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# Rapid thermalization of spin chain commuting Hamiltonians

Ivan Bardet<EMAIL_ADDRESS>Inria Paris, 75012 Paris, France  Ángela
Capel<EMAIL_ADDRESS>Department of Mathematics, Technische
Universität München, 85748 Garching, Germany  Munich Center for Quantum
Science and Technology (MCQST), 80799 München, Germany  Fachbereich
Mathematik, Universität Tübingen, 72076 Tübingen, Germany  Li Gao
<EMAIL_ADDRESS>Department of Mathematics, Technische Universität
München, 85748 Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), 80799 München, Germany  Department of Mathematics,
University of Houston, 77204 Houston, USA  Angelo Lucia<EMAIL_ADDRESS>David
Pérez-García<EMAIL_ADDRESS>Departamento de Análisis Matemático y Matemática
Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain  Instituto de
Ciencias Matemáticas, 28049 Madrid, Spain  Cambyse Rouzé<EMAIL_ADDRESS>Munich Center for Quantum Science and Technology (MCQST), 80799 München,
Germany

###### Abstract

We prove that spin chains weakly coupled to a large heat bath thermalize
rapidly at any temperature for finite-range, translation-invariant commuting
Hamiltonians, reaching equilibrium in a time which scales logarithmically with
the system size. From a physical point of view, our result rigorously
establishes the absence of dissipative phase transitions for Davies evolutions
over translation-invariant spin chains. The result has also implications in
the understanding of Symmetry Protected Topological phases for open quantum
systems.

##  I Introduction

Understanding how thermal noise affects quantum systems is a major open
problem in emerging quantum technologies. A key question there is how long
does it take for a system to thermalize? i.e. to converge to its thermal Gibbs
state. Or more specifically, which is the dependency of the thermalization
time, also known as mixing or decoherence time, on the temperature and the
system size?

In particular, it is important to identify those scenarios in which the mixing
time scales only logarithmically with the system size - such property is
usually called rapid mixing. From a negative point of view, in this regime
quantum properties that hold in the ground state but not in the thermal state
are suppressed too fast for them to be of any reasonable use. In the positive
side, thermal states with such short mixing time can be constructed very
efficiently with a quantum device that simulates the effect of the
corresponding thermal bath. Let us note that constructing thermal Gibbs states
is one of the main expected applications of a quantum computer, both as an
important self-standing problem [1], and also as a stepping stone in
optimization problems, via simulated annealing type algorithms [2, 3, 4, 5,
6]. On top of that, rapidly mixing systems have very desirable properties,
such as stability with respect to extensive perturbations in the noise
operator [7, 8].

Despite the importance of the question, very few mathematically rigorous
results are known in this direction. The reason is the lack of mathematical
techniques to tackle the problem. Indeed, the analogous results for classical
systems already required sophisticated mathematical tools, in particular, the
notion of log-Sobolev constant for the noise infinitesimal generator, whereas
estimates for the spectral gap of the generator are not enough to guarantee
such rapid mixing. Starting with the pioneer work of Glauber for the
particular case of classical 1D Ising model in 1963 [9], Holley and Stroock
[10] (building upon a previous result of Holley [11]) managed to prove the
rapid mixing property for all 1D classical models at any temperature. This was
done by showing that the log-Sobolev constant decreases at most
logarithmically with the system size. Later, Zegarlinski [12] improved their
result showing that the log-Sobolev constant was indeed bounded.

In the quantum regime all results have focused on systems with commuting
interactions. Note that this does not imply at all that the system is
classical. Indeed, such systems include all types of non-chiral topological
phases of matter. However most known results deal only with the spectral gap
of the generator, what can only guarantee a mixing time that grows
polynomially (and not logarithmically) with the system size [13]. For
instance, Alicki et al. [14] proved that the spectral gap has a uniform bound
independent of the system size for the quantum ferromagnetic 1D Ising model
and for Kitaev’s Toric Code in 2D at all temperatures. This result was
extended later for all abelian [15] and non-abelian [16] Kitaev’s quantum
double models in 2D, as well as for all 1D models with commuting interactions
[13].

In [17], a log-Sobolev inequality was also introduced in the quantum regime.
In particular, a bounded (or logarithmically growing) associated constant is
known to imply rapid mixing [18, 19]. Since then, several works have appeared
trying to estimate such constant for different noise models in many body
quantum systems with commuting interactions [18, 19, 20, 21]. Despite
considerable effort, the state of the art is that estimates have been obtained
either for rather artificial noise operators [22, 23] or only for sufficiently
high temperature [21].

In this Letter, we prove that at any temperature, 1D quantum systems with
commuting interactions are rapidly mixing. That is, they thermalize in a time
that scales only logarithmically with the system size. The approach to prove
it is to bound the log-Sobolev constant of the associated Davies generator,
which is the usual model for the action of a thermal bath in the weak coupling
limit.

