$\sigma_{1}\equiv\sigma_{1}^{(\alpha_{\tilde{l}})}:=\tr_{D_{\text{in}},(\alpha_{\tilde{r}})}\left[\tau_{CD_{\text{in}}}^{(\alpha_{\tilde{l}},\alpha_{\tilde{r}})}\right]$.
Here, we factored out the projections and rearrange the partial traces
suitably. The last equality is then just introducing a simplifying notation.
For simplicity we suppress the boundary conditions $(\alpha_{\tilde{l}})$
index on the states. Hence

$\displaystyle\left\|\left(E_{C}^{S(\alpha)}\circ
E_{D}^{S(\alpha)}-E_{CD}^{S(\alpha)}\right)(X)\right\|$
$\displaystyle\leq\max_{(\beta_{r})}\|\Tr_{C_{\text{in}}\setminus
D_{\text{in}},(\alpha_{\tilde{l}})}\left[(\sigma_{2}-\sigma_{1})N\right]\|$
(171) $\displaystyle=\max_{(\beta_{r})}\left|\Tr_{C_{\text{in}}\setminus
D_{\text{in}},(\alpha_{\tilde{l}})}[(\sigma_{2}-\sigma_{1})N]\right|$ (172)
$\displaystyle=\max_{(\beta_{r})}|\Tr_{C_{\text{in}}\setminus
D_{\text{in}},(\alpha_{\tilde{l}})}[(\mathbbm{1}-\sigma_{1}^{-\frac{1}{2}}\sigma_{2}\sigma_{1}^{-\frac{1}{2}})(\sigma_{1}^{\frac{1}{2}}N\sigma_{1}^{\frac{1}{2}})]|$
(173)
$\displaystyle\leq\max_{(\beta_{r})}\|(\mathbbm{1}-\sigma_{1}^{-\frac{1}{2}}\sigma_{2}\sigma_{1}^{-\frac{1}{2}})(\sigma_{1}^{\frac{1}{2}}N\sigma_{1}^{\frac{1}{2}})]\|_{1}$
(174)
$\displaystyle\leq\max_{(\beta_{r})}\|\sigma_{1}^{-\frac{1}{2}}\sigma_{2}\sigma_{1}^{-\frac{1}{2}}-\mathbbm{1}\|_{\infty}\|N\|_{\mathbb{L}_{1,\sigma_{1}}},$
(175)

where the equality in the second line follows since
$N=E_{D}^{S(\gamma_{l},\alpha_{\tilde{r}})}(X)$ is the identity on the
complementary Hilbert space to $(\mathcal{H}_{C_{\text{in}}\setminus
D_{\text{in}}}^{(\alpha_{\tilde{l}})})$. In the last inequality we used Hölder
and the definition of the $\mathbb{L}_{1,\sigma_{1}}$ norm. By definition of
$N$, we have

$\displaystyle\|N\|_{\mathbb{L}_{1,\sigma_{1}}}$
$\displaystyle=\|\sigma_{1}^{\frac{1}{2}}N\sigma_{1}^{\frac{1}{2}}\|_{1}\overset{N\geq
0}{=}\Tr[\sigma_{1}N]=\Tr[\tau_{CD_{\text{in}}}^{(\alpha_{\tilde{l}},\alpha_{\tilde{r}})}N]=\Tr[\tau_{CD_{\text{in}}}^{(\alpha)}E_{D}^{S(\alpha)}(X)]$
(176)
$\displaystyle=\Tr[E^{S(\alpha)}_{D*}(\tau_{CD_{\text{in}}}^{(\alpha)})X]=\Tr[\tau_{CD_{\text{in}}}^{(\alpha)}X]\overset{X\geq
0}{=}\|X\|_{\mathbb{L}_{1,\tau_{CD_{\text{in}}}^{(\alpha)}}}=1.$ (177)

Hence, to prove the theorem, we need to establish that
$\sigma_{1}\overset{\epsilon}{\sim}\sigma_{2}$ for any boundary condition
$(\alpha_{\tilde{l}},\beta_{r},\alpha_{\tilde{r}})$. We will do it with
$l\mapsto\epsilon(l)$ an exponentially decreasing function in
$l=\operatorname{dist}(C\setminus D,D\setminus C)$ with decay length $\xi$.
The two states can be written as

$\displaystyle\sigma_{1}$
$\displaystyle:=\tr_{D_{\text{in}},(\alpha_{\tilde{r}})}\left[\tau_{CD_{\text{in}}}^{(\alpha_{\tilde{l}},\alpha_{\tilde{r}})}\right]=\frac{1}{\Tr[...]}\tr_{\partial_{\text{out}}(CD)\cup
D_{\text{in}},(\alpha_{\tilde{r}})}\left[P^{(\alpha_{\tilde{l}})}P^{(\alpha_{\tilde{r}})}\sigma^{(CD\partial)}P^{(\alpha_{\tilde{l}})}P^{(\alpha_{\tilde{r}})}\right]$
(178)
$\displaystyle=\frac{1}{\Tr[...]}\tr_{\tilde{l}_{\text{out}}l_{\text{in}}D\tilde{r}}\left[P^{(\alpha_{\tilde{r}})}P^{(\alpha_{\tilde{l}})}\sigma^{(CD\partial)}P^{(\alpha_{\tilde{l}})}\right]\in\mathcal{B}(\mathcal{H}_{C_{\text{in}}\setminus
D_{\text{in}}}^{(\alpha_{\tilde{l}})}),$ (179) $\displaystyle\sigma_{2}$
$\displaystyle:=\tr_{F_{\text{in}},(\beta_{r})}\left[\tau_{C_{\text{in}}}^{(\alpha_{\tilde{l}},\beta_{r})}\right]=\frac{1}{\Tr[...]}\tr_{\partial_{\text{out}}(C)\cup
F_{\text{in}},(\beta_{r})}\left[P^{(\alpha_{\tilde{l}})}P^{(\beta_{r})}\sigma^{(C\partial)}P^{(\alpha_{\tilde{l}})}P^{(\beta_{r})}\right]$
(180)
$\displaystyle=\frac{1}{\Tr[...]}\tr_{\tilde{l}_{\text{out}}l_{\text{in}}Fr}\left[P^{(\beta_{r})}P^{(\alpha_{\tilde{l}})}\sigma^{(C\partial)}P^{(\alpha_{\tilde{l}})}\right]\in\mathcal{B}(\mathcal{H}_{C_{\text{in}}\setminus
D_{\text{in}}}^{(\alpha_{\tilde{l}})}).$ (181)

Both are full-rank states on $\mathcal{H}_{C_{\text{in}}\setminus
D_{\text{in}}}^{(\alpha_{\tilde{l}})}$ and thus have the same support. The
intuition now is as follows: since we have a Gibbs state that satisfies
exponential decay of covariance, strong local indistinguishability, and the
mixing condition (see Theorem 4.14), these two states should be approximately
the same in the bulk, where we compare them. This is because they might only
differ significantly on $D\setminus C$, but we look at them on $C\setminus D$.

From the assumptions of the Theorem we have that the family of Gibbs states
satisfy decay of covariance and hence by Theorem 4.14 also strong local
indistinguishability and the mixing condition. We start by applying the mixing
condition from (97) to each of the states, which guarantees the existence of
two exponentially decaying functions $\epsilon_{1},\epsilon_{2}$ in
$\operatorname{dist}(l,\tilde{r})>\operatorname{dist}(C\setminus D,D\setminus
C)$ and $\operatorname{dist}(l,r)=\operatorname{dist}(C\setminus D,D\setminus
C)$, respectively, s.t.

$\displaystyle\tr_{l_{\text{in}}D}\sigma^{CD\partial}\overset{\epsilon_{1}}{\sim}\tr_{l_{\text{in}}D\tilde{r}}\sigma^{CD\partial}\otimes\tr_{\tilde{l}ElD}\sigma^{CD\partial}$
(182)
$\displaystyle\overset{\autoref{prop:UsefulRelation}}{\implies}\tr_{\tilde{r}}P^{(\alpha_{\tilde{r}})}\tr_{l_{\text{in}}D}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}\overset{\epsilon_{1}}{\sim}\tr_{l_{\text{in}}D\tilde{r}}\sigma^{CD\partial}\Tr[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}],$
(183)
$\displaystyle\tr_{l_{\text{in}}F}\sigma^{C\partial}\overset{\epsilon_{2}}{\sim}\tr_{l_{\text{in}}Fr}\sigma^{C\partial}\otimes\tr_{\tilde{l}ElF}\sigma^{C\partial}$
(184)
$\displaystyle\overset{\autoref{prop:UsefulRelation}}{\implies}\tr_{r}P^{(\beta_{r})}\tr_{l_{\text{in}}F}\sigma^{C\partial}P^{(\beta_{r})}\overset{\epsilon_{2}}{\sim}\tr_{l_{\text{in}}Fr}\sigma^{C\partial}\Tr[P^{(\beta_{r})}\sigma^{C\partial}P^{(\beta_{r})}].$
(185)

Here the implication follows from Proposition 4.11 4) applied to the
projections $P^{(\alpha_{\tilde{r}})},P^{(\beta_{r})}$ respectively. Now, by
strong local indistinguishability (Theorem 4.14) there exists an exponentially
decaying function $\epsilon_{3}$ in
$\operatorname{dist}(l,r)=\operatorname{dist}(C\setminus D,D\setminus C)$,
s.t.

$\displaystyle\tr_{l_{\text{in}}D\tilde{r}}\sigma^{CD\partial}=\tr_{l_{\text{in}}FrG\tilde{r}}\sigma^{CD\partial}\overset{\epsilon_{3}}{\sim}\tr_{l_{\text{in}}Fr}\sigma^{C\partial}$
(186)

and from Theorem 4.14 one gets that uniform decay of covariance with decay
function, say
$\epsilon(\operatorname{dist})=Ke^{-\frac{\operatorname{dist}}{\xi}}$, implies
that
$\epsilon_{1}\propto\exp{\mathcal{O}(|\tilde{r}|+|\partial(El)|)}\epsilon(\operatorname{dist})$,
$\epsilon_{2}\propto\exp{\mathcal{O}(|r|+|\partial(El)|)}\epsilon(\operatorname{dist})$,
and
$\epsilon_{3}\propto\exp{\mathcal{O}(|\partial(El)|)}\mathcal{O}(|r|)\epsilon(\operatorname{dist})$.
Hence by transitivity and symmetry of the strong similarity relation, see
Proposition 4.11 1) and 2), it follows that

$\displaystyle\tr_{l_{\text{in}}D\tilde{r}}\left[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}\right]$
$\displaystyle\overset{\epsilon_{1}}{\sim}\tr_{l_{\text{in}}D\tilde{r}}\sigma^{CD\partial}\Tr[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}]\overset{\epsilon_{3}}{\sim}\tr_{l_{\text{in}}Fr}\sigma^{C\partial}\Tr[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}]$
(187)
$\displaystyle\overset{\epsilon_{2}(1-\epsilon_{2})^{-1}}{\sim}\tr_{l_{\text{in}}Fr}\left[P^{(\beta_{r})}\sigma^{C\partial}P^{(\beta_{r})}\right]\frac{\Tr[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}]}{\Tr[P^{(\beta_{r})}\sigma^{C\partial}P^{(\beta_{r})}]},$
(188)

and hence

$\displaystyle\widetilde{\sigma_{1}}:=\frac{\tr_{l_{\text{in}}D\tilde{r}}\left[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}\right]}{\Tr[P^{(\alpha_{\tilde{r}})}\sigma^{CD\partial}P^{(\alpha_{\tilde{r}})}]}\overset{\tilde{\eta}}{\sim}\frac{\tr_{l_{\text{in}}Fr}\left[P^{(\beta_{r})}\sigma^{C\partial}P^{(\beta_{r})}\right]}{\Tr[P^{(\beta_{r})}\sigma^{C\partial}P^{(\beta_{r})}]}=:\widetilde{\sigma_{2}},$
(189)

where $\tilde{\eta}$ is exponentially decreasing in
$\operatorname{dist}(C\setminus D,D\setminus C)$ with decay length $\xi$ and
prefactor scaling as $\exp(\mathcal{O}(|\partial(C\setminus D)|+|\partial
D\setminus C)|))$, since $\epsilon_{1},\epsilon_{2},\epsilon_{3}$ are. By
Proposition 4.11 $4^{\prime})$ it follows that this strong similarity of the
states also holds within each block $(\alpha_{\tilde{l}})$:

$\displaystyle\sigma_{1}=\frac{1}{\Tr[...]}\tr_{\tilde{l}_{\text{in}}}\left[P^{\alpha_{\tilde{l}}}\widetilde{\sigma_{1}}P^{\alpha_{\tilde{l}}}\right]\overset{\eta:=\tilde{\eta}(2+\tilde{\eta})}{\sim}\frac{1}{\Tr[...]}\tr_{\tilde{l}_{\text{in}}}\left[P^{\alpha_{\tilde{l}}}\widetilde{\sigma_{2}}P^{\alpha_{\tilde{l}}}\right]=\sigma_{2}.$
(190)

This establishes the bound for any boundary condition
$(\alpha_{\tilde{l}},\beta_{r},\alpha_{\tilde{r}})$, which concludes the
proof. Note that this also implies that the exponential decay rate of $\eta$
is the same as the one of the decay of covariance we assumed, which is the
correlation length of the Gibbs state $\xi$. ∎

##  5 Main results

In this section we give sufficient conditions for rapid thermalization of
Davies evolutions with unique fixed point the Gibbs state of a nearest-
neighbour, commuting Hamiltonian.

The results essentially show that a suitable
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$ decay (Definition 4.21), together with
some geometric features of the lattice, implies the existence of a constant or
log decreasing cMLSI constant $\alpha(\mathcal{L}^{D}_{\Gamma})$, which
directly implies rapid mixing. The main technical lemma is as follows.

###### Lemma 5.1

Let $\Lambda$ be a 2-colorable graph with finite growth constant, $\Phi$ a
uniformly bounded, nearest-neighbour, commuting potential, and $\beta>0$ some
inverse temperature. If the family of Schmidt generators associated to
$(\Lambda,\Phi,\beta)$ satisfies an exponentially decaying
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$ decay function $\eta_{C,D}(l)$ in
$l=\operatorname{dist}(C\setminus D,D\setminus C)$ whenever picking the
regions $C,D$ convex and such that

  * a

Case 1: $\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l^{2})$,

  * a

Case 2: $\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l)$,

then a family of Davies generators associated to $(\Lambda,\Phi,\beta)$
satisfies the MLSI with

  * a

Case 1: a system size independent MLSI constant
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$,

  * a

Case 2: a linear in system diameter decaying MLSI constant
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega((\operatorname{diam}(\Gamma))^{-1})_{|\Gamma|\to\infty}$.

The proof is deferred to Section 5.3. The intuition behind separating these
two cases is as follows. The bound on the MLSI requires certain choices of
coarse-grainings of the lattice into overlapping regions $C,D$. The choice of
how large those regions are as compared to their overlap is limited both by
the geometry of the lattice, and the decay of $\eta_{C,D}(l)$. Case 1 is the
one where it is possible to choose the overlap to be a vanishing fraction of
the regions, leading to a better scaling, while in Case 2 the overlap is a
finite fraction of the region. The diameter is the relevant quantity here,
because the decay of correlations is a function of the one dimensional
distance between two subregions and not the number of sites in the overlap in
general.

In the next two subsections, Theorem 5.1 is then applied to two types of decay
of $\eta_{C,D}(l)$, and for each of them we consider two different lattice
geometries, corresponding to the two cases of the theorem. Case 1: will apply
to the 1D quantum spin chain under gap and among others the $D$-dimensional
hypercubic lattices at high temperature. On the other hand, Case 2: will apply
to the 2D hypercubic lattice under the gap condition and $b$-ary trees at high
enough temperature. In particular:

  * •

In Section 5.1 we consider the case where the
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$ decay is implied by the existence of a
unformly bounded strictly positive gap
$\lambda(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$ of the
Davies Lindbladians established via the clustering results Theorem 4.14 and
Theorem 4.22, and [43, Corollary 27]. This shows rapid thermalization with a
constant decay rate of commuting 1D quantum systems here. This is visualised
in Figure 4. We also show that the same argument can also be applied to 2D
systems. There, we obtain a better bound on the thermalization time than one
would naïvely get from the gap, but still too slow to be rapidly mixing.

  * •

In Section 5.2, we consider the high temperature setting, where the
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$ is guaranteed to decay fast, see
Theorem 4.23. This setting notably includes $D$-dim hypercubic lattices and
$b$-ary trees, for which we hence show rapid mixing at high enough
temperatures.

