\section{Weak approximate tensorization of the relative entropy}\label{sec:strong-quasi-tensorization}

\subsection{A technical lemma}
In this section, we prove our main results concerning \textit{approximate tensorization of the relative entropy} (also known in the literature as \textit{quasi-factorization} \cite{cesi2001quasi}, \cite{[CLP18a]}, \cite{BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}). In particular, we relate the weak and strong constants to properties of the subalgebras. Let us first define in this subsection the notion of weak approximate tensorization:
\begin{definition}
	Let ${\mathcal{M}}\subset {\cal N}_1,\,{\cal N}_2\subset {\cal N}$ be finite-dimensional von Neumann algebras and $E^{{\mathcal{M}}},\,E_1 ,\, E_2$ associated conditional expectations onto ${\mathcal{M}}$, resp. ${\cal N}_1,\,{\cal N}_2$. These conditional expectations are said to satisfy a \textit{weak approximate tensorization} with constants $c \geq1$ and $d\geq 0$  if, for any state $\rho\in{\cal D}({\cal N})$:
	\begin{align}\tag{$\operatorname{AT(c,d)}$}\label{at}
	D(\rho\|E^{\mathcal{M}}_*(\rho))\le c\,\big(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\big)+d
	\end{align}
	The approximate tensorization is said to be \textit{strong} if $d=0$.
\end{definition}

\begin{remark}
	One can easily get similar inequalities for $k\ge 2$ algebras ${\mathcal{M}}\subset {\cal N}_1,\dots {\cal N}_k\subset {\cal N}$ by simply averaging over each inequality for two $k_1\ne k_2\in[k]$. Denoting by $c$ and $d$ as the maximum constants we get by considering two algebras ${\cal N}_{k_1}$ and ${\cal N}_{k_2}$ pairwise, we would thus obtain
	\begin{align}\label{at_multiple_k}
	D(\rho\|E^{\mathcal{M}}_*(\rho))\le \frac{2c}{k}\,\sum_{j=1}^k\,D(\rho\|E_{j*}(\rho))+d\,.
	\end{align}
	For sake of clarity, we will restrict to the case $k=2$ in the rest of the article.
	\end{remark}

In the next result, we derive a bound on the difference between $D(\rho\|E^{\mathcal{M}}_*(\rho))$ and the sum of the relative entropies $D(\rho\|E_{i*}(\rho))$, which is our key tool in finding constants $c$ and $d$ for which \ref{at} is satisfied. The result is inspired by the work of \cite{cesi2001quasi,[D02]} and makes use of the multivariate trace inequalities introduced in \cite{Sutter2017}:

\begin{lemma}\label{propAT}
	Let ${\mathcal{M}}\subset {\cal N}_1 ,\,{\cal N}_2\subset {\cal N}$ be finite-dimensional von Neumann algebras and $E^{{\mathcal{M}}},E_1 ,\, E_2$ their corresponding conditional expectations. Then the following inequality holds for any $\rho\in{\cal D}({\cal N})$, writing $\rho_j:=E_{j*}(\rho)$ and $\rho_{\mathcal{M}}:=E^{\mathcal{M}}_{*}(\rho)$:
	\begin{align}\label{mainequation}
	D(\rho\|\rho_{\mathcal{M}})\le \,D(\rho\|\rho_1)+D(\rho\|\rho_2)+\,\ln \left\lbrace \int_{-\infty}^\infty\,\mathop{\rm Tr}\nolimits\left[\rho_{1}\rho_{{\mathcal{M}}}^{\frac{-1-it}{2}}\rho_{2}\rho_{{\mathcal{M}}}^{\frac{-1+it}{2}}   \right]\,\beta_0(t)\,dt \right\rbrace \,,
	\end{align}
	with the probability distribution function
	\begin{equation*}
	\beta_0(t)= \frac{\pi}{2} (\cosh(\pi t)+ 1)^{-1}\,.
	\end{equation*}
\end{lemma}

\begin{proof}
	 The first step of the proof consists in showing the following bound:
	\begin{equation}\label{eq:step-31}
	D(\rho\|\rho_{\mathcal{M}}) \leq D(\rho\|\rho_1)+D(\rho\|\rho_2)+ \ln\mathop{\rm Tr}\nolimits [M]\,,
	\end{equation}
	where
	$  M = \exp \left[ - \ln \rho_{\mathcal{M}} + \ln \rho_1  + \ln \rho_2 \right] $.
	Indeed, given the conditional expectation of the statement of the theorem,  it follows that:
	\begin{align*}
	\displaystyle D(\rho\|\rho_{\mathcal{M}})- D(\rho\|\rho_1)- D(\rho\|\rho_2) &=\mathop{\rm Tr}\nolimits \left[ {\rho} \left( - \ln {\rho} \underbrace{ - \ln \rho_{\mathcal{M}} + \ln \rho_1+\ln \rho_2}_{\ln M} \right) \right] \\
	& = - D(\rho \| M).
	\end{align*}
	Moreover, since $\mathop{\rm Tr}\nolimits[M]\neq 1$ in general, from the non-negativity of the relative entropy of two states it follows that:
	\begin{equation*}
	D(\rho \| M) \geq  -\log \mathop{\rm Tr}\nolimits[M].
	\end{equation*}

	In the next step, we bound the error term making use of \cite[Theorem 7]{Lieb1973}  and \cite[Lemma 3.4]{Sutter2017}, concerning Lieb's extension of Golden-Thompson inequality and Sutter, Berta and Tomamichel's rotated expression for Lieb's pseudo-inversion operator using multivariate trace inequalities, respectively: applying Lieb's theorem to inequality (\ref{eq:step-31}), we have:
	\begin{align*}
	\mathop{\rm Tr}\nolimits [M] =& \mathop{\rm Tr}\nolimits \left[ \operatorname{exp}\left( - \ln  {\rho_{\mathcal{M}}} +
	\ln {\rho_1} +
	\ln {\rho_2}  \right) \right] \leq  \mathop{\rm Tr}\nolimits \left[ {\rho_1}\mathcal T_{\rho_{\mathcal{M}}} (\rho_2)\right],
	\end{align*}
	where $\mathcal T_{\rho_{\mathcal{M}}} $ is given by:
	\begin{equation*}
	\mathcal T_{\rho_{\mathcal{M}}} (X) := \int_0^\infty  (\rho_{\mathcal{M}} + t)^{-1} X (\rho_{\mathcal{M}} + t)^{-1} dt \, ,
	\end{equation*}
	and because of multivariate trace inequalities \cite{Sutter2017},
	\begin{equation*}
	\mathop{\rm Tr}\nolimits [M] \leq \int_{-\infty}^{+\infty}  \,\mathop{\rm Tr}\nolimits \left[ \rho_1  \, \rho_{\mathcal{M}}^{\frac{-1-it}{2}}  \rho_2 \, \rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right]\,\beta_0 (t) \, dt\,,
	\end{equation*}
	with $\beta_0$ as in the statement of the lemma.

\end{proof}

\begin{remark}
Note that, if a constant $d>0$  is such that
\begin{equation*}
\int_{-\infty}^\infty\,\mathop{\rm Tr}\nolimits\left[\rho_{1}\rho_{{\mathcal{M}}}^{\frac{-1-it}{2}}\rho_{2}\rho_{{\mathcal{M}}}^{\frac{-1+it}{2}}   \right]\,\beta_0(t)\,dt \leq d
\end{equation*}
for every $\rho \in {\cal D}({\cal N})$, then inequality \eqref{mainequation} constitutes a result of approximate tensorization AT($1,d$).
\end{remark}

 \Cref{propAT} essentially differs from those results of \cite{[CLP18a]} in the left-hand side of the inequality, since here now appears a conditional relative entropy instead of a usual relative entropy, as it did in the latter. This stronger result can be interpreted as a quantum generalization of the results in \cite{[D02],cesi2001quasi}. In these papers, the last term was further bounded by the relative entropy $D(\rho\|\rho_{\mathcal{M}})$ up to a small multiplicative error term, which lead to an approximate tensorization AT($c,0$) with a constant $c$ typically close to-albeit slightly larger than-$1$.

 In particular, consider the case of Gibbs measures on a configuration space $\Omega_{\Lambda}:= S^{\Lambda}$ where $\Lambda\subset\mathbb{Z}^D$ is a finite region of $\mathbb{Z}^D$ and where $S$ corresponds to the local configuration space (e.g. S=\{-1,+1\} for spin systems). Here ${\cal N}$ is the algebra of bounded measurable functions: ${\cal N}\equiv \mathbb{L}_\infty(\Omega_{\mathbb{Z}^D})$. We are then interested in conditional expectations arising from conditioning the Gibbs measures onto overlapping finite subregions $A$ and $B$ of $\Lambda$, so that ${\cal N}_1\equiv \mathbb{L}_\infty(\Omega_{A^c})$, $\mathcal{N}_2\equiv \mathbb{L}_\infty(\Omega_{B^c})$ and ${\mathcal{M}}\equiv  \mathbb{L}_\infty(\Omega_{(A\cup B)^c})$.

 In this case, the last term in (\ref{mainequation}) can be shown to be equal to $0$ at infinite temperature where there is no interaction between the different sites, and AT($1,0$) holds in this case. The finite temperature case can then be interpreted as an approximation of the latter, where the constant $c$ typically scales as $(1-\kappa\mathrm{e}^{-\xi d(A^c,B^c)})^{-1}$ for some positive universal constants $\kappa$ and $\xi$, under a condition of strong clustering of correlations in the invariant Gibbs measure on $\Lambda$. In particular, this condition is satisfied for 1D systems and at high temperature.

 In the quantum setting, it appears that the inequality AT($1,0$) does not hold in general even at infinite temperature, unless the Hamiltonian is classical. We differ a precise discussion on the quantum versions of the aforementioned conditions of clustering of correlations and their use to obtain results of approximate tensorization to Subsection \ref{sec:clustering}.

To conclude the first part of this section and show the first explicit result of approximate tensorization, in the next corollary, we rather directly bound the last term of (\ref{mainequation}) by a quantity that characterizes the conditional expectations involved. This result generalizes a former result of \cite{gao2017strong} for conditional expectations with respect to possibly non-tracial states.
\begin{corollary}\label{corollaruweak}
	With the notations of \Cref{propAT}, define the constant $$d:=\sup_{\rho\in{\cal D}({\cal N}_{2})}\inf\big\{\ln(\lambda)|\,E_{1*}(\rho)\le \lambda\eta\,\text{ for some }\,\eta\in{\cal D}({\mathcal{M}})  \big\}\equiv \sup_{\rho\in{\cal D}({\cal N}_{2})}\inf_{\eta\in{\cal D}({\mathcal{M}})}\,D_{\max}(E_{1*}(\rho)\|\eta)  \,.$$ Then the following weak approximate tensorization $\operatorname{AT}(1,d)$ holds:
	\begin{align*}
	D(\rho\|\rho_{\mathcal{M}})\le D(\rho\|\rho_1)+D(\rho\|\rho_2)+ d\,.
	\end{align*}
\end{corollary}
\begin{proof}
	We focus on the last term on the right-hand side of (\ref{mainequation}). First, note that:
\begin{equation*}
	\mathop{\rm Tr}\nolimits \left[ \rho_{1}\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\rho_{2}\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right] = \mathop{\rm Tr}\nolimits \left[ \rho\,E_1^*\left( \rho_{\mathcal{M}}^{\frac{-1-it}{2}}\rho_{2}\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right) \right] = \mathop{\rm Tr}\nolimits \left[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}E_{1*}(\rho_{2})\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right].
	\end{equation*}
	 We have by definition of $d$ that there exists a state $\eta\in{\cal D}({\mathcal{M}})$ such that for any $t\in\mathbb{R}$:
	\begin{align*}
	\mathop{\rm Tr}\nolimits \left[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}E_{1*}(\rho_{2})\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right] \le\,\mathrm{e}^{d} \mathop{\rm Tr}\nolimits \left[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\eta\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right]=\mathrm{e}^{d}\mathop{\rm Tr}\nolimits[\rho X_{\mathcal{M}}]\,,
	\end{align*}
	for some density $X_{\mathcal{M}}\in{\mathcal{M}}$ given by $\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\eta\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}$. Since ${\mathcal{M}}\subset {\cal N}$, $\mathop{\rm Tr}\nolimits[\rho X_{\mathcal{M}}]=\mathop{\rm Tr}\nolimits[\rho_{\mathcal{M}}\,X_{\mathcal{M}}]=\mathop{\rm Tr}\nolimits[\eta]=1$. The result follows.

