}$$C$$I$ Figure 3: Splitting of an
interval $I$ into three subintervals $I=ABC$, where $B$ shields $A$ from $C$.
Note that $B$ is further split into two parts
$B=B_{\mathrm{left}}B_{\mathrm{right}}$ with
$|B_{\mathrm{left}}|,|B_{\mathrm{right}}|\geq\lfloor\ell/2\rfloor$. Here we
consider $\ell=6$.

Proof of Statement 2. Using formula (20) for $\widetilde{E}_{A,BC}$, we can
rewrite the difference $\widetilde{E}_{A,BC}-E_{A,BC}$ as

$\left(\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}\,E_{A,B}-E_{A,BC}\,\operatorname{tr}_{C}\big{(}E_{B,C}\,\rho^{C}\big{)}\,\right)\,\left(\operatorname{tr}_{C}\big{(}E_{B,C}\,\rho^{C}\big{)}\right)^{-1},$

and thus, by Corollary 4.4

$\big{\|}\widetilde{E}_{A,BC}-E_{A,BC}\big{\|}\,\leq\,\mathcal{C}\left\|\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}\,E_{A,B}-E_{A,BC}\,\operatorname{tr}_{C}\big{(}E_{B,C}\,\rho^{C}\big{)}\,\right\|\,.$

Let us split the set $B$ into two parts
$B=B_{\mathrm{left}}B_{\mathrm{right}}$ with
$|B_{left}|,|B_{right}|\geq\lfloor\ell/2\rfloor$ (see Figure 3). Taking into
account that $E_{A,B_{\mathrm{left}}}$ and
$\operatorname{tr}_{C}\big{(}E_{B_{\mathrm{right}},C}\,\rho^{C}\big{)}$
commute, we can estimate

$\displaystyle\big{\|}\widetilde{E}_{A,BC}-E_{A,BC}\big{\|}\,$
$\displaystyle\leq\,\mathcal{C}\,\left\|\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}\,E_{A,B}-\operatorname{tr}_{C}\big{(}E_{B_{\mathrm{right}},C}\,\rho^{C}\big{)}\,E_{A,B_{\mathrm{left}}}\right\|$
$\displaystyle\hskip
8.5359pt+\mathcal{C}\left\|E_{A,B_{\mathrm{left}}}\,\operatorname{tr}_{C}\big{(}E_{B_{\mathrm{right}},C}\,\rho^{C}\big{)}-E_{A,BC}\,\operatorname{tr}_{C}\big{(}E_{B,C}\,\rho^{C}\big{)}\,\right\|$
$\displaystyle\leq\mathcal{C}\,\left\|\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}\right\|\,\left\|E_{A,B}-E_{A,B_{\mathrm{left}}}\right\|$
$\displaystyle\hskip
8.5359pt+\mathcal{C}\left\|\operatorname{tr}_{C}\big{(}E_{AB,C}\,\rho^{C}\big{)}-\operatorname{tr}_{C}\big{(}E_{B_{\mathrm{right}},C}\,\rho^{C}\big{)}\right\|\,\left\|E_{A,B_{\mathrm{left}}}\right\|$
$\displaystyle\hskip
8.5359pt+\mathcal{C}\left\|E_{A,B_{\mathrm{left}}}\right\|\,\left\|\operatorname{tr}_{C}\left(E_{B_{\mathrm{right}},C}\,\rho^{C}\right)-\operatorname{tr}_{C}\left(E_{B,C}\,\rho^{C}\right)\right\|$
$\displaystyle\hskip
8.5359pt+\mathcal{C}\,\left\|E_{A,B_{\mathrm{left}}}-E_{A,BC}\right\|\,\left\|\operatorname{tr}_{C}\big{(}E_{B,C}\rho^{C}\big{)}\right\|$
$\displaystyle\leq\mathcal{C}\,\big{\|}E_{AB,C}\big{\|}\,\big{\|}E_{A,B}-E_{A,B_{\mathrm{left}}}\big{\|}\,+\,\mathcal{C}\big{\|}E_{AB,C}-E_{B_{\mathrm{right}},C}\big{\|}\,\big{\|}E_{A,B_{\mathrm{left}}}\big{\|}$
$\displaystyle\hskip
8.5359pt+\mathcal{C}\big{\|}E_{A,B_{\mathrm{left}}}\big{\|}\,\big{\|}E_{B_{\mathrm{right}},C}-E_{B,C}\big{\|}\,+\,\mathcal{C}\,\big{\|}E_{A,B_{\mathrm{left}}}-E_{A,BC}\big{\|}\,\|E_{B,C}\|\,,$

where in the last inequality we have used again that the operator
$Q\longmapsto\operatorname{tr}_{C}\big{(}Q\,\rho^{C}\big{)}$ is contractive.
Finally, applying Corollary 3.4 and Remark 3.5 we can bound from above the
previous expression

$\displaystyle\big{\|}\widetilde{E}_{A,BC}-E_{A,BC}\big{\|}\,\leq\,\frac{4\,\mathcal{C}\,\mathcal{G}^{1+\lfloor\ell/2\rfloor}}{\left(\lfloor\,\lfloor\ell/2\rfloor/r\,\rfloor+1\right)!}\,.$

This easily yields the desired bound.∎

##  6 Correlations in finite intervals vs the infinite chain

Thermal states on the infinite chain $\mathbb{Z}$ are characterized in terms
of the KMS condition, after which they are also called _KMS states_. In our
setting, dealing with a finite-range interaction $\Phi$, it is well-known
that, for every finite temperature, the corresponding KMS state exists and is
unique [15, Theorem 6.2.47].

For every finite subinterval $I\subset\mathbb{Z}$, let us denote by
$\psi_{I}:\mathfrak{A}_{I}\longrightarrow\mathbb{C}$ the local Gibbs state as

$\psi_{I}(Q):=\operatorname{Tr}_{I}(\rho^{I}Q)=\frac{\operatorname{Tr}_{I}(e^{-H_{I}}\,Q)}{\operatorname{Tr}_{I}(e^{-H_{I}})}\,\,,\,\,Q\in\mathfrak{A}_{I}\,,\,$
(21)

and extended to a state on $\mathfrak{A}_{\mathbb{Z}}$ using the Hahn-Banach
Theorem as in [14, Proposition 2.3.24]. We are absorbing the inverse
temperature constant $\beta$ in the Hamiltonian to simplify. We denote by
$\psi_{\mathbb{Z}}$ the unique KMS state over the infinite chain (at inverse
temperature $\beta=1$). For every increasing and absorbing sequence
$I_{n}\nearrow\mathbb{Z}$, the sequence of states $\psi_{I_{n}}$ is weak∗
convergent to $\psi_{\mathbb{Z}}$, i.e.,

$\psi_{\mathbb{Z}}(Q)=\lim_{n\rightarrow\infty}\psi_{I_{n}}(Q)\,\,,\quad
Q\in\mathfrak{A}_{\mathbb{Z}}$ (22)

(see e.g. [15, Proposition 6.2.15 and its preceeding discussion]).

In his seminal paper, Araki [2] proved that if $\Phi$ is moreover translation
invariant, then it satisfies _exponential decay of correlations_ : There exist
constants $\mathcal{K}$, $\alpha>0$ (depending only on the range and the
strength of the interaction) such that for every pair
$Q_{A}\in\mathfrak{A}_{A}$ and $Q_{C}\in\mathfrak{A}_{C}$ with support in
finite intervals $A,C$,

$\big{|}\psi_{\mathbb{Z}}(Q_{A}Q_{C})-\psi_{\mathbb{Z}}(Q_{A})\psi_{\mathbb{Z}}(Q_{C})\big{|}\,\leq\,\mathcal{K}\,e^{-\alpha\,\operatorname{dist}(A,C)}\,\|Q_{A}\|\,\|Q_{C}\|\,.$
(23)

A natural question is whether these results imply that the interaction $\Phi$
satisfies the uniform clustering condition [13], as stated below, with a
function $\varepsilon(\ell)$ decaying exponentially fast in $\ell$ and
depending only on the range and interaction strength.

######  Definition 6.1 (Uniform clustering).

Let $\Phi$ be a local interaction on $\mathbb{Z}$. We say that it is _uniform
clustering_ if there is a positive and decreasing function
$\ell\mapsto\varepsilon(\ell)$ with the following property: for every finite
interval $I\subset\mathbb{Z}$ split into three subintervals $I=ABC$ with
$|B|\geq\ell$,

$\operatorname{Corr}_{\rho^{I}}(A:C)\leq\varepsilon(\ell)\,.$ (24)

In particular, we will say that it satisfies _exponential uniform clustering_
if $\varepsilon(\ell)$ can be taken to have exponential decay as $\ell$ tends
to infinity.

In this line, we can actually prove that the exponential decay of correlations
proven by Araki implies exponential uniform clustering. That is the content of
our next main result.

