# Conditional Independence of 1D Gibbs States with Applications to Efficient
Learning

Paul Gondolf<EMAIL_ADDRESS>Fachbereich Mathematik, Universität
Tübingen, 72076 Tübingen, Germany  Samuel O. Scalet<EMAIL_ADDRESS>Department
of Applied Mathematics and Theoretical Physics, University of Cambridge,
United Kingdom  Alberto Ruiz-de-Alarcón<EMAIL_ADDRESS>Fachbereich Mathematik, Universität Tübingen, 72076 Tübingen, Germany
Departamento de Análisis Matemático y Matemática Aplicada, Universidad
Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain.  Álvaro
M. Alhambra<EMAIL_ADDRESS>Instituto de Física Teórica UAM/CSIC, C/
Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain  Ángela Capel
<EMAIL_ADDRESS>Fachbereich Mathematik, Universität Tübingen,
72076 Tübingen, Germany

###### Abstract

We show that spin chains in thermal equilibrium have a correlation structure
in which individual regions are strongly correlated at most with their near
vicinity. We quantify this with alternative notions of the conditional mutual
information defined through the so-called Belavkin-Staszewski relative
entropy. We prove that these measures decay super-exponentially, under the
assumption that the spin chain Hamiltonian is translation-invariant. Using a
recovery map associated with these measures, we sequentially construct tensor
network approximations in terms of marginals of small (sub-logarithmic) size.
As a main application, we show that classical representations of the states
can be learned efficiently from local measurements with a polynomial sample
complexity. We also prove an approximate factorization condition for the
purity of the entire Gibbs state, which implies that it can be efficiently
estimated to a small multiplicative error from a small number of local
measurements. As a technical step of independent interest, we show an upper
bound to the decay of the Belavkin-Staszewski relative entropy upon the
application of a conditional expectation.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Previous work and future directions
  2. 2 Preliminaries
    1. 2.1 Basic notation
    2. 2.2 Entropy measures
    3. 2.3 Spin chains in 1D, local Hamiltonians and Gibbs states
    4. 2.4 Approximate factorisation of Gibbs states of local Hamiltonians in 1D
  3. 3 Main results
    1. 3.1 An upper bound on the DPI for the Belavkin Staszewski entropy
    2. 3.2 Superexponential decay of the BS-CMI
    3. 3.3 Approximate factorisation of the purity
  4. 4 Applications
    1. 4.1 Positive MPO approximations from recovery maps
    2. 4.2 Reconstructing the positive MPO from local tomography
    3. 4.3 Efficient estimation of the global purity
  5. A Proofs of approximate factorization of Gibbs states in 1D
    1. A.1 Proof of Lemma 2.4
    2. A.2 Proof of Lemma 2.5
  6. B Extension of results to exponentially-decaying interactions
    1. B.1 Exponential decay of the BS-CMI for exponentially-decaying interactions
    2. B.2 Efficient estimation of the global purity for exponentially-decaying interactions
    3. B.3 Learning of Gibbs states via MPO approximations for exponentially-decaying interactions
  7. C Lifting the upper bound on the DPI of the BS-entropy to channels

##  1 Introduction

It is by now widely established that tools and ideas from quantum information
theory very often give fresh perspectives to the study of complex quantum
many-body systems. A very notable way in which this happens is the systematic
characterization of the correlations among their constituents, which often
allows us to narrow down the complexity of the many-body states in specific
ways.

One of the main ways of constraining those correlations is the area law, which
states that the amount of information that two adjacent regions share of each
other is upper bounded by the size of their mutual boundary. This area law has
been shown for gapped ground states [67, 40, 2] using entanglement entropy,
and also for Gibbs states of finite temperatures and other classes of mixed
states [17, 34, 5] using the mutual information [83, 53, 70]. Importantly,
area laws can be linked with the efficiency of classical algorithms in the
form of tensor network methods [80, 56, 53, 37]. Another related way in which
many-body states are typically constrained is by a fast decay of correlations
of distant regions. This is often stated in terms of connected correlation
functions [41, 18, 6, 48, 35], but also with other quantifiers such as the
mutual information [15] (although they are often equivalent [21, 49, 16]), as
well as measures of quantum entanglement [55]. This decay means that distant
regions behave roughly independently of each other so that complex collective
effects (such as long-range order or entanglement) are not present.

There is a third, arguably more refined, way in which correlations can be
constrained in many-body systems: conditional independence [42, 57]. The idea
behind this notion is to quantify how much the (possibly small) correlations
between two separate regions $A$ and $C$ are mediated by a separating region
$B$ that shields one from the other, such as in Figure 1.

$A$$C$$B$$d(A,C)$ Figure 1: Regions $A$ and $C$ shielded by a region $B$.

In quantum systems, the notion of conditional independence of a state $\rho$
is typically expressed through the quantum conditional mutual information
(CMI) $I_{\rho}(A:C|B)$ [59], and its behaviour with the geometry of the
regions $A,B,C$. This quantity has the following equivalent definitions

$\displaystyle I_{\rho}(A:C|B)$
$\displaystyle=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_{B})$ (1)
$\displaystyle=I_{\rho}(A:BC)-I_{\rho}(A:B)$ (2)
$\displaystyle=H_{\rho}(A|B)-H_{\rho}(A|BC).$ (3)

That is, as a linear combination of subsystem entropies, or as a difference of
mutual informations or conditional entropies. When $I_{\rho}(A:C|B)=0$, we say
that $\rho_{ABC}$ is a quantum Markov state. Additionally, its smallness
allows us to establish that the state $\rho_{ABC}$ can be accurately
reconstructed from $\rho_{AB}$ by acting on $B$ alone with a CPTP map [42, 33,
74].

The CMI has often appeared in the context of quantum many-body systems. As an
analogue of the classical Hammersley-Clifford theorem [25], it is known that
states in which this quantity vanishes whenever $B$ shields $A$ from $C$
correspond to Gibbs states of commuting Hamiltonians [57, 20]. Additionally,
an exponential decay of the CMI as $B$ increasingly separates $A$ and $C$ has
been shown for various instances of Gibbs states [46, 52], and matrix product
states [76]. A fast decay with the distance between $A$ and $C$ guarantees the
accuracy of local reconstruction maps [33, 45], which is linked to the
efficiency of quantum algorithms for preparing Gibbs states [19], and of
learning phases of matter [63]. The CMI is also prominently featured in the
study of topological order at low energies [47, 73].

An arbitrarily small CMI is the main way of quantifying conditional
independence, but quantum information theory provides us with a framework to
systematically construct similar measures, for instance, through
generalizations of Eq. (2) and Eq. (3). While their operational meaning might
be entirely different [28, 10, 4], each possible definition comes with the
potential for applications. In this paper, we study several such alternative
notions of conditional independence. In contrast to the CMI, which is based on
the Umegaki relative entropy, ours are largely based on the Belavkin-
Staszewksi (BS) relative entropy [9]. We also study a measure in terms of
local Rényi-$2$ entropies inspired by recent results on efficient learning of
entropies and entanglement measures [79].

One of our main findings is that the decay of the BS-CMI in Gibbs states is
strictly faster than what is believed to hold for any measure of bipartite
correlations [6, 35, 15]: super-exponential, as opposed to just exponential.
This difference can be seen in analogy with the classical setting, in which
Gibbs distributions have exactly zero CMI, while still having an exponential
decay of correlations. In contrast, existing conjectures and results on the
quantum CMI only display an exponential decay [19, 46, 52]. Our work thus
suggests that in quantum Gibbs states there is a separation of the magnitude
of conditional independence and decay of correlations, much like in the
classical case.

We then study the application of our notions of conditional independence, and
their fast decay, to devise efficient learning schemes for properties of
quantum Gibbs states in one dimension. The problem of efficiently learning
quantum Gibbs states from local measurements has seen tremendous progress
recently, including methods whose efficiency is based on the decay of the CMI
[63]. In particular, how to efficiently learn the Hamiltonian with a small
number of samples and classical post-processing has been actively studied [7,
3, 39, 69, 50], including important experimental efforts [44], and efficient
protocols at all temperatures [8]. Results along these lines have also
appeared for matrix product states in 1D [26, 31]. With our techniques, based
on alternative notions of conditional independence, we shift the focus away
from Hamiltonian learning and instead study the problem of directly
reconstructing tensor network approximations to the state, as well as the
global purity.

###  1.1 Main results

In this paper, we focus on Gibbs states of local Hamiltonians in 1D at any
inverse positive temperature $\beta>0$. Given a finite chain $\Lambda=ABC$,
with $B$ shielding $A$ from $C$ as in Figure 1, and $\rho$ a Gibbs state of a
local, translation-invariant Hamiltonian on $\Lambda$, we investigate several
notions of conditional independence between $A$ and $C$ conditioned on $B$,
and show that they decay with the size of $B$.

1\. Superexponential decay of the BS Conditional Mutual Information

We first consider generalisations of the CMI involving the Belavkin-Staszewski
(BS) relative entropy [9], an upper bound for the Umegaki relative entropy,
which is given by Eq. (12). For three notions of BS Conditional Mutual
Information (BS-CMI), termed one-sided (Eq. (16)), two-sided (Eq. (17)) and
reversed (Eq. (18)), we establish superexponential decay and thereby
positively answer a conjecture from [15]. Note that this is faster than what
is expected to hold for the CMI in this context [46, 54, 52]. We prove that
all of these quantities exhibit the following scaling:

$\widehat{I}_{\rho}^{x}(A:C|B)\leq
c_{x}e^{\alpha_{x}(|A|+|B|)}\varepsilon(|B|),\quad
x\in\\{\operatorname{os},\operatorname{ts},\operatorname{rev}\\}$ (4)

where $A$ and $C$ are separated by $B$. The function
$\ell\mapsto\varepsilon(\ell)$ decays superexponentially with $\ell$, and
$c_{x}$ and $\alpha_{x}$ are universal for all intervals, only dependent on
the inverse temperature and the range and strength of the interaction that
gives rise to the local Hamiltonians as well as the local dimension. The proof
of this result is split into two parts, which constitute results of
independent interest.

1.1. Upper bound on the DPI for the BS-entropy

The first step of the proof is an upper bound on the data processing
inequality of the BS-entropy for conditional expectations. This bound is
expressed in terms of a difference measure that establishes the relationship
between $\rho$ and its asymmetric BS-recovery condition [12], denoted by
$\operatorname{\mathcal{B}}^{\sigma}_{\operatorname{\mathcal{E}}}(\rho):=\sigma\operatorname{\mathcal{E}}(\sigma)^{-1}\operatorname{\mathcal{E}}(\rho)$
and analogously with the roles of $\rho$ and $\sigma$ interchanged. More
precisely, in Section 3.1 we show that

$\widehat{D}(\rho\|\sigma)-\widehat{D}(\operatorname{\mathcal{E}}(\rho)\|\operatorname{\mathcal{E}}(\sigma))\leq\begin{cases}C(\rho,\sigma,\operatorname{\mathcal{E}})\,\|\rho\left(\operatorname{\mathcal{B}}^{\sigma}_{\operatorname{\mathcal{E}}}(\rho)\right)^{-1}-\operatorname{\mathds{1}}\|_{\infty}\\\\[4.30554pt]
C^{\prime}(\rho,\sigma,\operatorname{\mathcal{E}})\,\|\sigma\left(\operatorname{\mathcal{B}}^{\rho}_{\operatorname{\mathcal{E}}}(\sigma)\right)^{-1}-\operatorname{\mathds{1}}\|_{\infty}\end{cases}\,.$
(5)

This provides a complementary perspective on the strengthened data processing
inequality established in [12]. The reason is that it offers an upper limit on
the disparity of the BS-DPI, explicitly featuring a difference between the
states and their BS-recovery.

1.2. Decay of BS-CMI for Gibbs states of 1D local Hamiltonians

To prove Eq. (4), we combine the bounds of Eq. (5) with existing techniques of
decay of correlations on Gibbs states of 1D local Hamiltonians, such as those
in [15]. More specifically, in Section 2.4, we include a collection of
technical results for these objects based on tight estimates of Araki’s
expansionals [6], which we use, jointly with [15, Theorem 5.1], to prove:

$\norm{\rho_{ABC}\rho^{-1}_{BC}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}\leq\varepsilon(|B|)\,,$
(6)

for $\rho$ a Gibbs state on a possibly larger chain $\Lambda^{\prime}$ than
$\Lambda=ABC$, with $\Lambda\subset\Lambda^{\prime}$, and $\rho_{X}$ its
marginal on $X$ for $X\in\\{AB,B,BC,ABC\\}$. Noticing that the BS-recovery of
$\rho_{ABC}$ for the partial trace in $A$ is
$\rho_{AB}\rho_{B}^{-1}\rho_{BC}$, we combine Eq. (6) with Eq. (5) to conclude
Eq. (4).

2\. Efficient estimation of the global purity of 1D local Hamiltonians

In the next part of the paper, we consider another notion of conditional
independence, namely the purity. This quantity can be viewed as a version of
the CMI defined in terms of Rényi-$2$ entropies. For this purity, using
similar techniques and again results from [15] we resolve a conjecture from
[79] and show that in the same setting as above, it decays exponentially with
the size of $|B|$. We subsequently employ this to directly establish the
sample and computational efficiency of the scheme to learn the purity from
[79].

2.1. Approximate factorisation of the purity

In the first part of this result, contained in Section 3.3, we explicitly
prove that

$\left|\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{ABC}[{\rho_{ABC}}^{2}]\Tr_{B}[\rho_{B}^{2}]}-1\right|\leq
c_{p}e^{-\alpha_{p}|B|}\,,$ (7)

where $\rho_{ABC}$ is a Gibbs state and the other states are marginals
thereof. The constants $c_{p}$ and $\alpha_{p}>0$ are again universal and only
depend on the range, inverse temperature, and strength of the interaction, as
well as the local dimension.

2.2. Estimation of the global purity

Subsequently, in Section 4.3, we consider an $N$-partite system
$\Lambda=A_{1}\ldots A_{N}$ and consider an $N$-partite version of the purity
(removing the Rényi entropy of the total chain), given by

$P_{2}(\rho_{1:N})=\frac{\prod_{j=1}^{N-1}\Tr_{j:j+1}[\rho_{j:j+1}^{2}]}{\prod_{j=2}^{N-1}\Tr_{j}[\rho_{j}^{2}]}\,.$
(8)

Then, we show that, due to an iterated application of Eq. (7), we can obtain
an approximation of $\Tr[\rho_{1:N}^{2}]$ by $P_{2}(\rho_{1:N})$ up to an
error $\varepsilon$ as long as each region $A_{i}$ has size of order
$\mathcal{O}(\log N/\varepsilon)$. Considering next some local measurements of
each of the marginals of $\rho$ up to an error $\delta$, we show that there is
an algorithm that outputs an estimate of $\Tr[\rho_{1:N}^{2}]$ up to a
multiplicative error $\varepsilon$ with a number of samples and classical
post-processing cost of order $\operatorname{poly}(|\Lambda|/\varepsilon)$.

3\. Learning of Gibbs states of 1D local Hamiltonians via Matrix Product
Operator approximations

Finally, we apply our results for matrix product operator (MPO)
reconstructions of Gibbs states and their efficient learning. Again, we
consider a multipartite quantum state on an $N$-partite system. Specifically,
the results on the decay of BS-CMI allow us to give an efficient MPO
description of 1D Gibbs states. Furthermore, we show that this description can
also be learned efficiently from local tomography. We find a polynomial sample
and computational complexity in both the system size and inverse error.

3.1. Positive MPO approximations from recovery maps

Apart from the previously mentioned asymmetric recovery map, we introduce an
alternative, symmetric, and thereby positive recovery map

$\operatorname{\mathcal{R}}_{i}(X)=\rho_{i}^{1/2}(\rho_{i}^{-1/2}\rho_{i:{i+1}}\rho_{i}^{-1/2})^{1/2}\rho_{i}^{-1/2}X\rho_{i}^{-1/2}(\rho_{i}^{-1/2}\rho_{i:{i+1}}\rho_{i}^{-1/2})^{1/2}\rho_{i}^{1/2}\,.$
(9)

The recovery error is bounded by the BS-CMI with additional terms
corresponding to the lowest eigenvalue of marginals of thermal state and the
maximal mutual information [70], all of which can be appropriately bounded for
1D Gibbs states. This map is however not a quantum channel as it is not trace-
preserving. This poses a difficulty since the non-contractive map could
exponentially amplify recovery errors in earlier steps of the reconstruction.
We overcome this issue by proving a Lipschitz constant on a concatenation of
maps that is _independent_ of the level of concatenation. We prove a
representability result for the Gibbs state by the MPO obtained from
concatenating these maps

$\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|\leq\varepsilon$

with a subpolynomial bond dimension in $|\Lambda|/\varepsilon$.