\end{proof}

\begin{remark}
	In \cite{gao2019relative}, the authors showed that, for doubly stochastic conditional expectations (i.e. $E_{i*}=E_i$, $E^{\mathcal{M}}_*=E^{\mathcal{M}}$), the following equation holds: given the following block decomposition of the algebras ${\cal N}_2$ and ${\mathcal{M}}$:
	\begin{align*}
	{\cal N}_2\equiv \bigoplus_{l\in I_{{\cal N}_2}}\,\mathbb{M}_{m_l}\otimes {\mathds{1}}_{t_l}\,\qquad\qquad	{\mathcal{M}}\equiv \bigoplus_{k\in I_{{\mathcal{M}}}}\,\mathbb{M}_{n_k}\otimes {\mathds{1}}_{s_k}\,.
	\end{align*}
	\begin{align*}
	D({\cal N}_2\|{\mathcal{M}}):= \sup_{\rho\in{\cal D}({\cal N}_2)} \inf_{\eta\in{\cal D}({\mathcal{M}})}D_{\max}(\rho\| \eta)\equiv \max_{l\in I_{{\cal N}_2}}\ln \Big(\sum_{k\in I_{{\mathcal{M}}}}  \min(a_{kl},\,n_k)\,s_k/t_l \Big)\,,
	\end{align*}
	where $a_{kl}$ denotes the number of copies of the block $\mathbb{M}_{n_k}$ contained in the block $\mathbb{M}_{m_l}$. In the context of lattice spin systems, this typically corresponds to
	the	infinite temperature regime.
\end{remark}

\subsection{Approximate tensorization via noncommutative change of measure}

In this subsection, we prove a result of approximate tensorization using a noncommutative change of measure argument. Given a state $\sigma$ that is invariant for the conditional expectations $E^{\mathcal{M}}, E_1$ and $E_2$, we define the doubly stochastic conditional expectations ${E}^{(0),{\mathcal{M}}}, E_1^{(0)}$ and $E_2^{(0)}$ onto the same fixed-point algebras ${\mathcal{M}}\subset {\cal N}_1,{\cal N}_2\subset {\cal N} $. Then, the following proposition is a direct consequence of a recent noncommutative change of measure argument in  \cite{junge2019stability} under the assumption that strong approximate tensorization for the relative entropy holds at infinite temperature, a condition that we will further discuss for spin systems in Subsection \ref{subsec:preliminaries}.

\begin{proposition}\label{HSAT}
	As in \Cref{corollaruweak}, we define the constant
	\[d:=\sup_{\rho\in{\cal D}({\cal N}_{2})}\inf\big\{\ln(\lambda)|\,E_{1*}(\rho)\le \lambda\eta\,\text{ for some }\,\eta\in{\cal D}({\mathcal{M}})  \big\}\equiv \sup_{\rho\in{\cal D}({\cal N}_{2})}\inf_{\eta\in{\cal D}({\mathcal{M}})}\,D_{\max}(E_{1*}(\rho)\|\eta)  \,.\]
Let us assume that $\operatorname{AT}(1,d)$  holds at infinite temperature, i.e. for every $X \in {\cal B}({\cal H})$
\begin{equation}\label{eq:InfiniteTemperatureAT}
D( X \|E^{(0),{\mathcal{M}}}_{*}(X)) \leq D(X\|E^{(0)}_{1*}(X)) + D(X \|E^{(0)}_{2*}(X)) +d \, .
\end{equation}
	Then, the following result of $\operatorname{AT}(c,d')$  with $c=\frac{\lambda_{\max}(\sigma)}{\lambda_{\min}(\sigma)}$ and $d'= \lambda_{\max}(\sigma)\,d_{\cal H}\,d$ holds:
	\begin{align}\label{eq:NoncommChangMeasAT}
	D(\rho\|E^{\mathcal{M}}_{*}(\rho))\le \frac{\lambda_{\max}(\sigma)}{\lambda_{\min}(\sigma)}\,\big(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\big)+\lambda_{\max}(\sigma)\,d_{\cal H}\,d\,.
	\end{align}
\end{proposition}
\begin{proof}
	The proof of this result relies on the Holley-Stroock perturbative argument for the Lindblad relative entropy proved in \cite{junge2019stability}. This entropic distance is defined for two positive semi-definite operators $X,Y\in{\cal B}({\cal H})$ such that $Y$ is full rank as
	\[D_{\operatorname{Lin}}(X\|Y):=\mathop{\rm Tr}\nolimits[X(\log X-\log Y)]-\mathop{\rm Tr}\nolimits[X]+\mathop{\rm Tr}\nolimits[Y]\,.\]
	Next, we adapt the proof of Proposition 4.2 in \cite{junge2019stability} in order to relate the relative entropies $D_{\operatorname{Lin}}(\rho\|E_{*}^{\mathcal{M}}(\rho))$ and $D_{\operatorname{Lin}}(\rho\|E_{*}^{(0),{\mathcal{M}}}(\rho))$. Using this proposition, we directly see that for any positive, semidefinite operator $X$, we have
	\begin{align*}
	\frac{1}{\lambda_{\max}(\sigma)\,d_{\cal H}} \,D_{\operatorname{Lin}}(\Gamma_{\sigma}(X)\| E_{*}^{{\mathcal{M}}}(\Gamma_\sigma(X)) )\le  D_{\operatorname{Lin}}(X\|E_{*}^{(0),{\mathcal{M}}}(X))\le \frac{1}{\lambda_{\min}(\sigma)d_{{\cal H}}}\,D_{\operatorname{Lin}}(\Gamma_{\sigma}(X)\| E_{*}^{{\mathcal{M}}}(\Gamma_\sigma(X)) )\,,
	\end{align*}
	where $\lambda_{\min}(\sigma)$, resp. $\lambda_{\max}(\sigma)$, denotes the smallest, resp. largest, eigenvalue of the state $\sigma$. Analogous inequalities hold for $E_1$ and $E_2$.

	Now, using \eqref{eq:InfiniteTemperatureAT},  for $\rho:=\Gamma_\sigma(X)$ we have:
	\begin{align*}
	D(\rho\|E_{*}^{\mathcal{M}}(\rho))&\le \lambda_{\max}(\sigma)\,d_{\cal H}\,D_{\operatorname{Lin}}(X\|E_{*}^{(0),{\mathcal{M}}}(X))\\
	&\le \lambda_{\max}(\sigma)\,d_{\cal H}\,\big( D_{\operatorname{Lin}}(X\|E_{1*}^{(0)}(X))+D_{\operatorname{Lin}}(X\|E_{2*}^{(0)}(X))  +d\big)\\
	&\le \frac{\lambda_{\max}(\sigma)}{\lambda_{\min}(\sigma)}\,\big(D(\rho\|E_{1*}^{}(\rho))+D(\rho\|E_{2*}^{}(\rho)) \big) +\lambda_{\max}(\sigma)\,d_{\cal H}\,d\,.
	\end{align*}

\end{proof}

\begin{remark}
Similarly to what is done for classical spin systems in \cite{ledoux2001logSobolev}, the previous result can be rewritten in the following way. Consider the generalization of the relative entropy for $X=\Gamma_\sigma^{-1}(\rho)$ given by:
\[\operatorname{Ent}_{1,{\mathcal{M}}}(X):=D(\rho\|E_*^{\mathcal{M}}(\rho))\,,\]
with analogous expressions for ${\cal N}_1$ and ${\cal N}_2$ with their respective conditional expectations. Then, we can express this relative entropy as an infimum over $D_{\operatorname{Lin}}$. Indeed, Lemma 3.4 in \cite{junge2019stability} states that for all full-rank positive semi-definite $Y\in{\mathcal{M}}$,
 \begin{equation}\label{eq_proof_HS}
  D_{\operatorname{Lin}}(\rho\|\Gamma_{\sigma}(Y))=D_{\operatorname{Lin}}(\rho\|E_*^{\mathcal{M}}(\rho))+D_{\operatorname{Lin}}(E_*^{\mathcal{M}}(\rho)\|\Gamma_{\sigma}(Y))\,.
 \end{equation}
It shows in particular that $D_{\operatorname{Lin}}(\rho\|\Gamma_{\sigma}(Y))\geq D_{\operatorname{Lin}}(\rho\|E_*^{\mathcal{M}}(\rho))=\operatorname{Ent}_{1,{\mathcal{M}}}(X)$, with equality for $Y=E^{\mathcal{M}}(X)$. Thus we obtain
\begin{equation}\label{eq_var_entropy}
 \operatorname{Ent}_{1,{\mathcal{M}}}(X)=\underset{Y\in {\mathcal{M}}\,,Y>0}{\inf}\,D_{\operatorname{Lin}}(\rho\|\Gamma_{\sigma}(Y))\,
\end{equation}
and we can rewrite \eqref{eq:NoncommChangMeasAT} as
\begin{equation*}
 \operatorname{Ent}_{1,{\mathcal{M}}}(X) \leq \frac{\lambda_{\max}(\sigma)}{\lambda_{\min}(\sigma)}\,\big( \operatorname{Ent}_{1,{\cal N}_1}(X)+\operatorname{Ent}_{1,{\cal N}_2}(X) \big) +\lambda_{\max}(\sigma)\,d_{\cal H}\,d\,.
\end{equation*}

\end{remark}

\subsection{Approximate tensorization via Pinching map}
None of the previous results reduces to those of \cite{cesi2001quasi,[D02]} in the case of classical Gibbs measures over classical systems at finite temperature. In the following theorem, we take care of this issue by interpolating between these two extreme cases. Before stating the result, let us fix some notations. As before, we are interested in proving (weak) approximate tensorization results for the quadruple of algebras ${\mathcal{M}}\subset {\cal N}_1\,,\,{\cal N}_2\subset{\cal N}$. As a subalgebra of ${\cal B}({\cal H})$ for some Hilbert space ${\cal H}$, ${\mathcal{M}}$ bears the following block diagonal decomposition: given ${\cal H}=\bigoplus_{i\in I_{\mathcal{M}}}{\cal H}_i\otimes {\cal K}_i$:
	\begin{align}\label{eq_decomp}
	{\mathcal{M}}\equiv\bigoplus_{i\in I_{\mathcal{M}}}\,{\cal B}({\cal H}_i)\otimes {\mathds{1}}_{{\cal K}_i}\,,\qquad\text{ so that }\qquad\forall \rho\in{\cal D}({\cal N})\,,\, \rho_{\mathcal{M}}:=\sum_{i\in I_{\mathcal{M}}}\,\mathop{\rm Tr}\nolimits_{{\cal K}_i}[P_i\rho P_i]\,\otimes\tau_i\,,
	\end{align}
where $P_i$ corresponds to the projection onto the $i$-th diagonal block in the decomposition of ${\mathcal{M}}$, and each $\tau_i$ is a full-rank state on ${\cal K}_i$. We further make the observation that, since the restrictions of the conditional expectations $E_1$, $E_2$ and $E^{\mathcal{M}}$ on ${\cal B}({\cal H}_i\otimes{\cal K}_i)$ only act non-trivially on the factor ${\cal B}({\cal K}_i)$, we can define the conditional expectations ${E}_j^{(i)}$ and $({E}^{{\mathcal{M}}})^{(i)}$ acting on ${\cal B}( {\cal K}_i)$ and such that
\begin{equation}\label{eq_decom_cond}
 E_j|_{{\cal B}({\cal H}_i\otimes{\cal K}_i)}:={\rm{id}}_{{\cal B}({\cal H}_i)}\otimes {E}_j^{(i)}\,,\quad\text{resp.}\quad E^{\mathcal{M}}|_{{\cal B}({\cal H}_i\otimes{\cal K}_i)}:={\rm{id}}_{{\cal B}({\cal H}_i)}\otimes ({E}^{\mathcal{M}})^{(i)}\,.
\end{equation}
In order to get another form of approximate tensorization, we wish to compare the state $\rho$ with a classical-quantum state according to the decomposition given by ${\mathcal{M}}$. To this end we introduce the Pinching map with respect to the ${\cal H}_i$ in the decomposition of ${\mathcal{M}}$. First define $\rho_{{\cal H}_i}\equiv \mathop{\rm Tr}\nolimits_{{\cal K}_i}[P_i\,\rho\,P_i]$. Then each $\rho_{{\cal H}_i}$ can be diagonalized individually:
\[\rho_{{\cal H}_i}\equiv\sum_{\lambda^{(i)}\in\operatorname{Spec}(\rho_{{\cal H}_i})}\,\lambda^{(i)}\,\proj{\lambda^{(i)}}\,.\]
The Pinching map we are interested in is then:
\[\mathcal{P}_{\rho_{\mathcal{M}}}(X)\equiv \sum_{i\in I_{\mathcal{M}}}\,\sum_{\lambda^{(i)}\in\operatorname{Spec}(\rho_{{\cal H}_i})}\,\left(\proj{\lambda^{(i)}}\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i}\right)\,X\,\left(\proj{\lambda^{(i)}}\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i}\right)\,,\qquad X\in{\cal B}({\cal H})\,.\]
Remark that we have for all $\rho\in{\cal D}({\cal N})$:
\[\mathop{\rm Tr}\nolimits_{{\cal H}_i}[P_i\,\rho\,P_i]=\mathop{\rm Tr}\nolimits_{{\cal H}_i}[P_i\,\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)\,P_i]\,.\]