This result yields interesting consequences in the context of phase
transitions. It is well known that the phases of a given system in thermal
equilibrium can be classified according to its physical properties. Moreover,
changes of the system allow for the transformation between phases, sometimes
abruptly, which results in the appearance of a phase transition. This can also
occur in systems which are away from their thermal equilibrium. In this case,
due to dissipation, the environment drives the system to the aforementioned
equilibrium, which is represented by a state and depends on the system and the
environment parameters. As such parameters change, the properties of the
system might also change suddenly, yielding a so-called dissipative phase
transition [24, 25, 26, 27, 28], sometimes also referred to as noise-driven
quantum phase transitions [29] or simply quantum phase transitions driven by
dissipation [30]. Our result in particular implies that there are no
dissipative phase transitions for Davies evolutions over translation-invariant
spin chains.

The result also has implications in the context of Symmetry Protected
Topological (SPT) phases [31, 32, 33]. There has been a quite intensive study
of SPT in open quantum systems [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]
and there is yet no consensus on what is the fate of SPT in the presence of
temperature (see e.g. [36, 39] for negative results and [38, 45] for positive
ones). The 1D cluster state [46] \- which plays a key role in the paradigm of
measurement based quantum computation [47, 48] \- has a commuting Hamiltonian
and it is a non-trivial SPT phase under a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry [49]. Hence, our result applies to it and gives the first example of
a non-trivial interacting SPT phase with a provable decoherence time growing
only logarithmically with the system size for thermal noise at every non-zero
temperature, where in addition all relevant interactions in the problem can be
asked to preserve the symmetry, at least in a weak sense. The case of a
trivial Hamiltonian, which classically corresponds to the infinite-temperature
case, was already obtained in [41]. The result has the extra benefit of being
stable to extensive perturbations, a general property of quasi-local
dissipative evolutions with logarithmic decoherence time [7].

Our result does not apply however in the presence of a strong symmetry [50,
51], a key condition identified in [43] to preserve SPT in open quantum
systems, which emphasizes even more the totally different behavior between
weak and strong symmetries in the context of SPT phases in non-zero
temperature regimes.

The proof of our main result has two main steps. One step works in arbitrary
dimension and gives a way to upgrade a bound on the spectral gap of the Davies
generator to a bound of the log-Sobolev constant for commuting Hamiltonians.
The proof requires among other things the theory of operator spaces, that has
been already proven very useful in answering different questions within
quantum information theory [52]. The other step is to show that 1D systems
fulfill the hypothesis for such upgrade to hold.

We expect the first step to be of independent interest, since it opens the
possibility to upgrade to the log-Sobolev regime the recent result [16]
showing that the Davies generator of quantum double models in 2D have a
bounded gap.

##  II Mixing times for Davies maps

We now briefly recall the construction due to Davies [53], which under the
assumption of a weak-coupling limit with a thermal bath at inverse temperature
$\beta$, gives a description of the evolution of the system as a Markovian
master equation. The joint Hamiltonian of the system and the environment can
be decomposed as $H=H_{S}\otimes\mathds{1}_{E}+\mathds{1}_{S}\otimes
H_{E}+\lambda H_{I}$, where $H_{S}$ is the Hamiltonian of the system, $H_{E}$
the one of the bath, and $H_{I}$ is the coupling term between the two of them,
with the coupling constant $\lambda\geq 0$. We can decompose $H_{I}$ as
$H_{I}=\sum_{\alpha}S^{\alpha}\otimes B^{\alpha}$, where $S^{\alpha}$,
$B^{\alpha}$ are Hermitian. Renormalizing by the free evolution and sending
$\lambda\to 0$ while keeping $\tau=\lambda^{2}t$ constant, the reduced
evolution of the system is given by
$\rho(\tau)=\exp(\tau\mathcal{L})(\rho(0))$ [53]. Here, $\mathcal{L}$ is a
Lindbladian whose Lindblad operators, which we denote by $S^{\alpha}(\omega)$,
satisfy
$e^{itH_{S}}S^{\alpha}e^{-itH_{S}}=\sum_{\omega}S^{\alpha}(\omega)e^{-i\omega
t}$, where the sum is over the Bohr frequencies $\omega$ of the system
Hamiltonian $H_{S}$ (for more details, we refer the reader to our companion
paper [54]).