3.2. Reconstruction of positive MPO from local tomography

Considering the explicit form of the recovery map Eq. (9) in terms of the
marginals, we show how to learn this MPO representation and prove that the
approximation error to the state is robust to small tomographic errors. We
obtain the representation as an explicit formula of the estimated marginals.
As sub-logarithmically sized marginals are sufficient for the reconstruction,
using standard tomography results, the sample and computational cost incurred
to reconstruct the state to $\varepsilon$ error in 1-norm is again
$\operatorname{poly}(|\Lambda|/\varepsilon)$. A subpolynomial dependence in
system size is also possible using an inherently translation-invariant
formulation of the MPO, see Remark 4.10.

Bonus. Exponentially-decaying interactions

Many of the techniques used in our main results can or have been recently
extended to Gibbs states of Hamiltonians with exponentially-decaying
interactions, as seen in [64, 21, 16]. Therefore, a natural question is
whether all results presented so far extend to that framework. In Appendix B
we positively answer that question, with small but necessary modifications.
The decay of the BS-CMI is now only exponential in $|B|$ and holds only for
certain regimes of $\beta$. In contrast, the estimation of the global purity
follows the same as for finite-range interactions. Moreover, we can provide a
protocol to learn marginals of Gibbs states in 1D with translation-invariant,
exponentially-decaying interactions, by MPO approximations, albeit with a
slightly worse bond dimension than for finite-range. Nevertheless, as far as
we know, this is the first MPO approximation for Gibbs states in this
framework in the literature.

###  1.2 Previous work and future directions

It is interesting to compare our results to existing ones on the
representability of thermal states by tensor networks. In general, it is known
that representations with polynomial bond dimension exist for arbitrary
lattices [61, 81]. Furthermore, in one dimension, an MPO description of finite
Gibbs states with sub-polynomial bond dimension has been shown in [53]. A
feature common to all of these constructions is that they give explicit
formulas in terms of the Hamiltonian. While this easily allows for a
computation of the MPO given _knowledge of the systems’ interactions_ , it is
unclear how to _learn_ such a representation directly. To do that, a precise
analysis of the approximation quality given the errors in the estimated
Hamiltonian would be needed. Furthermore, even if such a result is achieved,
the best existing Hamiltonian learning results [8, 3, 39] take polynomial time
with degrees that are often impractically large. Defining our reconstruction
in terms of measurable marginals, we circumvent these complications and
provide a practical formula for the reconstruction. Another feature of our
construction is that it directly works in the thermodynamic limit.
Constructions explicitly involving the Hamiltonian would need to be truncated,
however, an approach in this direction has been put forward in [1].

While a recent result in [31, 68], proposes a way of learning finitely
correlated states (which can often be seen as equivalent to matrix product
operators) and the application of this framework to the learning of Gibbs
states, it is currently unclear whether this approach can yield an equivalent
result to ours. The constructions involve measurements on polynomially-sized
marginals, which can be learned efficiently in sample complexity, but possibly
with exponential circuit complexity.

Finally, on a more conceptual level, the proof technique could be of
independent interest in terms of extending the representability result to
larger classes of states. We provide sufficient information theoretic criteria
that guarantee efficient MPO approximations. It is an interesting question to
find out whether other types of states also satisfy these conditions. The
overall idea displays a strong analogy to the technique used in the proof of
representability of pure gapped ground states by matrix product states. There,
an information-theoretic area law for Rényi entropies [40] implies efficient
representability of the state by a matrix product state [80, 71] and is also
involved in the later rigorous proof of an efficient algorithm for the ground-
state energy [56].

Let us finally also comment on an alternative approach that might come to mind
when looking at our construction: Using a decay of the standard quantum
conditional mutual information and the corresponding recovery channels. The
existence of such recovery channels has been shown [33] and since they are
indeed quantum channels (CPTP-maps) their concatenation should be possible in
an analogous way. There are two obstructions to this route. Firstly, the
channels are not given by an equally simple explicit formula but involve an
optimization procedure, which could be cumbersome to deal with on the learning
side of our result [45, 75]. Secondly, no superexponential decay result is
known or believed to hold for the conditional mutual information.

##  2 Preliminaries

This section is dedicated to introducing and reviewing the basic terminology,
notions, and results of quantum systems, entropy measures, Gibbs states in
quantum spin chains, and their _approximate factorization_.

###  2.1 Basic notation

First, all vector spaces, tensor products and homomorphism spaces considered
in this paper are defined over the field of complex numbers
$\operatorname{\mathbb{C}}$. For a quantum system labelled $A$, we let
$\operatorname{\mathcal{H}}_{A}$ denote the Hilbert space. We generally adopt
the convention of using the capital letters $A,B,C,\ldots$ to label the
different subsystems constituting the system. The composition of subsystems is
denoted by juxtaposing the corresponding letters, as standard in the
literature. For a bipartite system $AB$ arising as the composition of two
systems $A$ and $B$, the Hilbert space is given by the tensor product Hilbert
space
$\operatorname{\mathcal{H}}_{AB}=\operatorname{\mathcal{H}}_{A}\otimes\operatorname{\mathcal{H}}_{B}$.
In the particular case of quantum spin chains, the juxtaposition of
consecutive letters representing intervals of qubits will typically presuppose
that these are, in addition, adjacent and non-overlapping intervals.

For any two finite-dimensional Hilbert spaces $\operatorname{\mathcal{H}}_{A}$
and $\operatorname{\mathcal{H}}_{B}$, we let
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A},\operatorname{\mathcal{H}}_{B})$
denote the set of bounded linear operators from
$\operatorname{\mathcal{H}}_{A}$ to $\operatorname{\mathcal{H}}_{B}$, endowed
with the operator norm $\norm{\,\cdot\,}$. The set of linear operators from
$\operatorname{\mathcal{H}}_{A}$ to itself is denoted by
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})\coloneq\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A},\operatorname{\mathcal{H}}_{A})$.
The trace map is generically denoted
$\Tr:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})\to\operatorname{\mathbb{C}}$,
sometimes with subindices if we want to emphasise the system it acts on. A
similar notational convention is adapted to denote the identity map
$\operatorname{\mathds{1}}$. In turn, for multipartite systems, we convey that
the partial trace over the subsystem $A$ is denoted
$\mathrm{tr}_{A}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{AB})\to\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{B})$.
The spaces of bounded linear operators can be equipped with the Schatten
$p$-norms, defined by $\norm{X}_{p}\coloneq(\operatorname{Tr}(|X|^{p}))^{1/p}$
for all $p\in[1,\infty)$, where $\absolutevalue{X}\coloneq(X^{*}X)^{1/2}$
stands for the operator square root. We recall that the Schatten $1$-norm
coincides with the usual trace norm, defined as the sum of the singular values
of the operator, and the Schatten $2$-norm is the Hilbert-Schmidt norm arising
from the Hilbert-Schmidt inner product on
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})$. Furthermore, in
the limiting case $p\to+\infty$ the Schatten $\infty$-norm recovers the
operator norm $\norm{\,\cdot\,}_{\infty}=\norm{\,\cdot\,}$, also characterized
as the largest singular value of the operator. We recall that all Schatten
$p$-norms are submultiplicative and unitarily invariant. Moreover, they
monotonically ordered in the sense that $\norm{X}_{p}\geq\norm{X}_{q}$ for any
$1\leq p\leq q\leq\infty$, and satisfy Hölder’s inequality
$\norm{XY}_{r}\leq\norm{X}_{p}\norm{Y}_{q}$ for all $p,q,r\in[1,\infty]$
satisfying $1/r=1/p+1/q$.

Over any system $A$ we consider the non-negative and normalised set
$\operatorname{\mathcal{S}}(\operatorname{\mathcal{H}}_{A})\subset\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})$
of states or density operators, i.e. non-negative endomorphisms $\rho$ such
that $\operatorname{Tr}[\rho]=1$. In particular, we consider the maximally
mixed state $\pi_{A}\coloneq d_{A}^{-1}\operatorname{\mathds{1}}_{A}$ on $A$,
where $d_{A}=\dim\operatorname{\mathcal{H}}_{A}$. In general, we reserve the
Greek letters $\rho$ and $\sigma$ for all states considered here. For any
multipartite mixed state $\rho_{AB}$, we denote the marginals that arise after
application of the partial trace with a subindex that indicates the system
they act on, i.e. $\rho_{A}\coloneq\tr_{B}[\rho_{AB}]$ and
$\rho_{B}\coloneq\tr_{A}[\rho_{AB}]$.

As we will be relying on the notion of conditional expectation, we shortly
want to introduce it here as well. For a von Neuman subalgebra
$\operatorname{\mathcal{N}}$ of
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$ there exists a unique
map, called conditional expectation, we will generally denote with
$\operatorname{\mathcal{E}}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})\to\operatorname{\mathcal{N}}$,
which satisfies the properties that it projects onto
$\operatorname{\mathcal{N}}$ orthogonally w.r.t the Hilbert-Schmidt inner
product. An important example and the use case of such maps in this paper are
partial traces followed by an embedding, i.e maps of the form
$\pi_{A}\otimes\tr_{A}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{AB})\to\operatorname{\mathds{1}}_{A}\otimes\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{B})\subseteq\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{AB})$,
where $X\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{AB})$ is
mapped to $\pi_{A}\otimes\tr_{A}[X_{AB}]$.

Conditional expectations form a subset within the broader set of quantum
channels, encompassing completely positive trace-preserving linear mappings.
Note at last that through Stinespring’s dilation theorem, every quantum
channel can be represented by a composition of an isometry with a partial
trace.

###  2.2 Entropy measures

We introduce of several entropy measures that will appear throughout the text,
and present some connections between them. Arguably, the most recognized
measure among them is the _von Neumann entropy_ of a state $\rho$, defined by
the expression

$S(\rho)\coloneq-\Tr[\rho\log(\rho)]\,.$ (10)

where we convey $0\,\log 0\equiv 0$. The von Neumann entropy captures the
entropy of a quantum state and is a quantum analogous to the classical Shannon
entropy [72]. Also inspired by the classical setting, one can generalize the
Kullback-Leibler divergence [51] to the quantum setting. The most prominent of
those generalizations is the _Umegaki relative entropy_ [78], for which given
two quantum states $\rho$ and $\sigma$, it is defined by the expression

$D(\rho\|\sigma)\coloneq\begin{cases}\Tr[\rho\log\rho-\rho\log\sigma]&\text{
if }\ker\sigma\subseteq\ker\rho\,,\\\ +\infty&\text{ otherwise}\,.\end{cases}$
(11)

A less prominent one, which has however gained attention in the last years as
e.g. a tool to estimate channel capacities [30] and the decay of correlation
measures for Gibbs state in 1D [15] is the Belavkin-Staszewski relative
entropy [9] or BS-entropy for short. For any two invertible mixed states
$\rho$ and $\sigma$, it is defined as

$\widehat{D}(\rho\mathbin{\|}\sigma):=\begin{cases}\Tr[\rho\log(\rho^{1/2}\sigma^{-1}\rho^{1/2})]&\text{
if }\ker\sigma\subseteq\ker\rho\,,\\\ +\infty&\text{ otherwise}\,.\end{cases}$
(12)

When talking about a generalization of an entropic measure to the quantum
setting, we mean that if the involved quantum states commute they can be
regarded as probability vectors, so the quantum measure reduces to its
classical analogue. For example, in the case of the Umegaki and BS-entropy,
one would recover the Kullback-Leibler divergence. This means that both of
them agree in the commuting case, while the BS-entropy is strictly bigger than
the Umegaki relative entropy in the case of non-commuting states [43]. Note
further that both divergences can be defined for general positive semidefinite
matrices, and we will use those definitions under slight abuse of notation.

Whilst not featured in our decay results, we will make use of the maximal
Rényi divergence in our results on MPO constructions [27]

$D_{\infty}(\rho\|\sigma)=\log\inf\left\\{\lambda:\rho\leq\lambda\sigma\right\\}\,,$

which arises as a limit of a wider family of sandwiched Rényi entropies [60].

As we are mostly concerned with the analysis of quantities involving the BS-
entropy, we skip the definitions of the analogues for the relative,
respectively von Neumann entropy, and only note that conditional entropy,
mutual information, and conditional mutual information were initially defined
in terms of the Shannon entropy and then rewritten using the Umegaki relative
entropy. Inspired by this latter representation, the BS-entropy analogues are
obtained by just replacing the Umegaki with the BS-entropy. We refer to [13,
14, 70, 84] for partial or complete definitions of these quantities.

For any bipartite mixed state $\rho_{AB}$, we define the _BS-conditional
entropy_ by

$\widehat{H}_{\rho}(A\mathbin{|}B)\coloneq-\widehat{D}(\rho_{AB}\mathbin{\|}\pi_{A}\otimes\rho_{B}),$
(13)

and the _BS-conditional mutual information_ by

$\widehat{I}_{\rho}(A\mathbin{|}B)\coloneq\widehat{D}(\rho_{AB}\mathbin{\|}\rho_{A}\otimes\rho_{B}).$
(14)

Equivalently, we also introduce the _maximal mutual information_ [70]

$I_{\infty}(A|B)\coloneq D_{\infty}(\rho_{AB}\|\rho_{A}\otimes\rho_{B}).$ (15)

The definition of the BS-conditional mutual information (BS-CMI) is relatively
more ambiguous and hence we recall the following definitions from [13]. Let us
consider a tripartite quantum system $ABC$, then, for a quantum state
$\rho_{ABC}$ we define the _one-sided BS-conditional mutual information_
between the systems $A$ and $C$ conditioned on the system $B$ by the
expression

${\widehat{I}}^{\mathrm{os}}_{\rho}(A\mathbin{:}C\mathbin{|}B)\coloneq\widehat{D}(\rho_{ABC}\mathbin{\|}\pi_{A}\otimes\rho_{BC})-\widehat{D}(\rho_{AB}\mathbin{\|}\pi_{A}\otimes\rho_{B})$
(16)

and the _two-sided BS-conditional mutual information_ as

${\widehat{I}}^{\mathrm{ts}}_{\rho}(A\mathbin{:}C\mathbin{|}B)\coloneq\widehat{D}(\rho_{ABC}\mathbin{\|}\rho_{A}\otimes\rho_{BC})-\widehat{D}(\rho_{AB}\mathbin{\|}\rho_{A}\otimes\rho_{B}).$
(17)

Lastly, we define the reversed BS-conditional mutual information

${\widehat{I}}^{\mathrm{rev}}_{\rho}(A\mathbin{:}C\mathbin{|}B):=\widehat{D}(\pi_{A}\otimes\rho_{BC}\mathbin{\|}\rho_{ABC})-\widehat{D}(\pi_{A}\otimes\rho_{B}\mathbin{\|}\rho_{AB}),$
(18)

where we recall that $\pi_{A}$ stands for the maximally mixed state on $A$.