\begin{theorem}\label{theo_AT_pinching}
 Define
	\begin{align}\label{cond_L1_clustering}
	&c_1:=\max_{i\in I_{\mathcal{M}}}\|E_{1}^{(i)}\circ E_{2}^{(i)}-(E^{\mathcal{M}})^{(i)}:\,\mathbb{L}_1(\tau_i)\to\mathbb{L}_\infty\|\,.
	\end{align}
	Then, the following inequality holds:
	\begin{align}\label{eqgeneral}
	D(\rho\|\rho_{\mathcal{M}})\le \frac{1}{(1-2{c_1})}\,\big(D(\rho\|\rho_1)+D(\rho\|\rho_2)+D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\|E_{1*}\circ E_{2*}(\eta)\big)+\,c_1 D(\eta\|\mathcal{P}_{\rho_{\mathcal{M}}}(\rho))\big)\,,
	\end{align}
	for any $\eta\in{\cal D}({\cal N})$ such that $\eta=\mathcal{P}_{\rho_{\mathcal{M}}}(\eta)$ and $\mathop{\rm Tr}\nolimits_{{\cal K}_i}[P_i\,\eta\,P_i]=\rho_{{\cal H}_i}$. Alternatively, we can get
	\begin{align}\label{eqchanged}
	D(\rho\|\rho_{\mathcal{M}})\le \frac{1}{(1-2{c_1})}\,\big(D(\rho\|\rho_1)+D(\rho\|\rho_2)\big)+D\big(\rho\|\mathcal{P}_{\rho_{\mathcal{M}}}(\rho) )\,.
	\end{align}
	Consequently, \ref{at} holds with
	\begin{align}
	 & c:=\frac{1}{(1-2{c_1})}\,, \nonumber\\
	 & d:= \frac{1}{(1-2{c_1})}\left(\underset{\rho\in{\cal D}({\cal N})}{\sup}\,\underset{\eta\in{\cal D}({\cal N})}\inf\,D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\|E_{1*}\circ E_{2*}(\eta)\big)+\,c_1 D(\eta\|\mathcal{P}_{\rho_{\mathcal{M}}}(\rho))\big)\right)\,, \label{eq_theo_AT_pinching}
	\end{align}
where the infimum in the second line runs over $\eta$ such that $\eta=\mathcal{P}_{\rho_{\mathcal{M}}}(\eta)$ and $\mathop{\rm Tr}\nolimits_{{\cal K}_i}[P_i\,\eta\,P_i]=\rho_{{\cal H}_i}$.
\end{theorem}
\begin{remark}
In particular, any state $\eta$ of the form $\eta:= \sum_{i\in I_{\mathcal{M}}}\,\rho_{{\cal H}_i}\otimes \tau_i'$, for an arbitrary family of subnormalized states $\tau_i'$, satisfies the conditions stated below \Cref{eqgeneral}.
\end{remark}

\begin{proof}
	The proof starts similarly to that of Corollary \ref{corollaruweak}. We once again simply need to bound the integral on the right hand side of (\ref{mainequation}). By considering $\eta$ as in the statement of the theorem and writing for the moment $\tilde{d}:=D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\|E_{1*}\circ E_{2*}(\eta)\big)$,  we obtain
	\begin{align*}
	\mathop{\rm Tr}\nolimits\Big[ \rho_{1}\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,\rho_{2}\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}\Big]=		\mathop{\rm Tr}\nolimits\Big[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,E_{1*}(\rho_{2})\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}\Big]\le \mathrm{e}^{\tilde{d}}  \,	\mathop{\rm Tr}\nolimits\Big[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,E_{1*}\circ E_{2*}(\eta)\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}\Big]\,.
	\end{align*}
	To simplify the notation, let us write: $\eta_{12}:=E_{1*}\circ E_{2*}(\eta)$. Now, note that the following holds:
		\begin{equation*}
	\mathop{\rm Tr}\nolimits \left[   \left( \rho  - \rho_{\mathcal{M}} \right)  \rho_{\mathcal{M}}^{\frac{-1-it}{2}}   \left( \eta_{12}   - \rho_{\mathcal{M}} \right)  \rho_{\mathcal{M}}^{\frac{-1+it}{2}}  \right]
	= \mathop{\rm Tr}\nolimits \left[ \rho \, \rho_{\mathcal{M}}^{\frac{-1-it}{2}}	\,  \eta_{12} \, \rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right] -1 -1 + 1,
	\end{equation*}
	since $ E^{\mathcal{M}}_*$, $E_{1*}$ and $E_{2*}$ are conditional expectations in the Schr\"{o}dinger picture and, thus, trace preserving. Therefore,
	\begin{align*}
	& \ln  \int_{-\infty}^{+\infty} \mathrm{e}^{\tilde{d}}  \, \mathop{\rm Tr}\nolimits \left[ \rho \, \rho_{\mathcal{M}}^{\frac{-1-it}{2}} \, \eta_{12} \, \rho_{\mathcal{M}}^{\frac{-1+it}{2}} \right] \,\beta_0 (t) \,dt \, \\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad}= \ln \int_{-\infty}^{+\infty} \mathrm{e}^{\tilde{d}}  \, \left( \mathop{\rm Tr}\nolimits \left[   \left( \rho   - \rho_{\mathcal{M}} \right)  \rho_{\mathcal{M}}^{\frac{-1-it}{2}}   \left( \eta_{12}   - \rho_{\mathcal{M}} \right) \rho_{\mathcal{M}}^{\frac{-1+it}{2}}  \right]  + 1  \right)\,\beta_0 (t) \,dt \\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad} \leq  \tilde{d} + \int_{-\infty}^{+\infty}  \mathop{\rm Tr}\nolimits \left[   \left( \rho   - \rho_{\mathcal{M}} \right)  \rho_{\mathcal{M}}^{\frac{-1-it}{2}}   \left( \eta_{12}  -\rho_{\mathcal{M}} \right) \rho_{\mathcal{M}}^{\frac{-1+it}{2}}  \right] \, \beta_0 (t)\,dt\,,
	\end{align*}
	where we have used that $	\ln(x +1)\le x$ for positive real numbers. 	 Defining $X:=\Gamma_{\rho_{\mathcal{M}}}^{-1}(\rho)$ and $Y_t:=\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,\eta\, \rho_{\mathcal{M}}^{\frac{-1+it}{2}}$, we note that
	\begin{equation}
	E_1 \circ E_2 [Y_t] = \rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,\eta_{12}\, \rho_{\mathcal{M}}^{\frac{-1+it}{2}},
	\end{equation}
and we can rewrite the previous expression as
	\begin{align}
	& \int_{-\infty}^{+\infty}  \mathop{\rm Tr}\nolimits \left[   \left(\rho   - \rho_{\mathcal{M}} \right) \rho_{\mathcal{M}}^{\frac{-1-it}{2}}   \left( \eta_{12}  - \rho_{\mathcal{M}}\right) \rho_{\mathcal{M}}^{\frac{-1+it}{2}}  \right]  \, \beta_0 (t)\,dt\nonumber \\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad}=  \int_{-\infty}^{+\infty} \mathop{\rm Tr}\nolimits \left[    \left( X  - E^{\mathcal{M}}[X] \right)  \Delta_{\rho_{\mathcal{M}}}^{-it/2}   \left( \eta_{12}   - \rho_{\mathcal{M}}  \right)  \right] \, \beta_0 (t)\,dt \nonumber\\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad}=  \int_{-\infty}^{+\infty}  \mathop{\rm Tr}\nolimits \left[    \left(X  - E^{\mathcal{M}}[X]  \right)  \rho_{\mathcal{M}}^{\frac{1}{2}}    \left( E_1 \circ E_2 [Y_t]   -E^{\mathcal{M}} [Y_t] \right)  \rho_{\mathcal{M}}^{\frac{1}{2}}  \right] \, \beta_0 (t)\,dt\nonumber  \\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad}= \int_{- \infty}^{+ \infty} \left\langle  X  - E^{\mathcal{M}}[X]  , \ E_1 \circ E_2 [Y_t]   -E^{\mathcal{M}} [Y_t]  \right\rangle_{\rho_{\mathcal{M}}}\, \beta_0 (t)\,dt\, ,
	\end{align}
thus obtaining the following inequality
	\begin{align*}
	\ln\,	\int_{-\infty}^{\infty}	\mathrm{e}^{\tilde{d}}  \, 	\mathop{\rm Tr}\nolimits\Big[ \rho_{1}\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}\,\eta_{12}\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}\Big]\,\beta_0(t)\,dt\le \tilde{d}+\int_{-\infty}^\infty \,\langle X-E^{\mathcal{M}}[X]\,,E_1\circ E_2[Y_t]-E^{\mathcal{M}}[Y_t]\rangle_{\rho_{\mathcal{M}}}\,\beta_0(t)\,dt .
	\end{align*}

	 Now, we focus on the integrand on the right-hand side of the above inequality. Denote for any $A\in{\cal B}({\cal H})$,
	\[A^{(\lambda,i)}:=\left(\proj{\lambda^{(i)}}\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i}\right)P_i\,A\,P_i\left(\proj{\lambda^{(i)}}\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i}\right)\,.\]
	We also write $A^{(\lambda,i)}=\proj{\lambda^{(i)}}\otimes A^{(\lambda,i)}$ by a slight abuse of notation. Then
	\begin{align*}
	\langle X-E^{\mathcal{M}}[X]\,,&\, E_1\circ E_2[Y_t]-E^{\mathcal{M}}[Y_t]\rangle_{\rho_{\mathcal{M}}}\\
	&= \sum_{i\in I_{\mathcal{M}}}\,\sum_{\lambda^{(i)}\in\operatorname{Spec}(\rho_{{\cal H}_i})}\,\lambda^{(i)}\,\langle X^{(\lambda,i)}-E^{\mathcal{M}}[X^{(\lambda,i)}],\,E_1\circ E_2[Y_t^{(\lambda,i)}]-E^{\mathcal{M}}[Y_t^{(\lambda,i)}]\rangle_{\tau_i}\,.
	\end{align*}

	Next, by Hölder's inequality each summand in the right-hand side above is upper bounded by
	\begin{align*}
	& \|({\rm{id}}-(E^{\mathcal{M}})^{(i)})[X^{(\lambda,i)}] \|_{\mathbb{L}_1(\tau_i)}\,\|(E_1^{(i)}\circ E_2^{(i)}-(E^{\mathcal{M}})^{(i)}) [Y_t^{(\lambda,i)}] \|_\infty\nonumber\\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad} \le c_1\,  \|({\rm{id}}-(E^{\mathcal{M}})^{(i)})[X^{(\lambda,i)}] \|_{\mathbb{L}_1(\tau_i)}\,\| ({\rm{id}}-(E^{\mathcal{M}})^{(i)})[Y_t^{(\lambda,i)}]\|_{\mathbb{L}_1(\tau_i)}\nonumber\\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad} = c_1\,\|\rho^{(\lambda,i)}-E^{\mathcal{M}}_{*}(\rho^{(\lambda,i)})\|_1\,\|\tau_i'-\tau_i\|_{1}\label{eqchange}\\
	&\phantom{adasdasdad}\phantom{adasdasdad}\phantom{adasdasdad} \le \frac{c_1}2\,\left(\|\rho^{(\lambda,i)}-E^{\mathcal{M}}_{*}(\rho^{(\lambda,i)})\|_1^2+\|\tau_i'-\tau_i\|_{1}^2\right)\,,\nonumber
	\end{align*}
where we use Young inequality in the last line. Using Pinsker's inequality and summing over the indices $i$ and $\lambda^{(i)}$, we find that
	\begin{align*}
	\ln\int_{-\infty}^\infty   \mathop{\rm Tr}\nolimits\left[ \rho\,\rho_{\mathcal{M}}^{\frac{-1-it}{2}}E_{1*}\circ E_{2*}(\mathcal{P}_{\mathcal{M}}(\rho))\,\rho_{\mathcal{M}}^{\frac{-1+it}{2}}    \right]  \beta_0(t)\,dt\le \tilde{d} +c_1\,D(\rho\|\rho_{\mathcal{M}})+ c_1\,D(\eta\|\mathcal{P}_{\rho_{\mathcal{M}}}(\rho))\,.
	\end{align*}
    \Cref{eqgeneral} follows after rearranging the term. In order to obtained \Cref{eqchanged}, we exploit that $\rho_{\mathcal{M}}$ is a fixed point of $\mathcal{P}_{\rho_{\mathcal{M}}}$ and therefore
    \[D(\rho\|\rho_{\mathcal{M}})=D(\rho\|\mathcal{P}_{\rho_{\mathcal{M}}}(\rho))+D(\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)\|\rho_{\mathcal{M}})\,.\]
    We can then apply \Cref{eqgeneral} to $\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)$ and remark that the weak constant vanish. The result follows after remarking that $\mathcal{P}_{\rho_{\mathcal{M}}}\circ E_*^{\mathcal{M}}=E_*^{\mathcal{M}}\circ\mathcal{P}_{\rho_{\mathcal{M}}}$ and applying the data-processing inequality to the map $\mathcal{P}_{\rho_{\mathcal{M}}}$.
\end{proof}

\begin{proposition}\label{propboundsd1d2}
	With the notations of \Cref{propAT} and \Cref{theo_AT_pinching},
	\begin{equation*}
	 \sup_{\rho\in{\cal D}({\cal N})}\,D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\| E_{1*}\circ E_{2*}(\eta)\big)\leq d_1 + d_2
	\end{equation*}
 where
	\begin{align*}
	 &d_1:=\sup_{\rho\in{\cal D}({\cal N})}\,D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\| E_{1*}\circ E_{2*}(\mathcal{P}_{\mathcal{M}}(\rho))\big)\,,\\
	&d_2:=\,\max_{i\in I_{\mathcal{M}}}\sup_{\rho^{(i)}\in{\cal D}(P_i{\cal N} P_i)}\,I_{\max}\big( {\cal H}_i:{\cal K}_i  \big)_{\rho^{(i)}}\,,
	\end{align*}
    where $\mathcal{P}_{\mathcal{M}}:=\sum_{i\in I_{\mathcal{M}}}P_i(\cdot)P_i$.