Figure 4: Relation between the main different notions of clustering (static
properties on Gibbs states) and their connection to thermalization of Davies
QMS (dynamical properties) in 1D systems. For those systems, we show that
system size invariant gap is equivalent to a system size independent MLSI
constant via the visualised chain of implications. Yellow arrows signify
implications proved in this work.

###  5.1 Rapid Thermalization from Gap

Here we show two direct applications of Lemma 5.1. The first is that for 1D
quantum spin chains with commuting nearest-neighbour Hamiltonians, having a
gap is a sufficient (and necessary) condition for rapid thermalization. We
also show how in 2D we obtain an improvement over previous results on mixing.
Before stating both results, we recall what was previously established in
Section 4.

Let $\Lambda$ be a graph with finite growth constant, $\Phi$ a uniformly
bounded, nearest-neighbour, commuting potential and $\beta$ a suitable inverse
temperature, such that the family of Davies Lindbladians associated to
$(\Lambda,\Phi,\beta)$ has a positive spectral gap, i.e.
$\inf_{\Gamma\subset\subset\Lambda}\lambda(\mathcal{L}^{D}_{\Gamma})>0$ .
Then, for any $\Gamma\subset\subset\mathbb{Z}$ or $\mathbb{Z}^{2}$ and
$C,D\subset\Gamma$ with $C\cap D\neq\emptyset$, by [43, Corollary 27], Theorem
4.14, and Theorem 4.22 we have q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-decay
with decay rate

$\displaystyle\eta_{\text{gap},C,D}(l)=\exp{\mathcal{O}(|\partial(C\setminus
D)|+|\partial(D\setminus
C)|)}\mathcal{O}\left(\exp\left(-\frac{\operatorname{dist}(C\setminus
D,D\setminus C)}{\xi}\right)\right),$ (191)

where $\xi$ is the thermal correlation length of the family of Gibbs states
associated to $(\Lambda,\Phi,\beta)$. Together with this scaling, two
judicious choices of regions $C,D$ (one for each case in Theorem 5.1) and
their corresponding overlap allows us to obtain the following proposition.

###### Proposition 5.2

It holds that $\eta_{\text{gap},C,D}(l)$ is exponentially decaying in $l$ when
picking $C,D$ convex with $\operatorname{dist}(C\setminus D,D\setminus C)=l$
and such that

  * a

Case 1: For $\Lambda=\mathbb{Z}$, we have
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l^{2})$ ,

  * a

Case 2: For $\Lambda=\mathbb{Z}^{2}$, and the thermal correlation length $\xi$
is $\xi<\xi_{\text{max}}(\Phi,\beta)=\mathcal{O}(1)$, we have
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l)$

###### Proof.

For $\Lambda=\mathbb{Z}$, the boundary of convex sets is $2$, since they are
intervals of the chain. Since this is a constant number, the decay holds
independently of the correlation length $\xi$. For $\Lambda=\mathbb{Z}^{2}$,
the boundary of convex sets scales linearly in their diameter. Hence so do
$|\partial(C\setminus D)|,|\partial(D\setminus C)|$ and thus, if the
correlation length $\xi$ is small enough to dominate over the linear growth in
the exponent, we get exponential decay of $l\mapsto\eta_{\text{gap},C,D}(l)$.
∎

Together with Lemma 5.1 this directly yields the main result of the section.

###### Theorem 5.3

The Davies generator
$\mathcal{L}^{D}_{\Lambda}:=\\{\mathcal{L}^{D}_{\Gamma}\\}_{\Gamma\subset\subset\Lambda}$
corresponding to a uniform, nearest-neighbour, commuting family of
Hamiltonians acting on the locally-finite dimensional quantum spin system
$\mathcal{H}_{\Lambda}$ satisfies a MLSI with constant

  1. $1)$

$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$ independent
of system size, when $\Lambda=\operatorname{\mathbb{Z}}$ is the spin-chain, or

  2. $2)$

$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega\left((\sqrt{|\Gamma|})^{-1}\right)_{|\Gamma|\to\infty}$
square-root decreasing in system size, when $\Lambda=\mathbb{Z}^{2}$ and the
correlation length of the fixed point (Gibbs state) is
$\xi<\xi_{\text{max}}(\Phi,\beta)=\mathcal{O}(1)$,

whenever the family of thermal states satisfies uniform exponential decay of
covariance or when the Davies Lindbladians are uniformly gapped, i.e.
$\inf_{\Gamma\subset\subset\Lambda}\lambda(\mathcal{L}^{D}_{\Gamma})>0$, which
implies the former.

The square root decay of the MLSI constant in the 2-dimensional setting in
Theorem 5.3 does not imply rapid mixing, since it only guarantees a polynomial
thermalization time. However, it still improves the bound on the mixing time
over the one we would get only from having a gap, i.e. we get
$t_{\text{mix}}(\epsilon)=\mathcal{O}(\sqrt{|\Gamma|}\log|\Gamma|,\log\frac{1}{\epsilon})$,
as opposed to an arbitrary
$t_{\text{mix}}(\epsilon)=\mathcal{O}(\text{poly}(|\Gamma|),\log\frac{1}{\epsilon})$.

Concerning the 1D result, we first of all note that we do not require $r=2$,
i.e. nearest-neighbour interactions assumption, since by coarse-graining, we
can always map a geometrically-$r$-local 1D system to a nearest neighbour
one.101010This, however, only works in the 1D case. The problem in higher
dimensions is that the coarse grained graph is no longer 2-colorable or
2-colorable under further coarse-graining. This result has important
implications in the 1D setting, where Theorem 5.3 not just gives rapid mixing
of the Davies dynamics but also an optimal scaling of the cMLSI constant from
either gap or exponential decay of covariance.

A priori, assuming gap is the stronger assumption, since in [43], it was
proved that existence of a system-size independent gap implies
$\mathbb{L}_{2}$-clustering, and thus exponential decay of covariance (or
$\mathbb{L}_{\infty}$-clustering). However, for the 1D systems considered
here,111111In fact, this holds more generally for uniformly bounded,
commuting, geometrically-local systems on any graph with a finite growth
constant. it is known that rapid thermalization implies existence of a
striclty positive gap of the generator [72, 43]. This means that in 1D we have
shown equivalence of strictly positive gap of generators, decay of covariance
of the invariant state, MLSI with uniformly strictly positive constant, and
rapid thermalization. Therefore, at least in 1D quantum spin chains, this
answers in the affirmative an open question from [43], namely whether for
commuting systems on $\mathbb{Z}$, existence of exponential decay of
covariance is sufficient to prove existence of a spin-system size invariant
strictly positive spectral gap for $\mathcal{L}^{D}_{\Gamma}$. This is due to
the fact that, as a combination of Theorem 4.14 and Theorem 5.3, exponential
decay of covariance of a thermal state of a uniform geometrically-local,
commuting Hamiltonian implies a strictly positive and system size invariant
MLSI constant, which implies gap. We refer the reader to Figure 4 for a more
clear depiction of these implications. We thus highlight the strength of our
result for the 1D setting in the following corollary.

###### Corollary 5.4 (1D constant MLSI at any temperature and locality)

Let $\Lambda=\mathbb{Z}$ be the one-dimensional quantum spin chain endowed
with a uniformly $J$-bounded, geometrically-$r$-local, commuting potential
$\Phi$ for some $r\in\mathbb{N}$ and let $\beta>0$ be any inverse temperature.
There exists a strictly positive constant $\alpha$ independent of the spin-
chain length that lower bounds the MLSI constant of any element of the Davies
generators associated to $(\mathbb{Z},\Phi,\beta)$, i.e.
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$. Hence for
any finite $\Gamma\subset\subset\Lambda$ we have

$\displaystyle D(\rho_{t}\|\sigma^{\Gamma})\leq
e^{-t\alpha(\mathcal{L}_{\Gamma}^{D})}D(\rho_{0}\|\sigma^{\Gamma}),$ (192)

and thus also rapid thermalization with mixing time
$t_{\text{mix}}(\epsilon)=\mathcal{O}\left(\log|\Gamma|,\log\frac{1}{\epsilon}\right)$.

This follows from the recent result in [46], where it is shown that
exponential decay of covariance holds for such systems with correlation length
$\xi=\exp\mathcal{O}(\beta)$ and any finite temperature $\beta>0$.
Alternatively, for translation-invariant systems, a uniform lower bounded gap
is known by [43]. We do not require $r=2$, i.e. nearest-neighbour
interactions, since by coarse-graining, we can always map a
geometrically-$r$-local 1D system to a nearest neighbour one.121212This,
however, only works in the 1D case. The problem in higher dimensions is that
the coarse grained graph is no longer 2-colorable or 2-colorable under further
coarse-graining.

Corollary 5.4 is a strict improvement over the best previous results [10, 9],
where $\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(\log|\Gamma|^{-1})$ is
logarithmically decreasing in the system size for translation-invariant
Hamiltonians. With our result, we obtain the same system-size independence as
with the so-called LSI constant in the classical setting, which is known to be
optimal.

###  5.2 Rapid Thermalization from High Temperature

In this section we consider rapid thermalization as a consequence of being at
high temperature, as given in Theorem 4.23. Being at high temperature means
that we can assume uniform-q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-clustering
with decay function

$\displaystyle\eta_{C,D;\textup{thermal}}(l)=\mathcal{O}(|C\cup
D|)\exp\left(-\frac{\operatorname{dist}(C\setminus D,D\setminus
C)}{\xi^{\prime}}\right).$ (193)

See [24][Theorem 6,7, Proposition 2] for the proof. The fact that we now have
a linear dependence $\mathcal{O}(|C\cup D|)$ of the pre-factor as opposed to
the exponential one in Eq. (191) means that we can define more favourable
choices of regions $C,D$ in the proofs, thus yielding stronger results. This
is of particular importance for $D>1$, where we prove rapid mixing under the
high temperature assumption.

We now present the following condition on the decay length $\xi^{\prime}$,
which will feature in the main theorem below.

###### Definition 5.5 (Geometric condition on decay length)

For an infinite graph $\Lambda$, recall $N(l):=\sup_{x\in\Lambda}|B_{l}(x)|$,
where $B_{l}(x):=\\{v\in\Lambda|\operatorname{dist}(x,v)\leq l\\}$ is the ball
around $x$ of radius $l$. We require the decay length $\xi^{\prime}$ of the
uniform-q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-clustering to satisfy

$\displaystyle\xi^{\prime}<\frac{l}{2\log N(l)}$ (194)

eventually in $l$, i.e. for $l\geq l_{0}$ for some $l_{0}\in\mathbb{N}$.

The decay length $\xi^{\prime}$ is not just a function of the graph alone, but
also of the potential $\Phi$ and the inverse temperature $\beta$. If condition
(194) holds, then $N(l)\exp\left(-\frac{l}{\xi^{\prime}}\right)$ (and in turn,
$\eta_{C,D}(l)$) is at least exponentially decaying in $l$.

Recall that hypercubic lattices are sub-exponential graphs, whereas $b-$ary
trees are exponential. Hence for infinite hypercubic lattices of dimension $D$
we have that $N(l)\propto l^{D}$ and thus this condition is trivially
fulfilled for any $\xi^{\prime}>0$. For $b$-ary trees we have that
$N(l)=\sum_{k=0}^{l}b^{k}=\frac{b^{l+1}-1}{b-1}\propto b^{l}$, and hence this
condition becomes $\xi^{\prime}<\frac{1}{2\log b}$, which is an implicit
condition on the temperature $\beta^{-1}$. Since it holds trivially for
$\beta=0$, it should also hold for a small enough constant $\beta>0$.

Having established this, we now have the following proposition, similarly to
Proposition 5.2.

###### Proposition 5.6

The function $\eta_{C,D;\textup{thermal}}(l)$ is exponentially decaying in $l$
when choosing $C,D$ convex such that

  * a

Case 1: For $\Lambda=\mathbb{Z}^{D}$ a $D-$dim hypercubic lattice,
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l^{2})$. 131313This
also holds for any subexponential graph, when requiring that we still have
exponential decay when on can choose
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l^{1+\delta})$ for
any $0<\delta\leq 1$. The proof still goes through under this slightly weakend
’Case 2’ condition, hence the main result directly extends from hypercubic to
subexponential graphs.

  * a

Case 2: For $\Lambda$ exponential graph,
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l)$ and the the
decay length $\xi^{\prime}$ is small enough (see e.g. Condition (194)).

###### Proof.

Let $\Lambda$ be a $D$-dim hyper-cubic graph. Then $|C\cup D|$ scales at worst
as $(\operatorname{diam}(C)+\operatorname{diam}(D))^{D}=\mathcal{O}(l^{2D})$,
so $\eta_{C,D;\textup{thermal}}(l)\leq\mathcal{O}(l^{2D})\exp(-l\xi^{-1})$.
This is eventually exponentially decaying.
For exponential graphs there exists a finite growth constant, say $b$, such
that we have that $|C\cup D|\leq
b^{\operatorname{diam}(C)+\operatorname{diam}(D)}=\exp(\mathcal{O}(l))$, and
thus $\eta_{C,D;\textup{thermal}}(l)\leq\exp(\mathcal{O}(l)-l\xi^{\prime-1})$
is exponentially decaying when $\xi^{\prime}$ is small enough. In the proof of
Lemma 5.11 we will only be requiring exponential decay for sets $C,D$ such
that $\operatorname{diam}(C\cup D)=2l=2\operatorname{dist}(C\setminus
D,D\setminus C)$. In this case we have that condition (194) is sufficient to
guarantee the claimed exponential decay, as it holds that $|C\cup D|\leq
b^{\operatorname{diam}(C\cup D)}\leq\exp(2l\log b)$, and thus
$\eta_{C,D;\textup{thermal}}(l)\leq\exp(2l\log b-l\xi^{\prime-1})$. ∎

This allows us to apply Theorem 5.1 directly. In this section, as opposed to
Proposition 5.2 above, Case 1 will involve generic sub-exponential lattices,
and Case 2 will be exponential trees. With those geometries, and given the
choice of regions that Proposition 5.6 allows for, the arguments from Sec. 5.3
below allow us to obtain the following main result in the high temperature
setting.

###### Theorem 5.7 (Davies MLSI and rapid mixing from high temperature)

Let $\Lambda$ be a 2-colorable graph with finite growth constant $\nu$ and
$\Phi$ a uniformly $J-$bounded, nearest-neighbour, commuting potential on it.
If the temperature is high enough, i.e. $\beta^{-1}>10eJ\nu$, the Davies
generators
$\mathcal{L}^{D}_{\Lambda}:=\\{\mathcal{L}^{D}_{\Gamma}\\}_{\Gamma\subset\subset\Lambda}$
associated to $(\Lambda,\Phi,\beta)$ acting on the locally-uniformly-finite
dimensional quantum spin system $\mathcal{H}_{\Lambda}$ satisfy a MLSI with
constant

  1. $1)$

$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$ independent
of system size, when $\Lambda$ is a sub-exponential graph, such as all
$\mathbb{Z}^{D}$, or

  2. $2)$

$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega\left((\log|\Gamma|)^{-1}\right)_{|\Gamma|\to\infty}$
logarithmically decreasing in system size, when $\Lambda$ is an exponential
graph and the correlation length of the fixed point (Gibbs state) satisfies
condition (194). For tree graphs, this condition is $\xi^{\prime}<(2\log
b)^{-1}$.

This means that in both cases the dynamics generated by the Davies generators
are rapidly mixing. Importantly, the result also holds when we have a
polynomial dependence on $|C\cup D|$ instead of the linear prefactor of
$\eta_{C,D;\textup{thermal}}(l)$. The requirement on the high temperature
comes solely through Theorem 4.23, hence if one could prove (193) from some
other starting point, such as having a gap, then one would equally obtain
Theorem 5.7.

The result constitutes a strict extension of the main Theorem of [24], since
we are able to prove rapid thermalization for Davies dynamics, as opposed to
for the more artificial setting of Schmidt dynamics. For the sub-exponential
graph such as hypercubic lattices, the constant bound on the MLSI yields an
optimal scaling of the thermalization time of the semi-group. It should be
possible to extend it to wider ranges of temperatures, as long as the right
notion of decay of correlations holds.

The result for trees is, to the authors knowledge, the first of its kind in
the quantum setting. Classically it is known, however, that the exponential
decay rate of the relative entropy towards the equilibrium is tree-size
independent [53]. We expect this to also hold in the quantum case, since the
proof there hinges upon a very analogous condition on the temperature as the
one here, which is implicit through $\xi^{\prime}<(2\log b)^{-1}$.