###  2.3 Spin chains in 1D, local Hamiltonians and Gibbs states

We write $\Lambda\Subset\operatorname{\mathbb{Z}}$ for $\Lambda$ being a
finite subset of $\operatorname{\mathbb{Z}}$, use $|\Lambda|$ to denote its
cardinality and
$\operatorname{diam}(\Lambda):=\max\\{x-y\;:\;x,y\in\Lambda\\}$ its diameter.
With every site $x\in\operatorname{\mathbb{Z}}$ we associate a finite-
dimensional Hilbert space
$\operatorname{\mathcal{H}}_{x}:=\operatorname{\mathbb{C}}^{d}$ with
corresponding linear operators
$\operatorname{\mathcal{A}}_{x}:=\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{x})$.
We generalise this concept to finite sets, where for
$\Lambda\Subset\operatorname{\mathbb{Z}}$, we define the Hilbert space
$\operatorname{\mathcal{H}}_{\Lambda}:=\bigotimes_{x\in\Lambda}\operatorname{\mathcal{H}}_{x}$
and the algebra of linear operators as
$\operatorname{\mathcal{A}}_{\Lambda}:=\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{\Lambda})=\bigotimes_{x\in\Lambda}\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{x})$.
The last equality holds due to the finite-dimensional nature of
$\operatorname{\mathcal{H}}_{\Lambda}$. For
$\Lambda^{\prime}\subseteq\Lambda\Subset\operatorname{\mathbb{Z}}$, we can
therefore consider $X\in\operatorname{\mathcal{A}}_{\Lambda^{\prime}}$ as an
element of $\operatorname{\mathcal{A}}_{\Lambda}$ by identifying $X$ with
$X\otimes\operatorname{\mathds{1}}_{\Lambda\backslash\Lambda^{\prime}}$. This
identification is kept implicit and allows us to define the algebra of local
observables for general $\Sigma\subseteq\operatorname{\mathbb{Z}}$ simply by

$\operatorname{\mathcal{A}}_{\Sigma}:=\overline{\bigcup\limits_{\Lambda\Subset\Sigma}\operatorname{\mathcal{A}}_{\Lambda}}^{\norm{\cdot}_{\infty}}\,,$
(19)

where the closure is taken with respect to the operator norm. Now an
interaction on $\Sigma\subseteq\operatorname{\mathbb{Z}}$ is defined as

$\Phi:\\{\Lambda\Subset\Sigma\\}\to\operatorname{\mathcal{A}}_{\operatorname{\mathbb{Z}}},\quad\Lambda\mapsto\Phi(\Lambda)\in\operatorname{\mathcal{A}}_{\Lambda}\quad\text{with}\quad\Phi(\Lambda)=\Phi(\Lambda)^{*}\,,$
(20)

and its corresponding local Hamiltonian on any $\Lambda\Subset\Sigma$ as

$H_{\Lambda}:=\sum\limits_{\Lambda^{\prime}\subseteq\Lambda}\Phi(\Lambda^{\prime})\,.$
(21)

In this paper, we confine our focus to finite-range interactions,
characterized by the existence of parameters $R>0$ and $J>0$ such that
$\Phi(\Lambda)=0$ whenever the diameter of $\Lambda$ exceeds $R$, and

$\sum\limits_{\Lambda\Subset\operatorname{\mathbb{Z}}\;:\;x\in\Lambda}\norm{\Phi(\Lambda)}_{\infty}\leq
J\,,$

for all $x\in\operatorname{\mathbb{Z}}$. In the literature, it is common to
alternatively impose the condition that for all
$\Lambda\Subset\operatorname{\mathbb{Z}}$,
$\norm{\Phi(\Lambda)}_{\infty}\leq\widetilde{J}$, which can be related to $J$
by observing that, due to the interaction vanishing if the diameter of the
input exceeds $R$,

$\sum\limits_{\Lambda\Subset\operatorname{\mathbb{Z}}\;:\;x\in\Lambda}\norm{\Phi(\Lambda)}_{\infty}=\sum\limits_{k=0}^{R}\sum\limits_{\Lambda\in\operatorname{\mathcal{P}}(\\{x-k,\ldots,x-k+R\\}\backslash\\{x\\})}\norm{\Phi(\\{x\\}\cup\Lambda)}_{\infty}\leq(R+1)2^{R}\widetilde{J}\,.$

In addition to the finite range, we further require interactions to exhibit
translation invariance.

Lastly, we define the Gibbs state for a Hamiltonian
$H\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$ with $H=H^{*}$
over an arbitrary (finite-dimensional) Hilbert space
$\operatorname{\mathcal{H}}$ at inverse temperature $\beta\in(0,\infty)$ as

$\rho_{\operatorname{\mathcal{H}}}^{\beta}[H]:=\frac{e^{-\beta
H}}{\Tr[e^{-\beta H}]}\,.$ (22)

In Section 3.2, i.e., when we prove superexponential decay of Gibbs states of
local translation invariant Hamiltonians on a quantum spin system, we drop the
index of the Hilbert space, absorb the inverse temperature into the
Hamiltonian and write for
$\Lambda\Subset\Sigma\subseteq\operatorname{\mathbb{Z}}$ just

$\rho^{\Lambda}:=\rho^{\beta}_{\operatorname{\mathcal{H}}_{\Lambda}}[H_{\Lambda}]\,.$
(23)

Notice that Gibbs states of finite $\beta$ are always full rank.

###  2.4 Approximate factorisation of Gibbs states of local Hamiltonians in 1D

We now present the technical lemmas we require regarding the decay of
correlations and uniformity of 1D Gibbs states. While they hold at all
temperatures for any finite-range interaction, the constants involved depend
on range, interaction strength and temperature. This dependence is mostly the
same for all involved quantities, so we introduce the following convenient
notation.

Let $\Lambda\Subset\mathbb{Z}$ be a finite interval. Let us split $\Lambda$
into two subintervals $X$ and $Y$ so that $\Lambda=X\cup Y$ or $\Lambda=XY$
for short and write

$E_{X,Y}\,:=\,e^{-\,H_{XY}}\,e^{\,H_{X}\,+\,H_{Y}}\,.$ (24)

Note that $E^{*}_{X,Y}=e^{\,H_{X}\,+\,H_{Y}}\,e^{-\,H_{XY}}$ and
$E^{-1}_{X,Y}=e^{-H_{X}\,-\,H_{Y}}\,e^{\,H_{XY}}$. The following proposition
is extracted from [15, 64] and contains an alternative formulation of Araki’s
results for estimates on expansionals [6].

######  Proposition 2.1 ([15, Corollary 3.4]).

Let $\Phi$ be an interaction of finite-range $R$ and strength $J$ over
$\operatorname{\mathbb{Z}}$ which is further translation invariant. Then the
following hold.

  1. (i)

There is an absolute constant $\mathcal{G}>1$ depending only on $J$, $R$ and
$\beta$ such that, for any finite interval $\Lambda=XY\Subset\mathbb{Z}$ split
into two subintervals $X$ and $Y$, we have:

$\norm{E_{X,Y}}\,,\,\norm{E_{X,Y}^{-1}}\,\,\leq\,\,\operatorname{\mathcal{G}}\,.$

  2. (ii)

There is a positive and decreasing function $\ell\mapsto\delta(\ell)$ with
superexponential decay and depending on $J$, $R$ and $\beta$ such that if we
add two intervals $\widetilde{X}$ and $\widetilde{Y}$ adjacent to $X$ and $Y$,
respectively, so that we get a larger interval $\widetilde{X}XY\widetilde{Y}$,
then

$\left\|E_{X,Y}^{-1}-E^{-1}_{\widetilde{X}X,Y\widetilde{Y}}\right\|,\left\|E_{X,Y}-E_{\widetilde{X}X,Y\widetilde{Y}}\right\|\,\leq\,\delta(\ell)\,.$

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

Note in addition that for $\Lambda^{\prime}\subset\Lambda\Subset\mathbb{Z}$
with local Hamiltonian $H_{\Lambda^{\prime}}$ supported in $\Lambda^{\prime}$
but lifted to $\operatorname{\mathcal{H}}_{\Lambda}$, the map

$\operatorname{\mathcal{A}}_{\Lambda}\to\operatorname{\mathcal{A}}_{\Lambda}\,,\quad
Q\mapsto\tr_{\Lambda^{\prime}}[e^{-H_{\Lambda^{\prime}}}Q]=\tr_{\Lambda^{\prime}}[e^{-\frac{1}{2}H_{\Lambda^{\prime}}}Qe^{-\frac{1}{2}H_{\Lambda^{\prime}}}]$
(25)

where $\tr_{\Lambda^{\prime}}$ denotes the partial trace in
$\Lambda^{\prime}$, has the following property:

$\norm{\tr_{\Lambda^{\prime}}[e^{-H_{\Lambda^{\prime}}}Q]}_{\infty}\leq\norm{\tr_{\Lambda^{\prime}}[e^{-H_{\Lambda^{\prime}}}]}_{\infty}\,\norm{Q}_{\infty}=\Tr_{\Lambda^{\prime}}[e^{-H_{\Lambda^{\prime}}}]\norm{Q}_{\infty}\,.$
(26)

As a consequence,

$\operatorname{\mathcal{A}}_{\Lambda}\to\operatorname{\mathcal{A}}_{\Lambda}\,,\quad
Q\mapsto\tr_{\Lambda^{\prime}}[\rho^{\Lambda^{\prime}}Q]$ (27)

is positive, unital and hence contractive. This observation is essential for
the following result, in which we are considering an interval $\Lambda$ split
into three subintervals $\Lambda=ABC$, where $B$ shields $A$ from $C$, such as
the one in Figure 2.

$A$$B$$C$$\Lambda$ Figure 2: Representation of an interval $\Lambda$ split
into three subintervals $\Lambda=ABC$, where $B$ shields $A$ from $C$.

######  Proposition 2.2.

In the context of Proposition 2.1, let
$\Lambda=ABC\Subset\operatorname{\mathbb{Z}}$ (we admit the possibility of
some subintervals being empty). Then, there is an absolute constant
$\mathcal{C}$ depending only on the strength $J$, range $R$ and inverse
temperature $\beta$ of interactions, such that

$\displaystyle\big{\|}\tr_{B}[\rho^{B}Q]\big{\|},\big{\|}\tr_{B}[\rho^{B}Q]^{-1}\big{\|}\,\leq\,\mathcal{C}\,\,,\quad\quad
Q\in\\{E_{B,C}^{*}\,,\,E_{B,C}\,,\,E_{A,B}^{*}\,,\,E_{A,B}\\}\,,$ (28)
$\displaystyle\big{\|}\tr_{AB}[\rho^{AB}Q]\big{\|}\,,\,\big{\|}\tr_{AB}[\rho^{AB}Q]^{-1}\big{\|}\,\leq\,\mathcal{C}\,\,,\quad\quad
Q\in\\{E_{A,B}^{*\,-1}\,,\,E_{A,B}^{\,-1}\\}\,,$ (29)
$\displaystyle\big{\|}\tr_{B}\big{[}\rho^{B}E_{A,B}^{*}E_{AB,C}^{*}\big{]}\big{\|}\,,\,\big{\|}\tr_{B}\big{[}\rho^{B}E_{A,B}^{*}E_{AB,C}^{*}\big{]}^{-1}\big{\|}\,\leq\,\mathcal{C}\,.$
(30)

These two propositions are instrumental in the proof of the following result,
which provides an approximate factorization of the Gibbs state of a finite-
range, translation-invariant Hamiltonian.

######  Theorem 2.3 ([15, Eq. (17) in Theorem 5.1]).

For $\Phi$ a finite-range, translation-invariant interaction over
$\operatorname{\mathbb{Z}}$ there exists a positive function
$\ell\mapsto\varepsilon(\ell)$ with superexponential decay in $\ell$,
depending only on $J$, $R$ and $\beta$ such that for every
$\Lambda\Subset\operatorname{\mathbb{Z}}$ split into three subintervals
$\Lambda=ABC$, where $B$ shields $A$ from $C$ and $|B|\geq\ell$ for its local
Gibbs state $\rho^{\Lambda}=:\rho_{ABC}$ it holds that

$\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}\leq\varepsilon(\ell)\,.$
(31)

The form of the superexponentially decaying function is given by

$\varepsilon(\ell)=\operatorname{\mathcal{C}}_{1}\frac{\operatorname{\mathcal{C}}_{2}^{1+\lfloor\ell/2\rfloor}}{(\lfloor\lfloor\ell/2\rfloor/R\rfloor+1)!}$

for constants $\operatorname{\mathcal{C}}_{1},\operatorname{\mathcal{C}}_{2}$
that only depend on $J$, $R$, and $\beta$.

Note that, in Eq. (31), only $\rho_{ABC}$ denotes a Gibbs state of a local
Hamiltonian while $\rho_{AB},\rho_{BC}$ and $\rho_{B}$ are marginals, obtained
by partially tracing out the systems missing in the index. We further
suppressed tensoring with identity, i.e. $\rho_{AB}$ for example has to be
understood as $\rho_{AB}\otimes\operatorname{\mathds{1}}_{C}$.

In the rest of the paper, we will need a refined version of this theorem for
Gibbs states on $\Lambda=A^{\prime}ABCC^{\prime}$, for which we only compare
the marginals in $AB$, $B$, $BC$ and $ABC$ as above. Before that, we need the
following technical lemmata. The first constitutes a generalization of the
second point in Proposition 2.2 for $A$ with two connected components and
arbitrary operators.

$A$$B$$C$$C^{\prime}$$A^{\prime}$$\Lambda$ Figure 3: Representation of an
interval $\Lambda$ split into five subintervals
$\Lambda=A^{\prime}ABCC^{\prime}$.

######  Lemma 2.4.

Let $\Lambda\Subset\mathbb{Z}$ be a finite interval split into five
subintervals $\Lambda=ABC$ as in Figure 3. Then, for $\rho=\rho^{\Lambda}$ the
Gibbs state on $\Lambda$, and for every $\beta>0$, the following holds:

$\norm{\tr_{AC}[\rho^{A}\otimes\rho^{C}Q]^{-1}}_{\infty}\,\leq\,\mathcal{C}\,$
(32)

for $Q=E_{AB,C}\,E_{A,BC}$ and a constant $\operatorname{\mathcal{C}}$ that
only depends on $J$, $R$, $\beta$ and $d$.

We defer the proof of Lemma 2.4 to Appendix A. Next, we show that the decay
provided by the measure of conditional independence provided in Theorem 2.3
can be reduced from the total Gibbs state to a large marginal of it. The proof
of this lemma is also ommited from the main text and can be found in Appendix
A.

######  Lemma 2.5.

Let $\Lambda\Subset\mathbb{Z}$ be a finite interval split into five
subintervals $\Lambda=A^{\prime}ABCC^{\prime}$ as in Figure 3. Then, for
$\rho=\rho^{\Lambda}$ the Gibbs state on $\Lambda$, and for every $\beta>0$,
the following holds

$\left\|\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}-\operatorname{\mathds{1}}\right\|_{\infty}<\mathcal{C}^{4}\,\left\|\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}-\operatorname{\mathds{1}}\right\|_{\infty}\,,$
(33)

for $\mathcal{C}>1$ depending on $R,J$ and $\beta$, and given in Proposition
2.2.

######  Remark 2.6.

Note that Eq. (33) can be interpreted as a modified data-processing inequality
for the partial trace for the mixing condition between two positive states
$\rho$ and $\sigma$,

$\norm{\eta\,\sigma^{-1}-\operatorname{\mathds{1}}}_{\infty}\,,$

with contraction coefficient upper bounded by $\mathcal{C}^{4}$, identifying
$\eta:=\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}$,
$\sigma:=\rho_{A^{\prime}ABCC^{\prime}}$, and taking the partial trace in
$A^{\prime}C^{\prime}$.

Combining now the findings of Theorem 2.3 and Lemma 2.5, we can conclude the
following:

######  Corollary 2.7.

For $\Phi$ a finite-range, translation-invariant interaction over
$\operatorname{\mathbb{Z}}$ there exists a positive function
$\ell\mapsto\tilde{\varepsilon}(\ell)$ with superexponential decay in $\ell$,
depending only on $J$, $R$ and $\beta$ such that for every
$\Lambda\Subset\operatorname{\mathbb{Z}}$ split into three subintervals
$\Lambda=A^{\prime}ABCC^{\prime}$, with $|B|\geq\ell$ for its local Gibbs
state $\rho^{\Lambda}=:\rho_{A^{\prime}ABCC^{\prime}}$ it holds that

$\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}\leq\tilde{\varepsilon}(\ell)\,.$
(34)

In addition to the above, the authors in [15, Theorem 6.2] further showed
exponential decay of correlations between spatially separated regions. This
will become relevant to us later in the form of local indistinguishability, as
per Theorem 2.9.

######  Proposition 2.8 (Decay of correlations).

For $\Phi$ a finite-range, translation-invariant interaction over
$\operatorname{\mathbb{Z}}$ there exist positive constants $c$, $\alpha$,
depending only on $J$, $R$ and $\beta$ such that for every
$\Lambda\Subset\operatorname{\mathbb{Z}}$ split into three subintervals
$\Lambda=ABC$, where $B$ shields $A$ from $C$ and for
$O_{A}\in\operatorname{\mathcal{A}}_{A}$ and
$O_{C}\in\operatorname{\mathcal{A}}_{C}$ it holds that

$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{A}\,O_{C}]-\Tr_{ABC}[\rho^{ABC}O_{A}]\Tr_{ABC}[\rho^{ABC}O_{C}]\right|$
$\displaystyle\leq\norm{O_{A}}_{\infty}\norm{O_{C}}_{\infty}\,c\,e^{-\alpha|B|}\,.$

The exponential decay of correlations can be used to prove local
indistinguishability of Gibbs states, a result that can be found in [15,
Proposition 7.1].