    Furthermore, given $i\in I_{\cal N}$, denote by $I^{(i)}_{\mathcal{M}}$ the set of indices corresponding to the minimal projectors in ${\mathcal{M}}$ contained in the $i$-th block of ${\cal N}$. Moreover, for each of the blocks $i$ of ${\cal N}$, of corresponding minimal projector $P^{{\cal N}}_i$, decompose $P^{\cal N}_i{\mathcal{M}} P^{\cal N}_i$ as follows: letting $P_i^{\cal N}{\cal H}:= \bigoplus_{j\in I^{(i)}_{\mathcal{M}}} \,{\cal H}^{(i)}_{j}\otimes {\cal K}^{(i)}_j$,
	\begin{align*}
	P_i^{\cal N}{\mathcal{M}} P_i^{\cal N}:=\bigoplus_{j\in I^{(i)}_{\mathcal{M}}} {\cal B}({\cal H}^{(i)}_j)\otimes {\mathds{1}}_{{\cal K}_j^{(i)}}\,.
	\end{align*}
	Then,
	\begin{align*}
	&d_1\le \max_{i\in I_{\cal N}}\ln(|I^{(i)}_{\mathcal{M}}|)\,\\
	&d_2\le   2\max_{i\in I_{\cal N}}\max_{j\in I^{(i)}_{\mathcal{M}}}\min\left\lbrace \ln (d_{{\cal H}^{(i)}_j}),\ln(d_{{\cal K}^{(i)}_j}) \right\rbrace \,.
	\end{align*}
\end{proposition}
\begin{proof}
	We first proceed by proving the bound $d\leq d_1+d_2$. For all $\rho\in{\cal D}({\cal N})$, we can use the chain rule on the max-relative entropy to obtain:
	\begin{align*}
	& D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\|E_{1*}\circ E_{2*}(\eta)\big) \\
	& ~~~~~~~~~~~~~~~~~~~~~~~~~ \leq D_{\max}\big(E_{1*}\circ E_{2*}(\rho)\|E_{1*}\circ E_{2*}(\mathcal{P}_{\mathcal{M}}(\rho))\big) +D_{\max}\big(E_{1*}\circ E_{2*}(\mathcal{P}_{\mathcal{M}}(\rho))\|E_{1*}\circ E_{2*}(\eta)\big)\\
	& ~~~~~~~~~~~~~~~~~~~~~~~~~  \leq d_1 + D_{\max}\big(\mathcal{P}_{\mathcal{M}}(\rho)\|\eta\big)\,,
	\end{align*}
    where the second inequality follows from the data processing inequality for $D_{\max}$. Then
    \[D_{\max}\big(\mathcal{P}_{\mathcal{M}}(\rho)\|\eta\big)\leq\max_{i\in I_{\mathcal{M}}}D_{\max}\big(\mathcal{P}_{\mathcal{M}}(\rho)^{(i)}\|\eta^{(i)}\big)\,,\]
	where we write $A^{(i)}:=P_i\,A\,P_i$ for any $A\in{\cal B}({\cal H})$. This last $D_{\max}$ is exactly $I_{\max}\big( {\cal H}_i:{\cal K}_i  \big)_{\rho^{(i)}}$ after minimizing on $\eta$.	We are left with proving the two separate bounds on $d_1$ and $d_2$ respectively. The first bound is a simple consequence of the data processing inequality for $D_{\max}$ and the Pinching inequality. The second bound is a consequence of Lemma B.7 in \cite{berta2011quantum}.
\end{proof}

\begin{remark}
	In the case of a classical evolution over a classical system, taking $\eta=\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)$ shows that $d=0$ in \Cref{eq_theo_AT_pinching}, and thus we get back the strong approximate tensorization of \cite{cesi2001quasi}. The estimation of the constant $c$ under a condition of clustering of correlations is discussed in the next section.
\end{remark}

\begin{example}[Pinching onto different bases]\label{example2} Take ${\cal H}={\mathbb{C}}^l$ and assume that the algebra ${\cal N}_1$ is the diagonal onto some orthonormal basis $|e^{(1)}_k\rangle$, whereas ${\cal N}_2$ is the diagonal onto the basis $|e^{(2)}_k\rangle$. Moreover, choose ${\mathcal{M}}$ to be the trivial algebra $\mathbb{C}{\mathds{1}}_\ell$. Hence for each $i\in\{1,2\}$, $E_i$ denotes the Pinching map onto the diagonal $\operatorname{span}\left\lbrace |e_k^{(i)}\rangle\langle e_k^{(i)}|\right\rbrace$ and $E^{\mathcal{M}}=\frac{{\mathds{1}}}{\ell}\mathop{\rm Tr}\nolimits[\cdot]$. Then, for any $X\ge0$:
	\begin{align*}
	\|(E_1\circ E_2-E^{\mathcal{M}})(X)\|_\infty&=\Big\|\sum_{k,k'}|e^{(1)}_k\rangle\langle e^{(1)}_k|e^{(2)}_{k'}\rangle\langle e^{(2)}_{k'}| X| e^{(2)}_{k'}\rangle\langle e^{(2)}_{k'}|e^{(1)}_k\rangle\langle e^{(1)}_k|-\frac{1}{\ell}\,|e^{(1)}_k\rangle\langle e^{(1)}_k|\,\langle e^{(2)}_{k'}|X|e^{(2)}_{k'}\rangle\Big\|_\infty\\
	&=\Big\|\sum_{k,k'}\,\big( |\langle e^{(1)}_k|e^{(2)}_{k'}\rangle |^2-\frac{1}{\ell} \big)|e^{(1)}_k\rangle\langle e^{(1)}_k|\,\langle e^{(2)}_{k'}| X| e^{(2)}_{k'}\rangle\Big\|_\infty\\
	&=\max_{k}\,\sum_{k'}\,\Big| |\langle e^{(1)}_k|e^{(2)}_{k'}\rangle |^2-\frac{1}{\ell} \Big| \quad\langle  e^{(2)}_{k'}| X| e^{(2)}_{k'}\rangle\\
	&\le \varepsilon\,\frac{1}{\ell}\mathop{\rm Tr}\nolimits[X]
	\end{align*}
	where $\varepsilon:=\ell \,\max_{k,k'}\,\Big| |\langle e^{(1)}_k|e^{(2)}_{k'}\rangle |^2-\frac{1}{\ell} \Big|$. Hence
	\begin{align*}
	\|(E_1\circ E_2-E^{\mathcal{M}}):\mathbb{L}_1(\ell^{-1}{\mathds{1}})\to \mathbb{L}_\infty\|\le \varepsilon \,,
	\end{align*}
	so that by chosing $\eta=\rho=\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)$ in Theorem \ref{theo_AT_pinching}, as long as $2\varepsilon<1$, for any $\rho\in {\cal D}(\mathbb{C}^\ell)$, AT($(1-2\varepsilon)^{-1},0$) holds:
	\begin{align}\label{our-bound}
	D(\rho\| \ell^{-1}{\mathds{1}})\le \frac{1}{1-2\varepsilon}\,(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho)))\,.
	\end{align}

This result is related to Example 4.5 of \cite{nick2019scoopingpaper}. There the author obtains an inequality that can be rewritten in the following form:
	\begin{align}\label{nick-bound}
	D(\rho\| \ell^{-1}{\mathds{1}})\le 4 \left(  \ln_{1-\delta} \left( \frac{2}{3 \ell + 5}   \right) + 1   \right)   \,(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho)))\,,
	\end{align}
where $\delta$ here is related with $\varepsilon$ in our example by:
\begin{equation*}
\delta \geq  \frac{1}{\ell} (1- \varepsilon).
\end{equation*}

The approximate tensorization derived in (\ref{our-bound}) can be used to find exponential convergence in relative entropy  of the primitive quantum Markov semigroup $\mathrm{e}^{t{\cal L}}$, where
\begin{align*}
{\cal L}(X):=E_1(X)+E_2(X)-2X\,.
\end{align*}
Indeed, for any state $\rho\in{\cal D}({\cal H})$, denoting by $\rho_t$ the evolved state $\mathrm{e}^{t{\cal L}}(\rho)$ up to time $t$, the fact that $D(\rho_t\|\ell^{-1}{\mathds{1}})\le \mathrm{e}^{-\alpha t}D(\rho\|\ell^{-1}{\mathds{1}})$ holds for some $\alpha >0$ is equivalent to the so-called modified logarithmic Sobolev inequality, which can be written for all $\rho\in{\cal D}({\cal H})$ by \cite[Lemma 3.4]{junge2019stability} as
\begin{align*}
  \alpha D(\rho\|\ell^{-1}{\mathds{1}})\le  D(\rho\|E_{1*}(\rho))+D(E_{1*}(\rho)\|\rho)+D(\rho\|E_{2*}(\rho))+D(E_{2*}(\rho)\|\rho)\,.
\end{align*}
By positivity of the relative entropy, it suffices to prove the existence of a constant $\alpha>0$ such that
\begin{align*}
\alpha D(\rho\|\ell^{-1}{\mathds{1}})\le D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\,.
\end{align*}
This last inequality is equivalent to (\ref{our-bound}) for $\alpha=1-2\varepsilon$.
\end{example}

\subsection{Application: tightened entropic uncertainty relations}

In this subsection, we use the results obtained above to derive some applications concerning entropic uncertainty relations.
Let us first recall that, given a function $f\in\mathbb{L}_2(\mathbb{R})$ and its Fourier transform ${\mathcal{F}}[f]$ with $\|f\|_{\mathbb{L}_2(\mathbb{R})}=\|{\mathcal{F}}[f]\|_{\mathbb{L}_2(\mathbb{R})}=1$, Weyl proved in \cite{weyl1950theory} the following uncertainty relation:
\begin{align}\label{uncertainty}
V(|f|^2)\,V(|{\mathcal{F}}[f]|^2)\ge \frac{1}{16\pi^2}\,,
\end{align}
where, given a probability distribution function $g$, $V(g)$ denotes its variance. The uncertainty inequality means that $|f|^2$ and $|{\mathcal{F}}[f]|^2$ cannot both be concentrated arbitrarily close to their corresponding means. An entropic strenghthening of (\ref{uncertainty}) was derived independently by Hirschmann \cite{hirschman1957note} and Stam \cite{stam1959some}, and tightened later on by Beckner \cite{beckner1975inequalities}:
\begin{align*}
H(|f|^2)+H(|{\mathcal{F}}[f]|^2)\ge \ln\frac{\mathrm{e}}{2}\,,
\end{align*}
where $H(g):=-\int_{\mathbb{R}} \,g(x)\ln g(x)\,dx$ stands for the differential entropy functional. In the quantum mechanical setting, this inequality has the interpretation that the total amount of uncertainty, as quantified by the entropy, of non-commuting observables (i.e. the position and momentum of a particle) is uniformly lower bounded by a positive constant independently of the state of the system. For an extensive review of entropic uncertainty relations for classical and quantum systems, we refer to the recent survey \cite{coles2017entropic}.

More generally, given two POVMs $\mathbf{X}:= \{X_x\}_{x}$ and $\mathbf{Y}:=\{Y_y\}_{y}$ on a quantum system $A$, and in the presence of side information $M$ that might help to better predict the outcomes of $\mathbf{X}$ and $\mathbf{Y}$, the following state-dependent tightened bound was found in \cite{frank2013extended} (see also \cite{berta2010uncertainty} for the special case of measurements in two orthonormal bases and \cite{maasen1988uncertainty} for the case without memory): for any bipartite state $\rho_{AM}\in{\cal D}({\cal H}_A\otimes {\cal H}_M)$,
\begin{align}\label{uncert}
S(X|M)_{(\Phi_{\mathbf{X}}\otimes {\rm{id}}_M)(\rho)}+S(Y|M)_{(\Phi_{\mathbf{Y}}\otimes {\rm{id}}_M)(\rho)}\ge- \ln c'+S(A|M)_\rho\,,
\end{align}
with $c'=\max_{x,y}\{\mathop{\rm Tr}\nolimits(X_x\,Y_x)\}$, where $\Phi_{\mathbf{Z}}$ denotes the quantum-classical channel corresponding to the measurement $\mathbf{Z}\in\{\mathbf{X},\mathbf{Y}\}$:
\begin{align*}
\Phi_{\mathbf{Z}}(\rho_A):=\sum_{z}\,\mathop{\rm Tr}\nolimits(\rho_A Z_z)\,|z\rangle\langle z|_Z\,.
\end{align*}

The above inequality has been recently extended to the setting where the POVMs are replaced by two arbitrary quantum channels in \cite{gao2018uncertainty}. In this section, we restrict ourselves to the setting of \cite{berta2010uncertainty}, so that the measurement channels reduce to the Pinching maps of \Cref{example2}. First of all, we notice that the relation (\ref{uncert}) easily follows from \Cref{corollaruweak}, as we show below.