Under the assumption that there are no operators commuting with every
$S^{\alpha}$ except the multiples of identity, one can show [55] that the
Gibbs state of $H_{S}$ at inverse temperature $\beta$, namely
$\sigma_{\beta}={Z_{\beta}}^{-1}\exp(-\beta H_{S})$ is the unique fixed point
of the evolution generated by $\mathcal{L}$, and moreover:

$\forall\rho,~{}~{}\exp(t\mathcal{L})(\rho)\overset{t\rightarrow\infty}{\underset{\,}{\longrightarrow}}\sigma_{\beta}.$
(1)

An important problem concerns the speed at which the convergence (1) occurs.
This is quantified by the mixing time of the dynamics: for $\epsilon>0$,

$\displaystyle t_{\operatorname{mix}}(\epsilon):=\inf\big{\\{}t\geq
0|\,\|e^{t\mathcal{L}}(\rho)-\sigma_{\beta}\|_{1}\leq\epsilon\big{\\}}\,,$ (2)

where $\|X\|_{1}:={\operatorname{tr}}|X|$ denotes the trace norm. One way of
controlling the mixing time is via the analysis of the spectral gap of
$\mathcal{L}$. It is well-known that, whenever the gap can be lower bounded by
a constant independent of system size $|\Lambda|$ [18],
$t_{\operatorname{mix}}(\epsilon)=\mathcal{O}(\sqrt{n})$. This is the case for
Davies generators over spin chains at any positive temperature [13]. Moreover,
Glauber dynamics, which can be interpreted as the classical analogues of
Davies generators, are known to thermalize logarithmically faster in 1D [10,
12] with
$t_{\operatorname{mix}}(\epsilon)=\mathcal{O}(\operatorname{polylog}(n))$.
This property of a local (quantum) Markovian evolution is known as rapid
mixing.

One way to prove rapid mixing is to consider the exponential decay of the
relative entropy between the evolved state at time $t$ and the invariant state
$\sigma_{\beta}$:

$\displaystyle D(e^{t\mathcal{L}}(\rho)\|\sigma_{\beta})\leq e^{-4\alpha
t}\,D(\rho\|\sigma_{\beta})\,.$ (3)

The constant $\alpha$ appearing in (3) is known as the modified logarithmic
Sobolev constant (MLSI constant) of the semigroup. By Pinsker’s inequality
together with the bound $D(\rho\|\sigma_{\beta})=\mathcal{O}(\log(n))$, one
can easily show that $\alpha=\Omega(\operatorname{polylog}(n)^{-1})$ implies
the rapid mixing property. This is precisely what we achieve in this article.

##  III Main result

We now state the main result of our paper, namely an exponential decay for the
entropy in the form of Equation (3). We consider a finite chain
$\Lambda=\llbracket 1,n\rrbracket$ and the Davies generator
$\mathcal{L}_{\Lambda}$ of a quantum Markov semigroup with unique invariant
state $\sigma\equiv\sigma_{\Lambda}^{\beta}:=\frac{\operatorname{e}^{-\beta
H_{\Lambda}}}{{\operatorname{tr}}[\operatorname{e}^{-\beta H_{\Lambda}}]}$,
the Gibbs state of a finite-range, translation-invariant, commuting
Hamiltonian at inverse temperature $\beta<\infty$.

######  Theorem 1.

In the setting introduced above, there exists
$\alpha_{\Lambda}=\Omega((\ln|\Lambda|)^{-1})$ such that, for all
$\rho\in\mathcal{D}(\mathcal{H}_{\Lambda})$ and all $t\geq 0$,

$\displaystyle D(e^{t\mathcal{L}_{\Lambda}}(\rho)\|\sigma)\leq
e^{-\alpha_{\Lambda}t}\,D(\rho\|\sigma)\,,$ (4)

a)$D(\rho\|\sigma)$b)$D_{A_{1}}(\rho\|\sigma)$+$D_{A_{2}}(\rho\|\sigma)$$D_{B_{1}}(\rho\|\sigma)$$D_{B_{2}}(\rho\|\sigma)$$\Lambda$$\times$c)$\sigma_{B^{c}}$$\sigma_{A^{c}}$$\sigma_{B^{c}}$$\sigma_{A^{c}}$d)$\underset{x\in\Lambda}{\sum}\,D\left(\rho\|E_{x}(\rho)\right)$
Figure 1:  Quasi-factorizations of the relative entropy used in the proof of
Theorem 1: First, we consider in a) the relative entropy between the evolved
state and the equilibrium in $\Lambda$ and reduce it to some conditional
relative entropies in smaller regions $\\{{A_{i},B_{i}\\}}$ as in b), in the
spirit of the results of [22, 20, 21], and a multiplicative error term in c),
depending on how correlations decay on $\sigma$ between $(\cup_{i}A_{i})^{c}$
and $(\cup_{i}B_{i})^{c}$, which can be interpreted as a mixing condition and
is controlled using Araki’s estimates [56] as in the recent [57]. In d), we
use operator spaces to lift the results of [58] to non-tracial conditional
expectations to further reduce the latter conditional relative entropies to
another form of on-site conditional relative entropies.

The proof of our result is schetched in Figure 1. More details are provided in
Appendix A. For a complete proof, we refer the reader to [54].