######  Theorem 2.9 (Local indistinguishability).

For $\Phi$ a finite-range, translation-invariant interaction over
$\operatorname{\mathbb{Z}}$ there exist positive constants $c^{\prime}$,
$\alpha^{\prime}$, depending only on $J$, $R$ and $\beta$ such that for every
$\Lambda\Subset\operatorname{\mathbb{Z}}$ split into three subintervals
$\Lambda=ABC$, where $B$ shields $A$ from $C$ and for
$O_{A}\in\operatorname{\mathcal{A}}_{A}$ and
$O_{C}\in\operatorname{\mathcal{A}}_{C}$ it holds that

$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{A}]-\Tr_{AB}[\rho^{AB}O_{A}]\right|$
$\displaystyle\leq\norm{O_{A}}_{\infty}\,c^{\prime}\,e^{-\alpha^{\prime}|B|}\,,$
$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{C}]-\Tr_{BC}[\rho^{BC}O_{C}]\right|$
$\displaystyle\leq\norm{O_{C}}_{\infty}\,c^{\prime}\,e^{-\alpha^{\prime}|B|}\,.$

All these results will be essential for our proof that the purity of a Gibbs
state of a translation-invariant, local Hamiltonian in 1D decays exponentially
fast with the size of the middle system (cf. Section 3.3).

######  Remark 2.10.

All results contained in this section can be extended to the context of short-
range (i.e. exponentially-decaying) interactions, as shown in [64, 21]. We
review all these extensions and provide the necessary new ones in Appendix B.

##  3 Main results

###  3.1 An upper bound on the DPI for the Belavkin Staszewski entropy

The data-processing inequality (DPI) of many entropy measures, namely the fact
that they cannot increase under the application of quantum channels, is one of
the fundamental properties in quantum Shannon theory, and the study of when
this property holds and how tight it is constitutes a very relevant problem in
this field. For the Umegaki relative entropy, it was originally shown by Petz
[65, 66] that saturation of the DPI is equivalent to recoverability of one
state in terms of the other, through the so-called Petz recovery map.
Strengthened versions for the relative entropy, namely lower bounds of the DPI
after applying a conditional expectation (or in general, a quantum channel)
have been obtained in terms of the distance to a recovery map in many
different quantities in the past few years [75, 23, 36], starting with the
breakthrough of [33]. These results have also been extended to the larger
families of standard f-divergences [22, 43]. Results in the converse
direction, i.e. upper bounds for the DPI, can be found e.g. in [13].

A similar question can be considered for the BS-entropy. This quantity also
satisfies a DPI, which was proven to be saturated if and only if [12]

$\widehat{D}(\rho\|\sigma)-\widehat{D}(\operatorname{\mathcal{E}}(\rho)\|\operatorname{\mathcal{E}}(\sigma))=0\Leftrightarrow\sigma=\operatorname{\mathcal{E}}(\sigma)\mathcal{E}(\rho)^{-1}\rho=:\operatorname{\mathcal{B}}^{\rho}_{\operatorname{\mathcal{E}}}(\sigma)\,.$
(35)

The term on the right-hand side above was coined BS-recovery condition. In the
same paper, a strengthened DPI for the BS-entropy (and any maximal
f-divergence) in terms of the distance between a state and its BS-recovery was
provided.

Here, as a technical result of independent interest, we give an upper bound on
the DPI of the BS-entropy. More specifically, we provide an upper bound for
$\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{E}}(X)\|\operatorname{\mathcal{E}}(Y))$
in terms of distance measures that explicitly feature the recovery condition
for the BS entropy. For this, we use the usual representation of the logarithm
we want to recall here. For $X>0$, we have that

$\log(X)=\int_{0}^{\infty}\frac{1}{t+1}-\frac{1}{t+X}\,dt\,.$ (36)

Using this representation we derive a simple technical lemma that later
becomes useful in the proof.

######  Lemma 3.1.

Let $X,Y\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$ with $X>0$,
$Y>0$ and
$\operatorname{\mathcal{E}}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})\to\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$
a conditional expectation. We further set $Z=X^{1/2}Y^{-1}X^{1/2}$,
$X_{\operatorname{\mathcal{E}}}:=\operatorname{\mathcal{E}}(X)$,
$Y_{\operatorname{\mathcal{E}}}:=\operatorname{\mathcal{E}}(Y)$ and finally
$Z_{\operatorname{\mathcal{E}}}=X_{\operatorname{\mathcal{E}}}^{1/2}Y_{\operatorname{\mathcal{E}}}^{-1}X_{\operatorname{\mathcal{E}}}^{1/2}$.
For $T>0$, we find that

$\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}-X\frac{1}{t+Z}]dt=\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}^{-1/2}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}X_{\operatorname{\mathcal{E}}}^{1/2}(Y^{-1}X-Y_{\operatorname{\mathcal{E}}}^{-1}X_{\operatorname{\mathcal{E}}})X^{1/2}\frac{1}{t+Z}X^{1/2}]\,dt\,.$

###### Proof.

We find by cyclicity of the trace and the fact that
$\operatorname{\mathcal{E}}$ is a conditional expectation that

$\displaystyle\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}\frac{1}{1+Z_{\operatorname{\mathcal{E}}}}-X\frac{1}{t+Z}]dt$
$\displaystyle=\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}^{1/2}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}X_{\operatorname{\mathcal{E}}}^{-1/2}X-X^{1/2}\frac{1}{t+Z}X^{-1/2}X]\,dt\,.$
$\displaystyle=\int_{0}^{T}\Tr[\left(X_{\operatorname{\mathcal{E}}}^{1/2}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}X_{\operatorname{\mathcal{E}}}^{-1/2}-X^{1/2}\frac{1}{t+Z}X^{-1/2}\right)X]\,dt\,.$

Noticing that

$X_{\operatorname{\mathcal{E}}}^{1/2}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}X_{\operatorname{\mathcal{E}}}^{-1/2}-X^{1/2}\frac{1}{t+Z}X^{-1/2}=X_{\operatorname{\mathcal{E}}}^{1/2}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}X_{\operatorname{\mathcal{E}}}^{-1/2}(XY^{-1}-X_{\operatorname{\mathcal{E}}}Y_{\operatorname{\mathcal{E}}}^{-1})X^{1/2}\frac{1}{t+Z}X^{-1/2}$

and inserting it into the above equation gives the claim. ∎

With this lemma at hand, we are set to prove the upper bound on the BS-entropy
DPI.

######  Theorem 3.2.

Let $X,Y\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$ with $X>0$,
$Y>0$ and
$\operatorname{\mathcal{E}}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})\to\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}})$
a conditional expectation. Then, we have

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{E}}(X)\|\operatorname{\mathcal{E}}(Y))$
$\displaystyle\leq\norm{X^{-1/2}YX^{-1/2}}_{\infty}\norm{X}_{1}\norm{\operatorname{\mathcal{E}}(X)^{1/2}}_{\infty}\norm{\operatorname{\mathcal{E}}(X)^{-1/2}}_{\infty}$
(37)
$\displaystyle\hphantom{=}\cdot\norm{\operatorname{\mathcal{E}}(Y)^{-1}\operatorname{\mathcal{E}}(X)}_{\infty}\norm{XY^{-1}\operatorname{\mathcal{E}}(Y)\operatorname{\mathcal{E}}(X)^{-1}-\operatorname{\mathds{1}}}_{\infty}\,.$

and

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{E}}(X)\|\operatorname{\mathcal{E}}(Y))$
$\displaystyle\leq\norm{X^{-1/2}YX^{-1/2}}_{\infty}\norm{X}_{1}\norm{\operatorname{\mathcal{E}}(X)^{1/2}}_{\infty}\norm{\operatorname{\mathcal{E}}(X)^{-1/2}}_{\infty}$
(38)
$\displaystyle\hphantom{=}\cdot\norm{Y^{-1}X}_{\infty}\norm{YX^{-1}\operatorname{\mathcal{E}}(X)\operatorname{\mathcal{E}}(Y)^{-1}-\operatorname{\mathds{1}}}_{\infty}\,.$

###### Proof.

Both inequalities follow from the same estimate through the application of
Hölder inequality. To obtain this expression we employ Equation 36 using the
notation of Lemma 3.1

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(X_{\operatorname{\mathcal{E}}}\|Y_{\operatorname{\mathcal{E}}})=\lim_{T\to\infty}\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}-X\frac{1}{t+Z}]\,dt\,.$

Further, using Lemma 3.1 and Hölder’s inequality we can upper bound

$\displaystyle\int_{0}^{T}\Tr[X_{\operatorname{\mathcal{E}}}\frac{1}{t+Z_{\operatorname{\mathcal{E}}}}-X\frac{1}{t+Z}]dt$
$\displaystyle\leq\int_{0}^{T}\norm{X_{\operatorname{\mathcal{E}}}^{-1/2}}\frac{1}{t+\norm{Z_{\operatorname{\mathcal{E}}}^{-1}}^{-1}}\norm{X_{\operatorname{\mathcal{E}}}^{1/2}}\norm{Y^{-1}X-Y_{\operatorname{\mathcal{E}}}^{-1}X_{\operatorname{\mathcal{E}}}}_{\infty}\norm{X^{1/2}}_{2}\frac{1}{t+\norm{Z^{-1}}_{\infty}^{-1}}\norm{X^{1/2}}_{2}\,dt\,.$

Note also that
$\norm{\frac{1}{t+Z}}_{\infty}=\frac{1}{t+\norm{Z^{-1}}_{\infty}^{-1}}$ and
analogously for the fraction including $Z_{\operatorname{\mathcal{E}}}$.
Utilising DPI for $(A,B)\mapsto\norm{A^{1/2}B^{-1}A^{1/2}}_{\infty}$ (see e.g.
[77, Proposition 4.7]), we can further estimate
$\frac{1}{t+\norm{Z_{\operatorname{\mathcal{E}}}^{-1}}^{-1}}\leq\frac{1}{t+\norm{Z^{-1}}^{-1}}$.
Finally, the fact that $\norm{X^{1/2}}_{2}^{2}=\norm{X}_{1}$, integrating both
sides, and then taking the limit $T\to\infty$ give

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(X_{\operatorname{\mathcal{E}}}\|Y_{\operatorname{\mathcal{E}}})\leq\norm{X_{\operatorname{\mathcal{E}}}^{-1/2}}_{\infty}\norm{X^{-1/2}YX^{-1/2}}_{\infty}\norm{X_{\operatorname{\mathcal{E}}}^{1/2}}_{\infty}\norm{Y^{-1}X-Y_{\operatorname{\mathcal{E}}}^{-1}X_{\operatorname{\mathcal{E}}}}_{\infty}\norm{X}_{1}.$

∎

######  Remark 3.3.

The above result can readily be extended to quantum channels instead of
conditional expectations. The statement and proof of that result can be found
in the appendix in Corollary C.1.

######  Remark 3.4.

As previously stated in the introduction, the above result complements the
findings of [12], providing a comprehensive perspective on the data processing
inequality of the BS-entropy. Using the notation of the theorem for quantum
states $\rho$ and $\sigma$, and the BS-recovery for $\sigma$ denoted by
$B^{\rho}_{\operatorname{\mathcal{E}}}(\sigma)=\sigma_{\operatorname{\mathcal{E}}}\rho_{\operatorname{\mathcal{E}}}^{-1}\rho$,
the following chain of inequalities hold:

$\displaystyle\left(\frac{\pi}{8}\right)^{4}\norm{\rho^{-1/2}\sigma\rho^{-1/2}}_{\infty}^{-4}$
$\displaystyle\norm{\rho_{\operatorname{\mathcal{E}}}^{-1}}_{\infty}^{-2}\norm{\operatorname{\mathcal{B}}^{\rho}_{\operatorname{\mathcal{E}}}(\sigma)-\sigma}_{2}^{4}$
$\displaystyle\leq\widehat{D}(\rho\|\sigma)-\widehat{D}(\operatorname{\mathcal{E}}(\rho)\|\operatorname{\mathcal{E}}(\sigma))\leq$
$\displaystyle\hphantom{\leq\widehat{D}(\rho\|\sigma)-\widehat{D}}\norm{\rho^{-1/2}\sigma\rho^{-1/2}}_{\infty}\norm{\rho_{\operatorname{\mathcal{E}}}^{-1}}_{\infty}^{1/2}\norm{\sigma^{-1}\rho}_{\infty}\norm{\sigma(\operatorname{\mathcal{B}}_{\operatorname{\mathcal{E}}}^{\rho}(\sigma))^{-1}-\operatorname{\mathds{1}}}_{\infty}\,.$

###  3.2 Superexponential decay of the BS-CMI

In this section, we prove one of our main results: the superexponential decay
of the three definitions of the BS-CMI. The proof crucially involves Theorem
3.2, as well as the results from Sec. 2.4. Before we do so, we need a
technical lemma concerned with norm estimates of functions of Gibbs states of
local translation invariant Hamiltonians on a spin chain.

######  Lemma 3.5.

Let $\Phi$ be a finite-range, translation invariant interaction over
$\operatorname{\mathbb{Z}}$ and $\Lambda\Subset\operatorname{\mathbb{Z}}$
split into three subintervals $\Lambda=ABC$, where $B$ shields $A$ from $C$
with local Gibbs state $\rho^{\Lambda}=\rho_{ABC}$, then

  1. 1.

$\norm{\rho_{A}\rho_{B}\rho_{AB}^{-1}}_{\infty}\leq\operatorname{\mathcal{C}}$,
$\norm{\rho_{AB}\rho_{A}^{-1}\rho_{B}^{-1}}_{\infty}\leq\operatorname{\mathcal{C}}$,
$\norm{\rho_{ABC}\rho_{B}^{-1}}_{\infty}\leq\operatorname{\mathcal{C}}$.

  2. 2.

$\norm{\rho_{B}^{-1}}_{\infty}\leq\operatorname{\mathcal{C}}e^{\alpha|B|}$.

where the constants $\operatorname{\mathcal{C}},\alpha$ only depend on
interaction strength $J$ and range $R$ of $\Phi$ and $\alpha$ additionally
depends on the local dimension $d$. Note that only $\rho_{ABC}$ is the Gibbs
state of a local Hamiltonian, while all other states are marginals of that
state. Furthermore, these conditions still hold if
$\rho_{ABC}=\tr_{A^{\prime}C^{\prime}}[\rho_{A^{\prime}ABCC^{\prime}}]$ where
$\rho_{A^{\prime}ABCC^{\prime}}$ is the Gibbs state on the larger system
$A^{\prime}ABCC^{\prime}$.

###### Proof.

Let us denote the partition function for a given interval by

$Z_{A}=\Tr[e^{-H_{A}}].$

Note that the perturbation formula [58, Lemma 3.6]

$\left|\log(\Tr[e^{H+P}])-\log(\Tr[e^{H}])\right|\leq\|P\|$

implies that ratios of partition functions are bounded by a constant
$\operatorname{\mathcal{C}}$ only depending on the interaction strength,
range, and inverse temperature

$\frac{Z_{A}Z_{B}}{Z_{AB}},\frac{Z_{AB}}{Z_{A}Z_{B}}\leq\operatorname{\mathcal{C}}$

For the first part, we identify $CC^{\prime}$ with $B^{\prime}$. We use an
argument analogous to Corollary 2.7.