\begin{example}\label{ex:example3}
Take ${\cal H}_{AM}={\cal H}_A \otimes {\cal H}_M$ a bipartite system and, as in the case of Example \ref{example2}, assume that the algebra ${\cal N}_1$ is the diagonal onto some orthonormal basis $|e^{(\mathcal{X})}_x\rangle$ in ${\cal H}_A $, whereas ${\cal N}_2$ is the diagonal onto the basis $|e^{(\mathcal{Y})}_y\rangle$ also in ${\cal H}_A $. Moreover, choose ${\mathcal{M}}$ to be the algebra $\mathbb{C}{\mathds{1}}_\ell \otimes \mathcal{B}({\cal H}_M)$. Hence for each alphabet $\mathcal{Z}\in\{\mathcal{X},\mathcal{Y}\}$, $E_\mathcal{Z}$ denotes the Pinching map onto the diagonal $\operatorname{span}\left\lbrace |e_z^{(\mathcal{Z})}\rangle\langle e_z^{(\mathcal{Z})}|\right\rbrace$, which we tensorize with the identity map in $M$, and $E^{\mathcal{M}} \otimes {\rm{id}}_M=\frac{1}{d_A}{\mathds{1}}_A \otimes\mathop{\rm Tr}\nolimits_A[\cdot] $. Then, for every $\rho_{AM} \in \mathcal{D}({\cal H}_{AM})$,
\begin{align*}
S(X|M)_{(E_{\mathcal{X}}\otimes {\rm{id}}_M)(\rho_{AM})} & = - D\left( (E_{\mathcal{X}}\otimes {\rm{id}}_M)(\rho_{AM}) \Big| \Big| \frac{{\mathds{1}}_A}{d_A} \otimes \rho_M \right) +\ln d_A \\
& = D(\rho_{AM} || (E_{\mathcal{X}}\otimes {\rm{id}}_M)(\rho_{AM})) - D(\rho_{AM} || (E^{\mathcal{M}}\otimes {\rm{id}}_M)(\rho_{AM})) +\ln d_A,
\end{align*}
where the last equality is derived from \cite[Lemma 3.4]{junge2019stability}. Hence, since
\begin{equation*}
D(\rho_{AM} || (E^{\mathcal{M}}\otimes {\rm{id}}_M)(\rho_{AM})) = - S(A| M)_{\rho_{AM}} + \ln d_A ,
\end{equation*}
by virtue of Corollary \ref{corollaruweak} we have
\begin{align*}
S(X|M)_{(E_{\mathcal{X}}\otimes {\rm{id}}_M)(\rho_{AM})} &+ S(Y|M)_{(E_{\mathcal{Y}}\otimes {\rm{id}}_M)(\rho_{AM})} \\
& \geq  D(\rho_{AM} || (E_{\mathcal{X}}\otimes {\rm{id}}_M)(\rho_{AM}))+  D(\rho_{AM} || (E_{\mathcal{Y}}\otimes {\rm{id}}_M)(\rho_{AM}))\\
& \; \; \; \;  - 2 D(\rho_{AM} || (E^{\mathcal{M}}\otimes {\rm{id}}_M)(\rho_{AM})) + 2 \ln d_A\\
& \geq S(A|M)_{\rho_{AM}} - d + \ln d_A ,
\end{align*}
where
\begin{equation*}
d:=\sup_{\rho\in{\cal D}({\cal N}_{2})}\inf\big\{\ln(\lambda)|\,E_{1*}(\rho)\le \lambda\eta\,\text{ for some }\,\eta\in{\cal D}({\mathcal{M}})  \big\} .
\end{equation*}

Now, taking into account the computations of Example \ref{example2}, notice that
\begin{equation*}
d= d_A \, \underset{x, \, y}{\text{max}}\, |\langle e^{(\mathcal{X})}_x|e^{(\mathcal{Y})}_{y}\rangle |^2 ,
\end{equation*}
obtaining thus expression (\ref{uncert}).
\end{example}

 However, close to the completely mixed state, this inequality is not tight  whenever $\mathbf{X}$ and $\mathbf{Y}$ are not mutually unbiased bases (i.e. $\exists x\in{\cal X},y\in\mathcal{Y}$ such that $|\langle X^x|Y^y\rangle|^2>\frac{1}{d_A}$). Here, we derive the following strengthening of \Cref{uncert} when $d_M=1$ as a direct consequence of  \Cref{theo_AT_pinching}:
\begin{corollary}\label{cor:2pinching}
	Given a finite alphabet $\mathcal{Z}\in \{\mathcal{X},\mathcal{Y}\}$, let $E_{\mathcal{Z}}$ denote the Pinching channels onto the orthonormal basis $\{|e^{(\mathcal{Z})}_z\rangle\}_{z\in\mathcal{Z}}$ corresponding to the measurement $\mathbf{Z}$. Assume further that $c_1= d_A\max_{x,y}\big| |\langle e^{(\mathcal{X})}_x|e^{(\mathcal{Y})}_y\rangle|^2-\frac{1}{d_A}   \big|<\frac{1}{2}$.	Then the following strenghtened entropic uncertainty relation holds for any state $\rho\in{\cal D}({\cal H}_{A})$,
	\begin{align}
	S(X)_{E_{\mathcal{X}}(\rho)}+S(Y)_{E_{\mathcal{Y}}(\rho)}\ge  (1+2c_1)\,S(A)_\rho+(1-2c_1)\ln d_A\,.
	\end{align}
\end{corollary}

\begin{proof}
Following the first lines of Example \ref{ex:example3} for $d_M=1$, we have
\begin{equation*}
S(X)_{E_{\mathcal{X}}(\rho)}+  S(Y)_{E_{\mathcal{Y}}(\rho)} = D(\rho || E_{\mathcal{X}}(\rho)) + D(\rho || E_{\mathcal{Y}}(\rho)) - 2 D(\rho || E^{\mathcal{M}}(\rho)) + 2 \ln d_A ,
\end{equation*}
where $E^{\mathcal{M}}=\frac{{\mathds{1}}}{\ell}\mathop{\rm Tr}\nolimits[\cdot]$. Then, by virtue of Theorem \ref{theo_AT_pinching}, and choosing $\eta=\rho=\mathcal{P}_{\rho_{\mathcal{M}}}(\rho)$, we have:
\begin{equation*}
S(X)_{E_{\mathcal{X}}(\rho)}+  S(Y)_{E_{\mathcal{Y}}(\rho)} \geq  (-1 - 2c_1) \, D(\rho || E^{\mathcal{M}}(\rho)) + 2 \ln d_A .
\end{equation*}

To conclude, just notice that
\begin{equation*}
D(\rho || E^{\mathcal{M}}(\rho)) = - S(A)_{\rho} + \ln d_A ,
\end{equation*}

\end{proof}

Analogously, we can study the case for three different orthonormal basis (see \cite{berta2010uncertainty}). For that, let us recall that given $ {\cal N}_1, {\cal N}_2, {\cal N}_3 \subset {\cal N}$ von Neumann subalgebras and ${\mathcal{M}} \subset {\cal N}_1 \cap {\cal N}_2 \cap {\cal N}_3$, if we consider their associated conditional expectations $E_i$ with respect to a state $\sigma$, and for each pair $({\cal N}_i, {\cal N}_j)$ a result of AT($c_{ij}, d_{ij}$) holds, then for every $\rho \in {\cal D}({\cal N})$:
\begin{equation}
D(\rho || E_*^{\mathcal{M}} (\rho)) \leq \frac{2}{3} \, \underset{i, \, j \in \qty{1,2,3}}{\text{max}} \qty{c_ij} \left( \, D(\rho || E_{1*} (\rho) ) + D(\rho || E_{2*} (\rho) ) + D(\rho || E_{3*} (\rho) ) \, \right) + \frac{d_{12} + d_{13} + d_{23}}{3}.
\end{equation}

\begin{corollary}\label{cor:3pinching}
	Given a finite alphabet $I \in \{\mathcal{X},\mathcal{Y}, \mathcal{Z}\}$, and using the same notation that in Corollary \ref{cor:2pinching}, assume that
	 $$c_1= d_A \underset{I, \, J \in \qty{\mathcal{X}, \mathcal{Y}, \mathcal{Z}}}{\text{max}} \left| |\langle e^{(I)}_i|e^{(J)}_j\rangle|^2-\frac{1}{d_A}   \right|<\frac{1}{2} .$$
	 Then the following strenghtened entropic uncertainty relation holds for any state $\rho\in{\cal D}({\cal H}_{A})$,
	\begin{align*}
	S(X)_{E_{\mathcal{X}}(\rho)}+S(Y)_{E_{\mathcal{Y}}(\rho)}+S(Z)_{E_{\mathcal{Z}}(\rho)}\ge  (2+2c_1)\,S(A)_\rho+(1-2c_1)\ln d_A\,.
	\end{align*}
\end{corollary}

 \subsection{Clustering of correlations}\label{sec:clustering}

The constant $c_1:=\max_{i\in I_{\mathcal{M}}}\|E_{1}^{(i)}\circ E_{2}^{(i)}-(E^{\mathcal{M}})^{(i)}:\,\mathbb{L}_1(\tau_i)\to\mathbb{L}_\infty\|$ appearing in \Cref{theo_AT_pinching}
provides a bound on the following covariance-type quantity: for any $i\in I_{\mathcal{M}}$ and any $X, Y\in \mathbb{L}_1(\tau_i)$,
\begin{align}
\operatorname{Cov}_{\mathbb{C}{\mathds{1}}_{{\cal K}_i},\tau_i}(E_1^{(i)}[X],E_2^{(i)}[Y])&:=\langle  E_1^{(i)}[X]-(E^{\mathcal{M}})^{(i)}[X],\,E_2^{(i)}[Y]-(E^{\mathcal{M}})^{(i)}[Y]\rangle_{\tau_i}\nonumber\\
&\le \,c_1\,\|X\|_{\mathbb{L}_1(\tau_i)}\,\|Y\|_{\mathbb{L}_1(\tau_i)}\,.\label{condL1sclust}
\end{align}
We will call the above property \textit{conditional $\mathbb{L}_1$ clustering of correlations}, and denote it by $\operatorname{cond\mathbb{L}}_1(c_1)$.  Conversely, one can show by duality of $\mathbb{L}_p$-norms that if $\operatorname{cond\mathbb{L}}_1(c_1')$ holds for some positive constant $c_1'$, then
$c_1\le c_1'$: for all $i\in I_{\mathcal{M}}$
\begin{align*}
\|E_{1}^{(i)}\circ E_{2}^{(i)}-(E^{\mathcal{M}})^{(i)}:\,\mathbb{L}_1(\tau_i)\to\mathbb{L}_\infty\|&=\sup_{X\in \mathbb{L}_1(\tau_i)}\,\|E_{1}^{(i)}\circ E_{2}^{(i)}[X]-(E^{\mathcal{M}})^{(i)}[X]\|_\infty\\
&=\sup_{X,Y\in \mathbb{L}_1(\tau_i)}\,\langle Y,\,E_1^{(i)}\circ E_2^{(i)}[X]-(E^{\mathcal{M}})^{(i)}[X]\rangle_{\tau_i}\\
&=\sup_{X,Y\in \mathbb{L}_1(\tau_i)}\,\langle E_1^{(i)}[Y]-(E^{\mathcal{M}})^{(i)}[Y],\, E_2^{(i)}[X]-(E^{\mathcal{M}})^{(i)}[X]\rangle_{\tau_i}\\
&\le c_1'\,.
\end{align*}

In \cite{Kastoryano2014}, the authors introduced a different notion of clustering of correlation in order to show the positivity of the spectral gap of Gibbs samplers\footnote{We formulate it in our general framework of finite-dimensional $*$-algebras.}.
\begin{definition}\label{defL2strong}
	We say that ${\mathcal{M}}\subset {\cal N}_1,{\cal N}_2\subset {\cal N}$ satisfies \emph{strong $\mathbb{L}_2$ clustering of correlations} with respect to the state $\sigma\in{\cal D}({\mathcal{M}})$ with constant $c_{2}>0$ if for all $X,Y\in{\cal N}$,
	\begin{align}\label{L2_clust}
	\operatorname{Cov}_{{\mathcal{M}},\sigma}(E_1[X],E_2[Y])\le \,c_{2}\,    \|X\|_{\mathbb{L}_2(\sigma)}\,\|Y\|_{\mathbb{L}_2(\sigma)}\,.
	\end{align}
	Equivalently, $\|E_1\circ E_2-E^{\mathcal{M}}:\,\mathbb{L}_2(\sigma)\to \mathbb{L}_2(\sigma)\|\le c_2$.
\end{definition}

 \Cref{defL2strong} does not depend on the state $\sigma\in {\cal D}({\mathcal{M}})$ chosen.
This is the content of the next theorem.

\begin{theorem}\label{theo_L2clust}
	Let ${\mathcal{M}}\subset {\cal N}_1,{\cal N}_2\subset{\cal N}\subset {\cal B}({\cal H})$ be von Neumann subalgebras of the algebra ${\cal B}({\cal H})$ so that ${\cal N}_1 \cap {\cal N}_2 \neq \emptyset$. Then, for any two states $\sigma,\sigma'\in{\cal D}({\mathcal{M}})$:
	\begin{equation*}
	\sup_{X\in{\cal N}}\,\frac{\operatorname{Cov}_{{\mathcal{M}},\,\sigma}(E_1[X],E_2[X])}{\|X\|_{\mathbb{L}_2(\sigma)}^2} = \sup_{X\in{\cal N}}\,\frac{\operatorname{Cov}_{{\mathcal{M}},\,\sigma'}(E_1[X],E_2[X])}{\|X\|_{\mathbb{L}_2(\sigma')}^2}
	\end{equation*}
\end{theorem}