######  Theorem 6.2.

Let $\Phi$ be finite-range interaction over $\mathbb{Z}$. If the infinite-
chain thermal state $\psi_{\mathbb{Z}}$ satisfies the exponential decay of
correlations condition, then there are constants $\widetilde{\mathcal{K}}$ and
$\widetilde{\alpha}$ $($depending only on $\mathcal{K}$, $\alpha$ from (23),
and on the range $r$ and strength $J$ of the interaction$)$ with the following
property: for every finite interval $I\subset\mathbb{Z}$ split into three
subintervals $I=ABC$ with $|B|\geq\ell\geq 0$ and every observable
$Q_{A}\in\mathfrak{A}_{A}$ and $Q_{C}\in\mathfrak{A}_{C}$,

$\big{|}\psi_{ABC}(Q_{A}Q_{C})\,-\,\psi_{ABC}(Q_{A})\,\psi_{ABC}(Q_{C})\big{|}\,\,\leq\,\,\|Q_{A}\|\,\|Q_{C}\|\,\widetilde{\mathcal{K}}\,e^{-\widetilde{\alpha}\ell}\,.$

Proof. We can assume, without loss of generality that $|A|,|B|,|C|\geq\ell$.
Indeed, denoting $\ell^{\prime}=\lfloor\ell/3\rfloor$, which has the same
scale as $\ell$, we can split $B$ into three subintervals $B=B_{1}B_{2}B_{3}$
and redefine $A^{\prime}:=AB_{1}$, $B^{\prime}:=B_{2}$ and
$C^{\prime}:=B_{3}C$ satisfying
$|A^{\prime}|,|B^{\prime}|,|C^{\prime}|\geq\ell^{\prime}$. Moreover, we are
going to add two adjacent intervals
$\widetilde{C}=\widetilde{C}_{1}\widetilde{C}_{2}$ with
$|\widetilde{C}_{1}|\geq\ell$ on the right side of $ABC$, and other two
intervals $\widetilde{A}=\widetilde{A}_{2}\widetilde{A}_{1}$ with
$|\widetilde{A}_{1}|\geq\ell$ on the left side of $ABC$. Thus we have a larger
finite interval

$\widetilde{I}\,=\,\widetilde{A}ABC\widetilde{C}\,=\,\widetilde{A}_{2}\widetilde{A}_{1}ABC\widetilde{C}_{1}\widetilde{C}_{2}\,.$

$\widetilde{A}$$\widetilde{A}_{2}$$\widetilde{A}_{1}$$A$$B$$C$$\widetilde{C}$$\widetilde{C}_{1}$$\widetilde{C}_{2}$$\widetilde{I}$
Figure 4: Splitting of an interval $I$ into three subintervals $I=ABC$ with
$|A|,|B|,|C|\geq\ell$, to which we further append $\widetilde{A}$ and
$\widetilde{C}$ at the left and right side, respectively. Subsequently, we
split $\widetilde{A}$ and $\widetilde{C}$ into two subintervals each, so that
each of them is of size at least $\ell$.

We will repeatedly use that for scalars
$a,a^{\prime},b,b^{\prime}\in\mathbb{C}$

$\left|\frac{a}{b}-\frac{a^{\prime}}{b^{\prime}}\right|\leq\frac{1}{|b|}|a-a^{\prime}|+\frac{|a^{\prime}|}{|b|\,|b^{\prime}|}\,|b-b^{\prime}|\,.$
(25)

Let us denote

$\Theta_{\widetilde{A},ABC,\widetilde{C}}\,:=\,e^{\frac{1}{2}H_{\widetilde{A}ABC\widetilde{C}}}\,e^{-\frac{1}{2}H_{\widetilde{A}}-\frac{1}{2}H_{ABC}-\frac{1}{2}H_{\widetilde{C}}}\,=\,\Theta_{\widetilde{A}ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC}\,,$

where

$\Theta_{\widetilde{A}ABC,\widetilde{C}}\,:=\,e^{\frac{1}{2}H_{\widetilde{A}ABC\widetilde{C}}}\,e^{-\frac{1}{2}H_{\widetilde{A}ABC}-\frac{1}{2}H_{\widetilde{C}}}\quad,\quad\Theta_{\widetilde{A},ABC}\,:=\,e^{\frac{1}{2}H_{\widetilde{A}ABC}}e^{-\frac{1}{2}H_{\widetilde{A}}\,-\frac{1}{2}H_{ABC}}\,,$

and in general

$\Theta_{X,Y}\,:=\,e^{\frac{1}{2}H_{XY}}\,e^{-\frac{1}{2}H_{X}-\frac{1}{2}H_{Y}}\,.$

Note that the norm and locality properties of these observables (and their
inverses) are known by Corollary 3.4. In particular, we can approximate,

$\Theta_{\widetilde{A},ABC,\widetilde{C}}\,\approx\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{C,\widetilde{C}_{1}}$
(26)

using the aforementioned corollary and the conditions
$|A|,|\widetilde{A}_{1}|,|B|,|\widetilde{B}_{1}|\geq\ell$. Indeed

$\begin{split}\big{\|}\Theta_{\widetilde{A},ABC,\widetilde{C}}\,-\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{C,\widetilde{C}_{1}}\big{\|}&=\big{\|}\Theta_{\widetilde{A}ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC}\,-\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{C,\widetilde{C}_{1}}\big{\|}\\\\[5.69054pt]
&\leq\mathcal{G}\big{\|}\Theta_{\widetilde{A},ABC}-\Theta_{\widetilde{A}_{1},A}\big{\|}+\mathcal{G}\,\big{\|}\Theta_{\widetilde{A}ABC,\widetilde{C}}-\Theta_{C,\widetilde{C}_{1}}\big{\|}\\\\[5.69054pt]
&\leq
2\,\mathcal{G}\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,.\end{split}$
(27)

Let $Q:=Q_{A}Q_{C}$. Next, let us rewrite

$\begin{split}\psi_{ABC}(Q)\,\,&=\,\,\frac{\mathrm{Tr}_{ABC}\left(e^{-H_{ABC}}\,Q\right)}{\mathrm{Tr}_{ABC}\left(e^{-H_{ABC}}\right)}\\\\[5.69054pt]
&=\,\frac{\mathrm{Tr}_{\widetilde{A}ABC\widetilde{C}}\left(e^{-H_{\widetilde{A}}-H_{ABC}-H_{\widetilde{C}}}\,Q\right)}{\mathrm{Tr}_{\widetilde{A}ABC\widetilde{C}}\left(e^{-H_{\widetilde{A}}-H_{ABC}-H_{\widetilde{C}}}\right)}\\\\[8.53581pt]
&=\,\frac{\mathrm{Tr}_{\widetilde{A}ABC\widetilde{C}}\left(e^{-H_{\widetilde{A}ABC\widetilde{C}}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}\,Q\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\right)}{\mathrm{Tr}_{\widetilde{A}ABC\widetilde{C}}\left(e^{-H_{\widetilde{A}ABC\widetilde{C}}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\right)}\\\\[8.53581pt]
&=\frac{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\Theta_{\widetilde{A},ABC,\widetilde{C}}\,Q\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\right)}{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\Theta_{\widetilde{A},ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\right)}\,.\end{split}$
(28)

This suggests, having (26) in mind, that we can estimate

$\psi_{ABC}(Q)\,\approx\,\frac{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\,\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}\,.$
(29)

Note that by Corollary 3.4,

$\mathcal{G}^{-4}\,\leq\,\psi_{\widetilde{A}ABC\widetilde{C}}\left(\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)\,,\,\,\psi_{\widetilde{A}ABC\widetilde{C}}\left(\Theta_{\widetilde{A},ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\right)\,\leq\,\mathcal{G}^{4}\,,$
(30)

Combining (28), (25) and (30), we can bound

$\displaystyle\left|\psi_{ABC}(Q)\,-\,\frac{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\,\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}{\psi_{\widetilde{A}ABC\widetilde{C}}\left(\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}\right|\,$
$\displaystyle\hskip
71.13188pt\leq\,\mathcal{G}^{4}\,\left\|\Theta_{\widetilde{A},ABC,\widetilde{C}}\,Q\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}-\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right\|\,+\,$
$\displaystyle\hskip
96.73918pt+\,\mathcal{G}^{12}\,\|Q_{A}\|\,\|Q_{C}\|\,\left\|\Theta_{\widetilde{A},ABC,\widetilde{C}}\,\Theta_{\widetilde{A},ABC,\widetilde{C}}^{\dagger}\,-\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right\|$
$\displaystyle\hskip
71.13188pt\leq\,2\,\mathcal{G}^{6}\,\|Q_{A}\|\,\|Q_{C}\|\,\left\|\Theta_{\widetilde{A},ABC,\widetilde{C}}-\Theta_{\widetilde{A}_{1},A}\,\Theta_{C,\widetilde{C}_{1}}\right\|\,+\,$
$\displaystyle\hskip
96.73918pt+\,2\,\mathcal{G}^{14}\,\|Q_{A}\|\,\|Q_{C}\|\,\left\|\Theta_{\widetilde{A},ABC,\widetilde{C}}\,-\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{C,\widetilde{C}_{1}}\right\|$
$\displaystyle\hskip 71.13188pt\leq
8\,\mathcal{G}^{15}\,\|Q_{A}\|\,\|Q_{C}\|\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,.$

Note that in the above approximation, the inner observables
$Q_{A},Q_{C},\Theta_{\widetilde{A}_{1},A},\Theta_{C,\widetilde{C}_{1}}$ are
independent of the size of $\widetilde{A}_{2}$ and $\widetilde{C}_{2}$. Hence,
we can take limit when $|\tilde{A}_{2}|,|\widetilde{C}_{2}|$ tend to infinity
so that

$\left|\psi_{ABC}(Q)-\frac{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}\right|\,\leq\,8\,\mathcal{G}^{15}\,\|Q_{A}\|\,\|Q_{C}\|\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,.$
(31)

We can next use the exponential decaying condition on $\psi_{\mathbb{Z}}$ and
(25) to estimate

$\displaystyle\left|\frac{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}\,-\,\frac{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right)}{\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right)}\,\frac{\psi_{\mathbb{Z}}\left(\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}{\psi_{\mathbb{Z}}\left(\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)}\right|\,$
$\displaystyle\hskip
42.67912pt\leq\,\mathcal{G}^{4}\,\left|\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)\right.$
$\displaystyle\hskip
85.35826pt\left.-\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right)\,\psi_{\mathbb{Z}}\left(\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)\right|$
$\displaystyle\hskip
71.13188pt+\mathcal{G}^{12}\,\|Q_{A}\|\,\|Q_{C}\|\,\left|\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)\right.$
$\displaystyle\hskip
176.407pt\left.-\psi_{\mathbb{Z}}\left(\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right)\,\psi_{\mathbb{Z}}\left(\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right)\right|$
$\displaystyle\hskip
42.67912pt\leq\mathcal{G}^{4}\,\mathcal{K}\,e^{-\alpha\,\ell}\,\left\|\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right\|\,\left\|\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right\|\,$
$\displaystyle\hskip
71.13188pt+\mathcal{G}^{12}\,\|Q_{A}\|\,\|Q_{C}\|\,\mathcal{K}\,e^{-\alpha\ell}\,\left\|\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\right\|\,\left\|\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\right\|$
$\displaystyle\hskip
42.67912pt\leq\,2\,\mathcal{G}^{16}\,\|Q_{A}\|\,\|Q_{C}\|\,\mathcal{K}\,e^{-\alpha\,\ell}\,.$