$\displaystyle\|\rho_{A}\rho_{B}\rho^{-1}_{AB}\|_{\infty}$
$\displaystyle\hskip
28.45274pt=\Big{\|}\tr_{A^{\prime}BB^{\prime}}[\rho^{A^{\prime}}\otimes\rho^{BB^{\prime}}E_{A^{\prime},ABB^{\prime}}E_{A,BB^{\prime}}]\tr_{A^{\prime}AB^{\prime}}[\rho^{A^{\prime}A}\otimes\rho^{B^{\prime}}E_{A^{\prime}A,BB^{\prime}}E_{B,B^{\prime}}]E_{A,B}^{-1}$
$\displaystyle\hskip
42.67912pt\tr_{A^{\prime}B^{\prime}}[\rho^{A^{\prime}}\otimes\rho^{B^{\prime}}E_{A^{\prime},ABB^{\prime}}E_{AB,B^{\prime}}]^{-1}\Big{\|}_{\infty}\frac{Z_{A^{\prime}A}Z_{BB^{\prime}}}{Z_{A^{\prime}ABB^{\prime}}}$

Using submultiplicativity of the norm, the first two partial traces are
bounded by contractivity of the conditional expectation and Proposition 2.1.
The same proposition also bounds $\|E_{A,B}^{-1}\|_{\infty}$ For the inverse
partial trace we use Proposition 2.2 for the appropriately identified
subsystems. Analogously,

$\displaystyle\|\rho_{AB}\rho_{A}^{-1}\rho_{B}^{-1}\|_{\infty}$
$\displaystyle\hskip
28.45274pt=\Big{\|}\tr_{A^{\prime}B^{\prime}}[\rho^{A^{\prime}}\otimes\rho^{B^{\prime}}E_{A^{\prime},ABB^{\prime}}E_{AB,B^{\prime}}]E_{A,B}\tr_{A^{\prime}BB^{\prime}}[\rho^{A^{\prime}}\otimes\rho^{BB^{\prime}}E_{A^{\prime},ABB^{\prime}}E_{A,BB^{\prime}}]^{-1}$
$\displaystyle\hskip
42.67912pt\tr_{A^{\prime}AB^{\prime}}[\rho^{A^{\prime}A}\otimes\rho^{B^{\prime}}E_{A^{\prime}A,BB^{\prime}}E_{B,B^{\prime}}]^{-1}\Big{\|}_{\infty}\frac{Z_{A^{\prime}ABB^{\prime}}}{Z_{A^{\prime}A}Z_{BB^{\prime}}}$

and, switching the notation back to a Gibbs state on the original
$A^{\prime}ABCC^{\prime}$,

$\displaystyle\|\rho_{ABC}\rho_{B}^{-1}\|_{\infty}$ $\displaystyle\hskip
28.45274pt=\Big{\|}\tr_{A^{\prime}C^{\prime}}[\rho^{A^{\prime}}\rho^{C^{\prime}}E_{A^{\prime},ABCC^{\prime}}E_{ABC,C^{\prime}}]E_{A,BC}E_{B,C}\frac{e^{-H_{A}}}{Z_{A}}\frac{e^{-H_{C}}}{Z_{C}}$
$\displaystyle\hskip
42.67912pt\tr_{A^{\prime}ACC^{\prime}}[\rho^{A^{\prime}A}\rho^{CC^{\prime}}E_{A^{\prime}A,BCC^{\prime}}E_{A^{\prime}AB,CC^{\prime}}]^{-1}\Big{\|}_{\infty}\frac{Z_{A^{\prime}}Z_{A}Z_{C}Z_{C^{\prime}}}{Z_{AA^{\prime}}Z_{CC^{\prime}}}$

are bounded in the same way, apart from the last equation, which also uses
$\|\rho^{A}\|_{\infty},\|\rho^{C}\|_{\infty}\leq 1$.

For the second point, it is sufficient to restrict to a system on $ABC$. We
start with an analogous computation

$\displaystyle\|\rho_{B}^{-1}\|_{\infty}=\left\|(\rho^{B})^{-1}\tr_{AC}[E_{A,BC}E_{B,C}\rho^{A}\rho^{C}]\right\|_{\infty}\frac{Z_{ABC}}{Z_{A}Z_{B}Z_{C}}.$

We use a similar bound except for the term $\|(\rho^{B})^{-1}\|_{\infty}$,
which can be bounded by considering the norm of the local Hamiltonian.

$\rho^{B}=\frac{e^{-H_{B}}}{\Tr[e^{-H_{B}}]}\geq\frac{e^{-\|H_{B}\|_{\infty}}}{d_{B}e^{\|H_{B}\|_{\infty}}}\operatorname{\mathds{1}}_{B}\geq
e^{-2\norm{H_{B}}_{\infty}}d_{B}^{-1}\operatorname{\mathds{1}}_{B}\,.$

Combining this with $\norm{H_{B}}_{\infty}\leq J|B|$ gives at last

$\norm{(\rho^{B})^{-1}}_{\infty}\leq e^{\alpha|B|}$

with $\alpha=2J+\log(d)$ and hence the claim. ∎

We are now set to prove the main result of the section.

######  Theorem 3.6.

For $\Phi$ a finite-range, translation invariant interaction over
$\operatorname{\mathbb{Z}}$ there exists a positive function
$\ell\mapsto\varepsilon(\ell)$ with superexponential decay in $\ell$,
depending only on $J$, $R$ and $\beta$ such that for every
$\Lambda\Subset\operatorname{\mathbb{Z}}$ split into consecutive subintervals
$\Lambda=A^{\prime}ABCC^{\prime}$, with $A^{\prime}$ and $C^{\prime}$ possibly
empty, for the marginal on $ABC$ of its local Gibbs state
$\tr_{A^{\prime}C^{\prime}}[\rho^{\Lambda}]=\rho_{ABC}$ it holds that

$\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)\leq
c_{\operatorname{ots}}e^{\alpha_{\operatorname{ots}}(|A|+|B|)}\varepsilon(|B|)\,,$
(39)

and further

$\widehat{I}^{\operatorname{ts}}_{\rho}(A:C|B)\leq
c_{\operatorname{ots}}e^{\alpha_{\operatorname{ots}}(|A|+|B|)}\varepsilon(|B|)\,.$
(40)

At last, we have

$\widehat{I}_{\rho}^{\operatorname{rev}}(A:C|B)\leq
c_{\operatorname{rev}}e^{\alpha_{\operatorname{rev}}(|A|+|B|)}\varepsilon(|B|)\,.$
(41)

Here, $c_{\operatorname{ots}}$, $\alpha_{\operatorname{ots}}$ and
$c_{\operatorname{rev}}$, $\alpha_{\operatorname{rev}}$ are constants only
depending on inverse temperature $\beta$, strength $J$ and range $R$ of $\Phi$
and local dimension $d$.

###### Proof.

We begin with the one-sided version and note that we can write

$\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)=D(\rho_{ABC}\|\pi_{A}\otimes\rho_{BC})-D(\operatorname{\mathcal{E}}(\rho_{ABC})\|\operatorname{\mathcal{E}}(\pi_{A}\otimes\rho_{BC}))$

with $\operatorname{\mathcal{E}}(\cdot):=\tr_{C}[\cdot]\otimes\pi_{C}$ a
conditional expectation. Using Theorem 3.2 we obtain

$\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)\leq\norm{\rho_{ABC}^{-1/2}\rho_{BC}\rho_{ABC}^{-1/2}}_{\infty}\norm{\rho_{ABC}}_{1}\norm{\rho_{AB}^{1/2}}_{\infty}\norm{\rho_{AB}^{-1/2}}_{\infty}\norm{\rho_{B}^{-1}\rho_{AB}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$

where we already simplified terms and cancelled constants. The fact that
$\norm{AB}_{\infty}\leq\norm{BA}_{\infty}$ for normal $AB$ [11, Proposition
IX.1.1] and that for quantum states
$\norm{\rho^{-p}}_{\infty}=\norm{\rho^{-1}}^{p}_{\infty}$, $p\in[0,\infty)$,
gives us

$\displaystyle\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)$
$\displaystyle\leq\norm{\rho_{BC}\rho_{ABC}^{-1}}_{\infty}\norm{\rho_{AB}^{-1}}_{\infty}^{1/2}\norm{\rho_{B}^{-1}\rho_{ABC}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq\norm{\rho_{A}^{-1}}_{\infty}\norm{\rho_{A}\rho_{BC}\rho_{ABC}^{-1}}_{\infty}\norm{\rho_{AB}^{-1}}_{\infty}^{1/2}\norm{\rho_{B}^{-1}\rho_{ABC}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq
c_{\operatorname{ots}}e^{\alpha_{\operatorname{ots}}(|A|+|B|)}\varepsilon(|B|)\,.$

In the last inequality, we used Theorem 2.3 with Lemma 2.5 and Lemma 3.5.

For the two-sided definition, we again use Theorem 3.2 with the same
conditional expectation and $\rho_{ABC}$ but with
$\sigma=\rho_{A}\otimes\rho_{BC}$ instead. Applying similar simplifications as
before, Theorem 2.3, Lemma 2.5, and Lemma 3.5, we obtain

$\displaystyle\widehat{I}^{\operatorname{ts}}_{\rho}(A:C|B)$
$\displaystyle\leq\norm{\rho_{A}\rho_{BC}\rho_{ABC}^{-1}}_{\infty}\norm{\rho_{AB}^{-1}}_{\infty}^{1/2}\norm{\rho_{A}^{-1}\rho_{B}^{-1}\rho_{AB}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq\norm{\rho_{A}\rho_{BC}\rho_{ABC}^{-1}}_{\infty}\norm{\rho_{AB}^{-1}}_{\infty}^{1/2}\norm{\rho_{A}^{-1}}_{\infty}\norm{\rho_{B}^{-1}\rho_{ABC}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq
c_{\operatorname{ots}}e^{\alpha_{\operatorname{ots}}(|A|+|B|)}\varepsilon(|B|)\,.$

For the reversed version, we set $X=\pi_{A}\otimes\rho_{BC}$, $Y=\rho_{ABC}$
with the conditional expectation fixed as before. Employing Theorem 3.2 in the
form of the second inequality we obtain after initial simplifications

$\displaystyle\widehat{I}^{\operatorname{rev}}_{\rho}(A:C|B)$
$\displaystyle\leq\norm{\rho_{BC}^{-1/2}\rho_{ABC}\rho_{BC}^{-1/2}}_{\infty}\norm{\rho_{B}^{-1}}^{1/2}_{\infty}\norm{\rho_{ABC}^{-1}\rho_{BC}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq\norm{\rho_{ABC}\rho_{A}^{-1}\rho_{BC}^{-1}}_{\infty}\norm{\rho_{B}^{-1}}^{1/2}\norm{\rho_{A}^{-1}}_{\infty}\norm{\rho_{A}\rho_{BC}\rho_{ABC}^{-1}}_{\infty}\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}$
$\displaystyle\leq
c_{\operatorname{rev}}e^{\alpha_{\operatorname{rev}}(|A|+|B|)}\varepsilon(|B|)\,.$

The last estimation again follows from Theorem 2.3, Lemma 2.5 and Lemma 3.5. ∎

Note that all the BS-CMI bounds depend on the length of one of the side
systems exponentially. At least in the case of the two-sided version, one can
relatively easily show a _constant_ upper bound on the DPI independent of the
dimension, which might hint towards an improved bound. Lastly, we would also
like to restate Remark 3.4 and translate it to the context of one-sided BS-CMI
to make the inequalities more accessible.

######  Remark 3.7.

In the case where $\rho_{ABC}$ is a state on a tripartite system (not
necessarily with any relation to a Gibbs state), the following chain of
inequalities holds:

$\displaystyle\left(\frac{\pi}{8}\right)^{4}\norm{\rho_{ABC}^{-1/2}\rho_{BC}\rho_{ABC}^{-1/2}}_{\infty}^{-4}$
$\displaystyle\norm{\rho_{AB}^{-1}}_{\infty}^{-2}\norm{\rho_{B}\rho_{AB}^{-1}\rho_{ABC}-\rho_{BC}}_{\infty}$
$\displaystyle\leq\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)\leq$
$\displaystyle\hphantom{\widehat{I}^{\operatorname{os}}_{\rho})}\norm{\rho_{ABC}^{-1/2}\rho_{BC}\rho_{ABC}^{-1/2}}_{\infty}\norm{\rho_{AB}^{-1}}_{\infty}^{1/2}\norm{\rho_{BC}^{-1}\rho_{ABC}}_{\infty}\norm{\rho_{BC}\rho_{ABC}^{-1}\rho_{AB}\rho_{B}^{-1}-\operatorname{\mathds{1}}}_{\infty}\,.$

######  Remark 3.8.

An analogue of Theorem 3.6 for exponentially-decaying interactions is provided
in Theorem B.5. The main difference is the exponential instead of the
superexponential decay with $|B|$ which only holds for specific regimes of
$\beta$.

###  3.3 Approximate factorisation of the purity

We now consider another possible measure of conditional independence given by
the approximate factorisation of the purity. This notion is inspired by [79],
where it was shown that given an interval $\Lambda\Subset\mathbb{Z}$ split as
$\Lambda=ABC$, if a state $\rho_{\Lambda}$ is prepared by a finite depth
circuit, then the following holds

$\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}=1$
(42)

whenever $|B|\geq 2\ell-1$, $\ell$ being the depth of the circuit. This may
suggest a definition of conditional independence based on a notion of CMI
defined in terms of Rényi-$2$ entropies, as, in analogy with Eq. (1),

$I_{2}(A:C|B):=\log\frac{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}=S_{2}(\rho_{AB})+S_{2}(\rho_{BC})-S_{2}(\rho_{B})-S_{2}(\rho_{\Lambda})\,,$
(43)

with $S_{2}(\rho)=-\log\Tr[\rho^{2}]$. It should not be confused with the
$2$-CMI that could be defined from Petz Rényi or sandwiched Rényi divergences.
However, notice that this quantity, contrary to $I(A:C|B)$,
${\widehat{I}}^{\mathrm{os}}_{\rho}(A\mathbin{:}C\mathbin{|}B)$,
${\widehat{I}}^{\mathrm{ts}}_{\rho}(A\mathbin{:}C\mathbin{|}B)$ and
${\widehat{I}}^{\mathrm{rev}}_{\rho}(A\mathbin{:}C\mathbin{|}B)$, is not
necessarily positive, and likely lacks most other relevant information-
theoretic properties.

Nevertheless, the authors of [79], motivated by the problem of efficiently
learning Rényi entropies, introduce an approximate factorisation condition for
this measure, which holds when the previous Eq. (42) fails up to, at most, an
exponentiallly small error in $|B|$. They proved that translation-invariant
Matrix Product Density Operators satisfy this property, and also conjectured
it for a larger class of states, and numerically verified it in some relevant
models. This approximate factorisation is equivalent to the exponential decay
in $|B|$ of Eq. (43). In this section, we prove that this property holds for
any translation-invariant, finite-range Hamiltonian in 1D at any inverse
temperature $\beta>0$.

######  Proposition 3.9.

Let $\Phi$ be a finite-range translation-invariant interaction over
$\mathbb{Z}$. Then, there exist positive constants $c_{p},\alpha_{p}$
depending only on the strength $J$, range of the interaction $r$ and inverse
temperature $\beta>0$ with the following property: For every
$\Lambda\Subset\mathbb{Z}$ split as $\Lambda=ABC$, where $B$ shields $A$ from
$C$ (see Figure 4), and for $\rho_{\Lambda}:=\rho^{\Lambda}$ the Gibbs state
on $\Lambda$,

$\left|\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}-1\right|\leq
c_{p}e^{-\alpha_{p}|B|}\,.$ (44)

###### Proof.

Consider $\ell\in\mathbb{N}$ such that $|B|\geq 3\ell$ and split $B$ into
$B_{1}$, $B_{2}$ and $B_{3}$ such that $|B_{1}|,|B_{2}|,|B_{3}|\geq\ell$ as in
Figure 4.

$A$$B_{1}$$B$$B_{2}$$B_{3}$$C$$\Lambda$ Figure 4: Representation of an
interval $\Lambda$ split into three subintervals $\Lambda=ABC$, with $B$
further split into $B_{1}$, $B_{2}$ and $B_{3}$ such that
$|B_{1}|,|B_{2}|,|B_{3}|\geq\ell$.

For any $X\subset\Lambda$, let us denote $Z_{X}:=\Tr_{X}[e^{-H_{X}}]$ and by
$\rho^{X}$ the Gibbs state on $X$, i.e. $\rho^{X}:=e^{-H_{X}}/Z_{X}$. Let us
further define

$\lambda_{ABC}:=\frac{Z_{ABC}Z_{B}}{Z_{AB}Z_{BC}}\,.$ (45)

Then, we can rewrite $\lambda_{ABC}$ in terms of expansionals as

$\displaystyle\lambda_{ABC}$
$\displaystyle=\frac{\Tr_{ABC}[e^{-H_{ABC}}]\Tr_{B}[e^{-H_{B}}]}{\Tr_{AB}[e^{-H_{AB}}]\Tr_{BC}[e^{-H_{BC}}]}$
$\displaystyle=\big{(}\Tr_{ABC}[\rho^{ABC}\underbrace{e^{H_{ABC}}e^{-H_{A}-H_{BC}}}_{E_{A,BC}^{-1\,*}}]\big{)}^{-1}\Tr_{AB}[\rho^{AB}\underbrace{e^{H_{AB}}e^{-H_{A}-H_{B}}}_{E_{A,B}^{-1\,*}}]\,.$

By Proposition 2.2, we have

$|\lambda_{ABC}|,|\lambda_{ABC}^{-1}|\leq\mathcal{C}^{2}\,,$ (46)

and by Step 2 in the proof of Proposition 8.1 of [15], there exist
$\tilde{c}_{1},\alpha_{1}>0$ such that

$|\lambda_{ABC}-1|\leq\tilde{c}_{1}e^{-\alpha_{1}\ell}\,.$ (47)

Note that the above two estimates can be used to obtain

$|\lambda_{ABC}^{-1}-1|=|\lambda_{ABC}^{-1}||1-\lambda_{ABC}|\leq\operatorname{\mathcal{C}}^{2}\tilde{c}_{1}e^{-\alpha_{1}\ell}=:c_{1}e^{-\alpha_{1}\ell}$
(48)

We will follow a similar but more complicated approach for

$\widehat{\lambda}_{ABC}^{-1}:=\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}\,.$
(49)

Let us analyse each of the terms involved in the previous expression
independently. For that, we will consider and compare the partition functions
and Gibbs states associated to $H_{\Lambda}$ and to $2H_{\Lambda}$, denoting
$\widetilde{Z}_{X}:=\Tr_{X}[e^{-2H_{X}}]$ and
$\widetilde{\rho}_{X}:=e^{-2H_{X}}/\widetilde{Z}_{X}$ for every
$X\subset\Lambda$.