We first prove a technical lemma.
\begin{lemma}\label{lemma:L2clustering-invariant}
	Given a conditional expectation $E:{\cal N}\to{\mathcal{M}}\subset {\cal N}\subset {\cal B}({\cal H})$ that is invariant with respect to two different full-rank states, $\rho$ and $\sigma$, the following holds:
	\begin{align*}
	\Gamma_\rho^{1/2}\circ E\circ \Gamma_{\rho}^{-1/2}=\Gamma_\sigma^{1/2}\circ E\circ \Gamma_\sigma^{-1/2}
	\end{align*}
\end{lemma}

\begin{proof}[Proof of \Cref{lemma:L2clustering-invariant}]
	Since we are in finite dimension, the von Neumann algebra ${\mathcal{M}}$ takes the following form:
	\begin{align*}
	{\mathcal{M}}=\bigoplus_i\,{\cal B}({\cal H}_i)\otimes{\mathds{1}}_{{\cal K}_i}\,,
	\end{align*}
	for some decomposition ${\cal H}:=\bigoplus_i\,{\cal H}_i\otimes{\cal K}_i$ of ${\cal H}$. Therefore, since $\rho$ and $\sigma$ are invariant stats of $E$, they can be decomposed as follows:
	\begin{align*}
	\rho=\bigoplus_{i}\, \rho_i\otimes\tau_i\,,~~~	\sigma=\bigoplus_{i}\, \sigma_i\otimes\tau_i\,,
	\end{align*}
	for given positive definite operators $\sigma_i$, $\rho_i$ and where $\tau_i$ is given by ${\mathds{1}}_{\mathcal{K}_i}/d_{\mathcal{K}_i}$. Hence,
	\begin{align*}
	\rho^{-1/4}\sigma^{1/4}=\bigoplus_i\,\rho_i^{-1/4}\sigma_i^{1/4}\otimes{\mathds{1}}_{{\cal K}_i}\in{\cal N}.
	\end{align*}
	Then, it is clear that the following string of identities hold for all $Y\in{\cal B}({\cal H})$:
	\begin{align*}
	\rho^{-1/4}\,\sigma^{1/4}\,E\big[\sigma^{-1/4}\rho^{1/4}\,Y\,\rho^{1/4}\sigma^{-1/4}\big]\,\sigma^{1/4}\rho^{-1/4}&=E\big[\rho^{-1/4}\,\sigma^{1/4}\sigma^{-1/4}\rho^{1/4}\,Y\,\rho^{1/4}\sigma^{-1/4}\sigma^{1/4}\rho^{-1/4}\big]\\
	&=E[Y]\,.
	\end{align*}
	The result follows after choosing $Y=\rho^{-1/4}X\rho^{-1/4}$.

\end{proof}

\begin{proof}[Proof of \Cref{theo_L2clust}]
We begin with proving that the property of strong $\mathbb{L}_2$ clustering of correlations is independent of the invariant state, thanks to \Cref{lemma:L2clustering-invariant}. Indeed, if we choose $Y:= \Gamma_\sigma^{-1/2}(X)$ and call $X':= \Gamma_{\sigma'}^{1/2}(Y)$, it is clear that
	\begin{equation*}
	\norm{X}_{\mathbb{L}_2(\sigma)}^2= \norm{Y}_2^2 \; \; \text{ and } \; \;\norm{Y}_2^2 = \norm{X'}_{\mathbb{L}_2(\sigma')}^2.
	\end{equation*}
	Therefore, we have the following chain of identities:
	\begin{align*}
	\sup_{X\in{\cal N}} \, \frac{\operatorname{Cov}_{{\mathcal{M}},\,\sigma}(E_1[X],E_2[X])}{\|X\|_{\mathbb{L}_2(\sigma)}^2} & = \sup_{X\in{\cal N}} \, \frac{\langle X,\,E_1\circ E_2[X]-E^{\mathcal{M}}[X] \rangle_{\sigma} }{\|X\|_{\mathbb{L}_2(\sigma)}^2}\\
	&=\sup_{Y\in{\cal N}} \, \frac{\langle \Gamma_{\sigma}^{-1/2}(X),\,E_1\circ E_2[\Gamma_{\sigma}^{-1/2}(X)]-E^{\mathcal{M}}[\Gamma_{\sigma}^{-1/2}(X)] \rangle_{\sigma} }{\|Y\|_{2}^2}\\
	&=\sup_{Y\in{\cal N}}\, \frac{\langle X,\,\Gamma_{\sigma}^{1/2}(E_1\circ E_2-E^{\mathcal{M}})[\Gamma_{\sigma}^{-1/2}(X)] \rangle_{\operatorname{HS}} }{\|Y\|_{2}^2}\\
	&=\sup_{Y\in{\cal N}} \, \frac{\langle \Gamma_{\sigma'}^{-1/2}(X),\,E_1\circ E_2[\Gamma_{\sigma'}^{-1/2}(X)]-E^{\mathcal{M}}[\Gamma_{\sigma'}^{-1/2}(X)] \rangle_{\sigma'} }{\|Y\|_{2}^2}\\
	&=\sup_{X'\in{\cal N}} \, \frac{\operatorname{Cov}_{{\mathcal{M}},\,\sigma}(E_1[X'],E_2[X'])}{\|X'\|_{\mathbb{L}_2(\sigma)}^2}\,,
	\end{align*}
	where we have used Lemma \ref{lemma:L2clustering-invariant} in the fourth line.
\end{proof}

It is easy to see that the above notion of strong $\mathbb{L}_2$ clustering of correlation implies that of a \textit{conditional $\mathbb{L}_2$ clustering}, denoted by $\operatorname{cond\mathbb{L}_2}(c_2)$, simply defined by replacing the $\mathbb{L}_1$ norms by $\mathbb{L}_2$ norms in \Cref{condL1sclust}, or equivalently by assuming that
\[
\max_{i\in I_{\mathcal{M}}}\|E_1^{(i)}\circ E_2^{(i)}-(E^{\mathcal{M}})^{(i)}:\,\mathbb{L}_2(\tau_i)\to \mathbb{L}_2(\tau_i)\|\le c_2\,.                                                                                \]
One can ask whether the converse holds. We prove it under the technical assumption that the composition of conditional expectations $E_1\circ E_2$ cancels off-diagonal terms in the decomposition of ${\mathcal{M}}$:

\begin{equation}\label{eq_d1=0}
E_{1*}\circ E_{2*}=E_{1*}\circ E_{2*}\circ\mathcal{P}_{\mathcal{M}}\,.
\end{equation}
We postpone to the next section a discussion on when this holds.

\begin{theorem}\label{theo_transference}
Assume that \Cref{eq_d1=0} holds. Then:
 \begin{enumerate}
  \item $d_1=0$ in \Cref{propboundsd1d2} and
  \item strong $\mathcal{L}_2$ clustering is equivalent to conditional $\mathcal{L}_2$ clustering.
 \end{enumerate}
\end{theorem}
\begin{proof}
Point 1. is straigthforward so we focus on point 2. As already mentioned, strong $ \mathbb L_2$ clsutering implies $\operatorname{cond\mathbb{L}_2}(c_2)$, so we only need to prove the only implication. Now assume that $\operatorname{cond\mathbb{L}_2}(c_2)$ holds with a constant $c_2$ and take $X\in{\cal D}({\cal N})$. We write $T=E_1\circ E_2-E^{\mathcal{M}}$. Remark that, according to the decomposition of ${\mathcal{M}}$ given in \Cref{eq_decomp} and exploiting \Cref{eq_d1=0}, $T$ acts on $X$ as:
 \begin{equation}\label{eq_decomp_T}
   T(X)=\sum_{i\in I_{\mathcal{M}}}\,\left(\mathbb I_{{\cal B}({\cal H}_i)}\otimes T^{(i)}\right)(P_i\,X\,P_i)\,,
  \end{equation}
 where $T^{(i)}$ acts on ${\cal B}({\cal K}_i)$ and where the $P_i$ are the orthogonal projections on ${\cal H}_i\otimes{\cal K}_i$.

 Consider now the Hilbert-Schmidt decomposition of $P_iXP_i$ with respect to $({\cal B}({\cal H}_i),\langle\cdot|\cdot\rangle_{\sigma_i})$ and $({\cal B}({\cal K}_i),\langle\cdot|\cdot\rangle_{\tau_i})$:
 \[P_i\,X\,P_i=\sum_\alpha\,f_\alpha^{(i)}\otimes g_\alpha^{(i)}\,.\]
 Thus we have
  \[\| P_i\,X\,P_i\|_{\mathbb L_2(\sigma_i\otimes\tau_i)}^2=\sum_\alpha\,\|f_\alpha^{(i)}\|_{\mathbb L_2(\sigma_i)}^2\,\|g_\alpha^{(i)}\|_{\mathbb L_2(\tau_i)}^2\,,\]
 and therefore
    \begin{align*}
    \|T(X)\|_{\mathbb L_2(\sigma_i\otimes\tau_i)}^2
    & = \sum_{i\in I_{\mathcal{M}}}\,\|\left(\mathbb I_{{\cal B}({\cal H}_i)}\otimes T^{(i)}\right)(P_i\,X\,P_i)\|_{\mathbb L_2(\sigma_i\otimes\tau_i)}^2 \\
    & = \sum_{i\in I_{\mathcal{M}}}\,\|\sum_\alpha f_\alpha^{(i)}\otimes T^{(i)}(g_\alpha^{(i)})\|_{\mathbb L_2(\sigma_i\otimes\tau_i)}^2 \\
    & = \sum_{i\in I_{\mathcal{M}}}\,\sum_\alpha \|f_\alpha^{(i)}\|_{\mathbb L_2(\sigma_i)}^2\,\|T^{(i)}(g_\alpha^{(i)})\|_{\mathbb L_2(\tau_i)}^2 \\
    & \leq c_2 \|\sum_{i\in I_{\mathcal{M}}}\,P_i\,X\,P_i\|_{\mathbb L_2(\sigma)}^2 \\
    & \leq c_2 \|X\|_{\mathbb L_2(\sigma)}^2\,,
    \end{align*}
 where in the third line we use that $(f_\alpha^{(i)})_\alpha$ is an orthogonal family for every $i\in I_{\mathcal{M}}$. This shows that
    \[\|E_1\circ E_2-E^{\mathcal{M}}\,:\,L_2(\sigma)\to L_2(\sigma)\|\leq c_2\,,\]
 which is equivalent to strong $L_2$ clustering.
\end{proof}

Similarly to \Cref{defL2strong}, one could define a notion of \textit{strong $\mathbb{L}_1$ clustering of correlation} with respect to a state $\sigma\in{\cal D}({\mathcal{M}})$:
\begin{align*}
\|E_1\circ E_2-E^{\mathcal{M}}:\,\mathbb{L}_1(\sigma)\to \mathbb{L}_\infty({\cal N})\|\equiv c_1(\sigma)\,<\infty.
\end{align*}
This would in particular imply $\operatorname{cond\mathbb{L}_1}(c_1(\sigma))$. With this notion, and from an argument very similar to that of the proof of \Cref{theo_AT_pinching}, we could show the following bound on the error term in \Cref{propAT}:
	\begin{align*}
	\ln \left\lbrace \int_{-\infty}^\infty\,\mathop{\rm Tr}\nolimits\left[\rho_{1}\rho_{{\mathcal{M}}}^{\frac{-1-it}{2}}\rho_{2}\rho_{{\mathcal{M}}}^{\frac{-1+it}{2}}   \right]\,\beta_0(t)\,dt \right\rbrace&\le  	 2\,\|E_{1}\circ E_2-E^{{\mathcal{M}}}:\,\mathbb{L}_1(\rho_{\mathcal{M}})\to\mathbb{L}_\infty\|\,D(\rho\|\rho_{\mathcal{M}})\,\\
	&\equiv 2 \,c_1(\rho_{\mathcal{M}})\,D(\rho\|\rho_{\mathcal{M}})\,.
	\end{align*}

From this, one would conclude a strong approximate tensorization result if one could find a uniform bound on $c_1(\sigma)$ for any $\sigma\in{\cal D}({\mathcal{M}})$. However, and as opposed to the case of strong $\mathbb{L}_2$ clustering, the constant $c_1(\sigma)$ depends on the state $\sigma$, and can in particular diverge: this is the case whenever there exists $i\in I_{\mathcal{M}}$ such that $\dim({\cal H}_i)<\infty$, and for a state $\sigma:=|\psi\rangle\langle\psi|_{{\cal H}_i}\otimes \tau_i$ that is pure on ${\cal H}_i$. This justifies our choice of $\operatorname{cond\mathbb{L}}_1$ as the better notion of $\mathbb{L}_1$ clustering in the quantum setting.

\begin{remark}
As a consequence of the previous proposition, we realize that the condition assumed in \cite{Kastoryano2014} of strong $\mathbb{L}_2$ clustering of correlation with respect to one invariant state, to prove positivity of the spectral gap for the Davies dynamics, would be analogous to assuming strong $\mathbb{L}_1$ clustering of correlation with respect to any invariant state. If we could reduce the problem mentioned above to finding a  bound on $c_1(\sigma)$ for only one $\sigma$, we would also be able to prove positivity of the MLSI for the Davies dynamics from here, although from the discussion above we know that it is not possible in general.
\end{remark}

\section{Conditional expectations on fixed-points of Markovian evolution}\label{sec:condexp}

In this section, we consider conditional expectations arising from Petz recovery maps and from Davies generators. The latter model the dynamics resulting from the weak coupling limit of a system in contact with a large heat-bath. The main result, \Cref{coro_equal_cond}, states that the corresponding conditional expectations coincide.