Therefore, by combining the two inequalities

$\left|\psi_{ABC}(Q)-\frac{\psi_{\mathbb{Z}}\left(\,\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\right)}{\psi_{\mathbb{Z}}\left(\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\right)}\,\frac{\psi_{\mathbb{Z}}\left(\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}{\psi_{\mathbb{Z}}\left(\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}\right|\\\
\leq\,8\,\mathcal{G}^{16}\,\|Q_{A}\|\,\|Q_{C}\|\,\left(\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{K}e^{-\alpha\ell}\,\right)\,.$
(32)

In particular, applying (32) with $Q_{C}=\mathbbm{1}_{C}$ we deduce

$\left|\psi_{ABC}(Q_{A})-\frac{\psi_{\mathbb{Z}}\left(\,\Theta_{\widetilde{A}_{1},A}\,Q_{A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\right)}{\psi_{\mathbb{Z}}\left(\,\Theta_{\widetilde{A}_{1},A}\,\Theta_{\widetilde{A}_{1},A}^{\dagger}\,\right)}\right|\,\leq
8\,\mathcal{G}^{16}\,\|Q_{A}\|\,\left(\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{K}e^{-\alpha\ell}\,\right)\,,$
(33)

and analogously

$\left|\psi_{ABC}(Q_{C})-\frac{\psi_{\mathbb{Z}}\left(\,\Theta_{C,\widetilde{C}_{1}}\,Q_{C}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}{\psi_{\mathbb{Z}}\left(\,\Theta_{C,\widetilde{C}_{1}}\,\Theta_{C,\widetilde{C}_{1}}^{\dagger}\,\right)}\right|\,\leq
8\,\mathcal{G}^{16}\,\|Q_{C}\|\,\left(\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{K}e^{-\alpha\ell}\,\right)\,.$
(34)

Finally, if we combine (32), (33) and (34), then we conclude

$\displaystyle\left|\psi_{ABC}(Q)-\psi_{ABC}(Q_{A})\psi_{ABC}(Q_{C})\right|$
$\displaystyle\leq\,24\,\mathcal{G}^{16}\|Q_{A}\|\,\|Q_{C}\|\,\left(\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{K}e^{-\alpha\ell}\,\right)\,.$

The proof of the theorem is now complete.∎

##  7 Local indistinguishability and clustering

In order to prove our third main result in Section 8, we first need to show
some conditions on the Hamiltonian under study. More specifically, we need to
prove that, whenever exponential uniform clustering holds (see Definition
6.1), the expectation values of local observables do not distinguish between
Gibbs states on different intervals in some sense if the intervals are large
enough.

The next result is proved for general lattices in [13, Theorem 5] by means of
the Quantum Belief Propagation estimates for perturbed Hamiltonians [25]. It
shows that uniform clustering implies another property of the Gibbs state
under study, namely local indistinguishability. For the sake of completeness,
we include a proof based on Araki’s expansionals from Proposition 3.2 and
Corollary 3.4.

######  Proposition 7.1 (Local indistinguishability).

Let $\Phi$ be a finite-range interaction on $\mathbb{Z}$. If $\Phi$ satisfies
uniform clustering with decay $\varepsilon(\ell)$ in (24), then for every
finite interval $I\subset\mathbb{Z}$ split into three subintervals $I=ABC$
with $|B|\geq 2\ell\geq 0$ and every pair of observables
$Q_{A}\in\mathfrak{A}_{A}$ and $Q_{C}\in\mathfrak{A}_{C}$ we have

$\displaystyle\left|\operatorname{Tr}_{ABC}(\rho^{ABC}\,Q_{A})-\operatorname{Tr}_{AB}(\rho^{AB}\,Q_{A})\right|\,\leq\,\|Q_{A}\|\,\frac{4\,\mathcal{G}^{3+\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{G}^{4}\,\|Q_{A}\|\,\varepsilon(\ell)\,,$
$\displaystyle\left|\operatorname{Tr}_{ABC}(\rho^{ABC}\,Q_{C})-\operatorname{Tr}_{BC}(\rho^{BC}\,Q_{C})\right|\,\leq\,\|Q_{C}\|\,\frac{4\,\mathcal{G}^{3+\ell}}{(\lfloor\ell/r\rfloor+1)!}+\mathcal{G}^{4}\,\|Q_{C}\|\,\varepsilon(\ell)\,.$

$A$$B_{1}$$B_{2}$$C$$I$ Figure 5: Splitting of an interval $I$ into four
subintervals $I=AB_{1}B_{2}C$ with $|B_{1}|,|B_{2}|\geq\ell$. Here we take
$\ell=4$.

Proof. We illustrate the proof of the first inequality, the other one is
completely analogous. We can split $B$ into two subintervals $B=B_{1}B_{2}$
such that $|B_{1}|,|B_{2}|\geq\ell$ (see Figure 5). Let us denote, following
the nomenclature from Corollary 3.4,

$\Xi_{X,Y}:=E^{\dagger}_{X,Y}(-\tfrac{1}{2})=e^{-\frac{1}{2}H_{X}-\frac{1}{2}H_{Y}}\,e^{\frac{1}{2}H_{XY}}\,.$

Note that $\Xi_{X,Y}=\Theta_{X,Y}^{\dagger}$ in the previous section. As a
consequence of the aforementioned result, as well as Remark 3.5, we have

$\left\|\,\Xi_{AB,C}\,\right\|\,,\,\,\|\,\Xi_{AB,C}^{-1}\,\|\,\leq\,\mathcal{G}\,\,,\quad\|\,\Xi_{AB,C}-\Xi_{B_{2},C}\,\|\leq\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,.$
(35)

Note that

$e^{-H_{AB}-H_{C}}\,=\,\Xi_{AB,C}\,e^{-H_{ABC}}\,\Xi_{AB,C}^{\dagger}\,.$

Hence, we can rewrite

$\displaystyle\operatorname{Tr}_{AB}(\rho^{AB}\,Q_{A})$
$\displaystyle=\frac{\operatorname{Tr}_{ABC}\left(e^{-H_{AB}-H_{C}}\,Q_{A}\right)}{\operatorname{Tr}_{ABC}\left(e^{-H_{AB}-H_{C}}\right)}$
$\displaystyle=\frac{\operatorname{Tr}_{ABC}\left(e^{-H_{ABC}}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)}{\operatorname{Tr}_{ABC}\left(e^{-H_{ABC}}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)}$
$\displaystyle=\frac{\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)}{\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)}\,.$

The denominator is controlled by (35),

$\mathcal{G}^{-2}\,\leq\,\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)\,\leq\,\mathcal{G}^{2}\,,$

so that

$\displaystyle\left|\operatorname{Tr}_{AB}(\rho^{AB}\,Q_{A})-\operatorname{Tr}_{ABC}(\rho^{ABC}\,Q_{A})\right|$
$\displaystyle\leq\,\left|\frac{\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)}{\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)}-\operatorname{Tr}_{ABC}(\rho^{ABC}Q_{A})\right|$
$\displaystyle\leq\mathcal{G}^{2}\,\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,Q_{A}\right)\right|$
$\displaystyle\leq\mathcal{G}^{2}\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,Q_{A}\,\Xi_{B_{2},C}\right)\right|$
$\displaystyle\;\;+\mathcal{G}^{2}\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,Q_{A}\,\Xi_{B_{2},C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,Q_{A}\right)\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right)\right|$
$\displaystyle\;\;+\mathcal{G}^{2}\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,Q_{A}\right)\right|\,\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)\right|\,.$

The first and third summands in the last expression can be bounded similarly:
Using (35), it follows that

$\displaystyle\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,Q_{A}\,F_{B_{2},C}\right)\right|$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq\,\left\|\Xi_{AB,C}^{\dagger}\,Q_{A}\,\Xi_{AB,C}-\Xi_{B_{2},C}^{\dagger}\,Q_{A}\,\Xi_{B_{2},C}\right\|\,$
$\displaystyle\quad\quad\quad\quad\quad\quad\leq\,\|Q_{A}\|\,\frac{2\mathcal{G}^{\ell+1}}{(\lfloor\ell/r\rfloor+1)!}\,,$

and

$\displaystyle\left|\operatorname{Tr}_{ABC}(\rho^{ABC}\,Q_{A})\right|\,\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right)\right|$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\leq\,\|Q_{A}\|\,\left\|\Xi_{AB,C}^{\dagger}\,\Xi_{AB,C}-\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right\|\,$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\leq\,\|Q_{A}\|\,\frac{2\mathcal{G}^{\ell+1}}{(\lfloor\ell/r\rfloor+1)!}\,.$

On the second summand we apply the uniform clustering condition. Note that
$Q_{A}$ and $F_{B_{2},C}$ commute, and that $Q_{A}$ and
$F_{B_{2},C}^{\dagger}F_{B_{2},C}$ have supports separated by $B_{1}$, thus

$\displaystyle\left|\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,Q_{A}\,\Xi_{B_{2},C}\right)-\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,Q_{A}\right)\operatorname{Tr}_{ABC}\left(\rho^{ABC}\,\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right)\right|$
$\displaystyle\phantom{asdasdadsasdasdasdadasdasdasasdasdasddasd}\leq\,\varepsilon(\ell)\,\|Q_{A}\|\,\left\|\Xi_{B_{2},C}^{\dagger}\,\Xi_{B_{2},C}\right\|$
$\displaystyle\phantom{asdasdadsasdasdasdadasdasdasasdasdasddasd}\leq\,\mathcal{G}^{2}\,\varepsilon(\ell)\,\|Q_{A}\|\,.$

Combining all these estimates we conclude the result.∎

######  Remark 7.2.

Whenever $\varepsilon(\ell)$ is exponentially decaying with $\ell$, the second
summand in each of the upper bounds shown in Proposition 7.1 is more
restrictive than the previous one $($which exhibits superexponential decay$)$.
Therefore, in such a case, the first inequality $($and analogously for the
second one$)$ in the previous theorem can be simplified to:

$\left|\operatorname{Tr}_{ABC}(\rho^{ABC}Q_{A})-\operatorname{Tr}_{AB}(\rho^{AB}Q_{A})\right|\,\leq\,\|Q_{A}\|\,\widetilde{\mathcal{C}}\,\operatorname{e}^{-\gamma\ell}\,,$

for certain positive constants $\widetilde{\mathcal{C}}$, $\gamma>0$.