  * •

For $\Tr_{\Lambda}[\rho_{\Lambda}^{2}]$, we have:

$\displaystyle\Tr_{\Lambda}[\rho_{\Lambda}^{2}]$
$\displaystyle=\frac{\Tr_{ABC}\big{[}e^{-2H_{ABC}}\big{]}}{Z_{ABC}^{2}}=\frac{\widetilde{Z}_{ABC}}{Z_{ABC}^{2}}\,.$

  * •

For $\Tr_{AB}[\rho_{AB}^{2}]$, we obtain:

$\displaystyle\Tr_{AB}[\rho_{AB}^{2}]$
$\displaystyle=\frac{1}{Z_{ABC}^{2}}\Tr_{AB}\big{[}\tr_{C}[e^{-H_{ABC}}]^{2}\big{]}$
$\displaystyle=\frac{Z_{C}^{2}}{Z_{ABC}^{2}}\Tr_{AB}\Big{[}e^{-H_{AB}}\tr_{C}\big{[}\rho^{C}\underbrace{e^{H_{AB}}e^{H_{C}}e^{-H_{ABC}}}_{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}\underbrace{e^{-H_{ABC}}e^{H_{C}}e^{H_{AB}}}_{E_{AB,C}}\rho^{C}\big{]}e^{-H_{AB}}\Big{]}$
$\displaystyle=\frac{\widetilde{Z}_{AB}Z_{C}^{2}}{Z_{ABC}^{2}}\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}E_{AB,C}\rho^{C}\big{]}\Big{]}\,.$

  * •

Similarly for $\Tr_{BC}[\rho_{BC}^{2}]$, we can write:

$\displaystyle\Tr_{BC}[\rho_{BC}^{2}]$
$\displaystyle=\frac{Z_{A}^{2}\widetilde{Z}_{BC}}{Z_{ABC}^{2}}\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,BC}^{*}}\big{]}\tr_{A}\big{[}E_{A,BC}\rho^{A}\big{]}\Big{]}\,.$

  * •

Finally, for $\Tr_{B}[\rho_{B}^{2}]$, we have:

$\displaystyle\Tr_{B}[\rho_{B}^{2}]$
$\displaystyle=\frac{1}{Z_{ABC}^{2}}\Tr_{B}[\tr_{AC}[e^{-H_{ABC}}]^{2}]$
$\displaystyle=\frac{Z_{A}^{2}Z_{C}^{2}}{Z_{ABC}^{2}}\Tr_{B}\Big{[}e^{-H_{B}}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}\underbrace{e^{H_{A}}e^{H_{B}}e^{-H_{AB}}}_{E_{A,B}^{*}}\underbrace{e^{H_{AB}}e^{H_{C}}e^{-H_{ABC}}}_{E_{AB,C}^{*}}\big{]}$
$\displaystyle\phantom{asasassdAasd}\tr_{AC}\big{[}\underbrace{e^{-H_{ABC}}e^{H_{AB}}e^{H_{C}}}_{E_{AB,C}}\underbrace{e^{-H_{AB}}e^{H_{A}}e^{H_{B}}}_{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}e^{-H_{B}}\Big{]}$
$\displaystyle=\frac{Z_{A}^{2}Z_{C}^{2}\widetilde{Z}_{B}}{Z_{ABC}^{2}}\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B}^{*}}{E_{AB,C}^{*}}\big{]}\tr_{AC}\big{[}{E_{AB,C}}{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}\,.$

Denoting

$\widetilde{\lambda}_{ABC}:=\frac{\widetilde{Z}_{ABC}\widetilde{Z}_{B}}{\widetilde{Z}_{AB}\widetilde{Z}_{BC}}$
(50)

and replacing all these values in Eq. (49), after noticing that all partition
functions $Z_{X}$ of the original Hamiltonian simplify, we obtain

$\widehat{\lambda}_{ABC}^{-1}=\widetilde{\lambda}_{ABC}^{-1}\frac{\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B}^{*}}{E_{AB,C}^{*}}\big{]}\tr_{AC}\big{[}{E_{AB,C}}{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}}{\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}E_{AB,C}\rho^{C}\big{]}\Big{]}\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,BC}^{*}}\big{]}\tr_{A}\big{[}E_{A,BC}\rho^{A}\big{]}\Big{]}}\,.$

Let us denote by $\chi_{ABC}$ the last fraction above. Then, it is clear that

$\displaystyle\left|\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}-1\right|$
$\displaystyle=\left|\widetilde{\lambda}_{ABC}^{-1}\chi_{ABC}-1\right|$
$\displaystyle\leq\left|\widetilde{\lambda}_{ABC}^{-1}-1\right|+\left|\widetilde{\lambda}_{ABC}^{-1}\right|\left|\chi_{ABC}-1\right|$
$\displaystyle\leq
c_{1}e^{-\alpha_{1}\ell}+\mathcal{C}^{2}\left|\chi_{ABC}-1\right|\,,$

where we are using Eq. (46) and Eq. (47) for the partition functions
associated to the $2H_{\Lambda}$. Note that the constants would change
slightly, just modifying their dependence from $\beta$ to $\beta/2$, but we
keep the same notation for simplicity. Therefore, it is only left to bound the
last term in the expression above. We will repeat the combination of
Proposition 2.1 and Theorem 2.9.

Let us consider a splitting of $B$ into $B_{1}$, $B_{2}$ and $B_{3}$ such that
$|B_{1}|,|B_{2}|,|B_{3}|\geq\ell$ (see Figure 4). First, by triangle
inequality, Hölder’s inequality, point (ii) of Proposition 2.1 and Proposition
2.2, we get

$\displaystyle\left|\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}E_{AB,C}\rho^{C}\big{]}\Big{]}-\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\Big{]}\right|$
(51) $\displaystyle\hskip
28.45274pt=\left|\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}\big{(}{E_{AB,C}^{*}}-{E_{B_{3},C}^{*}}\big{)}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\Big{]}\right.$
$\displaystyle\hskip
42.67912pt\left.+\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}\left(E_{AB,C}-{E_{B_{3},C}}\right)\rho^{C}\big{]}\Big{]}\right|$
$\displaystyle\hskip
28.45274pt\leq\mathcal{C}\norm{{E_{AB,C}^{*}}-{E_{B_{3},C}^{*}}}+\mathcal{C}\norm{{E_{AB,C}}-{E_{B_{3},C}}}$
$\displaystyle\hskip 28.45274pt\leq 2\mathcal{C}\delta(\ell)\,.$

Similarly for the analogous term tracing out $BC$,

$\left|\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,BC}^{*}}\big{]}\tr_{A}\big{[}E_{A,BC}\rho^{A}\big{]}\Big{]}-\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}E_{A,B_{1}}\rho^{A}\big{]}\Big{]}\right|\leq
2\mathcal{C}\delta(\ell)\,.$ (52)

The term tracing out $B$ is slightly more involved but follows the same lines:

$\displaystyle\left|\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B}^{*}}{E_{AB,C}^{*}}\big{]}\tr_{AC}\big{[}{E_{AB,C}}{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}\right.$
(53) $\displaystyle\hskip
28.45274pt\left.-\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B_{1}}^{*}}\otimes{E_{B_{3},C}^{*}}\big{]}\tr_{AC}\big{[}{E_{A,B_{1}}}\otimes{E_{B_{3},C}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}\right|$
$\displaystyle=\left|\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B}^{*}}{E_{AB,C}^{*}}\big{]}\tr_{AC}\big{[}{E_{AB,C}}{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}\right.$
$\displaystyle\hskip
28.45274pt\left.-\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}{E_{A,B_{1}}}\rho^{A}\big{]}\otimes\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}{E_{B_{3},C}}\rho^{C}\big{]}\Big{]}\right|$
$\displaystyle\hskip 28.45274pt=4\mathcal{C}^{3}\delta(\ell)\,.$

Additionally, note that

$\displaystyle\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}E_{AB,C}\rho^{C}\big{]}\Big{]}\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,BC}^{*}}\big{]}\tr_{A}\big{[}E_{A,BC}\rho^{A}\big{]}\Big{]}\leq\mathcal{C}^{4}\,.$
(54)

Now, combining Eq. (51), Eq. (52), Eq. (53) and Eq. (54), we have

$\displaystyle\left|\chi_{ABC}-1\right|$ $\displaystyle\hskip
14.22636pt\leq\mathcal{C}^{4}\left|\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{AC}\big{[}\rho^{A}\otimes\rho^{C}{E_{A,B}^{*}}{E_{AB,C}^{*}}\big{]}\tr_{AC}\big{[}{E_{AB,C}}{E_{A,B}}\rho^{A}\otimes\rho^{C}\big{]}\Big{]}\right.$
$\displaystyle\hskip
42.67912pt-\left.\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{AB,C}^{*}}\big{]}\tr_{C}\big{[}E_{AB,C}\rho^{C}\big{]}\Big{]}\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,BC}^{*}}\big{]}\tr_{A}\big{[}E_{A,BC}\rho^{A}\big{]}\Big{]}\right|$
$\displaystyle\hskip
14.22636pt\leq\mathcal{C}^{4}\left|\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}{E_{A,B_{1}}}\rho^{A}\big{]}\otimes\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}{E_{B_{3},C}}\rho^{C}\big{]}\Big{]}\right.$
$\displaystyle\hskip
42.67912pt-\left.\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\Big{]}\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}E_{A,B_{1}}\rho^{A}\big{]}\Big{]}\right|$
$\displaystyle\hskip
42.67912pt+4\mathcal{C}^{7}\delta(\ell)+2\mathcal{C}^{7}\delta(\ell)+2\mathcal{C}^{7}\delta(\ell)\,.$

To conclude, we will estimate the difference in the previous expression using
the local indistinguishability of Gibbs states as in Theorem 2.9. Indeed, note
that

$\displaystyle\left|\Tr_{BC}\Big{[}\widetilde{\rho}^{BC}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}E_{A,B_{1}}\rho^{A}\big{]}\Big{]}-\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}E_{A,B_{1}}\rho^{A}\big{]}\Big{]}\right|\leq\mathcal{C}^{2}\,c_{2}\,e^{-\alpha_{2}\ell}\,,$

as well as

$\displaystyle\left|\Tr_{AB}\Big{[}\widetilde{\rho}^{AB}\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\Big{]}-\Tr_{B}\Big{[}\widetilde{\rho}^{B}\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\Big{]}\right|\leq\mathcal{C}^{2}\,c_{2}\,e^{-\alpha_{2}\ell}\,,$

for certain constants $c_{2},\alpha_{2}>0$. Therefore, denoting

$\xi_{B_{1}}:=\tr_{A}\big{[}\rho^{A}{E_{A,B_{1}}^{*}}\big{]}\tr_{A}\big{[}E_{A,B_{1}}\rho^{A}\big{]}\;,\quad\xi_{B_{3}}:=\tr_{C}\big{[}\rho^{C}{E_{B_{3},C}^{*}}\big{]}\tr_{C}\big{[}E_{B_{3},C}\rho^{C}\big{]}\,,$

we have

$\displaystyle\left|\Tr_{B}\big{[}\widetilde{\rho}^{B}\xi_{B_{1}}\otimes\xi_{B_{3}}\big{]}-\Tr_{AB}\big{[}\widetilde{\rho}^{AB}\xi_{B_{3}}\big{]}\Tr_{BC}\big{[}\widetilde{\rho}^{BC}\xi_{B_{1}}\big{]}\right|$
$\displaystyle\hskip
142.26378pt\leq\left|\Tr_{B}\big{[}\widetilde{\rho}^{B}\xi_{B_{1}}\otimes\xi_{B_{3}}\big{]}-\Tr_{B}\big{[}\widetilde{\rho}^{B}\xi_{B_{3}}\big{]}\Tr_{B}\big{[}\widetilde{\rho}^{B}\xi_{B_{1}}\big{]}\right|+2\mathcal{C}^{4}\,c_{2}\,e^{-\alpha_{2}\ell}\,,$

and we conclude the proof by using exponential decay of correlations as in
Proposition 2.8 giving us an upper estimate
$\operatorname{\mathcal{C}}^{4}c_{3}e^{-\alpha_{3}\ell}$. Combining all of the
above estimates finally gives the claim:

$|\widehat{\lambda}_{ABC}^{-1}-1|\leq
c_{1}e^{-\alpha_{1}\ell}+8\operatorname{\mathcal{C}}^{9}\delta(\ell)+\operatorname{\mathcal{C}}^{10}(2c_{2}e^{-\alpha_{2}\ell}+c_{3}e^{-\alpha_{3}\ell})\,.$

∎

######  Remark 3.10.

Note that Proposition 3.9 also holds for exponentially-decaying interactions,
with the exact same proof, just by adapting to that case the technical tools
employed. This is the content of Proposition B.7.

##  4 Applications

This section is devoted to applications of the main results from the previous
section in the context of MPO approximations and learning of Gibbs states. In
Section 4.1 we show that a set of information-theoretic criteria, most
importantly the decay of the BS-CMI, imply an efficient MPO representation of
a state, which by the previous section exists for one-dimensional Gibbs
states. Using the explicit form of this reconstruction in Section 4.2 we show
that this representation can be learned by tomography of small marginals. In
Section 4.3 we outline the scheme for estimating the global purity given
Proposition 3.9.

###  4.1 Positive MPO approximations from recovery maps

In this section, we provide a sequential reconstruction using a symmetric
recovery map for the BS-CMI, newly introduced in this manuscript. The
motivation for this map arises from Eq. (35), where we recalled that the DPI
for the BS-entropy saturates if, and only if, each state can be recovered from
the other by the so-called (asymmetric) BS-recovery condition. This condition,
albeit appealing and useful for applications in the context of Gibbs states
(cf. Theorem 3.6), is operationally flawed because of its lack of positivity,
let alone hermiticity. The desire to have a positive map with a similar spirit
encourages us to define the following symmetric recovery map for a particular
case in which both states are defined in a tripartite space and the
conditional expectation considered is a partial trace:

$\displaystyle\operatorname{\mathcal{R}}(X)$
$\displaystyle=\rho_{B}^{1/2}(\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2})^{1/2}\rho_{B}^{-1/2}X\rho_{B}^{-1/2}(\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2})^{1/2}\rho_{B}^{1/2}\,.$
(55)

The definition of this map is not arbitrary, as it follows from the
combination of some bounds obtained in [12] and [23], as we will see in the
Lemma below. Interestingly, it constitutes a single-shot recovery error bound
that involves the inverse BS-CMI, the lowest eigenvalue of marginals and the
maximal mutual information.

######  Lemma 4.1.

Given a tripartite state $\rho_{ABC}$, the following bound on the distance
between the state and the recovery map from Eq. (55) holds

$\displaystyle\widehat{I}_{\rho}^{\operatorname{rev}}(A:C|B)$
$\displaystyle=\widehat{D}(\pi_{A}\otimes\rho_{BC}\|\rho_{ABC})-\widehat{D}(\pi_{A}\otimes\rho_{B}\|\rho_{AB})$
(56)
$\displaystyle\geq\left(\frac{\pi}{8}\right)^{4}\|\Gamma\|_{\infty}^{-2}\|\operatorname{\mathcal{R}}(\rho_{BC})-\rho_{ABC}\|_{1}^{4}\,.$
(57)

######  Remark 4.2.

Note that $\operatorname{\mathcal{R}}(X)$ is completely positive but not
trace-preserving, so not a quantum channel. Several choices of recovery map
are possible, see [12] for an alternative definition. The one in the above
Lemma, despite its more complicated form, is necessary for proving the bound
on the Lipschitz constants below and has not been considered before to the
best of our knowledge.