\subsection{Conditional expectations generated by a Petz recovery map}

Here, we further discuss the notion of conditional expectations that we will consider in the case of Gibbs states on spin systems. The discussion is largely inspired by some results in \cite{carlen2017recovery}.

Let $\sigma$ be a faithful density matrix on the finite-dimensional algebra ${\cal N}$ and let ${\mathcal{M}}\subset{\cal N}$ be a subalgebra. We denote by $E_\tau$  the conditional expectation onto ${\mathcal{M}}$ with respect to the completely mixed state (i.e. $E_\tau$ is self-adjoint with respect to the Hilbert-Schmidt inner product). We also adopt the following notations: we write $\sigma_{\mathcal{M}}=E_\tau(\sigma)$ and
\[\mathcal A_\sigma(X):=\sigma_{\mathcal{M}}^{-\frac12}\,E_\tau[\sigma^{\frac12}\,X\,\sigma^{\frac12}]\,\sigma_{\mathcal{M}}^{-\frac12}\,.\]
Remark that $\mathcal A_\sigma$ is also the unique map such that for all $X\in{\cal N}$ and all $Y\in{\mathcal{M}}$:
\[\mathop{\rm Tr}\nolimits[\sigma^{\frac12}\,X\,\sigma^{\frac12}\,Y]=\mathop{\rm Tr}\nolimits[\sigma_{\mathcal{M}}^{\frac12}\,\mathcal A_\sigma(X)\,\sigma_{\mathcal{M}}^{\frac12}\,Y]\,.\]
The adjoint of $\mathcal A_\sigma$ is the Petz recovery map of $E_\tau$ with respect to $\sigma$, denoted by ${\cal R}_\sigma$:
\begin{align*}
\mathcal{R}_{\sigma}(\rho_{\mathcal{M}}):=\sigma^{\frac{1}{2}}\sigma_{\mathcal{M}}^{-\frac{1}{2}}\rho_{\mathcal{M}}\sigma_{\mathcal{M}}^{-\frac{1}{2}}\sigma^{\frac{1}{2}}\,,
\end{align*}
where $\rho_{\mathcal{M}}:=E_\tau(\rho)$. It is proved in \cite{carlen2017recovery} that $\mathcal A_\sigma$ is a conditional expectation if and only if $\sigma\,X\,\sigma^{-1}\in{\mathcal{M}}$ for all $X\in{\mathcal{M}}$. In the general case, we denote by
\begin{equation}\label{eq_cond_Petz}
E_\sigma:=\lim_{n\to\infty}\mathcal A_\sigma^n
\end{equation}
the projection on its fixed-point algebra for the $\sigma$-KMS inner product, which is a conditional expectation as we assumed $\sigma$ to be faithful. That is, $E_\sigma$ is the orthogonal projection for $\langle\cdot,\cdot\rangle_\sigma$ on the algebra:
\[{\mathcal{F}}(\mathcal A_\sigma)=\{X\in{\cal N}\,;\,\mathcal A_\sigma(X)=X\}\,.\]

\begin{example}\label{ex_bipartite}
 Our main example is the case of a bipartite system $AB$. In this case, ${\cal N}={\cal B}({\cal H}_{AB})$ and ${\mathcal{M}}={\mathds{1}}_{{\cal H}_A}\otimes{\cal B}({\cal H}_{B})$. Let $\sigma=\sigma_{AB}$ be a faithful density matrix on $AB$. The partial trace with respect to ${\cal H}_A$ is an example of a conditional expectation $E_\tau$ which is not compatible with $\sigma_{AB}$, in general. With this choice, we obtain:
 \begin{align*}
  & \sigma_{\mathcal{M}}=\sigma_B\,,\\
  & \mathcal A_{\sigma_{AB}}(X)=\sigma_{B}^{-\frac12}\,\mathop{\rm Tr}\nolimits_A[\sigma_{AB}^{\frac12}\,X\,\sigma_{AB}^{\frac12}]\,\sigma_{B}^{-\frac12}\,,\qquad\forall X\in{\cal B}({\cal H}_{AB})\, , \\
  & {\cal R}_{\sigma_{AB}}(\rho_B)=\sigma_{AB}^{\frac12}\sigma_{B}^{-\frac12}\,\rho_B\,\sigma_{B}^{-\frac12}\sigma_{AB}^{\frac12}\,,\qquad\forall \rho_{AB}\in{\cal D}({\cal H}_{AB})\,,
 \end{align*}
 where here we identify an operator $X_B$ with ${\mathds{1}}_A\otimes X_B$ for sake of simplicity. An important remark is that, in general, $E_{\sigma_{AB}^*}$ is not a recovery map.

\end{example}

We are now ready to state a first technical proposition, whose content is mostly contained in \cite{carlen2017recovery}.

\begin{proposition}\label{prop_condexp}
 Let $\rho$ be a density matrix on ${\cal N}$. Then the following assertions are equivalent:
\begin{enumerate}
 \item $D(\rho\|\sigma)=D(\rho_{\mathcal{M}}\|\sigma_{\mathcal{M}})$;
 \item $\rho={\cal R}_\sigma(\rho_{\mathcal{M}})$;
 \item $\rho=E_{\sigma*}(\rho)$;
  \item $D(\rho\|E_{\sigma*}(\rho))=0$.
 \item $D(\rho\|\sigma)=D(E_{\sigma*}(\rho)\|\sigma)$;
\end{enumerate}
 \end{proposition}

Remark that $(1)\Leftrightarrow(2)$ is Petz condition for equality in the data processing inequality. The equivalence $(3)\Leftrightarrow(4)$ is obvious, and $(4)\Leftrightarrow(5)$ is a consequence of Lemma 3.4 in \cite{junge2019stability}:
\begin{equation}\label{very_nice_lemma}
D(\rho\|\sigma)-D(E_{\sigma*}(\rho)\|\sigma)=D(\rho\|E_{\sigma*}(\rho))\,.
\end{equation}
We shall now give a direct proof of $(2)\Leftrightarrow(3)$.

\begin{proof}[Proof of \Cref{prop_condexp}]
We only prove $(2)\Leftrightarrow(3)$. Note that for $X\in{\cal N}$, by definition $X=\mathcal A_\sigma(X)$ iff $X=E_\sigma(X)$. Then let $\rho$ be a density matrix on ${\cal N}$ and define $X=\sigma^{-\frac12}\,\rho\,\sigma^{-\frac12}$. We have:
\begin{align*}
 \rho={\cal R}_\sigma(\rho_{\mathcal{M}})
 & \Leftrightarrow X=\mathcal A_\sigma(X) \\
 & \Leftrightarrow X=E_\sigma(X) \\
 & \Leftrightarrow \rho= \sigma^{\frac12}\,X\,\sigma^{\frac12} = \sigma^{\frac12}\,E_\sigma(X)\,\sigma^{\frac12}=E_{\sigma*}(\sigma^{\frac12}\,X\,\sigma^{\frac12})=E_{\sigma*}(\rho)\,.
\end{align*}
where in the last line we use property 3 in Proposition \ref{prop_condexp}.
\end{proof}

It would be interesting to compare the two notions of ``conditional'' relative entropies $D(\rho\|\sigma)-D(\rho_{\mathcal{M}}\|\sigma_{\mathcal{M}})$ (introduced in \cite{[CLP18],[CLP18a],BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}) and $D(\rho\|E_{\sigma*}(\rho))$. This is the content of the following proposition.

\begin{proposition}\label{prop_compar_cond}
 For any state $\eta\in{\cal D}({\cal N})$ such that $E_{\sigma*}(\eta)=\eta$ and any state $\rho\in{\cal D}({\cal N})$, we have
 \begin{equation}\label{eq_prop_compar_cond1}
  D(\rho\|\sigma)-D(\rho_{\mathcal{M}}\|\sigma_{\mathcal{M}})=D(\rho\|\eta)-D(\rho_{\mathcal{M}}\|\eta_{\mathcal{M}})\,,
 \end{equation}
i.e. the difference of relative entropies does not depend on the choice of the invariant state for $E_\sigma$.Consequently,
\begin{equation}\label{eq_prop_compar_cond2}
 D(\rho\|\sigma)-D(\rho_{\mathcal{M}}\|\sigma_{\mathcal{M}})\leq D(\rho\|E_{\sigma*}(\rho))\,.
\end{equation}
\end{proposition}

\begin{proof}
 \Cref{eq_prop_compar_cond2} is a direct consequence of \Cref{eq_prop_compar_cond1} when applied to $\eta=E_{\sigma*}(\rho)$, so we focus on the first equation (remark that it can be seen as a counterpart of \Cref{very_nice_lemma} for the difference of relative entropies). To this end, we need the following state $\sigma_{\mathop{\rm Tr}\nolimits}$ defined in \cite{BarEID17} and heavily exploited in \cite{bardet2018hypercontractivity}:
 \[\sigma_{\mathop{\rm Tr}\nolimits}=E_{\sigma*}\Big(\frac{\mathds 1}{d_{\cal H}}\Big)\,.\]
 It has the property that for all $X\in{\mathcal{F}}(\mathcal A_\sigma)$, $[X,\sigma_{\mathop{\rm Tr}\nolimits}]=0$ (see Lemma 3.1 in \cite{BarEID17}). Then it is enough to prove that for all $\eta\in{\cal D}({\cal N})$ such that $E_{\sigma*}(\eta)=\eta$, we have:
 \[D(\rho\|\sigma_{\mathop{\rm Tr}\nolimits})-D(\rho_{\mathcal{M}}\|(\sigma_{\mathop{\rm Tr}\nolimits})_{\mathcal{M}})=D(\rho\|\eta)-D(\rho_{\mathcal{M}}\|\eta_{\mathcal{M}})\,.\]
 Now any such $\eta$ can be written $\eta=X\sigma_{\mathop{\rm Tr}\nolimits}$ with $X\in{\mathcal{F}}(\mathcal A_\sigma)$. Remark that by definition of ${\mathcal{F}}(\mathcal A_\sigma)$, $X\in{\mathcal{M}}$  so that $E_\tau(\eta)=X E_\tau(\sigma_{\mathop{\rm Tr}\nolimits})$. Using the commutation between $X$ and $\sigma_{\mathop{\rm Tr}\nolimits}$ and developping the RHS of the previous equation we get the result.
\end{proof}

\subsection{Davies semigroups}\label{subsec:clustering-of-correlations}

Let ${\cal H}$ be the Hilbert space representing a quantum system and let $H$ be a selfadjoint operator on it, representing the Hamiltonian of the system. The corresponding Gibbs state at inverse temperature $\beta$ is defined as
\begin{align}
\sigma=\frac{\mathrm{e}^{-\beta H}}{\mathop{\rm Tr}\nolimits[\mathrm{e}^{-\beta H}]}\,.
\end{align}