###  5.3 Proof of main results

In this section we prove Lemma 5.1. The structure is as follows. We first
reproduce an important result from [24] that tells us that exponential
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$ decay implies an approximate
tensorization statement in Theorem 5.9. This result, the work in [24], and
Lemma 77 are the ingredients we need to establish the main result 5.7 for
quantum systems on hypercubic latices of dimension $D>1$. The main part of the
proof in Section 5.3.2 then consists of a geometric argument where we apply
the aforementioned approximate tensorization to show the main result
separately for Case 1 and Case 2. This is done by proving the result for the
two prototypical instances of both cases: quantum spin chains and $b$-ary
trees, respectively. The derivations then generalise straightforwardly to all
the instances of Cases 1 and 2.

Denote with $\eta(l)\equiv\eta_{C,D}(\operatorname{dist}(C\setminus
D,D\setminus C))$ the decay function of the
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-clustering of the family of Schmidt
generators associated to $(\Lambda,\Phi,\beta)$. In what follows, it is
important that, by assumption, in Case 1 we have that $\eta(l)$ is
exponentially decaying when picking $C,D$ convex s.t.
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l^{2})$ and in Case
2 that $\eta(l)$ is exponentially decaying when picking $C,D$ convex s.t.
$\operatorname{diam}(C),\operatorname{diam}(D)=\mathcal{O}(l)$. The two cases,
as well as their consequences as MLSIs for the various models, are depicted in
Figure 5.

Figure 5: Sketch of the proof of Lemma 5.1. In a first part, we recall the
results from Section 4 that showed the equivalence between clustering
conditions in the case of nearest neighbour interactions. In a second part, at
the right side of the diagram, we derive the various cases of Lemma 5.1 as a
combination of several lemmas in a geometric argument to derive positive MLSI
constants for several classes of Hamiltonians.

####  5.3.1 Approximate tensorization for an almost classical state $\omega$

In this step of the proof, we connect the exponential decay from uniform
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-clustering from Definition 4.1 with a
decomposition of a relative entropy of a conditional expectation on some
region with respect to that of smaller overlapping regions, which we refer to
as “approximate tensorization”.

Given a 2-colorable graph $\Lambda$ and a fixed 2-coloring, we denote the set
of vertices with labels 0 as $\Lambda_{0}$ and the set of vertices with labels
1 as $\Lambda_{1}=\Lambda\setminus\Lambda_{0}$.

###### Definition 5.8

Given the 2-colorable graph $\Lambda$ with 2-coloring
$\Lambda=\\{\Lambda_{0},\Lambda_{1}\\}$, denote a finite subgraphs
$\Gamma\subset\subset\Lambda$ with induced 2-colorings
$\\{\Gamma_{i}:=\Gamma\cap\Lambda_{i}\\}_{i=0}^{1}=:\\{\Gamma_{0},\Gamma\setminus\Gamma_{0}\\}$.
For a quantum state $\rho\in\mathcal{D}(\mathcal{H}_{\Gamma})$, we define

$\omega:=E_{\Gamma_{0}*}^{S}(\rho)=(\bigcirc_{a\in\Gamma_{0}}E_{a*}^{S})(\rho),$
(195)

where the second equality follows from Proposition 3.6.

The importance of this “semiclassical” state is that in [24] it was shown to
satisfy the following approximate tensorization statement from the
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-decay we assume here.

###### Theorem 5.9 (Approximate tensorization for $\omega$)

Let $(\Lambda,\Phi,\beta)$ be as assumed in this section. Fix some
$\Gamma\subset\subset\Lambda$ and given some
$\rho\in\mathcal{D}(\mathcal{H}_{\Gamma})$ set $\omega:=E^{S}_{A*}(\rho)$ as
defined above. Let $C,D\subset\Gamma$ be two convex subsets, s.t. for
$\\#\in\\{R:=C\cup D$, $\partial C,\partial D,\partial R\\}$ we have
$\\#\cap\Lambda_{0}=\emptyset,$ and $l:=\operatorname{dist}(C\setminus
D,D\setminus C)>1$. Then it holds that

$D(\omega\|E_{C\cup
D*}^{S}(\omega))\leq\frac{1}{1-2\eta_{C,D}(l)}[D(\omega\|E_{C*}^{S}(\omega))+D(\omega\|E_{D*}^{S}(\omega))],$
(196)

where $\eta(l)$ is the decay function of the uniform
q$\mathbb{L}_{1}\to\mathbb{L}_{\infty}$-clustering.

The proof of this approximate tensorization statement, given either Theorem
4.22 or Theorem 4.23, is just an application of [24, Theorem 8]. This theorem
will allow us to execute what is sometimes called a “divide-and-conquer”
strategy for the state $\omega$. That is, approximate tensorization allows us
to bound the relative entropy distance between $\omega$ and the projection
onto its fixed point on some lattice region $CD$ by the ones on the smaller
regions $C$ and $D$, up to a suitable factor that depends on the decay of
correlations of the fixed point. We will then iterate this procedure to end up
with finite size regions, on which we can bound the relative entropies by a
local MLSI constant, and finally generalize this to arbitrary states $\rho$.

From this we will show that, as long as we have exponential decay in this
approximate tensorization argument for convex regions $C,D$ with the right
diameter, as per the two cases, we can establish a lower bound on the MLSI. In
Case 1, with diameter $\mathcal{O}(l^{2})$, we can establish a constant lower
bound on the MLSI constant
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega(1)_{|\Gamma|\to\infty}$ and for Case
2, where the convex regions $C,D$ have diameters $\mathcal{O}(l)$, we can
establish a logarithmic (for trees at high temperature) and square root (for
2D lattices also at high temperature) lower bound on the MLSI constant
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega((\operatorname{diam}|\Gamma|)^{-1})_{|\Gamma|\to\infty}$.
To get there, we will require a repeated application of Theorem 5.9 with
appropriate choices of regions $C,D$, as we show in the next subsection.

####  5.3.2 Geometric argument

We now construct the aforementioned geometric divide-and-conquer-strategy
employing a Cesaro averaging over different choices of regions $C,D$. The
target is an upper bound on the relative entropy between the state $\omega$
and its image under the Schmidt conditional expectation on the whole $\Gamma$.
This bound is based on Theorem 5.9, and will be expressed in terms of the sum
of the relative entropies between $\omega$ and its image under the Schmidt
conditional expectation on smaller coarse-grained sets of a fixed size, called
$\\{R_{k}\\}_{k}$.

For the geometric argument, we will explicitly define the construction of the
relevant coarse-grained-sets for the $b$-ary tree, for $b\in\mathbb{N}$. The
$1$-D case is then covered as the $b=1$ case in trees. Since the Cesaro
averaging and clustering works analogously for $b$-ary trees and $D$-dim
hypercubic lattices, we will for simplicity only conduct it for the former.
For the definition of the subsets and further details on the $D$-dimensional
hypercubic construction see Section 5.3.3 and also [24].

Denote with $K:=\\{k|x_{k}\in\Gamma_{0}\\}$ the index set of
$\Gamma_{0}=\Gamma\cap\Lambda_{0}$. The $\\{R_{k}\\}_{k}$ will be of a
sufficiently large but finite minimal size and s.t. $E^{S}_{R_{k}*}\circ
E^{S}_{\Gamma_{0}*}=E^{S}_{\Gamma_{0}*}\circ E^{S}_{R_{k}*}$ holds, which is
always the case if their boundaries satisfy $\partial
R_{k}\cap\Gamma_{0}=\emptyset$. As $\Gamma_{0}$ is the union of single
vertices each with distance 2 from each other, each $x_{j}\in\Gamma_{0}$ is
either an element of $R_{k}$, or $\operatorname{dist}(x_{j},R_{k})=2$, so the
commutation of the conditional expectations follows from Proposition 3.6. For
trees, we define the subsets as follows 141414Note that the constructions are
not unique..

###### Definition 5.10 (Construction of coarse-grained sets for $b$-ary tree)

Denote the infinite $b$-ary tree with $\mathbb{T}_{b}$. Denote with $B_{x,l}$
the subtree rooted at site $x\in\mathbb{T}_{b}$ of height $l$. We define the
following set of subsets

$\displaystyle\\{R_{k}\\}_{k\in K}:=\\{B_{x_{k},l_{0}}\cap\Gamma\\}_{k\in
K}=\\{B_{x,l_{0}}\cap\Gamma\\}_{x\in\Gamma_{0}},$ (197)

where $l_{0}\in 2\mathbb{N}$ is a a suitably large constant to be fixed later
on. The $\\{R_{k}\\}_{k}$ form a coarse-graining into subtrees based on each
vertex of the same label (i.e. in $\Gamma_{0}$) of finite fixed height
$l_{0}$. We consider the cMLSI constant of our evolution on these sets, and
extend these via a Cesaro averaging argument to the whole lattice.

Through these sets, we define the following quantity with respect to the state
$\omega$ from (195)

$\displaystyle D_{R}(\omega):=\sum_{R_{k}\subset
R}D(\omega\|E^{S}_{R_{k}*}(\omega)),$ (198)

where the sets $\\{R_{k}\\}_{k\in K}$ are the coarse grained sets as specified
above. Observe that $D_{R}(\omega)$ is monotonically increasing in $R$, i.e.
if $A\subset B\subset\Gamma$ are two subregions, then $D_{A}(\omega)\leq
D_{B}(\omega)$ by positivity of the relative entropy. It is also additive up
to boundary terms, i.e. if $A,B\subset\Gamma$ are two disjoint subregions,
then

$\displaystyle
D_{AB}(\omega)=D_{A}(\omega)+D_{B}(\omega)+\sum_{\underset{R_{k}\cap
A\neq\emptyset}{R_{k}\cap B\neq\emptyset}}D(\omega\|E^{S}_{R_{k}*}(\omega)).$
(199)

Next we define a function $L\mapsto C(L):\mathbb{N}\to\mathbb{R}$, such that
$C(L)$ is the smallest number for which

$\displaystyle D(\omega\|E^{S}_{\Gamma*}(\omega))\leq C(L)D_{\Gamma}(\omega),$
(200)

where $L:=\operatorname{height}(\Gamma)$. It follows from this definition that
$C(L)$ is monotonically non-decreasing. The main target of this section is
thus to upper bound $C(L)$. These bounds are given in the following key lemma,
which is proven by inductively applying the approximate tensorization and
averaging suitably over the choices of partitions $C,D$. It is the main
ingredient of the geometric part of the proof.

###### Lemma 5.11

For sub-trees of $\mathbb{T}_{b}$ of the form $\Gamma=B_{x_{j},L}$ for some
$j\in K$, (200) holds with

  1. a

Case 1: $C(L)=\mathcal{O}(1)_{|\Gamma|\to\infty}$ is uniformly upper bounded
by $C(\infty)<\infty$.

  2. a

Case 2:
$C(L)=\mathcal{O}(L)_{L\to\infty}=\mathcal{O}(\operatorname{diam}|\Gamma|)_{|\Gamma|\to\infty}=\mathcal{O}(\log|\Gamma|)_{|\Gamma|\to\infty}$.

This lemma can be viewed as an approximate tensorization statement on many
regions for a state of the form of $\omega$. That is, it upper bounds the
relative entropy between it and its Schmidt conditional expectation on the
whole lattice $\Gamma$ in terms of the relative entropies of it and the
Schmidt conditional expectations of the fixed finite size regions
$\\{R_{k}\\}_{k\in K}$.

In order to prove it, we will make a repeated use of the approximate
tensorization result Theorem 5.9. First breaking up a region $B_{x_{k},L}$
into the following regions, then breaking these up in a recursive way until we
reach a decomposition in terms of the initial defined $\\{R_{k}\\}_{k}$. To
apply it, we now proceed with the choice of regions $C,D$ to be used in the
result. For a set $B_{x_{k},L}$ define

$\displaystyle C_{k}^{\tilde{l}}:=B_{x_{k},\tilde{l}}\,,\hskip
28.45274ptD_{k}^{\tilde{l},l}:=\bigcup_{\underset{\operatorname{dist}(x_{m},x_{k})=\tilde{l}-l}{m\in
K}}B_{x_{m},L+l-\tilde{l}}\,.$ (201)

Hence we cover the subtree $B_{x_{k},L}$ of height $L$ by a subtree with the
same root of height $\tilde{l}$, called $C_{k}^{\tilde{l}}$, and a union of
disjoint subtrees of height $L+l-\tilde{l}$, called $D_{k}^{\tilde{l},l}$,
s.t. their overlap has height $l$ and is s.t. we can apply the approximate
tensorization result Theorem 5.9 with the function
$\eta_{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}}(l)$.

Importantly we require that each of these sets begins and ends with some
vertices of the same index 0, i.e. in $\Gamma_{0}$. In formulae, this is
$C_{k}^{\tilde{l}}\cup D_{k}^{\tilde{l},l}=B_{x_{k},L}$ and
$\operatorname{dist}\big{(}C_{k}^{\tilde{l}}\setminus
D_{k}^{\tilde{l},l},D_{k}^{\tilde{l},l}\setminus C_{k}^{\tilde{l}}\big{)}=l$
for all $0\leq\tilde{l}\leq L$. Hence each set $B_{x_{k},L}$ has the family of
non-trivial partitions
$\\{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}\\}_{\tilde{l}=1}^{L-1}$ and for each
of these it holds, due to Theorem 5.9, that

$\displaystyle
D\big{(}\omega\big{\|}E^{S}_{B_{x_{k},L}*}(\omega)\big{)}\leq\frac{1}{1-2\eta_{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}}(l)}\left[D\Big{(}\omega\Big{\|}E^{S}_{C_{k}^{\tilde{l}}*}(\omega)\Big{)}+D\Big{(}\omega\Big{\|}E^{S}_{D_{k}^{\tilde{l},l}*}(\omega)\Big{)}\right],$
(202)

For an example of these regions see Figure 6.

Figure 6: Example of a partition of $B_{0,6}\subset\mathbb{T}_{2}$ into the
regions $C^{\tilde{l}}_{0}$, $D^{\tilde{l},l}_{0}$ with height of the regions
$\tilde{l}=4$ and $l=6-\tilde{l}+2=4$ and the height of their overlap being
$l=2$. This is in $B_{0,6}$ as part of a 2-ary tree. The  red vertices are the
ones of index 0. Notice that each of these sets ‘begins’ and ‘ends’ in these
sets.

In the $1D$ setting, these can be written as
$C_{k}^{\tilde{l}}=[x_{k},x_{k}+\tilde{l}]$ and
$D_{k}^{\tilde{l},l}=[x_{k}+\tilde{l}-l,x_{k}+L]$, where the
$\operatorname{height}$ becomes $\operatorname{diam}$, which is just the
length of the intervals.

Now, recalling the definition of the two cases we are considering, we have the
following key observation.

  1. a

Case 1: We can pick the overlap
$l=\lfloor{\sqrt{L}}\rfloor=\mathcal{O}(\sqrt{L})$. Then
$\eta(l)\equiv\eta_{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}}(l)$ is
exponentially decaying in $L$ for any $\xi>0$, by assumption. Set
$l_{\text{min},1}$ to be the smallest $L$, s.t. $\eta(l)<\frac{1}{2}$ for all
$l\geq l_{\text{min},1}$.

  2. a

Case 2: We have to pick the overlap $l=\frac{L}{N}=\mathcal{O}(L)$ for some
$N\in\mathbb{N}\setminus\\{1\\}$ and require the decay length
($\xi,\xi^{\prime}$) to be small enough such that we have eventual exponential
decay in $\eta(l)\equiv\eta_{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}}(l)$ in
$l$. This suffices for $N=2$, but we will keep it general in the following
derivation. We set $l_{0}$ to be the smallest $l$ such that
$\eta(l)<\frac{1}{2}$ for all $l\geq l_{0}$.

We now write the proof for each of these cases separately.

###### Proof of Case 1 in Lemma 5.11.

The proof idea is to average the approximate tensorization result, Theorem
5.9, over all the above defined coverings
$\\{C_{k}^{\tilde{l}},D_{k}^{\tilde{l},l}\\}_{\tilde{l}}$, for different
$\tilde{l}$, to get the relative entropy between $\omega$ and its Schmidt
conditional expectation on the whole of $B_{x_{k},L}$ in terms of the relative
entropy between it and the Schmidt conditional expectation on subregions of
height $\epsilon L$, where $\epsilon<1$. This allows us to bound $C(L)$ in
terms of the smaller $C(\epsilon L)$ times a multiplicative factor, which
allows us to ultimately bound $C(L)$ by a constant through a recursive
process. Hence to bound $C(L)$ by $C(l_{0})$, where $l_{0}$ is a fixed finite
size, requires $\mathcal{O}(\log(L))$ steps. However, since in Case 1 the
decay is assumed strong enough, we will be able to upper bound this product
uniformly by a constant.