##  IV SPT phases

In this section, we discuss how our result applies to spin chains that are
preserved by certain symmetry of the system. We shall see that fast
thermalization still holds even in the case of interactions with the thermal
bath that preserve a symmetry of the system, thus negating the possibility of
preserving information encoded in a thermal state. We begin with a brief
recollection of properties of 1D SPT states before presenting our result.

###  IV.1 Definition

Topological order is known to be robust against perturbations [59], which
endows those systems with a very desirable protection (e.g. for its use in
quantum information processing tasks). However, topological order is usually
hard to engineer. In particular, it does not exist in 1D systems.

There is a somewhat related notion, where protection is guaranteed only if the
perturbations keep a symmetry present in the system. Systems with that
property are named Symmetry Protected Topologically (SPT) ordered [32, 31,
33]. SPT is a much less restricted property and there are several natural 1D
systems which belong to a non-trivial SPT phase, among which the
antiferromagnetic spin-$1$ Heisenberg model [60, 61].

In a series of papers [33, 62, 63, 64], Matrix Product State (MPS) theory was
used to classify all possible SPT phases in 1D via the second cohomology group
$H^{2}(G,U(1))$, where $G$ is the symmetry group of the system. The idea is
quite simple to explain. A MPS on a 1D system with local Hilbert space
dimension $d$ is given by a sequence of $d$ matrices $A^{i}$, $i=1,\ldots,d$
of size $D\times D$, where $D$ is called the virtual or bond dimension. For a
system of size $L$, the MPS defined by the $A^{i}$ is

$|\psi_{A}\rangle=\sum_{i_{1},\ldots,i_{L}}{\rm
tr}\left(A^{i_{1}}A^{i_{2}}\cdots A^{i_{L}}\right)|i_{1}\cdots
i_{L}\rangle\;.$

That is, amplitudes are given by the trace of the product of matrices. A MPS
is said to be _injective_ if there exists a finite $K$ such that
$\\{A^{i_{1}}\cdots A^{i_{K}}\mid i_{1},\dots,i_{K}=1\dots d\\}$ spans the
whole algebra of $D\times D$ matrices.

If a MPS is invariant under an on-site symmetry, i.e. $u_{g}^{\otimes
L}|\psi_{A}\rangle=|\psi_{A}\rangle$ for all $L$ and all $g\in G$ – $u_{g}$ is
a given linear representation of $G$ – then thanks to the Fundamental Theorem
of MPS [65, 66, 67], there exists a projective representation $V_{g}$ of $G$
so that $\sum_{j}(u_{g})_{ij}A^{j}=V_{g}A^{i}V_{g}^{\dagger}$. That is, the
action of the representation $u_{g}$ on the physical index of the $A^{i}$
(which is precisely the index $i$) corresponds to the adjoint action of the
representation $V_{g}$ on the virtual indices (which are the row and column
entries of the matrices $A^{i}$).

Non-trivial SPT phases correspond to the cases in which $V_{g}$ is not
equivalent to a linear representation. Since the second cohomology group
$H^{2}(G,U(1))$ enumerates all non-equivalent projective representations, it
classifies non-trivial SPT phases associated to a symmetry group $G$.

It is by now well-known that MPS efficiently approximate ground states of
locally interacting gapped Hamiltonians, what makes this classification, in
principle restricted to MPS, valid in full generality, as first observed in
numerical examples [61] and then proven in full rigor by Ogata in [68].

###  IV.2 Example: the cluster state

When the Hamiltonian has commuting terms, its ground state is an exact MPS and
the argument given above works with full rigor. This is the case of the 1D
cluster model, a Hamiltonian with 3-body terms between each particle and its
nearest neighbors given by $Z\otimes X\otimes Z$, and where we are assuming
periodic boundary conditions.

This Hamiltonian has a unique ground state which is precisely the 1D cluster
state, the MPS given by matrices $A^{0}=|0)(+|$, $A^{1}=|1)(-|$ [69] (we use
curved-brackets to indicate that the associated spaces correspond to the
virtual indices.)

If we block two sites, the corresponding matrices are
$A^{00}=\frac{1}{\sqrt{2}}|0)(+|$, $A^{01}=\frac{1}{\sqrt{2}}|0)(+|$,
$A^{10}=\frac{1}{\sqrt{2}}|1)(+|$, and $A^{11}=-\frac{1}{\sqrt{2}}|1)(-|$.

It is straightforward that the 1D cluster Hamiltonian (on an even-size chain)
commutes with the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry generated by
$Z\otimes\mathbbm{1}$ and $\mathbbm{1}\otimes Z$. The MPS, being its unique
ground state, must keep the same symmetry. Moreover, the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry transfers to the virtual system:
acting with $X\otimes\mathbbm{1}$ in the physical level corresponds to
conjugation by $X$ in the virtual level, whereas acting with
$\mathbbm{1}\otimes X$ in the physical level corresponds to conjugation by $Z$
in the virtual level.