###### Proof of Lemma 4.1.

We use a particular case of the strengthened data-processing inequality for
the BS-entropy [12, Theorem 5.3], where we choose the channel to be the
conditional expectation
$\operatorname{\mathcal{E}}(\cdot)=\tr_{C}[\cdot]\otimes\pi_{C}$,
$\sigma=\pi_{A}\otimes\rho_{BC}$ for the first argument, and $\rho=\rho_{ABC}$
for the second (note the different convention for the naming of arguments).
Employing now [12, Theorem 5.3] gives the bound

$\displaystyle\widehat{I}^{\operatorname{rev}}(A:C|B)$
$\displaystyle=\widehat{D}(\pi_{A}\otimes\rho_{BC}\|\rho_{ABC})-\widehat{D}(\pi_{A}\otimes\rho_{B}\|\rho_{AB})$
$\displaystyle\geq\left(\frac{\pi}{4}\right)^{4}\|\Gamma\|^{-2}_{\infty}\|\rho_{BC}^{1/2}\rho_{B}^{-1/2}\Gamma_{\Tr}^{1/2}\rho_{B}^{1/2}-\Gamma^{1/2}\rho_{BC}^{1/2}\|^{4}_{2}\,,$

where $\Gamma=\rho_{BC}^{-1/2}\rho_{ABC}\rho_{BC}^{-1/2}$ and
$\Gamma_{\Tr}=\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2}$. Noticing that

$\displaystyle\Tr[\operatorname{\mathcal{R}}(\rho_{BC})]$
$\displaystyle=\Tr_{ABC}[\rho_{B}^{1/2}(\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2})^{1/2}\rho_{B}^{-1/2}\rho_{BC}\rho_{B}^{-1/2}(\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2})^{1/2}\rho_{B}^{1/2}]$
$\displaystyle=\Tr_{AB}[\rho_{B}^{1/2}(\rho_{B}^{-1/2}\rho_{AB}\rho_{B}^{-1/2})\rho_{B}^{1/2}]=1$

in addition to $\Tr_{ABC}[\rho_{ABC}]=1$ allows us to employ [23, Lemma 2.2],
which states

$\|X^{*}X-Y^{*}Y\|_{1}\leq 2\|X-Y\|_{2}$

for operators $X,Y$ such that $\Tr[X^{*}X]=\Tr[Y^{*}Y]=1$. This allows us to
lower bound Eq. (4.1) and obtain a strengthened DPI for the inverse BS-CMI
involving the distance of $\rho_{ABC}$ to
$\operatorname{\mathcal{R}}(\rho_{BC})$ in $1$ norm, i.e. the claim of Lemma
4.1. ∎

As we will see in the next pages, the recovery map introduced in Eq. (55),
when applied iteratively, provides a set of information-theoretic criteria for
the existence of MPO descriptions of an arbitrary state. For that, let us
consider a sequence of recovery maps that can reconstruct a quantum state on a
spin chain. We assume an appropriately subdivided chain on subintervals
$A_{1},\ldots,A_{N}$ as in Figure 5.

$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$…$A_{n-1}$$A_{n}$$\Lambda$ Figure 5:
Representation of an interval $\Lambda$ split into multiple subintervals
$\Lambda=A_{1}A_{2}\ldots A_{n}$.

Their size is chosen so that the inverted BS-CMI is sufficiently small, as we
will see in the next subsection. For simplicity, let us denote by
$\rho_{i}:=\rho_{A_{i}}=\tr_{\Lambda\setminus A_{i}}[\rho]$ and by
$\rho_{i:j}:=\rho_{A_{i},A_{i+1},\ldots,A_{j}}=\tr_{\Lambda\setminus(A_{i}\cup\ldots\cup
A_{j})}[\rho]$.

A standard argument to extend a recovery result from a single recovery map to
a chain is to use the contractivity of the recovery map to show that the
errors behave additively. This is not possible in our setting, as the non-
trace-preserving maps are not contractive. However, we will be able to
overcome this technical difficulty by proving a Lipschitz property of a
concatenation of maps, independently of the level of concatenation. More
specifically, in Lemma 4.3 we show that nevertheless the errors only suffer a
constant amplification that does not grow exponentially in the length of the
recovery chain.

We denote the individual recovery maps
$\operatorname{\mathcal{R}}_{i}:\mathcal{B}(\mathcal{H}_{A_{i}})\to\mathcal{B}(\mathcal{H}_{A_{i}A_{i+1}})$,
$i=1,\ldots,N-1$ as

$\operatorname{\mathcal{R}}_{i}(X)=\rho_{i}^{1/2}(\rho_{i}^{-1/2}\rho_{i:{i+1}}\rho_{i}^{-1/2})^{1/2}\rho_{i}^{-1/2}X\rho_{i}^{-1/2}(\rho_{i}^{-1/2}\rho_{i:{i+1}}\rho_{i}^{-1/2})^{1/2}\rho_{i}^{1/2}\,.$
(58)

For simplicity, we will abbreviate the map when adding on a larger chain, and
replace $\mathds{1}^{\otimes i-1}\otimes\operatorname{\mathcal{R}}_{i}$ by
$\operatorname{\mathcal{R}}_{i}$.

######  Lemma 4.3.

Let $1\leq j<N$, be natural numbers and $X\in\mathcal{B}(\mathcal{H}_{A_{j}})$
be positive semidefinite. There is a constant independent of the number of
concatenated maps that bounds the Lipschitz constant of the concatenated map:

$\displaystyle\left\|\left(\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(X)\right\|_{1}\leq\|\rho_{j}^{-1}\|_{\infty}\|X\|_{1}\,.$

This also holds if the $\rho_{i},\rho_{i:{i+1}}$ are not consistent marginals
of a fixed global state, as long as they are local positive states and
$\tr_{j}[\rho_{j:{j+1}}]=\rho_{{j+1}}$.

###### Proof.

The map $\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\cdot)$ preserves
positivity being a concatenation of positive maps. Hence, we readily observe
that:

$\norm{\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(X)}_{1}\leq\norm{\rho_{j}^{-1/2}X\rho_{j}^{-1/2}}_{\infty}\norm{\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j})}_{1}\,.$

Further inspection reveals that
$\operatorname{\mathcal{R}}_{i}(\rho_{i})=\rho_{i:i+1}$ and
$[\tr_{i},\operatorname{\mathcal{R}}_{k}]=0$ for $k>i$. Leveraging these
observations, we can immediately conclude:

$\displaystyle\norm{\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j})}_{1}$
$\displaystyle=\Tr[\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j})]=\Tr[\tr_{j}[\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j})]]$
$\displaystyle=\Tr[\tr_{j}[\bigcirc_{i=j+1}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j:j+1})]]$
$\displaystyle=\Tr[\bigcirc_{i=j+1}^{N-1}\operatorname{\mathcal{R}}_{i}(\tr_{j}[\rho_{j:j+1}])]=\Tr[\bigcirc_{i=j+1}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j+1})]\,.$

Iterating this process yields
$\norm{\bigcirc_{i=j}^{N-1}\operatorname{\mathcal{R}}_{i}(\rho_{j})}_{1}=\Tr[\rho_{N}]=1$.
Applying Hölder’s inequality and the hierarchy of Schatten p-norms to simplify
$\norm{\rho_{j}^{-1/2}X\rho_{j}^{-1/2}}_{\infty}$ finally confirms the desired
result. ∎

We now combine Lemma 4.3 with Lemma 4.1 to decompose the overall recovery
error, obtaining a bound on the recovery error based on entropic quantities.
This is the main technical result of the section.

######  Theorem 4.4.

For a multipartite quantum state $\rho_{1:N}$, the recovery error on the chain
between the concatenated recovery map and the original state is bounded by

$\displaystyle\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|_{1}\leq$
$\displaystyle\frac{16(N-1)}{\pi}\sup_{i}D_{\infty}(\mathds{1}\|\rho_{i})\exp(I_{\infty}(A_{1}\ldots
A_{i-1}:A_{i})/2)$ (59) $\displaystyle\quad
I_{\rho}^{\operatorname{rev}}(A_{i}:A_{1}\ldots A_{i-2}|A_{i-1})^{1/4}$

Note that the recovery map representation
$\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})$ is
also a matrix product operator representation with bond dimension
$D=\dim(A_{i})^{3}$.

###### Proof.

We denote the overall recovery error by $\varepsilon$

$\varepsilon=\left\|\operatorname{\mathcal{R}}_{N-1}(\ldots\operatorname{\mathcal{R}}_{1}(\rho_{1})\ldots)-\rho_{1:N}\right\|_{1}=\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|_{1}.$
(60)

First, we split the total error $\varepsilon$ into the individual errors for
each recovery step, where each error is still amplified by the subsequent
recovery maps:

$\displaystyle\varepsilon$
$\displaystyle\leq\sum_{i=1}^{N-1}\left\|\left(\bigcirc_{j=i+1}^{N-1}\operatorname{\mathcal{R}}_{j}\right)(\rho_{A_{1}\ldots
A_{i+1}}-\operatorname{\mathcal{R}}_{i}(\rho_{A_{1}\ldots
A_{i}}))\right\|_{1}$ (61)
$\displaystyle\leq\sum_{i=1}^{N-1}\left\|\left(\bigcirc_{j=i+1}^{N-1}\operatorname{\mathcal{R}}_{j}\right)(X_{A_{i+1}}^{+})\right\|_{1}+\left\|\left(\bigcirc_{j=i+1}^{N-1}\operatorname{\mathcal{R}}_{j}\right)(X_{A_{i+1}}^{-})\right\|_{1}$

Here, $X_{i}=\rho_{1:{i+1}}-\operatorname{\mathcal{R}}_{i}(\rho_{A_{1}\ldots
A_{i}})$ with decomposition into positive and negative parts
$X_{i}=X_{i}^{+}-X_{i}^{-}$ and $X_{A_{i}}^{\pm}=\Tr_{A_{1}\ldots
A_{i-1}}[X_{i}^{\pm}]$ such that the last norm is taken on systems
$A_{i}\ldots A_{N}$ only. Furthermore, this means

$\|X_{A_{i}}^{\pm}\|_{1}=\|X_{i}^{\pm}\|_{1}\leq\|X_{i}\|_{1}.$

Finally, combining Eqs. (60) and (61), and using Lemma 4.3 we can upper bound
the error as

$\displaystyle\varepsilon$
$\displaystyle=\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|_{1}$
(62) $\displaystyle\leq
2(N-1)\sup_{i}\|\rho_{i}^{-1}\|_{\infty}\frac{8}{\pi}\sqrt{\|\Gamma_{i}\|_{\infty}}\widehat{I}_{\rho}^{\operatorname{rev}}(A_{i}:A_{1}\ldots
A_{i-2}|A_{i-1})^{1/4}\,,$

where in the last inequality we are using that, by Lemma 4.1,

$\norm{X_{i}}_{1}\leq\frac{8}{\pi}\sqrt{\|\Gamma_{i}\|_{\infty}}\widehat{I}_{\rho}^{\operatorname{rev}}(A_{i}:A_{1}\ldots
A_{i-2}|A_{i-1})^{1/4}\,,$

and we introduced
$\Gamma_{i}=\rho_{1:{i-1}}^{-1/2}\rho_{1:i}\rho_{1:{i-1}}^{-1/2}$. To obtain
the desired formulation, let us note that all the quantities involved can be
formulated in terms of divergences. The factor $\Gamma_{i}$ is related to the
maximal Rényi mutual information as follows.

$\displaystyle\|\Gamma_{i}\|_{\infty}$
$\displaystyle\leq\left\|\rho_{i}^{1/2}\right\|_{\infty}^{2}\left\|(\rho_{1:{i-1}}\otimes\rho_{i})^{-1/2}\rho_{1:i}(\rho_{1:{i-1}}\otimes\rho_{i})^{-1/2}\right\|_{\infty}$
$\displaystyle\leq\exp(I_{\infty}(A_{1}\ldots A_{i-1}:A_{i}))\,.$

Similarly,

$\|\rho_{i}^{-1}\|_{\infty}=\exp D_{\infty}(\mathds{1}\|\rho_{i})\,,$

concluding thus the proof. ∎

We now consider the implications of Theorem 4.4 for MPO representations Gibbs
states. Note that due to the expected exponential growth of
$D_{\infty}(\mathds{1}\|\rho_{i})$ in Eq. (59) with the size of the subsystem,
at least an exponential decay of
$\widehat{I}_{\rho}^{\operatorname{rev}}(A:C|B)$ with sufficient rate, for any
adjacent three systems $A,B,C$, is needed to employ the above result. In the
case of thermal states, however, Theorem 3.6 guarantees a superexponential
rate, which dominates over the other prefactors.

######  Corollary 4.5.

For a $n$-site marginal of a Gibbs state in one dimension with translation-
invariant interaction $\Phi$ and a given accuracy $\varepsilon$, there is an
MPO representation of bond dimension

$D=\exp(2\log(d)\operatorname{\mathcal{C}}_{1}\frac{\log(n/\varepsilon)+\operatorname{\mathcal{C}}_{2}}{\log(\log(n/\varepsilon))}),$
(63)

where $\operatorname{\mathcal{C}}_{1},\operatorname{\mathcal{C}}_{2}$ are
constants depending on $J$, $R$, and $\beta$.

It is given by the above construction, i.e.,

$\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|\leq\varepsilon$

choosing the $A_{i}$ as consecutive regions of at least $l$ spins

$|A_{i}|=l\geq\operatorname{\mathcal{C}}_{1}\frac{\log(n/\varepsilon)+\operatorname{\mathcal{C}}_{2}}{\log(\log(n/\varepsilon))}\,.$

######  Remark 4.6.

Due to the uniformity of the result in Theorem 3.6 this result applies equally
to a local Gibbs state, i.e. the Gibbs state of the Hamiltonian on $n$ sites,
as to the marginal of a Gibbs state on a larger or even infinite system.

###### Proof of Corollary 4.5.

Here, we follow a similar procedure to that of [32, Corollary 3.3]. First,
recall that Theorem 3.6 provides the needed decay for the BS-CMI in the case
of thermal states. Furthermore, in this setting by Lemma 3.5 the growth of
$D_{\infty}(\mathds{1}\|\rho_{i})$ is in fact bounded by an exponential.
Finally, the maximal mutual information obeys an area law for thermal states
[70, Theorem 2]

$I_{\infty}(A:B)\leq\operatorname{\mathcal{C}}$

for any adjacent intervals $A,B$ and a constant $\operatorname{\mathcal{C}}$
that only depends on $\beta$, $J$ and $R$.

We now proceed by partitioning our system into subsystems of size $|A_{i}|=l$
to be determined later. Combining all these results we achieve the desired
accuracy if

$\displaystyle\varepsilon^{4}\leq\operatorname{\mathcal{C}}_{1}\frac{n}{l}\frac{\operatorname{\mathcal{C}}_{2}^{\lfloor
l/2\rfloor+1}}{(\lfloor\lfloor l/2\rfloor/R\rfloor+1)!}\,.$

Note that the factor $\exp(\alpha_{\operatorname{rev}}(|A_{i}|+|A_{i+1}|))$
for every $i$ is included in $\operatorname{\mathcal{C}}_{2}^{\lfloor
l/2\rfloor+1}$. Using Stirling’s approximation we find that a sufficient
condition for this is given by

$\log(\frac{n\operatorname{\mathcal{C}}_{1}}{\varepsilon^{4}})\leq-\left(\frac{l}{2}+1\right)\log(\operatorname{\mathcal{C}}_{2})+\left(\frac{l}{2R}-\frac{1}{R}\right)\left(\log(\frac{l}{2R}-\frac{1}{R})-1\right)\,.$

While this inequality cannot be inverted analytically, a further relaxation
based on a dual description of the convex function $x\mapsto x\log(x)$ yields
the sufficient condition

$2R\frac{\log(\frac{n\operatorname{\mathcal{C}}_{1}\operatorname{\mathcal{C}}_{2}^{2}}{\varepsilon^{4}})+a}{\log(a)-R\log(\operatorname{\mathcal{C}}_{2})}+1\leq
l$

for any $a>0$. Choosing $a=\log(n/\varepsilon)\mathcal{C}_{2}^{R}$ and again
combining all constants the claim follows. ∎

######  Remark 4.7.

Corollary 4.5 can be adapted for exponentially-decaying interactions,
obtaining an MPO approximation for a Gibbs state in that setting for the first
time in the literature, to the best of our knowledge. The bond dimension is
slightly worse in that case though, as it arises from the decay of the BS-CMI,
which is only exponential in $|B|$ in this case, as opposed to the
superexponential behaviour in the finite-range case. This is the content of
Corollary B.9.