Fix $x_{j}\in\Gamma_{0}$ and let $\frac{1}{2}<\epsilon<1$. We enumerate a
maximal set of partitions of $B_{x_{j},L}$ into
$\\{C_{j}^{\tilde{l}},D_{j}^{\tilde{l},\lfloor{\sqrt{L}}\rfloor}\\}$, s.t.
$\operatorname{height}(C_{j}^{\tilde{l}})=\tilde{l},\operatorname{height}(D_{j}^{\tilde{l},\lfloor{\sqrt{L}}\rfloor})=L+\lfloor{\sqrt{L}}\rfloor-\tilde{l}\leq\epsilon
L$ and s.t. different partitions have disjoint overlaps, i.e.
$\left(C_{j}^{\tilde{l}_{1}}\cap
D_{j}^{\tilde{l}_{1},\lfloor{\sqrt{L}}\rfloor}\right)\cap\left(C_{j}^{\tilde{l}_{2}}\cap
D_{j}^{\tilde{l}_{2},\lfloor{\sqrt{L}}\rfloor}\right)=\emptyset$, whenever
$\tilde{l}_{1}\neq\tilde{l}_{2}$. This works as long as
$\sqrt{L}\lesssim(2\epsilon-1)L$ which gives another condition on the minimal
size of $l=\lfloor{\sqrt{L}}\rfloor\geq:l_{\text{min},2}$. There exist
$\frac{L}{\lfloor{\sqrt{L}}\rfloor}=\mathcal{O}(\sqrt{L})$ of these
partitions, since their overlap is of height
$\lfloor{\sqrt{L}}\rfloor=\mathcal{O}(\sqrt{L})$. To simplify notation, we
refer to these partitions $\\{C_{i},D_{i}\\}^{\mathcal{O}(\sqrt{L})}_{i=1}$,
where the partition index $i$ is not to be confused with the fixed root $j$.
We now average over all the approximate tensorization results of these
partitions to get

$\displaystyle D(\omega\|E^{S}_{B_{x_{j},L}*}(\omega))$
$\displaystyle\leq\frac{1}{\mathcal{O}(\sqrt{L})}\sum_{i=1}^{\mathcal{O}(\sqrt{L})}\frac{1}{1-2\eta_{C_{i},D_{i}}(\sqrt{L})}\left[D(\omega\|E^{S}_{C_{i}*}(\omega))+D(\omega\|E^{S}_{D_{i}*}(\omega))\right]$
(203)
$\displaystyle\leq\frac{1}{1-2\eta(\sqrt{L})}\frac{1}{\mathcal{O}(\sqrt{L})}\sum_{i=1}^{\mathcal{O}(\sqrt{L})}C(\operatorname{height}(C_{i}))D_{C_{i}}(\omega)+C(\operatorname{height}(D_{i}))D_{D_{i}}(\omega)$
(204) $\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\sqrt{L})}\frac{1}{\mathcal{O}(\sqrt{L})}\Bigg{(}\sum_{i=1}^{\mathcal{O}(\sqrt{L})}\left(2D_{C_{i}\cap
D_{i}}(\omega)+D_{C_{i}\setminus D_{i}\cup D_{i}\setminus
C_{i}}(\omega)\right).$ (205)
$\displaystyle\quad\quad+\sum_{\underset{R_{k}\cap(C_{i}\setminus
D_{i})\neq\emptyset}{R_{k}\cap(C_{i}\cap
D_{i})\neq\emptyset}}D(\omega\|E^{S}_{R_{k}*}(\omega))+\sum_{\underset{R_{k}\cap(D_{i}\setminus
C_{i})\neq\emptyset}{R_{k}\cap(C_{i}\cap
D_{i})\neq\emptyset}}D(\omega\|E^{S}_{R_{k}*}(\omega))\Bigg{)}$ (206)
$\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\sqrt{L})}\frac{1}{\mathcal{O}(\sqrt{L})}\left(2D_{\bigcup_{i=1}^{\mathcal{O}(\sqrt{L})}(C_{i}\cap
D_{i})}(\omega)+\sum_{i=1}^{\mathcal{O}(\sqrt{L})}D_{(C_{i}\setminus
D_{i})\partial\cup(D_{i}\setminus C_{i})\partial}(\omega)\right)$ (207)
$\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\sqrt{L})}\frac{\lfloor{\sqrt{L}}\rfloor}{L}\left(2+\frac{L}{\lfloor{\sqrt{L}}\rfloor}\right)D_{B_{x_{j},L}}(\omega),$
(208)

where in the second line we used that $\eta_{C_{i},D_{i}}(\sqrt{L})$ does not
depend on $i$, and then the definition of $C(\operatorname{height}(\cdot))$.
In the third line we used that
$C(\operatorname{height}(C_{i})),C(\operatorname{height}(D_{i}))\leq
C(\epsilon L)$ since
$\operatorname{height}(C_{i})=\tilde{l},\operatorname{height}(D_{i})=L+\lfloor{\sqrt{L}}\rfloor-\tilde{l}\leq\epsilon
L$, and also used Eq. (199). Hence it follows that $C(L)\leq C(\epsilon
L)\frac{1}{1-2\eta(\sqrt{L})}(1+\frac{2}{\sqrt{L}})=:C(\epsilon L)f(L)$. This
allows us to bound $C(L)$ through an iterative process, since by definition in
Eq. (200) and the sets $\\{R_{k}\\}_{k\in K}$, $C(l_{0})=1$. Using $C(L)\leq
C(\epsilon L)f(L)$, $M=\mathcal{O}(\log L)$ times, s.t.
$\epsilon^{M}L=l_{0}=:\lceil{\max\\{l_{\text{min},1},l_{\text{min},2}\\}}\rceil$
then gives

$\displaystyle C(L)\leq C(l_{0})\prod_{k=1}^{M}f(\epsilon^{k}L)\leq
C(l_{0})\prod_{k=0}^{\infty}f(l_{0}\epsilon^{-k})<\infty,$ (209)

where it can be checked by inspection that the infinite product converges to a
constant since $\eta(\sqrt{L})$ is exponentially decaying in $\sqrt{L}$ and
$(1+\frac{2}{\sqrt{L}})\to 1$ fast enough. Note that $C(l_{0})=1$, independent
of the $x_{j}$ which we fixed initially. Hence the result follows. ∎

###### Proof of Case 2 in Lemma 5.11.

The proof follows exactly the same idea and techniques as the one above. The
main difference is that we need to choose the overlap of the coverings $C,D$
to scale as $\mathcal{O}(L)$, where $L=\operatorname{height}{B_{x_{j},L}}$, to
still get decay in the approximate tensorization. Hence, the number of
partitions has to be constant in system size, which yields a constant
multiplicative factor in each inductive step. Since we again need
$\mathcal{O}(\log(L))$ steps in the iteration to bound $C(L)$, this gives the
$\mathcal{O}(L)$, where $L=\operatorname{height}(B_{x_{k},L})$ of the original
set, scaling of the bound on $C(L)$. The details of this proof can be found in
Appendix E for completeness. ∎

After this key lemma on approximate tensorization, in Sec. 5.3.4 below we will
generalise this statement to arbitrary states $\rho$ and the Davies maps
instead of the Schmidt conditional expectation. This will then allow us to
extend the existence of a local cMLSI constant, see Theorem 2.3, to the whole
lattice with the cost of $C(L)^{-1}$.

####  5.3.3 Extension to hypercubic lattices

The main idea behind the proof of Lemma 5.11, where we average over all
suitable partitions of a given set to obtain the desired upper bound,
naturally extends to hypercubic lattices ($D>1$). To adapt it, we need to adap
the definitions of both the coarse-grained sets $\\{R_{k}\\}_{k}$ and the
‘suitable partitions’. This then yields the second case of Theorem 5.3 and the
first case of Theorem 5.7.

First, for the $\\{R_{k}\\}_{k}$, instead of the definition in equation (197)
we have the following definition.

###### Definition 5.12 (Construction of Coarse-grain sets for $D$-dim
hypercubic lattice)

Denote with $x+A$ the set $A\subset\mathbb{Z}^{D}$ shifted by
$x\in\mathbb{Z}^{D}$. Then, define the sets

$\displaystyle\\{R_{k}\\}_{k\in K}:=$
$\displaystyle\\{(x_{k}+[-l_{0}-1,l_{0}-1]^{D}\cup(\Gamma_{0}\cap(x_{k}+[l_{0},l_{0}]^{D})))\cap\Gamma\\}_{k\in
K}$ (210) $\displaystyle=$
$\displaystyle\\{(x+[-l_{0}-1,l_{0}-1]^{D}\cup(\Gamma_{0}\cap(x+[l_{0},l_{0}]^{D})))\cap\Gamma\\}_{x\in\Gamma_{0}},$
(211)

where again $l_{0}\in\mathbb{N}$ is a suitably large constant to be fixed
later on. These are essentially jagged hyper-cubes of side-length $2l_{0}+1$
around a center $x_{k}\in\Gamma_{0}$. An example for one of these jagged
hyper-cubes in 2D is shown in Figure 7.

Figure 7: Example of a coarse-grain set from Definition 5.12 in 2D. Here, we
consider the $\Gamma_{0}$, sites of index 0, to be the sites of purple color,
and $\Gamma_{1}$ those of blue. The boundary of $R_{k}$ is completely composed
of sites in purple.

For the suitable partitions $C_{k},D_{k}$, the idea is to take jagged hyper-
rectangles with sufficient overlap as the sets ‘$C_{k},D_{k}$’. Then, we apply
the approximate tensorization statement $D$ times to get (202), but with
$2^{D}$ terms on the right hand side, where the conditioned relative entropy
is on sub-regions with a strictly smaller diameter. The approximate
tensorization statement we obtain then yields, through iterative application
as in the proof of Lemma 5.11, the analogous following Lemma.

###### Lemma 5.13

For sub-lattices of $\mathbb{Z}^{D}$ of form $x_{j}+[-L,L]^{D}$ for some $j\in
K$, (200) holds, with the coarse-grained sets defined in (211), with

  1. a

Case 1: $C(L)=\mathcal{O}(1)_{|\Gamma|\to\infty}$ is uniformly upper bounded
by a $C(\infty)<\infty$.

  2. a

Case 2:
$C(L)=\mathcal{O}(L)_{L\to\infty}=\mathcal{O}(\operatorname{diam}|\Gamma|)_{|\Gamma|\to\infty}=\mathcal{O}(|\Gamma|^{\frac{1}{D}})_{|\Gamma|\to\infty}$.

For the $2$-D graph in Case 2, which is the only use of this Case 2 in our
main results, this gives us
$C(L)=\mathcal{O}(L)_{L\to\infty}=\mathcal{O}(\sqrt{|\Gamma|})_{|\Gamma|\to\infty}$.
Case 1 instead yields the constant MLSI at high temperature.

####  5.3.4 Putting everything together

In the last steps, we generalise the result above for the state
$\omega=E_{\Gamma_{0}}^{S}(\rho)$ and the Schmidt conditional expectations to
arbitrary states $\rho$ and to the Davies conditional expectations. This
finalises the bounds on the MLSI for the Davies generators.

###### Proof of Lemma 5.1 (and hence of Theorem 5.3 and Theorem 5.7).

Let $(\Lambda,\Phi,\beta)$ be as assumed in Lemma 5.1 ,
$\Lambda=\\{\Lambda_{0},\Lambda_{1}\\}$ be a two colouring, and
$\\{E^{S}_{\Gamma}\\}_{\Gamma\subset\subset\Lambda},\\{E^{D}_{\Gamma}\\}_{\Gamma\subset\subset\Lambda}$
the families of Schmidt and Davies conditional expectations. Let
$\Gamma\subset\subset\Lambda$151515e.g. for the $b-$ary tree,
$b\in\mathbb{N}$, wlog $\Gamma=B_{0,L}$. be a complete connected finite
subgraph, where we set $\\{x_{k}\\}_{k\in
K}=\Gamma_{0}:=\Gamma\cap\Lambda_{0}$, and set
$\omega:=E^{S}_{\Gamma_{0}}(\rho)$ for a state
$\rho\in\mathcal{D}(\mathcal{H}_{\Gamma})$. We first apply the chain rule for
the relative entropy (41), with
$\sigma\equiv\sigma^{\Gamma}=E^{D}_{\Gamma*}(\rho)=E^{S}_{\Gamma_{0}*}(\sigma)$,
so that

$\displaystyle
D(\rho\|E^{D}_{\Gamma*}(\sigma))=D(\rho\|\sigma)=D(\rho\|E^{S}_{\Gamma_{0}*}(\rho))+D(E^{S}_{\Gamma_{0}*}(\rho)\|\sigma)=D(\rho\|\omega)+D(\omega\|\sigma).$
(212)

The first summand $D(\rho\|\omega)$ satisfies exact tensorization (a form of
strong subadditivity), since the Schmidt conditional expectations of two sets
with distance two between each other commute, $E^{S}_{\\{x_{k}\\}*}\circ
E^{S}_{\\{x_{j}\\}*}=E^{S}_{\\{x_{j}\\}*}\circ
E^{S}_{\\{x_{k}\\}*}=E^{S}_{\\{x_{k}\\}\cup\\{x_{j}\\}*}$ for all
$x_{k},x_{j}\in\Gamma_{0}$. See Proposition 3.6 and [39, 62]. This means that

$\displaystyle
D(\rho\|\omega)=D(\rho\|E^{S}_{\Gamma_{0}*}(\rho))\leq\sum_{x_{k}\in\Gamma_{0}}D(\rho\|E^{S}_{\\{x_{k}\\}*}(\rho))\overset{\text{Lemma
}\ref{lem:RelEntCondExpBound}}{\leq}\sum_{x_{k}\in\Gamma_{0}}D(\rho\|E^{D}_{\\{x_{k}\\}\partial*}(\rho)).$
(213)

We can also bound the second summand $D(\omega\|\sigma)$ as follows, using
Lemma 5.11, the DPI for the relative entropy, and finally Lemma 3.10.

$\displaystyle D(\omega\|\sigma)=D(\omega\|E^{S}_{\Gamma*}(\omega))$
$\displaystyle\overset{\text{Lemma
}\ref{lemma:OmegacMLSI}}{\leq}C\sum_{x_{k}\in\Gamma_{0}}D(\omega\|E^{S}_{R_{k}*}(\omega))=C\sum_{x_{k}\in\Gamma_{0}}D(E^{S}_{\Gamma_{0}*}(\rho)\|(E^{S}_{R_{k}*}\circ
E^{S}_{\Gamma_{0}*})(\rho))$ (214)
$\displaystyle\overset{DPI}{\leq}C\sum_{x_{k}\in\Gamma_{0}}D(\rho\|E^{S}_{R_{k}*}(\rho))\overset{\text{Lemma
}\ref{lem:RelEntCondExpBound}}{\leq}C\sum_{x_{k}\in\Gamma_{0}}D(\rho\|E^{D}_{R_{k}\partial*}(\rho)),$
(215)

where in the second line we used that $E^{S}_{R_{k}*}\circ
E^{S}_{\Gamma_{0}*}=E^{S}_{\Gamma_{0}*}\circ E^{S}_{R_{k}*}$, which holds by
the construction of the sets $R_{k}$ in (197).161616This is since $\partial
R_{k}\cap\Gamma_{0}=\emptyset$. Recall that $\Gamma_{0}$ is the union of
single vertices each with distance 2 from each other. Hence the claim follows
from Proposition 3.6. Recall that by Lemma 5.11 in Case 1 the constant $C$ is
independent of system size, whereas in Case 2 for trees it scales
logarithmically with system size. Importantly, the regions
$\\{\\{x_{k}\\}\partial,R_{j}\partial\\}_{k,j}$ are of fixed finite size.
Hence by Theorem 2.3 there exists cMLSI constants $\alpha_{0},\alpha_{1}>0$,
s.t. for any $j,k$

$\displaystyle\alpha_{0}D\left(\rho\|E^{D}_{\\{x_{k}\\}\partial*}(\rho)\right)\leq\operatorname{EP}_{\mathcal{L}^{D}_{\\{x_{k}\\}\partial}}(\rho),\hskip
28.45274pt\alpha_{1}D\left(\rho\|E^{D}_{R_{j}\partial*}(\rho)\right)\leq\operatorname{EP}_{\mathcal{L}^{D}_{R_{j}\partial}}(\rho).$
(216)

Putting everything above together, we have

$\displaystyle D(\rho\|E^{D}_{\Gamma*}(\rho)=D(\rho\|\sigma)$
$\displaystyle\leq\frac{1}{\min\\{\alpha_{0},\alpha_{1}\\}}\left(\sum_{x_{k}\in\Gamma_{0}}\operatorname{EP}_{\mathcal{L}^{D}_{\\{x_{k}\\}\partial}}(\rho)+C\sum_{k|x_{k}\in\Gamma_{0}}\operatorname{EP}_{\mathcal{L}^{D}_{R_{k}\partial}}(\rho)\right)$
(217)
$\displaystyle\leq\frac{2mC}{\min\\{\alpha_{0},\alpha_{1}\\}}\operatorname{EP}_{\mathcal{L}^{D}_{\Gamma}}(\rho),$
(218)

where in the last inequality we used the positivity and additivity of the
entropy production and the fact that, by construction, each site $x\in\Gamma$
is contained in at most a constant number, say $2m$, of regions
$R_{k}\partial$, since they are of fixed finite size. The same holds for
$\\{x_{k}\\}\partial$ with $\leq 2m$ regions. Thus, it follows that

$\displaystyle\alpha(\mathcal{L}^{D}_{\Gamma})\geq\frac{\min\\{\alpha_{0},\alpha_{1}\\}}{2mC}>0,$
(219)

so that the final scaling is $\Omega(C^{-1})$. Therefore in Case 1 from Lemma
5.11 we get
$\alpha(\mathcal{L}^{D}_{\Gamma})=\mathcal{O}(1)_{|\Gamma|\to\infty}$ whereas
in Case 2 we get
$\alpha(\mathcal{L}^{D}_{\Gamma})=\Omega((\operatorname{diam}|\Gamma|)^{-1})_{|\Gamma|\to\infty}$.
Note that for trees $\operatorname{diam}(B_{0,L})=\mathcal{O}(\log|B_{0,L}|)$.
∎

##  6 Further applications

We have established a constant lower bound on the cMLSI as our main result for
subexponential graphs. Here we explore the consequences of the system size
independence of this bound for physical and statistical properties of such
systems under dissipative evolution and of its steady state $\sigma$. Assume
$(\Lambda,\Phi,\beta)$ suitable, such that the system admits a system size
independent lower bound on the Davies cMLSI constant
$\alpha(\mathcal{L}^{D}_{\Gamma})$ as in 5.4.