Hence, the linear representation of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ in
the physical level corresponds to the projective representation given by the
Pauli group at the virtual level. The fact that such projective representation
is not equivalent to a linear one (equivalently, it corresponds to the only
non-trivial element of the second cohomology group
$H^{2}(\mathbb{Z}_{2}\times\mathbb{Z}_{2},U(1))$), is precisely what gives the
1D cluster state its non-trivial SPT character.

###  IV.3 Davies generator for SPT phases

In the case of an injective MPS invariant with respect to an on-site symmetry
$u_{g}$ for some group $G$, one can also construct a local two-body
frustration-free Hamiltonian $H_{S}$ which commutes with $u_{g}$, known as the
_parent Hamiltonian_

$[H_{S},u_{g}]=0\quad\forall g\in G.$

In this situation, we ask for the Davies thermalization process to also
respect the symmetry $G$. We will do this by requiring that the Davies
generator is _covariant_ with respect to the symmetry $u_{g}$: for every state
$\rho$ and every $g\in G$, it holds that

$\mathcal{L}(u_{g}^{\dagger}\rho
u_{g})=u_{g}^{\dagger}\mathcal{L}(\rho)u_{g}\quad\forall\rho,\forall g\in G.$
(5)

We remark that a sufficient condition for this to happen is for the jump
operators $S^{\alpha}$’s to commute with $u_{g}$ up to a phase:

$S^{\alpha}u_{g}=\omega^{\alpha}_{g}u_{g}S^{\alpha},\quad\omega^{\alpha}_{g}\in
U(1)\,.$ (6)

It is easy to construct many examples of covariant Davies generators. In fact,
this is always possible when the parent Hamiltonian of the MPS is a sum of
Pauli terms (tensor products of Pauli matrices), by choosing $S^{\alpha}$ to
be the one-site Pauli operators. This covers the case of the 1D cluster state.

We also remark that any such covariant generator $\mathcal{L}$ can be obtained
as the weak-coupling limit of the interaction with a thermal bath that is
_weakly symmetric_ , in the sense that there exists a representation $U_{g}$
of $G$ acting on the Hilbert space of the environment such that, for each
$g\in G$ and all $\alpha$, $[H_{E},U_{g}]=0$ and $[S^{\alpha}\otimes
B^{\alpha},u_{g}\otimes U_{g}]=0$. In fact, if this is not the case, one can
extend the original environment by a conjugate copy of the system:

$\tilde{B}^{\alpha}=B^{\alpha}\otimes\overline{S}^{\alpha},\quad
U_{g}=\mathds{1}\otimes\overline{u}_{g},$

obtaining a weakly-symmetric thermal bath.

###  IV.4 Thermalization of 1D SPT phases

Our main result applied to the 1D SPT phases state implies the following.

######  Corollary 1.

For every inverse temperature $\beta<\infty$, 1D SPT phases thermalize in time
logarithmic in the system size, even when the thermal bath is chosen to be
weakly symmetric.

##  V Discussion

In this work, we have shown that the Davies dynamics associated to any 1D spin
chain translation-invariant commuting Hamiltonian at finite temperature
satisfies a log-Sobolev inequality, and therefore the corresponding
thermalization process converges logarithmically fast in terms of the system
size (the _rapid mixing_ property). This also holds under the assumption that
the evolution is weakly symmetric with respect to a given symmetry, for
example in the case of SPT phases.

We expect our two-step proof strategy to be relevant in higher dimensions. We
leave the study of log-Sobolev constants for Davies generators of 2D double
models, whose gap was recently investigated in [16], to future work.

Finally, one could ask whether our result for SPT phases would apply to the
setting where the thermal bath is chosen to be _strongly symmetric_ , in the
sense that the representation $U_{g}$ acting on the Hilbert space of the
environment is the trivial one. This is not the case, given that this
condition prevents the thermal evolution to be ergodic, and in particular for
it to have a unique invariant state. This can be seen by noticing that strong
symmetry would imply that all $u_{g}$ are invariant, in the sense that
${\operatorname{tr}}\,[u_{g}\,\mathcal{L}(\rho)]=0$ for any $\rho.$ A
sufficient condition for this to happen is that $[S^{\alpha},u_{g}]=0$ for
each $\alpha$ and $g\in G$. In the presence of a full rank invariant state,
this condition is also necessary [70]. When $u_{g}$ is not irreducible (which
is the case for local on-site symmetries), this implies that $\mathcal{L}$ has
multiple invariant states and therefore it is not ergodic. This issue was
solved in [39] by restricting the initial state only to the subspace of
$u_{g}$-symmetric state, and studying the thermalization of the _symmetric
Gibbs ensemble_ (a non-full rank state). We leave open the question of whether
our techniques could be adapted to cover this case.