##  8 Decay of mutual information for Gibbs states of local Hamiltonians

In this section, we show that the BS-mutual information
$\widehat{I}_{\rho}(A:C)$ decays exponentially fast in the distance of the two
regions $A$, $C$, if $\rho$ is the Gibbs state of a local, finite-range,
translation-invariant Hamiltonian on an interval $I=ABC$. In the proof, we
make use of the fact that the interactions satisfy uniform clustering as in
Definition 6.1 with an exponential decay. We proved this fact in Theorem 6.2.

More specifically, given a local interaction $\Phi$, by virtue of Theorem 6.2
we know that there exist absolute positive constants $\widetilde{\mathcal{K}}$
and $\widetilde{\alpha}$ (depending only on $r$ and $J$) such that $\Phi$
satisfies uniform clustering as in Definition 6.1 with

$\epsilon(\ell)\,=\,\widetilde{\mathcal{K}}\,e^{-\widetilde{\alpha}\,\ell}\,,$
(36)

Now, we are ready to prove the main technical result of this section, which
will be used subsequently to prove an exponential decay of the BS-mutual
information:

######  Proposition 8.1.

Let $\Phi$ be a finite-range interaction over $\mathbb{Z}$ that satisfies
exponential uniform clustering. Then, there are absolute constants
$\widehat{\mathcal{K}}$ and $\widehat{\alpha}$ $($depending only on the range
$r$ and strength $J$ of the interaction, the local dimension $d$ and the
constants $\widetilde{\mathcal{K}}$ and $\widetilde{\alpha}$ appearing in
(36)$)$ with the following property: For each finite interval
$I=ABC\subset\mathbb{Z}$ split into three adjacent subintervals $A,B$ and $C$
with $|B|\geq 3\ell\geq 0$, and for $\rho=\rho^{I}$ the Gibbs state on $I$,

$\big{\|}\rho_{A}^{-1}\rho_{C}^{-1}\rho_{AC}-\mathbbm{1}\big{\|}\,\,\leq\,\,\widehat{\mathcal{K}}\,e^{-\widehat{\alpha}\,\ell}\,.$

Before we give the proof of Proposition 8.1, we will use it to prove the main
result of this section:

######  Theorem 8.2.

Let $\Phi$ be a finite-range interaction over $\mathbb{Z}$ that satisfies
exponential uniform clustering. Then, there are absolute constants
$\widehat{\mathcal{K}}$ and $\widehat{\alpha}$ $($depending only on the range
$r$ and strength $J$ of the interaction, the local dimension $d$ and the
constants $\widetilde{\mathcal{K}}$ and $\widetilde{\alpha}$ appearing in
(36)$)$ with the following property: For each finite interval
$I=ABC\subset\mathbb{Z}$ split into three adjacent subintervals $A,B$ and $C$
with $|B|\geq 3\ell\geq 0$ and for $\rho=\rho^{I}$ the Gibbs state on $I$,

$I_{\rho}(A:C)\leq\widehat{I}_{\rho}(A:C)\leq\widehat{I}^{q}_{\rho}(A:C)\leq\widehat{\mathcal{K}}e^{-\widehat{\alpha}\ell}\,,$

for every $q\geq 1$.

Proof. It is a direct consequence of Proposition 8.1 and Lemma 3.1.∎

In particular, the previous Theorem implies exponential decay of the BS-mutual
information in the setting studied by Araki [2]:

######  Corollary 8.3.

Let $\Phi$ be a finite-range and translation-invariant interaction over
$\mathbb{Z}$. Then, there are absolute constants $\mathcal{K}^{\prime}$ and
$\alpha^{\prime}$ $($depending only on the range $r$ and strength $J$ of the
interaction, the local dimension $d$ and the constants $\mathcal{K}$ and
$\alpha$ appearing in (23)$)$ with the following property: For each finite
interval $I=ABC\subset\mathbb{Z}$ split into three adjacent subintervals $A,B$
and $C$ with $|B|\geq 3\ell$ and for $\rho=\rho^{I}$ the Gibbs state on $I$,

$\widehat{I}_{\rho}(A:C)\leq\mathcal{K}^{\prime}e^{-\alpha^{\prime}\ell}\,.$

Proof. Due to [2], (23) holds true. As we have seen, this implies exponential
uniform clustering by Theorem 6.2. Thus, the assertion follows from Theorem
8.2.∎

Now we come to the proof of Proposition 8.1. Since its proof is rather
technical, we have split it into several steps that will be proved in
subsequent subsections.

The first step consists of rewriting the product
$\rho_{A}^{-1}\rho_{C}^{-1}\rho_{AC}$ using the notation for $E_{X,Y}$ that
was introduced in Corollary 3.4.

######  Step 1.

We can rewrite

$\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}=\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,\lambda_{ABC}\,,$
(37)

where

$\lambda_{ABC}\,\,=\,\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}^{-1}\,\operatorname{Tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{\dagger\,-1}\big{)}\,.$
(38)

The formulation given in Step 1 has the advantage that all the involved
factors and their inverses are, by Corollary 4.4, (uniformly) bounded by a
constant $\mathcal{C}$ depending only on the strength and range of the local
interaction, which proves the first part of the theorem.

Next, we need two estimates to conclude the proof of the theorem.

######  Step 2.

There are absolute constants
$\mathcal{K}^{\prime\prime},\alpha^{\prime\prime}>0$ satisfying

$|\lambda_{ABC}-1|\,\,\leq\,\,\mathcal{K}^{\prime\prime}\,e^{-\alpha^{\prime\prime}\,\ell}\,.$

######  Step 3.

There are absolute constants
$\mathcal{K}^{\prime\prime\prime},\alpha^{\prime\prime\prime}>0$ satisfying

$\left\|\operatorname{tr}_{B}(\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger})\,-\,\operatorname{tr}_{BC}(\rho^{BC}\,E_{A,BC}^{\dagger})\,\operatorname{tr}_{AB}(\rho^{AB}E_{AB,C}^{\dagger})\right\|\,\leq\mathcal{K}^{\prime\prime\prime}\,e^{-\alpha^{\prime\prime\prime}\,\ell}\,.$

Before proceeding into the proofs of the three steps, let us show how the
aforementioned theorem can be proved from them.

Proof of Proposition 8.1. As a combination of Corollary 4.4 with Step 1, we
have

$\displaystyle\left\|\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}\,-\,\mathbbm{1}\right\|\,\,$
$\displaystyle\quad\leq\,\,\left\|\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\right\|$
$\displaystyle\quad\quad\quad+\left\|\mathbbm{1}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\right\|$
$\displaystyle\quad\leq\,\,\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\right\|\,\left\|\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\right\|\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\right\|\,\left|\lambda_{ABC}-\mathbbm{1}\right|\,$
$\displaystyle\quad\quad\quad+\,\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\right\|\,\left\|\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\right\|\,$
$\displaystyle\quad\quad\quad\;\cdot\,\;\left\|\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\right\|$
$\displaystyle\quad\leq\,\mathcal{C}^{3}\,\left|\lambda_{ABC}-\mathbbm{1}\right|+\,\mathcal{C}^{2}\,\left\|\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\right\|\,.$

Therefore, applying Step 2 and Step 3,

$\left\|\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}\,-\,\mathbbm{1}\right\|\,\,\leq\,\,\mathcal{C}^{3}\,\mathcal{K}^{\prime\prime}\,e^{-\alpha^{\prime\prime}\,\ell}+\mathcal{C}^{2}\,\mathcal{K}^{\prime\prime\prime}\,e^{-\alpha^{\prime\prime\prime}\,\ell}\,.$

∎

We devote the rest of the section to proving the three steps stated above, as
well as some auxiliary ones that will be introduced afterwards for clarity.
Each step is proven below in a different subsection.

###  8.1 Proof of Step 1

First, we introduce some auxiliary terms that we combine with the initial ones
to obtain a more tractable expression.

$\displaystyle\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}\,$ $\displaystyle\hskip
5.69046pt=\,\operatorname{tr}_{BC}\big{(}e^{-H_{ABC}}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}e^{-H_{ABC}}\big{)}^{-1}\,\operatorname{tr}_{B}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{Tr}_{ABC}\big{(}e^{-H_{ABC}}\big{)}$
$\displaystyle\hskip
5.69046pt=\,\operatorname{tr}_{BC}\big{(}e^{-H_{ABC}}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}e^{-H_{ABC}}\big{)}^{-1}\,e^{-H_{A}-H_{C}}\,e^{H_{A}+H_{C}}\,\operatorname{tr}_{B}\big{(}e^{-H_{ABC}}\big{)}\,\operatorname{Tr}_{ABC}\big{(}e^{-H_{ABC}}\big{)}$
$\displaystyle\hskip
5.69046pt=\,\operatorname{tr}_{BC}\big{(}e^{H_{A}}\,e^{-H_{ABC}}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}e^{H_{C}}\,e^{-H_{ABC}}\big{)}^{-1}\,\,\operatorname{tr}_{B}\big{(}e^{H_{A}+H_{C}}\,e^{-H_{ABC}}\big{)}\,\operatorname{Tr}_{ABC}\big{(}e^{-H_{ABC}}\big{)}$
$\displaystyle\hskip
5.69046pt=\operatorname{tr}_{BC}\left(e^{-H_{BC}}\,e^{H_{A}+H_{BC}}\,e^{-H_{ABC}}\right)^{-1}\,\operatorname{tr}_{AB}\left(e^{-H_{AB}}\,e^{H_{AB}+H_{C}}\,e^{-H_{ABC}}\right)^{-1}$
$\displaystyle\hskip
22.76228pt\cdot\operatorname{tr}_{B}\left(e^{-H_{B}}\,e^{H_{A}+H_{B}+H_{C}}\,e^{-H_{ABC}}\right)\,\operatorname{Tr}_{ABC}\big{(}e^{-H_{ABC}}\big{)}\,.$