###  6.1 Exponential convergence to Gibbs states in the thermodynamic limit

A direct consequence of the spin-system size $|\Gamma|$ independence of the
cMLSI constant $\alpha(\mathcal{L}^{D}_{\Gamma})$ is that in the thermodynamic
limit we have an exponential decay of the relative entropy density between the
initial state at time $t$ and the thermal state, with a fixed decay rate
$\alpha$. This illustrates how quantum materials in the thermodynamic limit
thermalize when weakly coupled to an external environment.

###### Corollary 6.1

For local Hamiltonians $D(\rho_{t}\|\sigma^{\Gamma})=\mathcal{O}(|\Gamma|)$,
and hence

$\displaystyle\lim_{\Gamma\uparrow\Lambda}\frac{1}{|\Gamma|}D(\rho_{t}\|\sigma^{\Gamma})\leq
e^{-\alpha
t}\lim_{\Gamma\uparrow\Lambda}\frac{1}{|\Gamma|}D(\rho\|\sigma^{\Gamma}).$
(220)

###### Proof.

For a local Hamiltonian we have
$\|H_{\Gamma}\|\leq\sum_{\underset{\text{diam}(X)\leq
r}{X\subset\Gamma}}\|\Phi_{X}\|=\mathcal{O}(|\Gamma|)$ and hence

$\displaystyle
d^{|\Gamma|}e^{-\beta\mathcal{O}(|\Gamma|)}=\Tr[\mathbbm{1}e^{-\beta\|H\|}]\leq\Tr[e^{-\beta
H}]\leq\Tr[\mathbbm{1}e^{\beta\|H\|}]=d^{|\Gamma|}e^{\beta\mathcal{O}(|\Gamma|)},$
(221)

hence taking logarithms gives $\log Z_{\Gamma}=\log\Tr[e^{-\beta
H_{\Gamma}}]=\mathcal{O}(|\Gamma|)$ and we can bound

$\displaystyle D(\rho\|\sigma^{\Gamma})\leq-\Tr\left[\rho\log\frac{e^{-\beta
H}}{Z_{\Gamma}}\right]=\log Z_{\Gamma}+\beta\Tr[\rho
H_{\Gamma}]=\mathcal{O}(|\Gamma|).$ (222)

The result follows directly from MLSI with system size independent constant
$\alpha$ by dividing through $|\Gamma|$ and taking the limit. ∎

###  6.2 Local Mixing

The relative entropy $D(e^{t\mathcal{L}}(\rho)\|\sigma)$, as it appears in the
rapid mixing results, is rarely an adequate figure of merit in many-body
experiments, since it is associated with expectation values of observables
with support across the whole system. Even if two states are very different in
relative entropy, that difference may be invisible to reasonable read-out
capabilities.

Instead, the measurements that typically take place there are of observables
with support on a small number of subsystems, such as local magnetization or
currents. These, however, are always expected to thermalize in
$\mathcal{O}(1)$ time irrespective of the initial conditions, much faster than
global observables.

That this is the case can be shown to follow already from the relative entropy
decay. The proof only requires rapid mixing with a constant rate and the Lieb-
Robinson bound for local Lindbladians which, under the assumption that
$N(l)=e^{o(l)}$, reads as follows [13].

###### Theorem 6.2

Let $\mathcal{L}$ be the Lindbladian defined on the whole lattice $\Lambda$.
Also, let $Y$ be a region of the lattice, and $V_{l}$ a larger region such
that $Y\subset V_{l}\subset\Lambda$ and
$l\equiv\textup{dist}(Y,\Lambda\setminus V_{l})$. For any observable $O_{Y}$
with $\textup{supp}(O_{Y})\subset Y$,

$\|e^{t\mathcal{L}^{\dagger}}(O_{Y})-e^{t\mathcal{L}_{V_{l}}^{\dagger}}(O_{Y})\|\leq\|O_{Y}\|e^{-\Omega(l-vt)},$
(223)

where $\mathcal{L}_{V_{l}}=\sum_{\textup{supp}(\mathcal{L}_{X})\subset
V_{l}}\mathcal{L}_{X}$.

This means that the time evolution of $O_{Y}$ is to a good approximation
generated by the jump operators around its vicinity. Notice that in [13] it is
assumed that $N(l)=\text{poly}(l)$, but sub-exponential is also enough. With
this, the result on local thermalization is as follows.

###### Corollary 6.3

Assuming that the system has rapid mixing with constant decay rate $\alpha$,
as in Corollary 5.4, we have that, for any region $A$,

$D(\tr_{A^{c}}[e^{t\mathcal{L}}(\rho)]\|\sigma_{A})\leq\mathcal{O}\left(|A|+t^{\kappa+1}\right)\times
e^{-\alpha t},$ (224)

where $\sigma_{A}=\tr_{A^{c}}[\sigma]$ is the local marginal of the Gibbs
state.

###### Proof.

Our starting point is a local Lindbladian of the form of Eq. (23), as a sum of
local terms.

We now define three regions $A,B,C$ such that $A$ is our subsystem of
interest, and $B$ shields $A$ from $C$ with $l_{0}=\text{dist}(A,C)$, and also
such that $A\cup B\cup C=\Lambda$. We also define the Gibbs state on these
regions as $\tau_{\nu}=\frac{e^{-\beta H_{\nu}}}{Z_{\nu}}$, with $H_{\nu}$ the
subset of Hamiltonian terms with support in the respective $\nu\in\\{A,B,C\\}$
and $Z_{\nu}=\text{Tr}[e^{-\beta H_{\nu}}]$. Notice that these states are
different from the respective marginals of the global Gibbs state
$\sigma_{\nu}=\tr_{\nu^{c}}[\sigma]$. Also $\tau_{AB},Z_{AB}$ are those
defined on $A\cup B$.

Consider an arbitrary initial state on the lattice $\rho$, with
$\rho_{AB}=\tr_{C}[\rho]$. A direct calculation shows that

$D(\rho_{AB}\otimes\tau_{C}\|\sigma)=D(\rho_{AB}\|\tau_{AB})+\log{\frac{Z}{Z_{AB}Z_{C}}}+\text{Tr}[(\rho_{AB}\otimes\tau_{C})H_{I}],$
(225)

where $H_{I}=H-H_{AB}-H_{C}$, the interaction between $C$ and $AB$. Notice
that $D(\rho_{AB}\|\tau_{AB})=\mathcal{O}(|A|+|B|)$, so that

$D(\rho_{AB}\otimes\tau_{C}\|\sigma)\leq
D(\rho_{AB}\|\tau_{AB})+\mathcal{O}\left(|\partial_{BC}|\right)\leq\mathcal{O}(|A|+|B|),$
(226)

where $|\partial_{BC}|$ is the size of the boundary between $B$ and $C$, which
by definition is smaller than $|B|$.

We now consider the time evolution of this state. By data processing, we have

$D(e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})\|\sigma)\geq
D(\tr_{BC}[e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})]\|\sigma_{A}).$ (227)

We can compare this state with the one evolved under $\mathcal{L}_{AB}$,

$\displaystyle\|\tr_{BC}[e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})]-\tr_{B}[e^{t\mathcal{L}_{AB}}(\rho_{AB})]\|_{1}$
$\displaystyle=\max_{\|O_{A}\|=1}\left|\Tr[O_{A}(e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})-e^{t\mathcal{L}_{AB}}(\rho_{AB})\otimes\tau_{C})]\right|$
(228)
$\displaystyle=\max_{\|O_{A}\|=1}\left|\text{Tr}[(e^{t\mathcal{L}^{\dagger}}(O_{A})-e^{t\mathcal{L}_{AB}^{\dagger}}(O_{A}))\rho_{AB}\otimes\tau_{C}]\right|$
(229)
$\displaystyle\leq\max_{\|O_{A}\|=1}\|e^{t\mathcal{L}^{\dagger}}(O_{A})-e^{t\mathcal{L}_{AB}^{\dagger}}(O_{A})\|\leq
e^{-\Omega(l_{0}-vt)},$ (230)

where we have used the definition of the $1$-norm and the Lieb-Robinson bound.
Similarly, we also obtain

$\displaystyle\|\tr_{BC}[e^{t\mathcal{L}}(\rho)]-\tr_{BC}[e^{t\mathcal{L}_{AB}}(\rho_{AB}\otimes\tau_{C})]\|_{1}\leq
e^{-\Omega(l_{0}-vt)},$ (231)

so that by the triangle inequality,

$\|\tr_{BC}[e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})]-\tr_{BC}[e^{t\mathcal{L}}(\rho)]\|_{1}\leq
e^{-\Omega(l_{0}-vt)}.$ (232)

From continuity, this translates into a bound on relative entropy

$\displaystyle\left|D(\tr_{BC}[e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})]\|\sigma_{A})-D(\tr_{BC}[e^{t\mathcal{L}}(\rho)]\|\sigma_{A})\right|\leq\varepsilon\log\|\sigma_{A}^{-1}\|+2h\left(\frac{\varepsilon}{1+\varepsilon}\right),$
(233)

where $\varepsilon=e^{-\Omega(l_{0}-vt)}$ and $h(p)$ is the binary entropy.
Notice that $\log\|\sigma_{A}^{-1}\|=\mathcal{O}\left(|A|\right)$.

Additionally, by assumption, $\mathcal{L}$ is such that it obeys rapid mixing
with constant $\alpha$ independent of system size. Hence, applied to the
initial state $\rho_{AB}\otimes\tau_{C}$, it implies that

$D(e^{t\mathcal{L}}(\rho_{AB}\otimes\tau_{C})\|\sigma)\leq
D(\rho_{AB}\otimes\tau_{C}\|\sigma)e^{-\alpha t}.$ (234)

Let us choose $l_{0}$ such that $e^{-\Omega(l_{0}-vt)}\leq e^{-\alpha t}$,
which is possible as long as the radius of $B$ increases linearly as
$l_{0}\propto t$. Data processing, together with Eq. (233) and Eq. (226), then
imply that

$\displaystyle
D(\tr_{BC}[e^{t\mathcal{L}}(\rho)]\|\sigma_{A})\leq\mathcal{O}\left(|A|+t^{\kappa+1}\right)\times
e^{-\alpha t},$ (235)

which finishes the proof. ∎

Eq. (224) captures the most easily measurable character of the thermalization
process, which occurs even in the thermodynamic limit. If we look at an
individual subsystem, both the environment and the remainder of the system
$A^{c}$ can be seen as the entire bath that the subsystem $A$ is coupled to.
There is a crucial difference, however, between the two parts of this extended
bath: while $A$ is weakly coupled to the external environment, it is strongly
coupled to $A^{c}$. This means that, as Eq. (224) shows, the thermalization
happens towards the state $\sigma_{A}$, often called the _mean force_ Gibbs
state [74], instead of to the bare Gibbs state $\sigma^{A}=\frac{e^{-\beta
H_{A}}}{Z_{A}}$ with $H_{A}=\sum_{\text{supp}(h_{X})\subset A}h_{X}$. A
similar global-to-local reduction of rapid mixing was already proven in [28]
for the $1$-norm.

###  6.3 Wasserstein distance and transportation cost inequalities

A distance on $\mathcal{D}(\mathcal{H}_{\Gamma})$, which is important in what
follows is the quantum Wasserstein distance of order 1 [32] between two finite
dimensional quantum states $\rho,\sigma\in\mathcal{D}(\mathcal{H}_{\Gamma})$.
It is defined as

$\displaystyle W_{1}(\rho,\sigma)$
$\displaystyle\equiv\|\rho-\sigma\|_{W_{1}}:=$ (236)
$\displaystyle\frac{1}{2}\min\left\\{\sum_{i\in\Lambda}\|X^{(i)}\|_{1}\Bigg{|}\Tr[X^{(i)}]=0,X^{(i)*}=X^{(i)},\tr_{i}X^{(i)}=0\
\forall i\in\Gamma,\rho-\sigma=\sum_{i\in\Lambda}X^{(i)}\right\\}.$

Its dual norm with respect to the Hilbert-Schmidt inner product distance is
the Lipschitz distance [32], i.e. for any self-adjoint observable
$A\in\mathcal{B}(\mathcal{H}_{\Gamma})$

$\displaystyle\|A\|_{L}:=\max\\{\Tr[AX]|\Tr[X]=0,X=X^{*},\|X\|_{W_{1}}\leq
1\\}=2\max_{i\in\Gamma}\min_{A^{(i)}\in\mathcal{B}(\mathcal{H}_{\Gamma\setminus\\{i\\}})}\|A-\mathbbm{1}_{i}\otimes
A^{(i)}\|,$ (237)

where $\mathbbm{1}_{i}\in\mathcal{B}(\mathcal{H}_{i})$ is the identity on
system $i$ and $A^{(i)}$ does not act on system $i$. Thus by definition it
holds that $|\Tr[AX]|\leq\|X\|_{W_{1}}\|A\|_{L}$ for suitable $X,A$. For a
thorough overview and some properties see [32]. From the existence of an MLSI
constant $\alpha$ independent of the spin-system size $|\Gamma|$, one can show
that the following transport cost inequality holds with a transport cost
$c^{\prime}=c\frac{|\Gamma|}{\alpha}$ linear in spin-system size $|\Gamma|$
[60].

$\displaystyle\|\rho-\sigma\|_{W_{1}}\leq\sqrt{c^{\prime}D(\rho\|\sigma)}.$
(238)

For a proof of this see [60, Proposition 16 and Theorem 5] and [67, Theorems 3
and 4]. The fact that the transport cost scales linearly in $|\Gamma|$ has the
following important consequences.

####  6.3.1 Tighter bounds on the entropy difference and convergence via
relative entropy

The first application of this concerns bounding the von-Neumann entropy
difference between two quantum states $\rho,\sigma$ by their relative entropy.