## Acknowledgements

We are grateful for discussions with C. de Groot and A. Turzillo. IB is
supported by French A.N.R. grant: ANR-20-CE47-0014-01 “ESQuisses”. AC was
partially supported by an MCQST Distinguished Postdoc and the Seed Funding
Program of the MCQST (EXC-2111/Projekt-ID: 390814868). AL acknowledges support
from the BBVA Fundation and the Spanish “Ramón y Cajal” Programme
(RYC2019-026475-I / AEI / 10.13039/501100011033). This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No 648913).
DPG acknowledges support from MINECO (grant MTM2017-88385-P) and from
Comunidad de Madrid (grant QUITEMAD-CM, ref. P2018/TCS-4342). CR acknowledges
financial support from a Junior Researcher START Fellowship from the MCQST
(EXC-2111/Projekt-ID: 390814868).

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##  Appendix A Techniques and proof ideas

In this appendix, we expose the main idea behind the proof of our main result,
namely the exponential decay of the relative entropy in Equation (4). For
that, let us first rewrite such an equation in the following equivalent form:

$2\alpha
D(\rho\|\sigma)\leq-{\operatorname{tr}}[\mathcal{L}_{\Lambda}(\rho)\left(\ln\rho-\ln\sigma\right)]$
(7)

Moreover, let us recall that we say that the Markovian evolution
$(\operatorname{e}^{t\mathcal{L}_{\Lambda}})_{t\geq 0}$ satisfies a complete
modified logarithmic Sobolev inequality (CMLSI in short) if the
$(\operatorname{e}^{t\mathcal{L}_{\Lambda}}\otimes\operatorname{id}_{R})_{t\geq
0}$ has a positive MLSI constant $\alpha$ for any reference system
$\mathcal{H}_{R}$.

The proof of Theorem 1 relies on a reduction argument from the MLSI constant
of Equation (7) in $\Lambda$ to the complete MLSI constants in smaller sub-
chains $A,B\subset\Lambda$. With this reduction we intend to overcome the
issue that a MLSI constant in $\Lambda$ could in principle scale with the
system size. By relating it to complete MLSI constants in segments of fixed
size, we expect that the original MLSI constant only depends on the size of
such segments and, thus, is controllable.

This idea was originally introduced in [71, 72] in the context of classical
spin systems, with the aim of simplifying the traditional strategy for proving
positivity of MLSI constants. Lately, a quantum analogue of such a procedure
has been developed in [22, 20, 73, 74, 21, 58]. The key element of this
strategy are the so-called results of quasi-factorization (or approximate
tensorization) of the relative entropy. The proof of Theorem 1 is also based
on a result of this form.

The intuition behind this argument is the following: If both sides of
inequality (7) were linear with respect to the lattice where they are
considered, it would be a simple exercise to reduce an MLSI constant in
$\Lambda=A\cup B$ to the two MLSI constants in $A$ and $B$. The RHS of
Equation (7) actually satisfy such linearity. The generator that appears there
is indeed local, in the sense that
$\mathcal{L}_{\Lambda}={\sum}_{x\in\Lambda}\mathcal{L}_{x}$. Therefore, one
can study the entropy production of $\rho$ in $A\subset\Lambda$ by restricting
to the terms of the generator in $A$, i.e.

$\operatorname{EP}_{A}(\rho):=-{\operatorname{tr}}\left[\mathcal{L}_{A}(\rho)\left(\ln\rho-\ln\sigma\right)\right]\,,$
(8)

for $\mathcal{L}_{A}(\rho)={\sum}_{x\in A}\mathcal{L}_{x}(\rho)$. This yields
the direct consequence that for two non-overlapping regions
$A,B\subseteq\Lambda$,

$\displaystyle\operatorname{EP}_{A\cup
B}(\rho)=\operatorname{EP}_{A}(\rho)+\operatorname{EP}_{B}(\rho)\,.$ (9)

However, the LHS of Equation (7) is more subtle, as the relative entropy is
clearly not linear with respect to the lattice where the states have support.
Moreover, our aim at this stage is to reduce the area of action of the
relative entropy, rather than that of the density matrices involved. This is
the reason for introducing a couple of conditional relative entropies in
$A\subset\Lambda$ [22, 74]: for $\rho,\sigma\in\mathcal{D}(\mathcal{H})$:

$\displaystyle D_{A}(\rho\|\sigma)$
$\displaystyle:=D(\rho\|\sigma)-D(\rho_{A^{c}}\|\sigma_{A^{c}})\,,$ (10)
$\displaystyle\mathbb{D}_{A}(\rho\|\sigma)$
$\displaystyle:=D(\rho\|E_{A}(\rho))\,,$ (11)

where $E_{A}$ is the conditional expectation such that
$e^{t\mathcal{L}_{A}}\to E_{A}$ as $t\to\infty$. These conditional relative
entropies satisfy the following inequality [74]:

$D_{A}(\rho\|\sigma)\leq\mathbb{D}_{A}(\rho\|\sigma)$ (12)

With these notions at hand, we provide an elaborate result of quasi-
factorization of the relative entropy in terms of some conditional relative
entropies in overlapping subsystems and a multiplicative error term which we
can control. This is done in the two steps detailed below.