Next, we multiply by
$\operatorname{Tr}_{AB}(e^{-H_{AB}})\,,\,\operatorname{Tr}_{BC}(e^{-H_{BC}})\,,\,\operatorname{Tr}_{B}(e^{-H_{B}})$
and their inverses, so that after rearranging terms we arrive at

$\begin{split}\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}&=\operatorname{tr}_{BC}\left(\rho^{BC}\,e^{H_{A}+H_{BC}}\,e^{-H_{ABC}}\right)^{-1}\,\operatorname{tr}_{AB}\left(\rho^{AB}\,e^{H_{AB}+H_{C}}\,e^{-H_{ABC}}\right)^{-1}\\\\[5.69054pt]
&\hskip
184.9429pt\cdot\operatorname{tr}_{B}\left(\rho^{B}\,e^{H_{A}+H_{B}+H_{C}}\,e^{-H_{ABC}}\right)\,\lambda_{ABC}\,,\end{split}$

where $\lambda_{ABC}$ is the scalar

$\displaystyle\lambda_{ABC}$
$\displaystyle:=\,\,\operatorname{Tr}_{ABC}\left(e^{-H_{ABC}}\right)\,\operatorname{Tr}_{BC}\left(e^{-H_{BC}}\right)^{-1}\,\operatorname{Tr}_{AB}\left(e^{-H_{AB}}\right)^{-1}\,\operatorname{Tr}_{B}\big{(}e^{-H_{B}}\big{)}$
$\displaystyle=\,\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}\,e^{H_{ABC}}\,e^{-H_{A}-H_{BC}}\big{)}^{-1}\,\operatorname{Tr}_{AB}\left(\rho^{AB}\,e^{H_{AB}}\,e^{-H_{A}-H_{B}}\right)\,.$

Recall the notation $E_{\Lambda,\Lambda^{c}}$ introduced in Corollary 3.4. We
can then rewrite

$\rho_{A}^{-1}\,\rho_{C}^{-1}\,\rho_{AC}\,=\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}^{-1}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,\lambda_{ABC}$

and

$\lambda_{ABC}\,\,=\,\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}^{-1}\,\operatorname{Tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{\dagger\,-1}\big{)}\,.$

∎

###  8.2 Proof of Step 2

Note that we can bound:

$\displaystyle\left|\lambda_{ABC}-1\right|$
$\displaystyle=\left|\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}^{-1}\,\operatorname{Tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{\dagger\,-1}\big{)}-1\right|$
$\displaystyle\leq\left|\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}^{-1}\right|\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{\dagger\,-1}\big{)}-\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}\right|$
$\displaystyle\leq\mathcal{C}\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}\,E_{A,B}^{\dagger\,-1}\big{)}-\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}\right|\,,$

where the last inequality comes from Corollary 4.4. Since $|B|\geq 3\ell$, we
can split the interval $B$ into two subintervals $B=B_{1}B_{2}$, such that
$|B_{1}|,|B_{2}|\geq\ell$. Firstly, we add and subtract some intermediate
terms, which allow us to bound

$\displaystyle\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}\right|$
$\displaystyle\hskip
142.26378pt\leq\,\,\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B_{1}}^{\dagger\,-1}\big{)}\right|\,$
$\displaystyle\hskip
142.26378pt\quad\quad+\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B_{1}}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,B_{1}}^{\dagger\,-1}\big{)}\right|$
$\displaystyle\hskip
142.26378pt\quad\quad+\left|\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,B_{1}}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,BC}^{\dagger\,-1}\big{)}\right|\,.$

Note that we can enlarge $A$ and $C$ trivially to ensure $|A|$, $|C|>\ell$ if
needed (see Remark 3.5). Using that
$Q\mapsto\operatorname{Tr}_{\Lambda}(\rho^{\Lambda}Q)$ is contractive, and
estimates from Corollary 3.4, we deduce that the first and third summands can
be bounded

$\displaystyle\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B_{1}}^{\dagger\,-1}\big{)}\right|\,\,$
$\displaystyle\leq\,\,\left\|E_{A,B}^{\dagger\,-1}\,-\,E_{A,B_{1}}^{\dagger\,-1}\right\|\,\,\leq\,\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,,$
$\displaystyle\left|\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,B_{1}}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{ABC}(\rho^{ABC}E_{A,BC}^{\dagger\,-1})\right|\,\,$
$\displaystyle\leq\,\,\left\|E_{A,B_{1}}^{\dagger\,-1}\,-\,E_{A,BC}^{\dagger\,-1}\right\|\,\,\leq\,\,\frac{\mathcal{G}^{\ell}}{(\lfloor\ell/r\rfloor+1)!}\,.$

To bound the second summand, we can then use Proposition 7.1, since
$|B_{2}|\geq\ell$, which yields that

$\displaystyle\left|\operatorname{Tr}_{AB}\big{(}\rho^{AB}E_{A,B_{1}}^{\dagger\,-1}\big{)}\,-\,\operatorname{Tr}_{ABC}\big{(}\rho^{ABC}E_{A,B_{1}}^{\dagger\,-1}\big{)}\right|\,\,$
$\displaystyle\leq\,\,\|E_{A,B_{1}}^{\dagger\,-1}\|\,\widetilde{\mathcal{C}}\,e^{-\gamma\,\ell}\,\,\leq\,\,\mathcal{G}\,\widetilde{\mathcal{C}}\,e^{-\gamma\,\ell}\,.$

∎

###  8.3 Proof of Step 3

Since $|B|\geq 3\ell$, we can split $B$ into three subintervals
$B=B_{1}B_{2}B_{3}$ such that $|B_{j}|\geq\ell$ for each $j=1,2,3$, as in the
next figure:

$A$$B_{1}$$B_{3}$$B_{2}$$C$$I$ Figure 6: An interval $I$ split into five
subintervals $I=AB_{1}B_{2}B_{3}C$ with $|B_{1}|,|B_{2}|,|B_{3}|\geq\ell$.
Here we take $\ell=3$.

Let us first add and subtract some extra terms, and apply the triangle
inequality to get four more tractable summands to estimate:

$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}\right\|\,\,$
$\displaystyle\hskip
19.91684pt\leq\,\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}^{\dagger}\,E_{B_{3},C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
28.45274pt+\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}^{\dagger}\,E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{B_{3},C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
28.45274pt+\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{B_{3},C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
28.45274pt+\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}\right\|\,.$

The first and fourth summands can be easily handled by using the fact that
$Q\mapsto\operatorname{tr}_{\Lambda}(\rho^{\Lambda}Q)$ is contractive and
estimates from Corollary 3.4 and Remark 3.5. Indeed,

$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}^{\dagger}\,E_{B_{3},C}^{\dagger}\big{)}\right\|\,\,$
$\displaystyle\hskip
56.9055pt\leq\,\,\left\|E_{A,B}^{\dagger}\,E_{AB,C}^{\dagger}\,-\,E_{A,B_{1}}^{\dagger}\,E_{B_{3},C}^{\dagger}\right\|$
$\displaystyle\hskip
56.9055pt\leq\,\,\left\|E_{A,B}^{\dagger}-E_{A,B_{1}}^{\dagger}\right\|\,\left\|E_{AB,C}^{\dagger}\right\|\,+\,\left\|E_{A,B_{1}}^{\dagger}\right\|\,\left\|E_{AB,C}^{\dagger}-E_{B_{3},C}^{\dagger}\right\|$
$\displaystyle\hskip
56.9055pt\leq\,\,2\,\frac{\mathcal{G}^{\ell+1}}{(\lfloor\ell/r\rfloor+1)!}\,,$

and analogously

$\displaystyle\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{AB,C}^{\dagger}\big{)}\right\|\,\,$
$\displaystyle\hskip
56.9055pt\leq\,\,\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,BC}^{\dagger}\big{)}\right\|\,\left\|\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{B_{3},C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
85.35826pt+\,\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\|\,\|\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{B_{3},C}^{\dagger}\big{)}-\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{AB,C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
56.9055pt\leq\,\,\left\|\,E_{A,B_{1}}^{\dagger}\,-\,E_{A,BC}^{\dagger}\right\|\,\left\|\,E_{B_{3},C}^{\dagger}\right\|+\,\left\|\,E_{A,B_{1}}^{\dagger}\right\|\,\left\|\,E_{B_{3},C}^{\dagger}-\,E_{AB,C}^{\dagger}\right\|$
$\displaystyle\hskip
56.9055pt\leq\,\,2\,\frac{\mathcal{G}^{\ell+1}}{(\lfloor\ell/r\rfloor+1)!}\,.$

Bounding the second and third summands requires some extra work, so we are
going to do it in two more steps.

######  Step 4 (Auxiliary for Step 3).

There are absolute constants $\mathcal{K}^{*}$ and $\alpha^{*}$ such that

$\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{B_{3},C}^{\dagger}\big{)}\right\|\,\leq\mathcal{K}^{*}\,e^{-\alpha^{*}\,\ell}\,.$

######  Step 5 (Auxiliary for Step 3).

There are absolute constants $\mathcal{K}^{\sharp}$ and $\alpha^{\sharp}$ such
that

$\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\,E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{B_{3},C}^{\dagger}\big{)}\right\|\,\leq\mathcal{K}^{\sharp}\,e^{-\alpha^{\sharp}\,\ell}\,.$

The proofs for these steps are provided in the next subsections.∎

###  8.4 Proof of Step 4

We need two auxiliary results:

######  Lemma 8.4.

Let $\widetilde{A}ABC\widetilde{C}$ be a finite subsystem of $\mathbb{Z}$ and
let us fix $m$, $\ell\in\mathbb{N}$ such that $|\widetilde{A}|$, $|A|\leq m$,
$|\widetilde{C}|$, $|C|\leq m$ and $|B|\geq\ell$. Then, for all observables
$Q_{\widetilde{A}A}\in\mathfrak{A}_{\widetilde{A}A}$ and
$Q_{C\widetilde{C}}\in\mathfrak{A}_{C\widetilde{C}}$,