###### Corollary 6.4

Let $\rho\in\mathcal{D}(\mathcal{H}_{\Gamma})$ be arbitrary and
$\sigma\equiv\sigma^{\Gamma}$ the invariant Gibbs state in $\Gamma$. The
following two bounds hold

$\displaystyle i)\ $ $\displaystyle|S(\rho)-S(\sigma)|\leq
g\left(c\sqrt{\frac{|\Gamma|}{\alpha}}\sqrt{D(\rho\|\sigma)}\right)+c\sqrt{\frac{|\Gamma|}{\alpha}}\log{(d^{2}|\Gamma|)}\sqrt{D(\rho\|\sigma)}$
$\displaystyle\hskip
65.44142pt=\mathcal{O}(\sqrt{|\Gamma|}\log|\Gamma|)_{|\Gamma|\to\infty}\sqrt{D(\rho\|\sigma)},$
(239) $\displaystyle ii)\ $
$\displaystyle|S(\rho_{t})-S(\sigma)|=\mathcal{O}\left(|\Gamma|\log|\Gamma|,\frac{e^{-\frac{\alpha
t}{2}}}{\sqrt{\alpha}}\right),$ (240)

where $g(t)=(t+1)\log(t+1)-t\log t=o(t)_{t\to\infty}$,
$\rho_{t}:=e^{t\mathcal{L}_{\Lambda*}}(\rho)$, and $c$ is some constant
independent of $|\Gamma|$ depending only on the locality of the Lindbladian
$\mathcal{L}^{D}_{\Lambda}$.

These inequalities represent a
$\mathcal{O}\left(\frac{\sqrt{|\Gamma|}}{\log|\Gamma|}\right)$ improvement
compared to Pinsker’s inequality (see [32]) as

$\|\rho-\sigma\|_{W_{1}}\leq\frac{|\Gamma|}{2}\|\rho-\sigma\|_{1}\leq\frac{|\Gamma|}{\sqrt{2}}\sqrt{D(\rho\|\sigma)}\,,$
(241)

or to the following inequality from [65]

$|S(\rho)-S(\sigma)|\leq\sqrt{3}\log(d^{|\Gamma|})\sqrt{D(\rho\|\sigma)}=\mathcal{O}(|\Gamma|)\sqrt{D(\rho\|\sigma)}\,.$
(242)

The difference is that these apply to arbitrary quantum states $\rho,\sigma$,
whereas our result in Eq. (239) requires $\sigma$ to be the Gibbs state of a
suitable Hamiltonian. Note that these entropy difference bounds are optimal in
their scaling in $|\Gamma|$ up to logarithmic correction, since the entropy
difference is an extensive quantity, i.e. it scales as
$\mathcal{O}(|\Gamma|)$.

###### Proof.

We first use the following continuity bound from [32, Theorem 1], that states
that for any two density matrices
$\rho,\sigma\in\mathcal{D}(\mathcal{H}_{\Gamma})$

$\displaystyle|S(\rho)-S(\sigma)|\leq
g(\|\rho-\sigma\|_{W_{1}})+\|\rho-\sigma\|_{W_{1}}\log(d^{2}|\Gamma|),$ (243)

where $d$ is the local Hilbert space dimension, $\|\cdot\|_{W_{1}}$ the
quantum Wasserstein distance of order 1 (see (236) for the definition), and
$g(t)=(t+1)\log(t+1)-t\log t$. By the transport cost inequality from [60,
Proposition 16, Theorem 5] and [67, Theorems 3 and 4]171717Note that in that
paper a different normalization convention for the quantum Wasserstein
distance was used, as compared to here or the other references in this
section, which holds under the assumptions of the Corollary, we get Eq. (238).
Combining this with the inequality just above gives i). For ii) use i), the
bound $D(\rho\|\sigma^{\Gamma})\leq\mathcal{O}(|\Gamma|)$ from above, and the
MLSI in its integrated form $D(\rho_{t}\|\sigma)\leq e^{-\alpha
t}D(\rho\|\sigma)$. ∎

####  6.3.2 Gaussian concentration bound

A system size independent cMLSI constant also yields the following Gaussian
concentration bound.

###### Corollary 6.5

Let $O\in\mathcal{B}(\mathcal{H}_{\Lambda})$ be a $k$-local observable, i.e.

$\displaystyle O=\sum_{\underset{|X|\leq k}{X\subset\Lambda}}o_{X},$ (244)

such that for all $i\in\Lambda$, $\sum_{X\subset\Lambda:X\ni i}\|o_{X}\|\leq
g$, where each $o_{X}$ acts only non trivially on sites $X\subset\Lambda$. Let
$\sigma$ be the Gibbs state of some geometrically-$2$-local, uniformly
bounded, commuting Hamiltonian with uniform exponential decay of correlations,
at any fixed inverse temperature $\beta>0$. Then for $r\geq 0$ it holds that

$\displaystyle\mathbb{P}_{\sigma}(|O-\Tr[O\sigma]|\geq s)\leq
2\exp\left[-\frac{\alpha s^{2}}{\mathcal{O}(|\Gamma|)}\right].$ (245)

This means that quantum spin systems with geometrically-local, commuting
Hamiltonians with uniform exponential decay of correlations give rise to a
sub-Gaussian random variable in their thermal equilibrium states for any
observables of the above form. This bound reproduces the scaling of
independent random variables from Hoeffding’s inequality, and is thus optimal.
Eq. (245) constitutes a tightening in terms of its $|\Gamma|$-dependence and
generalization to a larger class of observables and $s$-values of the previous
best known Gaussian concentration bound in [4]181818The statement there [4,
Theorem 4.2] is equivalent to $\mathbb{P}_{\sigma}(|O-\Tr[O\sigma]|\geq s)\leq
2\exp\left[-\frac{\alpha s}{\mathcal{O}(\sqrt{|\Gamma|})}\right]$..

###### Proof.

This follows from Theorem 5.3 and [60, Theorem 7 and Lemma 7], when using the
transport cost $c^{\prime}=c\frac{|\Gamma|}{\alpha}$ in Eq. (238) and the fact
that $\|\Delta^{\frac{1}{2}}_{\sigma}(O)\|_{L}\leq
4gC=\mathcal{O}(1)_{|\Gamma|\to\infty}$, since

$\displaystyle\|\Delta^{\frac{1}{2}}_{\sigma}(O)\|_{L}$ $\displaystyle\leq
2\max_{i\in\Lambda}\left\|\Delta_{\sigma}^{\frac{1}{2}}(O)-\mathbbm{1}^{(i)}_{d}\otimes\hat{\text{tr}}_{i}\Delta_{\sigma}^{\frac{1}{2}}(O)\right\|$
(246) $\displaystyle=2\max_{i\in\Lambda}\left\|\sum_{X:|X|\leq
k}\Delta_{\sigma}^{\frac{1}{2}}(o_{X})-\mathbbm{1}^{(i)}_{d}\otimes\hat{\text{tr}}_{i}\sum_{X:|X|\leq
k}\Delta_{\sigma}^{\frac{1}{2}}(o_{X})\right\|$ (247)
$\displaystyle=2\max_{i\in\Lambda}\left\|\sum_{X:X\partial\ni i,|X|\leq
k}\left(\Delta_{\sigma}^{\frac{1}{2}}(o_{x})-\mathbbm{1}^{(i)}_{d}\otimes\hat{\text{tr}}_{i}\Delta_{\sigma}^{\frac{1}{2}}(o_{x})\right)\right\|$
(248) $\displaystyle\leq 2\max_{i\in\Lambda}\sum_{X:X\partial\ni i,|X|\leq
k}2\left\|\Delta_{\sigma}^{\frac{1}{2}}(o_{x})\right\|\leq
4\max_{i\in\Lambda}\sum_{X:|X|\leq k,X\partial\ni
i}\|o_{X}\|\left\|\text{exp}\left(\frac{\beta}{2}\sum_{\overset{B:B\cap
X\neq\emptyset}{\text{diam}(B)\leq r}}\Phi_{B}\right)\right\|$ (249)
$\displaystyle\leq 4g\text{exp}\left(\frac{\beta Jc_{r,k,\nu}}{2}\right)\equiv
4gC,$ (250)

where $\hat{\text{tr}}_{i}$ is the normalized partial trace. The last two
inequalities follow from

$\displaystyle\|\Delta_{\sigma}^{\frac{1}{2}}(o_{X})\|$
$\displaystyle=\|\sigma^{\frac{1}{2}}o_{X}\sigma^{-\frac{1}{2}}\|=\|e^{-\frac{\beta}{2}\sum_{B\cap
X\neq\emptyset}\Phi_{B}}o_{X}e^{\frac{\beta}{2}\sum_{B\cap
X\neq\emptyset}\Phi_{B}}\|$ (251)
$\displaystyle\leq\|o_{X}\|\|\text{exp}\left(\frac{\beta}{2}\sum_{\overset{B:B\cap
X\neq\emptyset}{|X|\leq k,\text{diam}(B)\leq
r}}\Phi_{B}\right)\|\leq\|o_{X}\|\text{exp}\left(\frac{\beta
Jc_{r,k,\nu}}{2}\right).$ (252)

Here $c_{r,k,\nu}$ is a constant which depends only on the locality of $O$,
the geometric locality $r$ of the Hamiltonian, and the growth constant $\nu$
of the graph. ∎

####  6.3.3 Ensemble equivalence under (long-range) Lipschitz observables

The canonical ensemble with inverse temperature $\beta$ is given by the Gibbs
state $\sigma_{\beta}$, where we now write the temperature dependence
explicitly.

Let $\sigma_{E,\delta}$ be the microcanonical ensemble with average energy $E$
defined on an energy-shell width $\delta$, so that if $P$ is the spectral
measure of the Hamiltonian
$H=\sum_{E}EP(E)\equiv\sum_{m}E_{m}\outerproduct{E_{m}}{E_{m}}$ then

$\displaystyle\sigma_{E,\delta}:=\frac{P((E-\delta,E])}{\Tr[P((E-\delta,E])]}=\frac{1}{\mathcal{N}_{E,\delta}}\sum_{E_{m}\in(E-\delta,E]}\outerproduct{E_{m}}{E_{m}}.$
(253)

Now, let $E=\text{arg}\max_{E\in\mathbb{R}}\left(e^{-\beta
E}\mathcal{N}_{E,\delta}\right)$, where
$\mathcal{N}_{E,\delta}:=\Tr[P((E-\delta,E])]$ is the number of eigenstates in
the energy interval $(E-\delta,E]$. This defines the average energy of the
microcanonical ensemble corresponding to the energy of $\sigma_{\beta}$ [71,
48].

Two ensembles represented respectively by the families of states
$\\{\sigma_{1}^{\Gamma},\sigma_{2}^{\Gamma}\\}_{\Gamma\subset\subset\Lambda}$
are said to be equivalent if, in the thermodynamic limit, they produce the
same expectation values on averaged geometrically-local observable
$\frac{O}{|\Gamma|}=\frac{1}{|\Gamma|}\sum_{i=1}^{|\Gamma|}O_{i}$, with
$\|O_{i}\|\leq g$ [48]. That is, if for any such observable

$\displaystyle\left|\Tr\left[\sigma_{1}^{\Gamma}\frac{O}{|\Gamma|}\right]-\Tr\left[\sigma_{2}^{\Gamma}\frac{O}{|\Gamma|}\right]\right|=\frac{1}{|\Gamma|}|\Tr[(\sigma_{1}^{\Gamma}-\sigma_{2}^{\Gamma})O]|\overset{|\Gamma|\rightarrow\infty}{\longrightarrow}0.$
(254)

In [49] it was shown that the microcanonical and canonical ensembles are
equivalent in this sense when the system satisfies suitable concentration
bounds, such as Eq. (245). We extend this notion of equivalence in the 1D case
to the more general class of Lipschitz observables, defined as
$O\in\mathcal{B}(\mathcal{H}_{\Gamma})$, s.t. $\|O\|_{L}<\infty$. These
notably include long-range locally bounded, $k-$local observables of the form
of Eq. (244).

###### Corollary 6.6 (Corollary 2 of [60] applied to our setting)

For any Lipschitz observable $O\in\mathcal{A}_{\Lambda}$, i.e.
$O\in\mathcal{A}_{\Lambda}$ s.t. $\|O\|_{L}<\infty$ and for
$\sigma_{E,\delta},\sigma_{\beta}$ the micro- and canonical ensemble states
with the same energy $E=\textup{arg}\max_{E\in\mathbb{R}}\left(e^{-\beta
E}\mathcal{N}_{E,\delta}\right)$, respectively, it holds that

$\displaystyle\frac{1}{|\Gamma|}|\Tr[\sigma_{E,\delta}O]-\Tr[\sigma_{\beta}O]|\leq\|O\|_{L}\hskip
2.84526pto(1)_{|\Gamma|\to\infty},$ (255)

for all $\sigma_{E},\delta$ such that $\delta=e^{-\mathcal{O}(|\Gamma|)}$.

###### Proof.

This follows directly from [60, Corollary 2] when employing the linearity of
the transport cost in the system size. The proof follows from the transport
cost inequality, which is used to bound the relative entropy between the
microcanonical and canonical ensembles, which yields the result. ∎

##  7 Outlook

In this paper, we have addressed the problem of estimating the speed of
convergence of certain quantum dissipative evolutions governed by Lindbladians
to their steady states, given by Gibbs states of local, commuting
Hamiltonians. We have particularly considered a large class of Davies
generators with nearest neighbour, commuting Hamiltonians, and derived for
them a positive (and, in some cases, system-size independent) MLSI from either
a positive spectral gap in the Lindbladian, or at high enough temperature.

After the completion of this work, the knowledge about existence of positive
MLSIs, and therefore rapid mixing, for various physically relevant systems
stands as follows:

  * •

For 1D systems, for Davies generators with k-local, commuting, translation-
invariant Hamiltonians, at any finite, inverse temperature $\beta$, a log-
decreasing MLSI
$\alpha(\mathcal{L}_{\Gamma}^{D})=\Omega((\log|\Gamma|)^{-1})_{|\Gamma|\rightarrow\infty}$
was derived in [10, 9]. Here, we have lifted that to a constant MLSI
$\alpha(\mathcal{L}_{\Gamma}^{D})=\Omega(1)_{|\Gamma|\rightarrow\infty}$ and
extended it to non-translation-invariant Hamiltonians. Both results yield
rapid mixing.

  * •

For 2D systems, for Davies generators with k-local, commuting, Hamiltonians,
at any finite, inverse temperature $\beta$, a positive spectral gap was shown
to be equivalent to $\mathbb{L}_{2}$-clustering in [43]. Here, we have shown
that, whenever the correlation length is small enough, this is further
equivalent to a square-root decreasing MLSI
$\alpha(\mathcal{L}_{\Gamma}^{D})=\Omega(\sqrt{|\Gamma|}^{-1})_{|\Gamma|\rightarrow\infty}$,
yielding a $\mathcal{O}(\sqrt{|\Gamma|}\log|\Gamma|)$-mixing time.

  * •

For high D systems and their Schmidt generators with nearest neighbour,
commuting, Hamiltonians, a positive, constant MLSI
$\alpha(\mathcal{L}_{\Gamma}^{D})=\Omega(1)_{|\Gamma|\rightarrow\infty}$ was
shown to hold at high-enough temperature in [24]. Here, we have extended this
to Davies generators of nearest neighbour, commuting, Hamiltonians at high-
enough temperature, concluding for them rapid mixing.

  * •

For trees, we have provided the first results in the direction of rapid
mixing, showing a log-decreasing MLSI
$\alpha(\mathcal{L}_{\Gamma}^{D})=\Omega((\log|\Gamma|)^{-1})_{|\Gamma|\rightarrow\infty}$
whenever the correlation length is small enough.

This shows definite progress in our understanding of the mixing times of
Davies generators with commuting Hamiltonians, for which we aim to eventually
have a complete picture. In this respect, there are still unsolved natural
questions, such as:

  * •

Can we extend the results for high D systems to geometrically-$k$-local
Hamiltonians? This would require a totally new approach, as the Schmidt
generators cannot be defined beyond nearest neighbour interactions.

  * •

Can we provide a more accurate result for specific models? For instance, for
Davies generators associated to quantum double models, a positive spectral gap
was shown to hold at any positive temperature in [51], extending the prior [2,
50]. Therefore, it might be possible that a similar behaviour translates to
the mixing time, yielding rapid mixing at every positive temperature.

The natural extension of this problem to the context of non-commuting
Hamiltonians is much less understood. As the Davies generator loses its
desirable properties whenever the Hamiltonian considered is non-commuting,
specially the locality, an alternative, physically-relevant Lindbladian needs
to be considered in this case. The recent [27] introduced a possible quasi-
local Lindbladian that by-passes the problems induced by the Davies, which
some of the authors of this manuscript recently showed to be gapped in [68].
The question whether this can be lifted to a positive MLSI, or any other means
to derive rapid mixing, even for any specific case (e.g. 1D systems) remains
open. An alternative approach to estimate mixing times of Lindbladians has
just been introduced in [34], but it is unclear whether it could be employed
in this context.