###  4.2 Reconstructing the positive MPO from local tomography

Since Theorem 4.4 provides a way to reconstruct an entire state $\rho$ from
the smaller marginals, it also offers a reconstruction of classical
representations of $\rho$, in the form of MPO, from tomographic data on small
regions. What is required from the tomographic data is to produce a classical
representation of the marginals with small tomography errors, which can be
done with standard results [38, 62]. To prove that our final MPO
reconstruction of $\rho$ is accurate, we thus need to show that the
reconstruction process, which happens at the level of classical post-
processing, is stable to the small errors in the local tomography. Here, since
in our setting the Hamiltonian is translation invariant, the cost of learning
a state from measurements is limited to the cost of learning the state in a
single region (although the precision might depend on the final size that we
want to reconstruct).

Our starting point is a measurement scheme for general quantum states that we
will apply to the marginals of the Gibbs state. Since we do not aim to give
constants in our runtime explicitly, any polynomial time measurement scheme is
sufficient for our purposes. Sample optimal measurement schemes for arbitrary
quantum states with quadratic sample complexity in the inverse error and
dimension have been developed in [38, 62]. The situation regarding the
classical runtime of these schemes is however unclear. Nevertheless, the
following result is an immediate consequence of measuring each coefficient of
the density matrix individually to sufficient precision and using the
equivalence of matrix norms up to dimensional factors.

######  Lemma 4.8.

There exists a measurement scheme that produces a classical representation of
a quantum state $\hat{\rho}$ such that with probability $\geq 1-c$

$\norm{\rho-\hat{\rho}}_{1}\leq\delta\,,$ (64)

with several samples and classical runtime bounded by

$\displaystyle n_{s}$ $\displaystyle\leq
C_{s}\operatorname{poly}\left(d,\frac{1}{\delta}\right)\log(1/c)\,,$
$\displaystyle t$ $\displaystyle\leq
C_{t}\operatorname{poly}\left(d,\frac{1}{\delta}\right)\log(1/c)\,.$

With this, we can now state the cost of local tomography that we require to
produce the accurate MPO representation of the whole state from the previous
section. The proof essentially shows how the errors in the marginals from
Lemma 4.8 propagate in the final error of the MPO approximation, which allows
us to estimate the final costs of samples and classical runtime.

######  Theorem 4.9.

Let us consider a $n$-site marginal of a Gibbs state in one dimension with
translation-invariant interaction, and define the approximate reconstruction
map as in Eq. (58),

$\hat{\operatorname{\mathcal{R}}}_{i}(X)=(\hat{\rho}_{i})^{1/2}((\hat{\rho}_{i})^{-1/2}\hat{\rho}_{i:i+1}(\hat{\rho}_{i})^{-1/2})^{1/2}(\hat{\rho}_{i})^{-1/2}X(\hat{\rho}_{i})^{-1/2}((\hat{\rho}_{i})^{-1/2}\hat{\rho}_{i:i+1}(\hat{\rho}_{i})^{-1/2})^{1/2}(\hat{\rho}_{i})^{1/2}\,,$
(65)

where each $\hat{\rho}_{i}$ is an approximation of the corresponding marginal
of $\rho$. Then, for a given accuracy $\varepsilon$, the MPO representation
$\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})$
has bond dimension as in Eq. (63) and is such that, with probability $\geq
1-c$,

$\left\|\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})-\rho_{1:N}\right\|_{1}\leq\varepsilon\,.$

The sample complexity and classical post-processing time for finding this MPO
are $\textup{poly}\left(n/\varepsilon\right)\log(1/c)$.

###### Proof.

We assume the repartitioned subsystems $A_{1},\ldots,A_{N}$ each of size
$|A_{i}|=l$ as of Corollary 4.5. We need to show that the accuracy of the
reconstruction map $\operatorname{\mathcal{R}}_{i}(X)$ is robust to small
errors in the estimated marginals $\rho_{i},\rho_{i:i+1}$. We repeatedly apply
those approximate reconstructions, and the error induced by each is shown in
Eq. (62).

To construct a good approximation to
$\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})$ we
require estimates of the marginals $\hat{\rho}_{i}$ and $\hat{\rho}_{i:i+1}$,
which we take to be such that with probability $\geq 1-c$

$\norm{\rho_{i:i+1}-\hat{\rho}_{i:i+1}}_{1}\leq\delta,$ (66)

for which we need $\textup{poly}\left(d^{l},\delta^{-1}\right)\log(N/c)$
samples. Then, for consistency, we take the single-region marginals as the
reductions $\hat{\rho}_{{i+1}}=\tr_{i}[\hat{\rho}_{i:i+1}]$. To account for
the possibility of failure in the tomography scheme, we choose a confidence
bound of $c/N$ for each marginal such that, by a union bound, all marginals
fulfil Eq. (66).

To obtain an accurate approximation to the recovery map, we need our estimates
to have well-behaved inverses. Let us note that according to Lemma 3.5

$\left\|\rho_{i:i+1}^{-1}\right\|_{\infty}\leq\operatorname{\mathcal{C}}e^{\alpha
l}\,.$

Thereby, explicitly requiring

$\delta\leq\frac{1}{2}\left\|\rho_{i:i+1}^{-1}\right\|_{\infty}^{-1}=\frac{1}{2\operatorname{\mathcal{C}}}e^{-\alpha
l}\,,$ (67)

we obtain the equivalent bounds for the approximate marginals

$\left\|\rho_{i:i+1}^{-1}\right\|_{\infty},\left\|\rho_{i}^{-1}\right\|_{\infty},\left\|(\hat{\rho})_{i:i+1}^{-1}\right\|_{\infty},\left\|(\hat{\rho})_{i}^{-1}\right\|_{\infty}\leq
2\operatorname{\mathcal{C}}e^{\alpha l}\,.$

With this and the definition of $\hat{\operatorname{\mathcal{R}}}_{i}$ from
Eq. (65), our target MPO approximation is
$\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)({\hat{\rho}}{}_{1})$.
To control the approximation error, we repeatedly apply the triangle
inequality so that

$\displaystyle\norm{\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)({\hat{\rho}}{}_{1})-\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})}_{1}\leq\sum_{j=1}^{N-1}$
$\displaystyle\norm{\left(\bigcirc_{i=j+1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)\left(\operatorname{\mathcal{R}}_{j}-\hat{\operatorname{\mathcal{R}}}_{j}\right)\left(\bigcirc_{i=1}^{j-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)({\hat{\rho}}_{1})}_{1}$
$\displaystyle+\norm{\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\hat{\rho}_{1}-\rho_{1})}_{1}.$

Applying Lemma 4.3 to each term in the sum, and Eq. (66) we have that

$\displaystyle\norm{\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)({\hat{\rho}}{}_{1})-\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})}_{1}\leq\sum_{j=1}^{N-1}$
$\displaystyle\norm{\rho_{j+1}^{-1}}_{\infty}\norm{\left(\operatorname{\mathcal{R}}_{j}-\hat{\operatorname{\mathcal{R}}}_{j}\right)\left(\bigcirc_{i=1}^{j-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})}_{1}$
$\displaystyle+2\delta\norm{\rho_{1}^{-1}}.$

To bound the summands
$\norm{\left(\operatorname{\mathcal{R}}_{j}-\operatorname{\mathcal{R}}^{\prime}_{j}\right)(X)}_{1}$,
we require the following two continuity bounds for the square root and its
inverse, as proven in [24],

$\displaystyle\norm{\sqrt{\rho}-\sqrt{{\hat{\rho}}{}}}_{\infty}$
$\displaystyle\leq\max\\{\norm{\rho^{-1}}_{\infty},\norm{({\hat{\rho}}{})^{-1}}_{\infty}\\}\cdot\sqrt{8}\norm{\rho-{\hat{\rho}}{}}_{\infty},$
$\displaystyle\norm{\rho^{-1/2}-({\hat{\rho}}{})^{-1/2}}_{\infty}$
$\displaystyle\leq\max\\{\norm{\rho^{-1}}_{\infty},\norm{({\hat{\rho}}{})^{-1}}_{\infty}\\}\cdot\norm{\rho^{-1/2}}_{\infty}\norm{({\hat{\rho}}{})^{-1/2}}_{\infty}\cdot\sqrt{8}\norm{\rho-{\hat{\rho}}{}}_{\infty}.$

Repeated applications of these and the triangle inequality, and using that
$\norm{\cdot}\leq\norm{\cdot}_{1}\leq d\norm{\cdot}$ then yield

$\displaystyle\norm{\left(\operatorname{\mathcal{R}}_{j}-\hat{\operatorname{\mathcal{R}}}_{j}\right)(X)}_{1}\leq
C\delta\norm{X}_{1}\left(\norm{\rho_{j}^{-1}}_{\infty}\norm{(\hat{\rho}_{j})^{-1}}_{\infty}\right)^{D}$

for some numerical constants $C,D$. Again, notice that by Lemma 4.3,

$\norm{\left(\bigcirc_{i=1}^{j-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})}_{1}\leq\norm{\hat{\rho}_{2}^{-1}}_{\infty}.$

Putting everything together, and introducing new numerical constants
$C^{\prime},D^{\prime}$, we thus obtain

$\norm{\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})-\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})}_{1}\leq
C^{\prime}N\delta d^{\alpha D^{\prime}l}.$

This means (recalling Eq. (67)) that if we set

$\delta=\min\left\\{\frac{\varepsilon d^{-\alpha
D^{\prime}l}}{C^{\prime}N},\frac{1}{2\operatorname{\mathcal{C}}}e^{-\alpha
l}\right\\}$

by Corollary 4.5 and the triangle inequality,

$\norm{\left(\bigcirc_{i=1}^{N-1}\hat{\operatorname{\mathcal{R}}}_{i}\right)(\hat{\rho}_{1})-\rho_{1:N}}_{1}\leq
2\varepsilon.$

Given the choice of $\delta$, the sample complexity is bounded by
$\textup{poly}\left(d^{l},N/\varepsilon\right)\log(1/c)$. The time complexity,
apart from the contribution due to the tomography itself, consists of
elementary matrix functions and multiplications which also run in time
$\textup{poly}\left(d^{l},N\right)$. Noting the sublogarithmic dependence of
$l$ on $n/\varepsilon$ in Corollary 4.5 and that $N\leq n$, the result
follows. ∎

######  Remark 4.10.

Compared to the previous section which gave a _subpolynomial_ dependence in
system size and inverse error, we only achieve polynomial dependence here. For
the inverse error this is due to the polynomial dependence in the tomography
scheme and cannot be avoided. However, for the system size the situation is
slightly different. Instead of explicitly giving all $N$ channels it can be
noted that for _infinite_ translation-invariant systems, they are all
identical and only measuring one of the marginals and reconstructing one of
the channels can be done in subpolynomial time in $n$. Similarly, for _finite_
translation-invariant only the sample complexity can be improved to
subpolynomial in $n$ by measuring on all subsystems simultaneously for each
sample. However, writing out the MPO for the $n$ systems takes linear time in
$n$ by definition again in both cases.

###  4.3 Efficient estimation of the global purity

Beyond MPO reconstructions of the entire state, the results on the
factorization of the purity from Section 3.3 allow us to efficiently estimate
the global purity of the global state $\rho_{1:N}$ from local purity
estimations.

For completeness, we now reproduce a simplified version of the approximation
scheme of the purity of 1D Gibbs states in [79], where a more detailed
analysis of the polynomial scaling of resources and pre-factors can be found.
Our main contribution here is that Proposition 3.9 is the missing ingredient
for a fully rigorous polynomial bound on the sample complexity of their
estimation algorithm. Instead, in [79], the exponential decay in Eq. (44) was
only studied numerically. To simplify the discussion, we do not consider the
randomized measurement toolbox [29], which potentially yields a slightly more
favourable scaling and more amenable classical post-processing, and instead
use the result from Lemma 4.8.

######  Theorem 4.11.

Under the conditions of Proposition 3.9, there is an algorithm that outputs an
estimate $\hat{P}_{2}(\rho_{1:N})$ such that

$\left|\frac{\hat{P}_{2}(\rho_{1:N})}{\Tr[\rho_{1:N}^{2}]}-1\right|\leq\varepsilon,$
(68)

with a number of samples and classical post-processing cost given by
$\textup{poly}(n/\varepsilon)$.

###### Proof.

Consider adjacent marginals of the state $\rho_{i}$ on regions of size $l$ and
adjacent pairs $\rho_{i:i+1}$. An estimate for the purity is as follows

$P_{2}(\rho_{1:N})=\frac{\prod_{j=1}^{N-1}\Tr_{j:j+1}[\rho^{2}_{j:j+1}]}{\prod_{j=2}^{N-1}\Tr_{j}[\rho^{2}_{j}]}.$

Then, from an iterated application of Proposition 3.9, and the straightforward
estimate for scalars $x,y$

$|xy-1|\leq|x||y-1|+|x-1|\,,$

we have that (see also Lemma 5 in [79])

$\displaystyle\left|\frac{P_{2}(\rho_{1:N})}{\Tr[\rho_{1:N}^{2}]}-1\right|\leq\left(1+\kappa
e^{-\alpha l}\right)^{N}-1.$

Therefore, with the choice $l=\mathcal{O}\left(\log{N/\varepsilon}\right)$ we
achieve error $\varepsilon$.

The numbers $\Tr_{j:j+1}[\rho^{2}_{j:j+1}]$ and $\Tr_{j}[\rho^{2}_{j}]$ can be
obtained to multiplicative error $\delta$ from
$\textup{poly}(2^{l},\delta^{-1})$ local measurements and the scheme in Lemma
4.8, since

$\frac{\Tr_{j}[\rho^{2}_{j}]}{\Tr_{j}[\hat{\rho}^{2}_{j}]}-1\leq\Tr_{j}[\hat{\rho}^{2}_{j}]^{-1}\norm{\rho_{j}-\hat{\rho}_{j}}_{1}\leq
2^{l}\delta\,,$

and thus our estimation of the global purity is

$\hat{P}_{2}(\rho_{1:N})=\frac{\prod_{j=1}^{N-1}\Tr_{j:j+1}[\hat{\rho}^{2}_{j:j+1}]}{\prod_{j=2}^{N-1}\Tr_{j}[\hat{\rho}^{2}_{j}]}\,.$

With a similar argument as above, this is such that

$\left|\frac{\hat{P}_{2}(\rho_{1:N})}{P_{2}(\rho_{1:N})}-1\right|\leq\left(1+4^{l}\delta\right)^{2N}-1\,.$

Therefore, choosing
$2^{l}\delta=e^{-\Omega(l)}=\mathcal{O}(1/\text{poly}(N/\varepsilon))$ we
again obtain the approximation error $\varepsilon$. The result follows from
the estimates on the sample and computational cost in Lemma 4.8, and noticing
that $n\geq N$. ∎

Notice that it is also possible to estimate the purity via the estimation of
the MPO representation in Theorem 4.9, since

$\left|\Tr[\left(\bigcirc_{i=1}^{N-1}\hat{\mathcal{R}}_{i}\right)(\hat{\rho}_{1})^{2}]-\Tr[\rho_{1:N}^{2}]\right|\leq\norm{\left(\bigcirc_{i=1}^{N-1}\hat{\mathcal{R}}_{i}\right)(\hat{\rho}_{1})-\rho_{1:N}}_{1}.$

This, however, is a much worse estimate than Eq. (68), since the purity is
typically an exponentially small number. Thus, additive approximations (as
opposed to multiplicative ones) are much less meaningful.

######  Acknowledgements.

AC, PG and AR-d-A acknowledge the support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID
470903074 - TRR 352. AC, PG and AR-d-A also acknowledge funding by the Federal
Ministry of Education and Research (BMBF) and the Baden-Württemberg Ministry
of Science as part of the Excellence Strategy of the German Federal and State
Governments (FKZ:PRO-CAPEL CUEVAS-2023-02 and “Use of correlation decay and
complexity theory in the study of quantum Gibbs states”). SOS acknowledges
support from the UK Engineering and Physical Sciences Research Council (EPSRC)
under grant number EP/W524141/1. AMA acknowledges support from the Spanish
Agencia Estatal de Investigación through the grants “IFT Centro de Excelencia
Severo Ochoa SEV-2016-0597” and “Ramón y Cajal RyC2021-031610-I”, financed by
MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR.

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