$\displaystyle\left\|\operatorname{tr}_{ABC}\big{(}\rho^{ABC}Q_{\widetilde{A}A}\big{)}\,-\,\operatorname{tr}_{AB}\big{(}\rho^{AB}Q_{\widetilde{A}A}\big{)}\right\|\,\,\leq\,\,\,\|Q_{\widetilde{A}A}\|\,d^{2m}\,\mathcal{K}_{1}\,e^{-\alpha_{1}\,\ell}\,.$
$\displaystyle\left\|\operatorname{tr}_{ABC}\big{(}\rho^{ABC}Q_{C\widetilde{C}}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}Q_{C\widetilde{C}}\big{)}\right\|\,\,\leq\,\,\,\|Q_{C\widetilde{C}}\|\,d^{2m}\,\mathcal{K}_{2}\,e^{-\alpha_{2}\,\ell}\,.$

Proof. We only prove the first inequality, the other is analogous. Let
$Q_{A\widetilde{A}}$ be an observable with support in a subsystem
$A\widetilde{A}$ of $\mathbb{Z}$. Considering
$\mathfrak{A}_{\widetilde{A}A}=\mathfrak{A}_{\widetilde{A}}\otimes\mathfrak{A}_{A}$
as the tensor product of two Hilbert spaces ($\mathfrak{A}_{A}$ of dimension
less than or equal to $d^{2m}$) endowed with the Hilbert-Schmidt scalar
product, we infer that $Q_{A\widetilde{A}}$ can be written using the operator
Schmidt decomposition as

$Q_{A\widetilde{A}}=\sum_{j=1}^{d^{2m}}\,\sqrt{\lambda_{j}}\,\,Q_{A}^{(j)}\otimes
Q_{\widetilde{A}}^{(j)}\quad,\quad Q_{A}^{(j)}\in\mathfrak{A}_{A}\quad,\quad
Q_{\widetilde{A}}^{(j)}\in\mathfrak{A}_{\widetilde{A}}\,,$ (39)

where the factors $Q_{A}^{(j)}$, resp. $Q_{\widetilde{A}}^{(j)}$, are
orthonormal, i.e.,

$\operatorname{Tr}(Q_{A}^{(j)\,\dagger}\,Q_{A}^{(k)})\,\,=\,\,\operatorname{Tr}(Q_{\widetilde{A}}^{(j)\,\dagger}\,Q_{\widetilde{A}}^{(k)})\,\,=\,\,\delta_{jk}\quad\text{for
every }j,k\,,$ (40)

and the Schmidt coefficients $\lambda_{j}$ satisfy the bound

$\sum_{j}|\lambda_{j}|\,\,\leq\,\,\|Q_{A\widetilde{A}}\|_{2}^{2}\,\,\leq\,\,d^{2m}\|Q_{A\widetilde{A}}\|^{2}\,.$
(41)

We can finally estimate

$\displaystyle\left\|\operatorname{tr}_{ABC}(\rho^{ABC}Q_{\widetilde{A}A})\,-\,\operatorname{tr}_{AB}(\rho^{AB}Q_{\widetilde{A}A})\right\|\,\,$
$\displaystyle\hskip
28.45274pt\leq\,\,\sum_{j=1}^{d^{2m}}\,\sqrt{\lambda_{j}}\,\big{\|}Q_{\widetilde{A}}^{(j)}\big{\|}\cdot\big{\|}\operatorname{tr}_{ABC}(\rho^{ABC}Q_{A}^{(j)})\,-\,\operatorname{tr}_{AB}(\rho^{AB}Q_{A}^{(j)})\big{\|}$
$\displaystyle\hskip
28.45274pt\leq\,\,\,\sum_{j=1}^{d^{2m}}\sqrt{\lambda_{j}}\,\big{\|}Q_{\widetilde{A}}^{(j)}\big{\|}\,\big{\|}Q_{A}^{(j)}\big{\|}\,\widetilde{\mathcal{C}}\,e^{-\gamma\,\ell}$
$\displaystyle\hskip 28.45274pt\leq
d^{2m}\,\|Q_{\widetilde{A}A}\|\,\widetilde{\mathcal{C}}\,e^{-\gamma\,\ell}\,,$

where in the second inequality we are using Proposition 7.1 and, in the last
line, the Cauchy-Schwarz inequality. Note that $\|Q^{(j)}_{A}\|$,
$\|Q_{\widetilde{A}}^{(j)}\|\leq 1$ since the factors are orthonormal with
respect to the Hilbert-Schmidt inner product.∎

######  Proposition 8.5.

Let $AB_{1}B_{2}B_{3}C$ be a subsystem of $\mathbb{Z}$ and let us fix
$\ell\in\mathbb{N}$ satisfying $|B_{1}|$, $|B_{3}|\geq\ell$. Then

$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}^{\dagger}\big{)}-\operatorname{tr}_{BC}\big{(}\rho^{BC}E_{A,B_{1}}^{\dagger}\big{)}\right\|\,\,\leq\,\,\widetilde{\mathcal{K}}_{2}\,e^{-\alpha_{2}\,\ell}\,,$
$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{B_{3},C}^{\dagger}\big{)}-\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{B_{3},C}^{\dagger}\big{)}\right\|\,\,\leq\,\,\widetilde{\mathcal{K}}_{1}\,e^{-\alpha_{1}\,\ell}\,.$

Proof. We just prove the first inequality, the other is analogous. We can
assume that $A$ corresponds to the interval $A=[1-a,0]$ and $B_{1}=[1,b]$ for
some $a,b\in\mathbb{N}$. Then, using Proposition 3.2 (see also Corollary 3.4)
we can rewrite it as a finite series

$E_{A,B_{1}}^{\dagger}\,\,=\,\,\sum_{m=1}^{\infty}\widetilde{E}_{m}\,,$

where each $\widetilde{E}_{m}$ has support in $[1-m,m]\cap AB_{1}$ and
$\|\widetilde{E}_{1}\|\leq\mathcal{G}$,

$\|\widetilde{E}_{m}\|\leq\frac{\mathcal{G}^{m-1}}{(\lfloor(m-1)/r\rfloor+1)!}\qquad\forall
m\geq 2\,.$

We can then use Lemma 8.4 and Corollary 3.4 to bound

$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}^{\dagger\,-1}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}E_{A,B_{1}}^{\dagger\,-1}\big{)}\right\|\,\,$
$\displaystyle\leq\,\,\sum_{m=1}^{\infty}\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\widetilde{E}_{m}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\widetilde{E}_{m}\big{)}\right\|$
$\displaystyle\leq\,\,\sum_{m=1}^{\infty}\,\|\widetilde{E}_{m}\|\,d^{2m}\,\mathcal{K}_{2}\,e^{-\alpha_{2}\,\ell}$
$\displaystyle\leq\,\,d^{2}\,\mathcal{K}_{2}\,e^{-\alpha_{2}\,\ell}\mathcal{G}+d^{2}\sum_{m=1}^{\infty}\,\frac{(\mathcal{G}d^{2})^{m}}{([m/r]+1)!}\,\mathcal{K}_{2}\,e^{-\alpha_{2}\,\ell}\,.$

It is easy to see that the series converges to a finite constant. We conclude
the proof by denoting

$\widetilde{\mathcal{K}}_{2}:=d^{2}\mathcal{K}_{2}\left(\mathcal{G}+\sum_{m=1}^{\infty}\,\frac{(\mathcal{G}d^{2})^{m}}{([m/r]+1)!}\right)\,.$

∎

We can finally prove the step.

Proof of Step 4. Using Proposition 8.5 and estimates from Corollary 3.4

$\displaystyle\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\,\operatorname{tr}_{AB}\big{(}\rho^{AB}E_{B_{3},C}^{\dagger}\big{)}\right\|\,\,$
$\displaystyle\hskip
85.35826pt\leq\,\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{A,B_{1}}^{\dagger}\big{)}\,-\,\operatorname{tr}_{BC}\big{(}\rho^{BC}\,E_{A,B_{1}}^{\dagger}\big{)}\right\|\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}E_{B_{3},C}^{\dagger}\big{)}\right\|$
$\displaystyle\hskip
85.35826pt+\left\|\operatorname{tr}_{BC}\big{(}\rho^{BC}E_{A,B_{1}}^{\dagger}\big{)}\right\|\,\left\|\operatorname{tr}_{B}\big{(}\rho^{B}\,E_{B_{3},C}^{\dagger}\big{)}\,-\,\operatorname{tr}_{AB}\big{(}\rho^{AB}\,E_{B_{3},C}^{\dagger}\big{)}\right\|\,\,$
$\displaystyle\hskip
85.35826pt\leq\,\,\,\widetilde{\mathcal{K}}_{1}\,e^{-\alpha_{1}\,\ell}\,\|E_{B_{3},C}^{\dagger}\|\,+\,\|E_{A,B_{1}}^{\dagger}\|\,\widetilde{\mathcal{K}}_{2}\,e^{-\alpha_{2}\,\ell}\,$
$\displaystyle\hskip
85.35826pt\leq\,\,2\,\mathcal{G}\,{\mathcal{K}}^{**}\,e^{-{\alpha^{**}}\,\ell}\,.$

∎

###  8.5 Proof of Step 5

Again we need an auxiliary lemma in the same spirit of Lemma 8.4.