The problem of estimating mixing times of Lindbladians has been well explored
in the quantum community in the past few years, with a notable increase of
interest very recently. The reason for this partly stems from an upraise in
the tools at our disposal to tackle this problem, and the appearance of more
applications derived from it. In particular, rapid mixing constitutes a
fundamental approach for quantum Gibbs sampling, as stated in [43, 25, 27,
24], which nevertheless can also be explored with alternative approaches as in
[7]. It has also been recently shown [73, 42] that systems with rapid mixing
are promising candidates for quantum advantage experiments in the context of
analogue open quantum simulation.

_Acknowledgments:_ A.C. acknowledges the support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
470903074 – TRR 352. A.M.A. acknowledges support from the Spanish Agencia
Estatal de Investigacion through the grants “IFT Centro de Excelencia Severo
Ochoa CEX2020-001007-S” and “Ramón y Cajal RyC2021-031610-I”, financed by
MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR.
This project was funded within the QuantERA II Programme that has received
funding from the EU’s H2020 research and innovation programme under the GA No
101017733.

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##  Appendix A Proof of properties of the relation from Definition 4.8

This section contains the proofs of Proposition 4.11, Corollary 4.12, and
Proposition 4.9.

###### Proof of Proposition 4.11.

  1. $1)$

The reflexivity is trivial.

  2. $2)$

For symmetry, notice that the spectra and thus the spectral radii of
$B^{-\frac{1}{2}}AB^{-\frac{1}{2}}-\mathbbm{1}$ and
$A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}-\mathbbm{1}$ are the same. Since they
are both normal, we have
$\|B^{-\frac{1}{2}}AB^{-\frac{1}{2}}-\mathbbm{1}\|\leq\epsilon$ which proves
the second implication. Now the first implication follows via

$\displaystyle\|B^{\frac{1}{2}}A^{-1}B^{\frac{1}{2}}-\mathbbm{1}\|=\|(B^{-\frac{1}{2}}AB^{-\frac{1}{2}})^{-1}-\mathbbm{1}\|$
$\displaystyle=\Big{\|}\sum_{k=1}^{\infty}{(-1)^{k}}(B^{-\frac{1}{2}}AB^{-\frac{1}{2}}-\mathbbm{1})^{k}\Big{\|}$
(256)
$\displaystyle\leq\sum_{k=1}^{\infty}\|B^{-\frac{1}{2}}AB^{-\frac{1}{2}}-\mathbbm{1}\|^{k}$
(257) $\displaystyle\leq\frac{\epsilon}{1-\epsilon},$ (258)

where in the last line we used that by assumption
$\|B^{-\frac{1}{2}}AB^{-\frac{1}{2}}-\mathbbm{1}\|\leq\epsilon$.

  3. $3)$

For transitivity, notice that

$\displaystyle\|A^{\frac{1}{2}}C^{-1}A^{\frac{1}{2}}-\mathbbm{1}\|$
$\displaystyle=\|A^{\frac{1}{2}}B^{-\frac{1}{2}}(B^{\frac{1}{2}}C^{-1}B^{\frac{1}{2}}-\mathbbm{1}+\mathbbm{1})B^{-\frac{1}{2}}A^{\frac{1}{2}}-\mathbbm{1}\|$
(259)
$\displaystyle\leq\|A^{\frac{1}{2}}B^{-\frac{1}{2}}(B^{\frac{1}{2}}C^{-1}B^{\frac{1}{2}}-\mathbbm{1})B^{-\frac{1}{2}}A^{\frac{1}{2}}\|+\|A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}-\mathbbm{1}\|$
(260)
$\displaystyle\leq\|(B^{\frac{1}{2}}C^{-1}B^{\frac{1}{2}}-\mathbbm{1})(B^{-\frac{1}{2}}AB^{-\frac{1}{2}})\|+\|A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}-\mathbbm{1}\|$
(261) $\displaystyle\leq\epsilon_{2}(1+\epsilon_{1})+\epsilon_{1}=\eta.$ (262)

For the second inequality, note that
$\|XYX^{*}\|=|\text{spec}(XYX^{*})|=|\text{spec}(X^{*}XY)|\leq\|X^{*}XY\|$
holds, for $X,Y$ some operators with $Y$ self-adjoint, where
$|\text{spec}(X)|$ denotes the spectral radius of $X$. Using this with
$X=A^{\frac{1}{2}}B^{-\frac{1}{2}}$ and
$Y=B^{\frac{1}{2}}C^{-1}B^{\frac{1}{2}}-\mathbbm{1}$ then gives the second
inequality. The third inequality follows from the assumptions and the fact
that $\|B^{-\frac{1}{2}}AB^{-\frac{1}{2}}\|\leq 1+\epsilon_{1}$ by the second
implication of symmetry.

  4. $4)$

For the tensor multiplicativity, notice that

$\displaystyle\|(A\otimes
F)^{\frac{1}{2}}(\tilde{A}\otimes\tilde{F})^{-1}(A\otimes
F)^{\frac{1}{2}}-\mathbbm{1}\|$
$\displaystyle=\|A^{\frac{1}{2}}\tilde{A}^{-1}A^{\frac{1}{2}}\otimes
F^{\frac{1}{2}}\tilde{F}^{-1}F^{\frac{1}{2}}-\mathbbm{1}\otimes\mathbbm{1}\|$
(263)
$\displaystyle\leq\|(A^{\frac{1}{2}}\tilde{A}^{-1}A^{\frac{1}{2}}-\mathbbm{1})\otimes(F^{\frac{1}{2}}\tilde{F}^{-1}F^{\frac{1}{2}}-\mathbbm{1})\|$
(264)
$\displaystyle\;\;\;+\|(A^{\frac{1}{2}}\tilde{A}^{-1}A^{\frac{1}{2}}-\mathbbm{1})\otimes\mathbbm{1}\|+\|\mathbbm{1}\otimes(F^{\frac{1}{2}}\tilde{F}^{-1}F^{\frac{1}{2}}-\mathbbm{1})\|$
(265)
$\displaystyle\leq\epsilon_{1}\epsilon_{2}+\epsilon_{1}+\epsilon_{2}=\eta.$
(266)

  5. $5)$

The following chain of implications holds

$\displaystyle\|D^{\frac{1}{2}}E^{-1}D^{\frac{1}{2}}-\mathbbm{1}\|\leq\epsilon$
$\displaystyle\Leftrightarrow(1-\epsilon)\,\mathbbm{1}\leq
D^{\frac{1}{2}}E^{-1}D^{\frac{1}{2}}\leq(1+\epsilon)\,\mathbbm{1}$ (267)
$\displaystyle\Leftrightarrow\frac{1}{1+\epsilon}\,\mathbbm{1}\leq
D^{-\frac{1}{2}}ED^{-\frac{1}{2}}\leq\frac{1}{1-\epsilon}\,\mathbbm{1}$ (268)
$\displaystyle\Leftrightarrow\frac{D}{1+\epsilon}\leq
E\leq\frac{D}{1-\epsilon}\,.$ (269)

Since $P$ is self-adjoint the above implies that the following holds:

$\displaystyle\frac{PDP}{1+\epsilon}\leq
PEP\leq\frac{PDP}{1-\epsilon}\implies\frac{\tr_{\mathcal{H}^{\prime}}PDP}{1+\epsilon}\leq\tr_{\mathcal{H}^{\prime}}PEP\leq\frac{\tr_{\mathcal{H}^{\prime}}PDP}{1-\epsilon}\,,$
(270)

where we identified $P$ with $\mathbbm{1}\otimes P$ for sake of clarity. The
above holds also in reverse on
$\operatorname{supp}{\tr_{\mathcal{H}^{\prime}}(PDP)}=\operatorname{supp}{\tr_{\mathcal{H}^{\prime}}(PEP)}$.
The fact that the (operator algebraic) supports of these two are the same is
directly implied by the previous sandwich inequality and positivity. So it
implies that

$\|(\tr_{\mathcal{H}^{\prime}}PDP)^{\frac{1}{2}}(\tr_{\mathcal{H}^{\prime}}PEP)^{-1}(\tr_{\mathcal{H}^{\prime}}PDP)^{\frac{1}{2}}-\mathbbm{1}_{\operatorname{supp}(\tr_{\mathcal{H}^{\prime}}(PDP)})\|\leq\epsilon\,,$
(271)

where $(\cdot)^{-1}$ represents the generalized inverse.

  6. $6)$

First note that if $D\overset{\epsilon}{\sim}E$, then
$D\overset{\mu}{\sim}\lambda E$ for $\lambda>0$, where
$\mu=\frac{\epsilon}{\lambda}+|1-\frac{1}{\lambda}|$, since

$\displaystyle\|D^{\frac{1}{2}}(\lambda
E)^{-1}D^{\frac{1}{2}}-\mathbbm{1}\|\leq\lambda^{-1}\,\|D^{\frac{1}{2}}E^{-1}D^{\frac{1}{2}}-\mathbbm{1}\|+\|\lambda^{-1}\mathbbm{1}-\mathbbm{1}\|.$
(272)

Now since
$D\overset{\epsilon}{\sim}E\implies\tr_{\mathcal{H}^{\prime}}PDP\overset{\epsilon}{\sim}\tr_{\mathcal{H}^{\prime}}PEP\Leftrightarrow\frac{\tr_{\mathcal{H}^{\prime}}PDP}{1+\epsilon}\leq\tr_{\mathcal{H}^{\prime}}PEP\leq\frac{\tr_{\mathcal{H}^{\prime}}PDP}{1-\epsilon}\implies
1+\epsilon\geq\frac{\Tr[PDP]}{\Tr[PEP]}\geq 1-\epsilon$ by the proof of
property 5). It follows that

$\displaystyle\|\mathscr{X}\|$
$\displaystyle\equiv\Big{\|}\left(\frac{\tr_{\mathcal{H}^{\prime}}PDP}{\Tr[PDP]}\right)^{\frac{1}{2}}\left(\frac{\tr_{\mathcal{H}^{\prime}}PEP}{\Tr[PEP]}\right)^{-1}\left(\frac{\tr_{\mathcal{H}^{\prime}}PDP}{\Tr[PDP]}\right)^{\frac{1}{2}}-\mathbbm{1}\Big{\|}$
(273)
$\displaystyle=\|(\tr_{\mathcal{H}^{\prime}}PDP)^{\frac{1}{2}}(\tr_{\mathcal{H}^{\prime}}PEP)^{-1}(\tr_{\mathcal{H}^{\prime}}PDP)^{\frac{1}{2}}\lambda-\mathbbm{1}\|,$
(274)

with $1-\epsilon\leq\lambda^{-1}=\frac{\Tr[PDP]}{\Tr[PEP]}\leq 1+\epsilon$.
Hence
$\|\mathscr{X}\|\leq\epsilon\lambda^{-1}+|1-\lambda^{-1}|\leq\epsilon(1+\epsilon)+\epsilon=\epsilon(2+\epsilon)$.

∎

###### Proof of Corollary 4.12.

The corollary is easily proved by induction using the transitivity of the
relation, spelled out here for convenience. Assume
$A_{i}\overset{\epsilon}{\sim}A_{i+1}$ for all $i$ and set
$A_{0}\overset{\eta_{k}}{\sim}A_{k}$, then by transitivity we have the
recursion

$\displaystyle\eta_{k+1}\leq\eta_{k}(1+\epsilon)+\epsilon.$ (275)

For $k=1$, noting that $\eta_{1}=\epsilon$, this gives
$\eta_{2}=\epsilon(1+\epsilon)+\epsilon=\epsilon^{2}+2\epsilon=(1+\epsilon)^{2}-1$.
For any $k$ we have by induction that

$\displaystyle\eta_{k+1}\leq\eta_{k}(1+\epsilon)+\epsilon=((1+\epsilon)^{k}-1)(1+\epsilon)+\epsilon=(1+\epsilon)^{k+1}-1.$
(276)

∎

###### Proof of Proposition 4.9.

Assume that $\operatorname{supp}(\omega)\subset\operatorname{supp}(\tau)$,
else the statement clearly holds. We may now WLOG assume the supports to be
equal. Set $\epsilon=\|\omega^{\frac{1}{2}}\tau^{-1}\omega^{\frac{1}{2}}\|$,
then the lower bound follows from

$\displaystyle
D_{\text{max}}(\omega\|\tau)=\log\|\omega^{\frac{1}{2}}\tau^{-1}\omega^{\frac{1}{2}}\|\leq\log\|\omega^{\frac{1}{2}}\tau^{-1}\omega^{\frac{1}{2}}-\mathbbm{1}+\mathbbm{1}\|\leq\log(1+\epsilon)\leq\epsilon.$
(277)

For the upper bound see that for
$X:=\omega^{\frac{1}{2}}\tau^{-1}\omega^{\frac{1}{2}}\geq 0$ we have

$\displaystyle\|\mathbbm{1}-X\|$ $\displaystyle=\|\mathbbm{1}-\exp\log
X\|=\left\|\mathbbm{1}-\sum_{n=0}^{\infty}\frac{(\log
X)^{n}}{n!}\right\|=\left\|\sum_{n=1}^{\infty}\frac{(\log
X)^{n}}{n!}\right\|=\left\|\sum_{n=1}^{\infty}\frac{(\log X)^{n}}{n!}\right\|$
(278) $\displaystyle=\left\|\log X\sum_{n=0}^{\infty}\frac{1}{n}\frac{(\log
X)^{n}}{n!}\right\|\leq\|\log X\|\exp(\|\log X\|),$ (279)

where the last inequality follows from subadditivity aubmultiplicativity of
the norm and $\frac{1}{n}\leq 1$. This is the desired upper bound, hence we
are done. ∎

##  Appendix B Proof of Lemma 4.17

###### Proof of Lemma 4.17..

Let us prove each item separately:

  1. $1)$

If $\Lambda$ and $\Phi$ are chosen as in the statement of the lemma, then
there exists a constant $c_{r,\nu}$ depending only on $r,\nu$ such that

$\displaystyle\|E_{A,B}^{\pm 1}\|$ $\displaystyle=\|\operatorname{e}^{\mp\beta
H_{AB}}\operatorname{e}^{\pm\beta H_{A}\pm\beta H_{B}}\|$ (280)
$\displaystyle=\Big{\|}\exp\\{\mp\beta\sum_{X\cap A\neq\emptyset,X\cap
B\neq\emptyset}\Phi_{X}\Big{\\}}\Big{\|}$ (281)
$\displaystyle\leq\exp\\{\Big{(}\beta J\sum_{\overset{X\cap
A\neq\emptyset,X\cap B\neq\emptyset}{\text{diam}(X)\leq r}}1\Big{)}\Big{\\}}$
(282) $\displaystyle\leq\exp(\beta
Jc_{r,\nu}\min\\{|\partial_{B}A|,|\partial_{A}B|\\})=:K_{A,B}=\exp{(\mathcal{O}(\beta|\partial_{A,B}|))},$
(283)

where we are denoting $\partial_{A}B:=(\partial B),\cap A$,
$\partial_{B}A:=(\partial A),\cap B$, and
$\partial_{A,B}:=\min\\{|\partial_{B}A|,|\partial_{A}B|\\}$. Hence it also
holds

$\displaystyle\|E_{A,B}^{\pm 1}E_{AB,C}^{\pm 1}\|\leq\|E_{A,B}^{\pm
1}\|\|E_{AB,C}^{\pm 1}\|\leq K_{A,B}K_{B,C}.$ (284)

  2. $2)$

Consider the map

$\begin{array}[]{cccc}\mathcal{B}(\mathcal{H}_{\Gamma})&\to&\mathcal{B}(\mathcal{H}_{\Gamma\setminus
B})&\\\
Q&\mapsto&\tr_{B}[\sigma^{B}Q]&=\frac{1}{\Tr[e^{-H_{B}}]}\tr_{B}[e^{-\frac{1}{2}H_{B}}Qe^{-\frac{1}{2}H_{B}}]\,,\end{array}$
(285)

which is positive and unital. Note that if $Q>0$ is strictly positive, then so
is $Q^{-1}>0$ and $\lambda_{\min}(Q)\mathbbm{1}=\|Q^{-1}\|^{-1}\mathbbm{1}\leq
Q\leq\|Q\|\mathbbm{1}=\lambda_{\max}(Q)\mathbbm{1}$. Applying the
aforementioned map to this inequality immediately gives that

$\displaystyle\|Q^{-1}\|^{-1}\mathbbm{1}_{B^{c}}\leq\tr_{B}[\sigma^{B}Q]\leq\|Q\|\mathbbm{1}_{B^{c}},$
(286)
$\displaystyle\|Q\|^{-1}\mathbbm{1}_{B^{c}}\leq\tr_{B}[\sigma^{B}Q]^{-1}\leq\|Q^{-1}\|\mathbbm{1}_{B^{c}},$
(287)

since inversion of two commuting operators is order reversing. We conclude by
taking norms.