###  A.1 Global-to-local reduction

We first recall the following result of quasi-factorization of the relative
entropy [22]: Consider a splitting of $\Lambda$ into two subchains $A$ and $B$
as in the following figure.

$A$$B$$\Lambda$ Figure 2: Possible splitting of a lattice $\Lambda$ into two
subregions $A,B$ with non-overlapping complements and such that $\Lambda=A\cup
B$.

Given $\sigma\in\mathcal{D}(\mathcal{H}_{\Lambda})$, let us denote

$\displaystyle h(\sigma_{A^{c}B^{c}})$ (13)
$\displaystyle:=\sigma_{A^{c}}^{-1/2}\otimes\sigma_{B^{c}}^{-1/2}\sigma_{A^{c}B^{c}}\sigma_{A^{c}}^{-1/2}\otimes\sigma_{B^{c}}^{-1/2}-\mathds{1}_{A^{c}B^{c}}\,.$

Then, for all states $\rho,\sigma\in\mathcal{D}(\mathcal{H}_{\Lambda})$ such
that $\left\|h(\sigma_{A^{c}B^{c}})\right\|_{\infty}<1/2$, the following
holds:

$\displaystyle D(\rho\|\sigma)$ (14)
$\displaystyle\leq\frac{1}{1-2\,\|h(\sigma_{A^{c}B^{c}})\|_{\infty}}\,\big{[}D_{A}(\rho\|\sigma)+D_{B}(\rho\|\sigma)\big{]}\,.$

This inequality between the relative entropy of two states in $\Lambda$ and
the respective conditional relative entropies in $A$ and $B$ would allow us to
reduce the MLSI constant in $\Lambda$ to the complete MLSI constants in $A$
and $B$. However, since $A$ and $B$ grow with $\Lambda$, this would still
yield a bad scaling of the MLSI constant with the system size. To overcome
this issue, we use a more involved geometry, by splitting our chain $\Lambda$
into small segments $\left\\{A_{i}\right\\}$ and $\left\\{B_{i}\right\\}$ so
that they present a pairwise overlap and all of them are of the same size, and
consider $A:=\bigcup_{i}A_{i}$, $B:=\bigcup_{i}B_{i}$. This construction is as
in the following picture.

$A_{1}$$B_{1}$$A_{2}$$B_{2}$$\Lambda$ Figure 3: Splitting of $\Lambda$ into
two regions $A$ and $B$, which are the union of small intervals
$\cup_{i=1}^{m}A_{i}$ and $\cup_{j=1}^{m}B_{j}$, such that $A_{i}\cap
B_{i}\neq\emptyset\neq B_{i}\cap A_{i+1}$ for all $i=1,\ldots m-1$ and
subregions $A_{i}\cap A_{j}=\emptyset=B_{i}\cap B_{j}$ for all $i\neq j$.

With this geometry at hand, we can further upper bound the conditional
relative entropies in the RHS of Equation (14) by the conditional relative
entropies on each $A_{i}$ and $B_{i}$ as a consequence of $\sigma$ being the
Gibbs state of a local, commuting Hamiltonian, and, in particular, a quantum
Markov chain [20]:

$\displaystyle D(\rho\|\sigma)\leq$ (15)
$\displaystyle\frac{1}{1-2\,\|h(\sigma_{A^{c}B^{c}})\|_{\infty}}\underset{i=1}{\overset{m}{\sum}}\big{[}D_{A_{i}}(\rho\|\sigma)+D_{B_{i}}(\rho\|\sigma)\big{]}\,.$

Next, we need to control the multiplicative error term in the RHS of Equation
(15). Intuitively, the quantity $\|h(\sigma_{A^{c}B^{c}})\|_{\infty}$ should
decrease with the size of the overlap between $A$ and $B$, as their
complements get more separated. This can be proven to hold as a consequence of
the recent [57, Proposition 8.1], based on the so-called Araki’s expansionals
[56]. Indeed, for the geometry $\Lambda=A\cup B$ introduced above, we prove
that $\|h(\sigma_{A^{c}B^{c}})\|_{\infty}$ is uniformly bounded by a constant
away from $1/2$ as long as we take the number of segments
$\left\\{A_{i},B_{i}\right\\}$ to be $2m$, with
$m=\Omega(\left|\Lambda\right|/\ln(\left|\Lambda\right|))$, and the size of
each segment
$\left|A_{i}\right|=\left|B_{i}\right|=\mathcal{O}(\ln(\left|\Lambda\right|))$.