######  Lemma 8.6.

Let $AB_{1}B_{2}B_{3}C$ be a subsystem of $\mathbb{Z}$ with $|A|$,
$|B_{1}|\leq n$, $|B_{2}|\geq\ell$ and $|C|$, $|B_{3}|\leq m$ for some
$n,m,\ell\in\mathbb{N}$. Then, for every pair of observables
$Q_{AB_{1}}\in\mathfrak{A}_{AB_{1}}$ and $Q_{B_{3}C}\in\mathfrak{A}_{B_{3}C}$
we deduce that

$\big{\|}\operatorname{tr}_{B}\big{(}\rho^{B}Q_{AB_{1}}Q_{B_{3}C}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}Q_{AB_{1}}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}Q_{B_{3}C}\big{)}\big{\|}\,\leq
d^{2n+2m}\,\,\|Q_{AB_{1}}\|\,\,\|Q_{B_{3}C}\|\,\,\breve{\mathcal{K}}\,e^{-\breve{\alpha}\ell}\,.$

Proof. Using the operator Schmidt decomposition, we can write

$\displaystyle
Q_{AB_{1}}\,=\,\sum_{a=1}^{d^{2n}}\,\sqrt{\lambda_{a}}\,Q_{A}^{(a)}\otimes
Q_{B_{1}}^{(a)}\quad\quad,\quad\quad
Q_{B_{3}C}\,=\,\sum_{c=1}^{d^{2m}}\,\sqrt{\mu_{c}}\,\,Q_{B_{3}}^{(c)}\otimes
Q_{C}^{(c)}\,,$

where for every $a$

$Q_{A}^{(a)}\in\mathfrak{A}_{A}\;\;,\quad\|Q_{A}^{(a)}\|\,\leq\,1\;\;,\quad
Q_{B_{1}}^{(a)}\in\mathfrak{A}_{B_{1}}\;\;,\quad\|Q_{B_{1}}^{(a)}\|\,\leq\,1\;\;,\quad\sum_{a=1}^{d^{2n}}|\lambda_{a}|\leq
d^{2n}\|Q_{AB_{1}}\|^{2}\,,$

and for every $c$

$Q_{B_{3}}^{(c)}\in\mathfrak{A}_{B_{3}}\;\;,\quad\|Q_{B_{3}}^{(c)}\|\,\leq\,1\;\;,\quad
Q_{C}^{(c)}\in\mathfrak{A}_{C}\;\;,\quad\|Q_{C}^{(c)}\|\,\leq\,1\;\;,\quad\sum_{c=1}^{d^{2m}}|\mu_{c}|\leq
d^{2m}\|Q_{B_{3}C}\|^{2}\,.$

Therefore

$\displaystyle\big{\|}\operatorname{tr}_{B}\big{(}\rho^{B}Q_{AB_{1}}Q_{B_{3}C}\big{)}\,-\,\operatorname{tr}\big{(}\rho^{B}Q_{AB_{1}}\big{)}\,\operatorname{tr}_{B}\big{(}\rho^{B}Q_{B_{3}C}\big{)}\big{\|}\,\,$
$\displaystyle\hskip 56.9055pt\hskip
56.9055pt\leq\,\,\sum_{a=1}^{d^{2n}}\,\sum_{c=1}^{d^{2m}}\,\sqrt{\lambda_{a}}\,\,\sqrt{\mu_{c}}\,\,\|Q_{A}^{(a)}\|\,\|Q_{C}^{(c)}\|$
$\displaystyle\hskip 56.9055pt\hskip
113.81102pt\cdot\big{\|}\operatorname{tr}_{B}\big{(}\rho^{B}Q_{B_{1}}^{(a)}Q_{B_{3}}^{(c)}\big{)}\,-\,\operatorname{tr}_{B}\big{(}\rho^{B}Q_{B_{1}}^{(a)}\big{)}\cdot\operatorname{tr}_{B}\big{(}\rho^{B}Q_{B_{3}}^{(c)}\big{)}\big{\|}$
$\displaystyle\hskip 56.9055pt\hskip
56.9055pt\leq\,\,\sum_{a=1}^{d^{2n}}\,\sum_{c=1}^{d^{2m}}\,\sqrt{\lambda_{a}}\,\,\sqrt{\mu_{c}}\,\,\big{\|}Q_{A}^{(a)}\big{\|}\,\big{\|}Q_{C}^{(c)}\big{\|}\,\big{\|}Q_{B_{1}}^{(a)}\big{\|}\,\big{\|}Q_{B_{3}}^{(c)}\big{\|}\,\,\widetilde{\mathcal{K}}\,e^{-\widetilde{\alpha}\ell}$
$\displaystyle\hskip 56.9055pt\hskip
56.9055pt\leq\,\,d^{2n+2m}\,\,\|Q_{AB_{1}}\|\,\,\|Q_{B_{3}C}\|\,\,\widetilde{\mathcal{K}}\,e^{-\widetilde{\alpha}\ell}\,,$

where we have used the uniform exponential decay of correlations and the
Cauchy-Schwarz inequality.∎

######  Proposition 8.7.

Let $AB_{1}B_{2}B_{3}C$ be a system with $|B_{2}|\geq\ell$ for some
$\ell\in\mathbb{N}$. Then,

$\left\|\operatorname{Tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}E_{B_{3},C}\big{)}\,-\,\operatorname{Tr}_{B}\big{(}\rho^{B}E_{A,B_{1}}\big{)}\cdot\operatorname{Tr}_{B}\big{(}\rho^{B}E_{B_{3},C}\big{)}\right\|\,\,\leq\,\,\breve{\mathcal{K}}^{\prime}\,e^{-\breve{\alpha}^{\prime}\ell}\,.$

Proof. Let us identify $A$ and $B_{1}$ with $[1-a,0]$ and $[1,b_{1}]$ for some
$a,b_{1}\in\mathbb{N}$. We can find, by Proposition 3.2 (see also Corollary
3.4), a decomposition

$E_{A,B_{1}}=\sum_{n=1}^{\infty}\widetilde{E}_{n}$

where

$\widetilde{E}_{n}\,\in\,\mathfrak{A}_{[1-n,n]\cap
AB_{1}}\,,\;\;\|\widetilde{E}_{n}\|\,\,\leq\,\,\frac{\mathcal{G}^{n-1}}{(\lfloor(n-1)/r\rfloor+1)!}\,\,\forall
n\geq 2\,,\;\;\|\widetilde{E}_{1}\|\leq\mathcal{G}.$

Analogously we can identify $B_{3}$ and $C$ with intervals $[1-b_{3},0]$ and
$[1,c]$ for certain $b_{3},c\in\mathbb{N}$ and find a decomposition

$E_{A,B_{1}}=\sum_{m=1}^{\infty}\widetilde{F}_{m}$

where

$\widetilde{F}_{m}\,\in\,\mathfrak{A}_{[1-m,m]\cap
B_{3}C}\,,\;\;\|\widetilde{F}_{m}\|\,\,\leq\,\,\frac{\mathcal{G}^{m-1}}{(\lfloor(m-1)/r\rfloor+1)!}\,\,\forall
m\geq 2\,,\;\;\|\widetilde{F}_{1}\|\leq\mathcal{G}.$

Note that whenever $1\leq n,m\leq\ell$, then $\widetilde{E}_{n}$ and
$\widetilde{F}_{m}$ have support in $AB_{1}$ and $B_{3}C$, respectively, where
we can enlarge $A$ and $C$ if necessary. Let us assume $\mathcal{G}\geq 1$,
making the constant possibly larger. Since both supporting intervals are
separated by an interval $B_{2}$ with $|B_{2}|\geq\ell$, we can apply Lemma
8.6 to bound, for $1\leq n,m\leq\ell$

$\displaystyle\left\|\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n}\widetilde{F}_{m})\,-\,\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n})\cdot\operatorname{tr}_{B}(\rho^{B}\widetilde{F}_{m})\right\|\,\,$
$\displaystyle\leq\,\,d^{2n+2m}\,\|\widetilde{E}_{n}\|\,\|\widetilde{F}_{m}\|\,\breve{\mathcal{K}}\,e^{-\breve{\alpha}\ell}$
$\displaystyle\leq\,\,\breve{\mathcal{K}}\,e^{-\breve{\alpha}\,\ell}\,\frac{(\mathcal{G}d^{2})^{n}}{(\lfloor(n-1)/r\rfloor+1)!}\,\frac{(\mathcal{G}d^{2})^{m}}{(\lfloor(m-1)/r\rfloor+1)!}\,.$

If $n>\ell$ or $m>\ell$, we will estimate in a rudimentary way

$\displaystyle\left\|\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n}\widetilde{F}_{m})\,-\,\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n})\cdot\operatorname{tr}_{B}(\rho^{B}\widetilde{F}_{m})\right\|\,\,$
$\displaystyle\leq\,\,2\,\|\widetilde{E}_{n}\|\,\|\widetilde{F}_{m}\|$
$\displaystyle\leq\,\,2\,\frac{\mathcal{G}^{n-1}}{(\lfloor(n-1)/r\rfloor+1)!}\,\,\frac{\mathcal{G}^{m-1}}{(\lfloor(m-1)/r\rfloor+1)!}\,.$

Combining these bounds

$\displaystyle\left\|\operatorname{tr}_{B}(\rho^{B}E_{A,B_{1}}E_{B_{3},C})\,-\,\operatorname{tr}_{B}(\rho^{B}E_{A,B_{1}})\cdot\operatorname{tr}_{B}(\rho^{B}E_{B_{3},C})\right\|\,\,$
$\displaystyle\hskip
113.81102pt\leq\,\sum_{n=1}^{\infty}\,\sum_{m=1}^{\infty}\,\left\|\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n}\,\widetilde{F}_{m})\,-\,\operatorname{tr}_{B}(\rho^{B}\widetilde{E}_{n})\cdot\operatorname{tr}_{B}(\rho^{B}\widetilde{F}_{m})\right\|$
$\displaystyle\hskip
113.81102pt\leq\,\breve{\mathcal{K}}\,e^{-\breve{\alpha}\,\ell}\,\left(\sum_{n=1}^{\ell}\,\frac{(\mathcal{G}d^{2})^{n}}{(\lfloor(n-1)/r\rfloor+1)!}\right)^{2}$
$\displaystyle\hskip
142.26378pt+4\,\left(\sum_{n=1}^{\infty}\,\frac{\mathcal{G}^{n}}{(\lfloor(n-1)/r\rfloor+1)!}\right)\left(\sum_{n=\ell+1}^{\infty}\,\frac{\mathcal{G}^{n}}{(\lfloor(n-1)/r\rfloor+1)!}\right)$
$\displaystyle\hskip
113.81102pt\leq\,\breve{\mathcal{K}}\,\left(\,e^{-\breve{\alpha}\,\ell}+\sum_{n=\ell+1}^{\infty}\,\frac{\mathcal{G}^{n}}{(\lfloor
n/r\rfloor+1)!}\right).$

The fact that the series decays superexponentially in $\ell$ can be seen by
considering the remainder term of the Taylor expansion of the exponential
function.∎

Proof of Step 5. This follows directly from an application of Proposition
8.7.∎

##  9 Final remarks and Conclusions

In this paper, we have proven some new results related to the decay of
correlations between spatially separated regions for Gibbs states of local,
finite-range, translation-invariant Hamiltonians in 1D. The main tools
employed in their proofs have been derived from Araki’s expansionals and the
BS-entropy.