  3. $3)$

This is just a special case of $2)$ given $1)$, since each of the $E_{A,B}$
are strictly positive, as the potential $\Phi$ is commuting, e.g. by self-
adjointness and the spectral theorem.

∎

##  Appendix C Proof of strong local indistinguishability in 1D

This section is devoted to the proof of Theorem 4.16.

###### Proof of Theorem 4.16:.

The proof follows a similar path as the proof of [16, Proposition 8.1] and
uses local indistinguishability (see e.g. [18, Theorem 5], [47, Corollary 2]),
which follows from the assumptions of the theorem. Thus, we know that

$\|\tr_{BC}\sigma^{ABC}-\tr_{B}\sigma^{AB}\|_{1}\leq
K^{\prime}e^{-a^{\prime}l}\,,$ (288)

with $K^{\prime},a^{\prime}>0$ depending only on $r,J\beta$.

Using the notation $E_{X,Y}:=e^{-H_{XY}}e^{H_{X}+H_{Y}}$ for $X,Y\subset I$
disjoint, we rewrite:

$\displaystyle(\tr_{BC}[\sigma^{ABC}])(\tr_{B}[\sigma^{AB}])^{-1}$ (289)
$\displaystyle\hskip
42.67912pt=\tr_{BC}[e^{-H_{ABC}}]\tr_{B}[e^{-H_{AB}}]^{-1}\Tr[e^{-H_{ABC}}]^{-1}\Tr[e^{-H_{AB}}]$
(290) $\displaystyle\hskip
42.67912pt=\tr_{BC}[e^{-H_{ABC}}]e^{H_{A}}e^{-H_{A}}\tr_{B}[e^{-H_{AB}}]^{-1}\Tr[e^{-H_{ABC}}]^{-1}\Tr[e^{-H_{AB}}]$
(291) $\displaystyle\hskip
42.67912pt=\tr_{BC}[e^{-H_{BC}}e^{-H_{ABC}}e^{H_{A}}e^{H_{BC}}]\tr_{B}[e^{-H_{B}}e^{-H_{AB}}e^{H_{A}}e^{H_{B}}]^{-1}\Tr[e^{-H_{ABC}}]^{-1}\Tr[e^{-H_{AB}}]$
(292) $\displaystyle\hskip
42.67912pt=\tr_{BC}[\sigma^{BC}E_{A,BC}]\tr_{B}[\sigma^{B}E_{A,B}]^{-1}\underbrace{\frac{\Tr[e^{-H_{AB}}]\Tr[e^{-H_{BC}}]}{\Tr[e^{-H_{ABC}}]\Tr[e^{-H_{B}}]}}$
(293) $\displaystyle\hskip
42.67912pt\equiv\tr_{BC}[\sigma^{BC}E_{A,BC}]\tr_{B}[\sigma^{B}E_{A,B}]^{-1}\hskip
12.80365pt\cdot\hskip 12.80365pt\lambda_{ABC}^{-1}.$ (294)

In the second line we multiplied by $\mathbbm{1}=e^{H_{A}}e^{-H_{A}}$, which
in the third line we can separate and pull into the partial traces, since
neither of them trace out the region $A$. Thus we may rewrite

$\displaystyle\|(\tr_{BC}\sigma^{ABC})(\tr_{B}\sigma^{AB})^{-1}-$
$\displaystyle\mathbbm{1}\|$ (295) $\displaystyle\leq$
$\displaystyle\|(\tr_{BC}\sigma^{ABC})(\tr_{B}\sigma^{AB})^{-1}-(\tr_{BC}[\sigma^{BC}E_{A,BC}])(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\|$
(296)
$\displaystyle+\|(\tr_{BC}[\sigma^{BC}E_{A,BC}])(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}-\mathbbm{1}\|$
(297) $\displaystyle\leq$
$\displaystyle\|\tr_{BC}[\sigma^{BC}E_{A,BC}]\|\|(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\||\lambda_{ABC}^{-1}-1|$
(298)
$\displaystyle+\|(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\|\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{B}[\sigma^{B}E_{A,B}]\|\,.$
(299)

By [16, Corollary 4], it holds that $\|(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\|\leq
C$ and $\|\tr_{BC}[\sigma^{BC}E_{A,BC}]\|\leq C$, for the same constant $C$
depending only on $r,J,\beta$. Furthermore in [16, Step 2], it is proven that
$\lambda_{ABC}-1$ decays exponentially, thus by the geometric series so does
$\lambda_{ABC}^{-1}-1$, i.e. there exist
$K^{\prime\prime},a^{\prime\prime}>0$, depending only on $r,J\beta$ s.t.

$|\lambda_{ABC}^{-1}-1|\leq K^{\prime\prime}e^{-a^{\prime\prime}l}\,.$ (300)

Hence, it remains to bound
$\|(\tr_{BC}[\sigma^{BC}E_{A,BC}])-(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\|$
exponentially in $l$. To do this we adopt a similar strategy to [16, Step 3].
Let us split $B=B_{1}B_{2}$ into to halves, such that $|B_{1}|=|B_{2}|=l$, and
write

$\displaystyle\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{B}[\sigma^{B}E_{A,B}]\|\leq$
$\displaystyle\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{BC}[\sigma^{BC}E_{A,B_{1}}]\|$
(301)
$\displaystyle+\|\tr_{BC}[\sigma^{BC}E_{A,B_{1}}]-\tr_{B}[\sigma^{B}E_{A,B_{1}}]\|$
(302)
$\displaystyle+\|\tr_{B}[\sigma^{B}E_{A,B_{1}}]-\tr_{B}[\sigma^{B}E_{A,B}]\|\,.$
(303)

Here we just used the triangle inequality of the operator norm twice. For the
first and third summand in Equation (301) use that the map
$Q\mapsto\tr_{X}[\sigma^{X}Q]$ is a contraction in $B(\mathcal{H}_{\Gamma})\to
B(\mathcal{H}_{\Gamma\setminus X})$ by the Russo Dye theorem (see e.g lemma
4.17 or [16, Section 3.4]), and thus

$\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{BC}[\sigma^{BC}E_{A,B_{1}}]\|\leq\|E_{A,BC}-E_{A,B_{1}}\|\,,$
(304)

where the right-hand side is exponentially decaying in $|B_{1}|=l$ by [16,
Corollary 3.4 and Remark 3.5]. The same holds true for

$\|\tr_{B}[\sigma^{B}E_{A,B_{1}}]-\tr_{B}[\sigma^{B}E_{A,B}]\|\leq\|E_{A,B_{1}}-E_{A,B}\|\,.$
(305)

For the second summand in Equation (301) we use [16, Proposition 8.5], which
gives

$\|\tr_{BC}[\sigma^{BC}E^{*}_{A,B_{1}}]-\tr_{B}[\sigma^{B}E^{*}_{A,B_{1}}]\|\leq\tilde{K}e^{-\tilde{a}l}$
(306)

for some $\tilde{K},\tilde{a}>0$, depending only on $r,J,\beta$. The same
holds true for the adjoints, which is exactly the second summand. In total we
have that there exist $K^{\prime\prime\prime},a^{\prime\prime\prime}>0$
depending only on $r$,$J,\beta$ such that

$\displaystyle\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{B}[\sigma^{B}E_{A,B}]\|\leq
K^{\prime\prime\prime}e^{-a^{\prime\prime\prime}l}.$ (307)

Now putting all of the above together we have our desired result.

$\displaystyle\|(\tr_{BC}[\sigma^{ABC}])(\tr_{B}[\sigma^{AB}])^{-1}-\mathbbm{1}\|$
$\displaystyle\leq\|\tr_{BC}[\sigma^{BC}E_{A,BC}]\|\|(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\||\lambda_{ABC}^{-1}-1|$
(308)
$\displaystyle\quad+\|(\tr_{B}[\sigma^{B}E_{A,B}])^{-1}\|\|\tr_{BC}[\sigma^{BC}E_{A,BC}]-\tr_{B}[\sigma^{B}E_{A,B}]\|$
(309) $\displaystyle\leq
C^{2}K^{\prime\prime}e^{-a^{\prime\prime}l}+CK^{\prime\prime\prime}e^{-a^{\prime\prime\prime}l}\,.$
(310)

∎

##  Appendix D Proof of properties of Schmidt conditional expectations

###### Proof of Lemma 3.7.

If $\operatorname{dist}(A_{1},A_{2})\geq 2$, then by definition of the
algebras $\mathcal{N}_{A_{1}}$ and $\mathcal{N}_{A_{2}}$ have only
$\mathbbm{1}=\mathcal{N}_{\emptyset}=\mathcal{N}_{A_{1}\cap A_{2}}$ in common.
Since it holds, that $\partial A_{1}\cup\partial A_{2}=\partial(A_{1}\cup
A_{2})$ their union is given by

$\displaystyle\mathcal{N}_{A_{1}}\cup\mathcal{N}_{A_{2}}$
$\displaystyle=\mathcal{B}(\mathcal{H}_{A_{1}})\otimes\mathcal{B}(\mathcal{H}_{A_{2}})\otimes\mathbbm{1}_{\mathcal{H}_{(A_{1})^{c}}}\otimes\mathbbm{1}_{\mathcal{H}_{(A_{2})^{c}}}\bigotimes_{i\in
I_{A_{1}}}\bigotimes_{j\in
J^{(i)}\setminus\\{0\\}}\mathcal{A}^{j}_{b_{j}}\otimes\bigotimes_{i\in
I_{A_{2}}}\bigotimes_{j\in J^{(i)}\setminus\\{0\\}}\mathcal{A}^{j}_{b_{j}}$
(311) $\displaystyle=\mathcal{B}(\mathcal{H}_{A_{1}\cup
A_{2}})\otimes\mathbbm{1}_{\mathcal{H}_{(A_{1}\cup
A_{2})^{c}}}\otimes\bigotimes_{i\in I_{A_{1}\cup A_{2}}}\bigotimes_{j\in
J^{(i)}\setminus\\{0\\}}\mathcal{A}^{j}_{b_{j}}$ (312)
$\displaystyle=\mathcal{N}_{A_{1}\cup A_{2}}$ (313)

For the other case we can WLOG assume $A_{1}\subset A_{2}$, hence $A_{1}\cup
A_{2}=A_{2}$ and $A_{1}\cap A_{2}=A_{1}$. Then clearly
$\mathcal{N}_{A_{1}}\cup\mathcal{N}_{A_{2}}=\mathcal{N}_{A_{2}}=\mathcal{N}_{A_{1}\cup
A_{2}}$. Similarly
$\mathcal{N}_{A_{1}}\cap\mathcal{N}_{A_{2}}=\mathcal{N}_{A_{1}}=\mathcal{N}_{A_{1}\cap
A_{2}}$. ∎

##  Appendix E Proof of Case 2 in Lemma 5.11

###### Proof of Case 2 in Lemma 5.11.

Let $N\in\mathbb{N}\setminus\\{1\\}$ and
$\frac{1}{2}<\epsilon<1,s.t.\frac{1}{2}+\frac{1}{2N}\geq\epsilon$. We
enumerate a maximal set of partitions of $B_{x_{j},L}$ into
$\\{C_{j}^{\tilde{l}},D_{j}^{\tilde{l},\frac{L}{N}}\\}$, s.t.
$\operatorname{height}(C_{j}^{\tilde{l}})=\tilde{l},\operatorname{height}(D_{j}^{\tilde{l},\frac{L}{N}})=L+\frac{L}{N}-\tilde{l}\leq\epsilon
L$ and s.t. different partitions have disjoint overlaps, i.e.
$\left(C_{j}^{\tilde{l}_{1}}\cap
D_{j}^{\tilde{l}_{1},\frac{L}{N}}\right)\cap\left(C_{j}^{\tilde{l}_{2}}\cap
D_{j}^{\tilde{l}_{2},\frac{L}{N}}\right)=\emptyset$, whenever
$\tilde{l}_{1}\neq\tilde{l}_{2}$. There exist $\frac{L}{\frac{L}{N}}=N$ of
these partitions, since their overlap is of ’width’ $\frac{L}{N}$. Call these
partitions $\\{C_{i},D_{i}\\}^{N}_{i=1}$. Now we average over all the
approximate tensorization results of these partitions to get

$\displaystyle D(\omega\|E^{S}_{B_{x_{j},L}*}(\omega))$
$\displaystyle\leq\frac{1}{N}\sum_{i=1}^{N}\frac{1}{1-2\eta_{C_{i},D_{i}}(\frac{L}{N})}\left[D(\omega\|E^{S}_{C_{i}*}(\omega))+D(\omega\|E^{S}_{D_{i}*}(\omega))\right]$
(314)
$\displaystyle\leq\frac{1}{1-2\eta(\frac{L}{N})}\frac{1}{N}\sum_{i=1}^{N}C(\operatorname{height}(C_{i}))D_{C_{i}}(\omega)+C(\operatorname{height}(D_{i}))D_{D_{i}}(\omega)$
(315) $\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\frac{L}{N})}\frac{1}{N}\Bigg{(}\sum_{i=1}^{N}\left(2D_{C_{i}\cap
D_{i}}(\omega)+D_{C_{i}\setminus D_{i}\cup D_{i}\setminus
C_{i}}(\omega)\right.$ (316)
$\displaystyle\quad+\sum_{\underset{R_{k}\cap(C_{i}\setminus
D_{i})\neq\emptyset}{R_{k}\cap(C_{i}\cap
D_{i})\neq\emptyset}}D(\omega\|E^{S}_{R_{k}*}(\omega))+\sum_{\underset{R_{k}\cap(D_{i}\setminus
C_{i})\neq\emptyset}{R_{k}\cap(C_{i}\cap
D_{i})\neq\emptyset}}D(\omega\|E^{S}_{R_{k}*}(\omega))\Bigg{)}$ (317)
$\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\frac{L}{N})}\frac{1}{N}\left(2D_{\bigcup_{i=1}^{N}(C_{i}\cap
D_{i})}(\omega)+\sum_{i=1}^{N}D_{(C_{i}\setminus
D_{i})\partial\cup(D_{i}\setminus C_{i})\partial}(\omega)\right)$ (318)
$\displaystyle\leq C(\epsilon
L)\frac{1}{1-2\eta(\frac{L}{N})}\frac{1}{N}\left(2+N\right)D_{B_{x_{j},L}}(\omega),$
(319)

where in the second line we used the definition of
$C(\operatorname{height}(\cdot))$ and in the third line we used that
$\operatorname{height}(C_{i}),\operatorname{height}(D_{i})\leq C(\epsilon L)$
since
$\operatorname{height}(C_{i})=\tilde{l},\operatorname{height}(D_{i})=L+\frac{L}{N}-\tilde{l}\leq\epsilon
L$.

Hence it follows that $C(L)\leq C(\epsilon
L)\frac{1}{1-2\eta(\frac{L}{N})}(1+\frac{2}{N})=:C(\epsilon
L)\tilde{f}(L)(1+\frac{2}{N})$. Repeating this $M=\mathcal{O}(\log L)$ times
s.t. $\epsilon^{M}L=l_{0}=:L_{\text{min},1}$ gives

$\displaystyle C(L)$ $\displaystyle\leq
C(l_{0})(1+\frac{2}{N})^{M}\prod_{k=1}^{M}\tilde{f}(\epsilon^{k}L)\leq
C(l_{0})(1+\frac{2}{N})^{M}\prod_{k=0}^{\infty}\tilde{f}(l_{0}\epsilon^{-k})$
(320)
$\displaystyle=\mathcal{O}\left(\left(1+\frac{2}{N}\right)^{\mathcal{O}(\log
L)}\right)=\mathcal{O}(L),$ (321)

where the infinite product converges since $\eta\left(\frac{L}{N}\right)$ is
exponentially decaying in $L$ under the condition on the decay length
$\xi,\xi^{\prime}$. Finally, notice that for exponential graphs (i.e. trees)
we have that $|B_{x_{j},L}|=b^{L}$ and hence
$\mathcal{O}(L)=\mathcal{O}(\log|B_{x_{j},L}|)$. Again, note that it is
independent of the $x_{j}$ which we fixed, so the result follows. The optimal
$N$ for which this works is $N=2$, hence it suffices if we have decay when
$\operatorname{diam}(C\cup
D)=\operatorname{diam}B_{x_{j},L}=L=2l=2\operatorname{dist}(C\setminus
D,D\setminus C)$, which is the one we have in Case 2. ∎