To conclude this step, we combine the previous inequalities with Equation (12)
to obtain:

$\boxed{D(\rho\,\|\,\sigma)\leq\mathcal{C}\sum_{i=1}^{2m}D\big{(}\rho\,\big{\|}E_{X_{i}}(\rho)\big{)}}$
(16)

for some universal constant $\mathcal{C}\geq 1$ and a covering
$\\{X_{i}\\}_{i=1}^{2m}$ of the chain $\Lambda$ by intervals $X_{i}$ of size
$|X_{i}|=O(\ln(\left|\Lambda\right|))$.

###  A.2 Quasi-local control of the constant

In the next step, we reduce the conditional expectations on the regions
$X_{i}$ in the relative entropies in the RHS of Equation (16) to single-site
conditional expectations on each of the sites composing each $X_{i}$. For
that, we extend a previous result of quasi-factorization (also called
approximate tensorization) for tracial conditional expectations from the
recent [58] to the non-tracial setting .

More specifically, for each $X_{i}$ we prove the following inequality

$\boxed{D\big{(}\rho\|{E}_{X_{i}}(\rho)\big{)}\leq\mathcal{K}_{X_{i}}\sum_{j\in
X_{i}}\,D\big{(}\rho\|{E}_{j}(\rho)\big{)}}$ (17)

where $\mathcal{K}_{X_{i}}$ scales logarithmically with $\left|\Lambda\right|$
whenever $|X_{i}|=\mathcal{O}(\ln\left|\Lambda\right|)$. Inspired from
LaRacuente’s work [75], we observe such constant $\mathcal{K}_{X_{i}}$ can be
determined by the norm of the distance between the product of single-site
conditional expectations ${E}_{j}$ and the conditional expectation
${E}_{X_{i}}$ for the region $X_{i}$. Indeed, using operator space theory we
show that $\mathcal{K}_{X_{i}}$ is essentially the smallest integer $k$ such
that $(\prod_{j\in X_{i}}E_{j})^{k}$ is close enough to ${E}_{X_{i}}$ as maps
between the amalgamated $L_{p}$ space introduced in [76]. In finite
dimensions, the norm of $(\prod_{j\in X_{i}}E_{j})^{k}-{E}_{X_{i}}$ goes to
$0$ as long as we can control the norm of $\prod_{j\in
X_{i}}E_{j}-{E}_{X_{i}}$ on the $L_{2}$ spaces by a universal constant
$\lambda<1$. This is obtained as a consequence of the non-closure of the
spectral gap for 1D commuting Gibbs samplers proven in [13] and the
detectability lemma [77, 78]. While the latter was already used in [13] to
prove that a certain condition of exponential decay of correlations implies
the non-closure of the gap, our main contribution here consists in leveraging
this proof to get the approximate tensorization (17) with constant
$\mathcal{K}_{X_{i}}\sim\Omega(\frac{|X_{i}|}{\ln\lambda})$. For this, we
extend some of the results of [58] to the Gibbs setting by developing the
operator space structure of non-tracial amalgamated $L_{p}$ spaces.

The amalgamated $L_{p}$ norm has been previously used [79, 80] in the study of
MLSI constants of quantum Markov semigroups. They are generalizations of
Pisier’s vector noncommutative $L_{p}$ spaces [81], which are closely related
to sandwiched Rényi condition entropy. For more information on this, we refer
the interested reader to our companion paper [54]. To the authors’ best
knowledge, this is the first time such techniques were used in the analysis of
dissipative many-body quantum systems, and we strongly believe that they will
find more applications elsewhere.

###  A.3 Merging global and quasi-local analysis

To conclude, we combine Equations (16) and (17) to obtain the following result
of quasi-factorization of the relative entropy:

$D(\rho\,\|\,\sigma)\leq\mathcal{K}\sum_{x\in\Lambda}D\big{(}\rho\,\big{\|}E_{x}(\rho)\big{)}$
(18)

for $\mathcal{K}$ scaling logarithmically with $\left|\Lambda\right|$. Now, we
recall the positivity of the complete MLSI for the local generators
$\mathcal{L}_{x}$ proven in [58, Theorem 3.3]: there exist a constant
$\alpha_{0}>0$ such that for all $x\in\Lambda$ and any
$\rho\in\mathcal{D}(\mathcal{H}_{\Lambda})$,

$\displaystyle\alpha_{0}\,D(\rho\|E_{x}(\rho))\leq\operatorname{EP}_{\\{x\\}}(\rho)\,.$
(19)

Combining Equations (18), (19) and the linearity of the entropy production
(8), we directly get

$\displaystyle
D(\rho\,\|\,\sigma)\leq\mathcal{K}\alpha_{0}^{-1}\operatorname{EP}_{\Lambda}(\rho)\,,$
(20)

and the result follows.