First, we have shown that the distance of a Gibbs state in the previous
setting from being BS-recoverable between three adjacent intervals decays
superexponentially with the size of the middle interval. This result can be
compared to [30, Theorem 4], where the distance from a Gibbs state in the same
context to a recovery map is shown to decay subexponentially instead.

Furthermore, we have proven that such Gibbs states exhibit exponential decay
of correlations over any finite interval, as opposed to Araki’s original
result, which concerns exponential decay of correlations over an infinite
chain. This result has subsequently allowed us to show exponential decay of
the mutual information for any Gibbs state of a local, finite-range,
translation-invariant Hamiltonian in 1D, for any temperature. These results
allow us to close a cycle by obtaining the equivalence between several well-
known properties in the context of decay of correlations (see Figure 7),
namely:

  * (i)

Exponential decay of correlations for thermal states on the infinite chain.

  * (ii)

Exponential uniform clustering.

  * (iii)

Exponential decay of the mutual information.

It might be interesting to remark here the relation between exponential
uniform clustering and local indinguishability as it was introduced by Brandão
and Kastoryano in [13] as a consequence of the exponential uniform clustering
condition. Particularizing [13, Theorem 5] to 1D, we have that for every
finite subsets $A\subset X\subset Y$ with $\operatorname{dist}(A,Y\setminus
X)\geq\ell$ and every observable $Q$ with support in $A$

$|\operatorname{Tr}_{X}(\rho^{X}Q)\,-\,\operatorname{Tr}_{Y}(\rho^{Y}Q)|\,\leq\,|\partial
X|\,\mathcal{K}\,e^{-\alpha\ell}\,\|Q\|$ (42)

where $\partial X$ is the set of nodes in $X$ with a neighbouring site in
$X^{c}$, and where $\mathcal{K},\alpha$ are constants independent of the
supporting sets. This local indistinguishability property is more general than
the one we consider in Proposition 7.1 (the supporting sets are not
necessarily intervals, but arbitrary finite subsets). Indeed, it can be easily
shown that this condition implies exponential uniform clustering, hence both
conditions are equivalent. For that, consider a finite interval
$I\subset\mathbb{Z}$ split into $I=ABC$, where $B$ shields $A$ from $C$, and
for which $|B|>3\ell$ for some $\ell\in\mathbb{N}$ with $\ell\geq r$. We can
then split $B=B_{1}B_{2}B_{3}$, where $B_{2}$ shields $B_{1}$ from $B_{3}$ and
$|B_{j}|\geq\ell$ for $j=1,2,3$. Let $\widetilde{A}=AB_{1}$,
$\widetilde{C}=B_{3}C$ and $X=\widetilde{A}\cup\widetilde{C}$. Then, for each
$Q_{A}\in\mathfrak{A}_{A}$ and $Q_{C}\in\mathfrak{A}_{C}$,

$|\operatorname{Tr}_{I}(\rho^{I}Q_{A}Q_{C})\,-\,\operatorname{Tr}_{X}(\rho^{X}Q_{A}Q_{C})|\,\leq\,4\,\mathcal{K}\,e^{-\alpha\ell}\,\|Q_{A}\|\,\|Q_{C}\|\,.$
(43)

We observe that $H_{X}=H_{\widetilde{A}}+H_{\widetilde{C}}$ since the distance
between $\widetilde{A}$ and $\widetilde{C}$ is larger than $\ell\geq r$, so
that $\rho^{X}=\rho^{\widetilde{A}}\rho^{\widetilde{C}}$ and hence

$\operatorname{Tr}_{X}(\rho^{X}Q_{A}Q_{C})=\operatorname{Tr}_{\widetilde{A}}(\rho^{\widetilde{A}}Q_{A})\,\operatorname{Tr}_{\widetilde{C}}(\rho^{\widetilde{C}}Q_{C})\,.$
(44)

Moreover, using again the local indistinguishability condition (42), we can
estimate

$\big{|}\operatorname{Tr}_{\widetilde{A}}(\rho^{\widetilde{A}}Q_{A})-\operatorname{Tr}_{I}(\rho^{I}Q_{A})\big{|}\,\leq\,2\mathcal{K}e^{-\alpha\ell}\,\|Q_{A}\|\,\,,\,\,\big{|}\operatorname{Tr}_{\widetilde{C}}(\rho^{\widetilde{C}}Q_{C})-\operatorname{Tr}_{I}(\rho^{I}Q_{C})\big{|}\,\leq\,2\mathcal{K}e^{-\alpha\ell}\,\|Q_{C}\|\,.$
(45)

Combining (43),(44), and (45) we conclude that

$\left|\operatorname{Tr}_{I}(\rho^{I}Q_{A}Q_{C})-\operatorname{Tr}_{I}(\rho^{I}Q_{A})\,\operatorname{Tr}_{I}(\rho^{I}Q_{C})\right|\,\leq\,8\mathcal{K}e^{-\alpha\ell}\,\|Q_{A}\|\,\|Q_{C}\|\,.$

This finishes the proof of the statement.

Exponential Uniform Clustering Definition 6.1
$\rho_{AC}\approx_{1}\rho_{A}\otimes\rho_{C}$ uniformly Exponential Decay of
the Mutual Info. High Temperature Kliesch et al. [31]Kuwahara et al. [33]
$\rho_{AC}^{-1}(\rho_{A}\otimes\rho_{C})\approx_{\infty}\mathbbm{1}$ uniformly
Lemma 3.1 Local Indistinguishability (with exponential rate) Prop. 8.1
Exponential Decay of Correlations Infinite chain KMS state Positive
Temperature Translation Invariant Araki [2]Th. 6.2 Figure 7: Here we show the
connections between several properties and results in 1D that have appeared in
the paper. We represent by red ellipsoids some assumptions on the temperature
and interactions on the system under study, by blue rounded rectangles some
results on decay of correlations and by green rhombi some conditions on
approximation of observables. The dashed arrows stand for results that were
previously known (or easily obtained from previously-known results), and the
solid ones, for original outcomes of this paper.

Regarding the decay of mutual information for Gibbs states, it was proven in
[29] that systems satisfying the property of rapid mixing exhibit exponential
decay of correlations. In particular, Theorem 12 in that reference shows that,
under the assumption of a positive _modified logarithmic Sobolev constant_
(MLSI constant) for a certain dissipative system, the mutual information
between spatially separated regions on a Gibbs state decays exponentially with
the distance between those regions, with a mutiplicative factor depending
polynomially on the system size. This result, together with the main findings
of [16] (in which it is shown that for local, finite-range, commuting
Hamiltonians in 1D or with nearest-neighbour interactions, at high enough
temperature, there is a certain evolution, satisfying the conditions of [29,
Theorem 12], with the Gibbs state of such a Hamiltonian as a fixed point, that
has a positive MLSI constant) yield an exponential decay for the mutual
information on Gibbs states in their setting. However, the main drawback of
this result is the polynomial dependence on the system size. Our Theorem 8.2
is a clear improvement to the latter in 1D, since we have no dependence with
the system size and our result holds for non-commuting Hamiltonians and for
any temperature.

One motivation to find a decay rate for the mutual information on Gibbs states
of certain systems is that of employing them to prove area laws. Indeed, it
might be possible to extend Theorem 8.2 to larger dimensions for high enough
temperature by using [13, Theorem 5], which is based on Quantum Belief
Propagation [25], instead of our Proposition 7.1 based on Araki’s
expansionals. Recall that the former presents the advantage with respect to
the latter that it holds for any dimension, and the drawback that it only
works for high-enough temperature. The possible extension of our result might
then yield in particular exponential decay of the mutual information on Gibbs
states of local, finite-range, non-commuting Hamiltonians at high-enough
temperature, a setting already explored in [33] for the conditional mutual
information. Note that the authors of [33] point out that their results extend
to the mutual information. However, as we would actually be estimating
geometric Rényi divergences, it might also be possible to use it to obtain an
area law for the mutual information that arises from such quantities. This
direction has recently been explored in [42].

To conclude, we expect that the results and techniques developed in the
current manuscript find applications in the fields of quantum information
theory and quantum many-body systems beyond those already mentioned here. For
instance, to illustrate the potential of Theorem 8.2, it can actually be used
in the context of quantum functional inequalities to prove examples of
positivity of a modified logarithmic Sobolev inequality (MLSI). Indeed, the
norm that is upper bounded in Proposition 8.1 can be interpreted as a mixing
condition for a Gibbs state, as stated in [5, Assumption 1]. In that paper,
the authors show that such a condition needs to be assumed in order to prove
that the heat-bath dynamics in 1D has a positive MLSI constant. Our
Proposition 8.1 can then be used to show that such an assumption holds for any
Gibbs state of a local, finite-range, translation-invariant Hamiltonian in 1D,
for any temperature. Jointly with some other recent techniques, it yields that
Davies generators in 1D converging to such a Gibbs state, at any temperature,
have a positive MLSI constant, and hence exhibit rapid mixing [3, 4].

Acknowledgements: The authors would like to thank Álvaro M. Alhambra, in
particular for suggesting the use of the geometric Rényi divergences that
improved the presentation of Section 3.1; Yoshiko Ogata, whose comments
inspired the proof of Theorem 6.2; and also Kohtaro Kato, Angelo Lucia and
David Pérez-García for insightful discussions. AB acknowledges support from
the VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059) and
from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within
the European Union’s Horizon 2020 Programme (QuantAlgo project) via the
Innovation Fund Denmark. AC is partially supported by a MCQST Distinguished
PostDoc fellowship, by the Seed Funding Program of the MCQST and by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC-2111 390814868. APH is partially supported
by the grants “Juan de la Cierva Formación” (FJC2018-036519-I) and 2021-MAT11
(ETSI Industriales, UNED), as well as by the Spanish Ministerio de Ciencia e
Innovación project PID2020-113523GB-I00, by Comunidad de Madrid project
QUITEMAD-CM P2018/TCS4342, and by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 648913).

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