Next, consider the Hamiltonian $H^{\operatorname{HB}}$ of the heat bath, as well as a set of system-bath interactions $\{ S_{\alpha}\otimes B_{\alpha} \}$, for some label $\alpha$. Here, we do not assume anything on the $S_\alpha$'s. The Hamiltonian of the universe composed of the system and its heat-bath is given by
\begin{align}
H=H_\Lambda+H^{\operatorname{HB}}+\sum_{\alpha\in\Lambda}S_{\alpha}\otimes B_{\alpha}\,.
\end{align}
Assuming that the bath is in a Gibbs state, by a standard argument (e.g. weak coupling limit, see \cite{[SL78]}), the evolution on the system can be approximated by a quantum Markov semigroup whose generator is of the following form:
\begin{align}\label{eq_lindblad}
{\cal L}^{\operatorname{D},\beta}(X)=\sum_{\omega,\alpha}\,\chi^{\beta}_{\alpha}(\omega)\,\Big(  S_{\alpha}^*(\omega)XS_{\alpha}(\omega)-\frac{1}{2}\,\big\{  S_{\alpha}^*(\omega)S_{\alpha}(\omega),X \big\}   \Big)\,.
\end{align}
The Fourier coefficients of the two-point correlation functions of the environment $\chi_{\alpha}^\beta$ satisfy the following KMS condition:
\begin{align}\label{eq_KMS}
\chi_{\alpha}^\beta(-\omega)=\mathrm{e}^{-\beta\omega}\,\chi_{\alpha}^\beta(\omega)\,.
\end{align}
The operators $S_{\alpha}(\omega)$ are the Fourier coefficients of the system couplings $S_{\alpha}$, which means that they satisfy the following equation for any $t\in\mathbb{R}$:
\begin{align}\label{eq!}
\mathrm{e}^{-itH}\,S_{\alpha}\mathrm{e}^{it H}=\sum_\omega\mathrm{e}^{it\omega}S_{\alpha}(\omega)\,\qquad\Leftrightarrow  \qquad  S_\alpha(\omega)=\sum_{\varepsilon-\varepsilon'=\omega}P_\varepsilon\,S_\alpha\,P_{\varepsilon'}\,.
\end{align}
where the sum is over a finite number of frequencies. This implies in particular the following useful relation:
\begin{align}\label{eq_eigenvector1}
\Delta_{\sigma}(S_{\alpha}(\omega))=\mathrm{e}^{\beta\omega}\,S_{\alpha}(\omega)\,.
\end{align}
The above identity means that the operators $S_{\alpha}(\omega)$ form a basis of eigenvectors of $\Delta_\sigma$. Next, we define the conditional expectation onto the algebra ${\mathcal{F}}({\cal L})$ of fixed points of ${\cal L}$ with respect to the Gibbs state $\sigma=\sigma^\beta$ as follows \cite{Kastoryano2014}:
\begin{align}\label{Davies_cond}
E^{\operatorname{D},\beta}:=\lim_{t\to \infty}\mathrm{e}^{t{\cal L}^{\operatorname{D},\beta}}\,.
\end{align}
Our first result is a characterization of the fixed-point algebra in the Davies case.
\begin{proposition}\label{prop_fixedpoint_Davies}
 One has
 \begin{equation}\label{eq_prop_fixedpoint_Davies}
  {\mathcal{F}}({\cal L}^{\operatorname{D},\beta})=\{\sigma^{it}\,S_\alpha\,\sigma^{-it}\,;\,\forall\,t\geq0,\,\forall \alpha\}'\,,
 \end{equation}
 where the notation $\{ \cdot \}'$ denotes the centralizer of the set.
\end{proposition}
\begin{proof}
	We recall that ${\mathcal{F}}({\cal L}^{\operatorname{D,\beta}})=\{S_\alpha(\omega)\}'$. Hence, since $\sigma^{it}S_\alpha \sigma^{-it}$ can be expressed as a linear combination of the $S_\alpha(\omega)$'s by \Cref{eq!}, it directly follows that
	\begin{align*}
	{\mathcal{F}}({\cal L}^{\operatorname{D,\beta}})\subseteq \{\sigma^{it}\,S_\alpha\,\sigma^{-it}\,;\,t\geq0\}'
	\end{align*}
	To prove the opposite direction, we let $X\in  \{\sigma^{it}\,S_\alpha\,\sigma^{-it}\,;\,t\geq0\}'$. This means in particular that, for all $t\in\mathbb{R}$, and all $\alpha$:
	\begin{align}\label{eq:sumomega}
	[X, \sigma^{it}S_{\alpha}\sigma^{-it}]=\sum_{\omega}\,\mathrm{e}^{it\omega}\,[X,S_{\alpha}(\omega)]=0\,.
	\end{align}
	Since the equation holds for all $t\in\mathbb{R}$, we can differentiate it $N\equiv|\{\omega\}|$ times at $0$  to get that, for any $0\le n\le N-1$:
	\begin{align*}
	\sum_{\omega}\,\omega^n\,[X,S_\alpha(\omega)]=0\,.
	\end{align*}
Using an arbitrary labelling of the $N$ distinct frequencies $\omega_1,...,\omega_N$, the resulting $N$ linear equations can be rewritten as
\begin{align*}
\begin{pmatrix}
1 & 1& 1 & \dots & 1 \\
\omega_1& \omega_2 &\omega_3 & \dots & \omega_N \\
\omega_1^2&\omega_2^2 &\omega_3^2 & \dots & \omega_N^2 \\
\hdotsfor{5} \\
\omega_1^{N-1} & \omega_2^{N-1} &\omega_3^{N-1} & \dots &\omega_N^{N-1}
\end{pmatrix}\,\begin{pmatrix}
[X,S_\alpha(\omega_1)]\\
[X,S_\alpha(\omega_2)]\\
[X,S_\alpha(\omega_3)]\\
\hdotsfor{1}\\
[X,S_\alpha(\omega_N)]
\end{pmatrix}=0
\end{align*}
Since all the frequencies $\omega_i$ are distinct, their Vandermonde matrix is invertible. Hence, $[X,S_\alpha(\omega)]=0$ for all $\omega$, so that $X\in{\mathcal{F}}({\cal L}^{\operatorname{D},\beta})$.
\end{proof}
Combining this result with a result from \cite{carlen2017recovery}, we can finally show that the conditional expectations in the Davies and the Petz cases are the same.
\begin{theorem}\label{coro_equal_cond}
 Define the algebra ${\mathcal{M}}=\{S_\alpha\}'$. Define $E^{\operatorname{D},\beta}$ as above and $E_{\sigma}$ as in \Cref{eq_cond_Petz} with respect to the inclusion ${\mathcal{M}}\subset{\cal B}({\cal H})$. Then both conditional expectations are equal.
\end{theorem}

\begin{proof}
 First, we remark that both conditional expectations are self-adjoint with respect to the $\sigma$-KMS inner product. Therefore, by uniqueness of the conditional expectation, it is enough to prove that ${\mathcal{F}}({\cal L}^{\operatorname{D},\beta})={\mathcal{F}}(A_{\sigma})$. The analysis of the algebra ${\mathcal{F}}(A_{\sigma})$ was carried out in \cite{carlen2017recovery}\footnote{Compared to \cite{carlen2017recovery}, the role of $\rho$ and $\sigma$ is exchanged. The result nevertheless stays the same, as can be readily checked from their proof.}. In particular, they proved (Theorem 3.3) that ${\mathcal{F}}(A_{\sigma})$ is the largest $*$-sub-algebra of ${\mathcal{M}}$ left-invariant by the modular operator. From this characterization, it is easy to see that ${\mathcal{F}}({\cal L}^{\operatorname{D},\beta})\subseteq {\mathcal{F}}({\cal A}_\sigma)$: indeed ${\mathcal{F}}({\cal L}^{\operatorname{D},\beta})=\{S_\alpha(\omega)\}'\subseteq \{S_\alpha\}'\equiv {\mathcal{M}}$. Moreover, for any $X\in {\mathcal{F}}({\cal L}^{\operatorname{D},\beta})$
 \begin{align*}
 [\Delta_\sigma(X),S_\alpha(\omega)]&=\sigma\,X\,\sigma^{-1}\,S_\alpha(\omega)-S_\alpha(\omega)\sigma X\sigma^{-1}\\
 &=\mathrm{e}^{-\beta\omega }\,\big( \sigma XS_{\alpha}(\omega)\sigma^{-1}-\sigma\,S_\alpha(\omega)X\sigma^{-1}  \big)\\
 &=\mathrm{e}^{-\beta\omega }\,\sigma\,[X,S_\alpha(\omega)]\,\sigma^{-1}\\
 &=0\,.
 \end{align*}
 It remains to show that any $*$-sub-algebra $\mathcal{V}$ of ${\mathcal{M}}$ which is invaraint by $\Delta_\sigma$ is contained in ${\mathcal{F}}({\cal L}^{\operatorname{D},\beta})$. This directly follows from (\ref{eq_prop_fixedpoint_Davies}): since for any $X\in\mathcal{V}$, $\Delta(X)\in\mathcal{V}$, we have that
 \begin{align*}
 [X,\sigma^{it}\,S_\alpha\,\sigma^{-it}]&=X\,\sigma^{it}\,S_\alpha\,\sigma^{-it}-\sigma^{it}\,S_\alpha\,\sigma^{-it}X\\
 &=\sigma^{it}\Delta_{\sigma}^{-it}(X)S_\alpha\sigma^{-it}-\sigma^{it}S_\alpha \Delta_{\sigma}^{-it}(X)\sigma^{-it}\\
 &=\sigma^{it}\,[\Delta_{\sigma}^{-it}(X),S_\alpha]\,\sigma^{-it}\\
 &=0\,
 \end{align*}
 and the result follows.
\end{proof}

\section{Introduction}
In the last few decades, entropy has been proven to be a fundamental object in various fields of mathematics and theoretical physics. Its quantum analogue characterizes the optimal rate at which two different states of a system can be discriminated when an arbitrary number of copies of the system is available. Given two states $\rho,\sigma$ of a finite-dimensional von Neumann algebra ${\cal N}\subset {\cal B}({\cal H})$, it is given by
\begin{align*}
D(\rho\|\sigma):=
\mathop{\rm Tr}\nolimits[\rho\,(\ln\rho-\ln\sigma)]\,,
\end{align*}
whenever $\mathop{\rm supp}\nolimits(\rho)\subset\mathop{\rm supp}\nolimits(\sigma)$, where $\mathop{\rm Tr}\nolimits$ denotes the unnormalized trace on ${\cal B}({\cal H})$. When $\sigma:={\mathds{1}}_{{\cal H}}/d_{\cal H}$ is the completely mixed state of ${\cal B}({\cal H})$, the relative entropy can be written in terms of the von Neumann entropy $S(\rho):=-\mathop{\rm Tr}\nolimits[\rho\ln\rho]$ of the state $\rho$:
\begin{align*}
D(\rho\|{\mathds{1}}_{\cal H}/d_{\cal H})=-S(\rho)+\ln(d_{\cal H})\,.
\end{align*}

Probably the most fundamental property of entropy is the following \textit{strong subadditivity} inequality (SSA) \cite{lieb1973SSA}: given a tripartite system ${\cal H}_{ABC}:={\cal H}_A\otimes {\cal H}_B\otimes {\cal H}_C$, and a state $\rho\equiv \rho_{ABC}$ on ${\cal H}_{ABC}$, the following holds
\begin{align}\tag{SSA}
S(\rho_{ABC})+S(\rho_B)\le S(\rho_{AB})+S(\rho_{BC})\,,
\end{align}
where for any subsystem $D$ of $ABC$, $\rho_{D}:=\mathop{\rm Tr}\nolimits_{D^c}[\rho_{ABC}]$ denotes the marginal state on $D$. Restated in terms of the quantum relative entropy, SSA takes the following form:
\begin{align}\label{SSA}
D\left(\rho_{ABC}\Big\|\rho_B\otimes \frac{{\mathds{1}}_{AC}}{d_{{\cal H}_{AC}}}\right)\le D\left(\rho_{ABC}\Big\| \rho_{AB}\otimes \frac{{\mathds{1}}_{C}}{d_{{\cal H}_C}}  \right)+D\left(\rho_{ABC}\Big\| \rho_{BC}\otimes \frac{{\mathds{1}}_{A}}{d_{{\cal H}_{A}}}  \right)\,.
\end{align}

In the present paper, we consider the following more general framework: let ${\mathcal{M}}\subset {\cal N}_1,{\cal N}_2\subset{\cal N}$ be four von Neumann subalgebras of the algebra of linear operators acting on a finite-dimensional Hilbert space ${\cal H}$, and let $E^{\mathcal{M}},E_1,E_2$ be corresponding conditional expectations onto ${\mathcal{M}},{\cal N}_1,{\cal N}_2$, respectively. When the quadruple $({\mathcal{M}},{\cal N}_1,{\cal N}_2,{\cal N})$ forms a \textit{commuting square}, that is when $E_{1}\circ E_2=E_2\circ E_1=E^{\mathcal{M}}$, the following generalization of SSA occurs: for any state $\rho$ on ${\cal N}$,
\begin{align}\label{eqcoarsegrain}
D(\rho\|E^{{\mathcal{M}}}_*(\rho))\le D(\rho\| E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\,,
\end{align}
where the maps $E^{\mathcal{M}}_*,E_{1*}$, $E_{2*}$ are the Hilbert-Schmidt duals of $E^{\mathcal{M}}, E_{1}, E_2$, also known as \textit{coarse-graining maps} \cite{petz2008}. One can easily recover the previous (SSA) inequality from (\ref{eqcoarsegrain}) by taking ${\cal N}\equiv{\cal B}({\cal H}_{ABC})$, and the coarse graining maps to be the partial traces onto the subalgebras ${\cal N}_1\equiv{\cal B}({\cal H}_{AB}) $, ${\cal N}_2\equiv {\cal B}({\cal H}_{BC})$ and ${\mathcal{M}}\equiv {\cal B}({{\cal H}_B})$.

In the context of interacting lattice spin systems, conditional expectations arising e.g. from the large time limit of a dissipative evolution on subregions of the lattice generally do not satisfy the commuting square assumption. In this case, approximations of the SSA were found in the classical case (i.e. when all algebras are commutative), and when ${\mathcal{M}}\equiv \mathbb{C}{\mathds{1}}_{{\cal H}}$ \cite{cesi2001quasi}. These inequalities, termed as \textit{approximate tensorization of the relative entropy} (also known in the literature as \textit{quasi-factorization of the relative entropy}, \cite{cesi2001quasi}, \cite{[CLP18a]}, \cite{BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}), take the following form
\begin{align*}
D(\rho\|\sigma)\le \frac{1}{1-2c}\,\big(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\big)\,,
\end{align*}
where $\sigma:= E^{{\mathcal{M}}}_*(\rho)$ and $c:=\|E_{1}\circ E_{2}-E^{\mathcal{M}}:\,\mathbb{L}_1(\sigma)\to \mathbb{L}_\infty({\cal N})\|$ is a constant that measures the distance from being a commuting square for the quadruple $({\mathcal{M}},{\cal N}_1,{\cal N}_2,{\cal N})$. Typically, $c=0$ at infinite temperature, and remains small for conditional expectations onto far apart regions and at high enough temperature. Such an inequality was recently generalized to the quantum setting in \cite{[CLP18],[CLP18a],BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}. In \cite{gao2017strong}, a different extension of SSA in the case of noncommuting squares was proposed, with an additive error term that measures the distance from being a commuting square.

\paragraph{Main result:}In this paper, we take one step further and prove a \textit{weak approximate tensorization} for the relative entropy, which amounts to the existence of positive constants $c\geq 1$ and $d\ge 0$ such that (see \Cref{theo_AT_pinching} )\footnote{The definition of (strong) approximate tensorization recently arose in the paper \cite{nick2019scoopingpaper}, where it was coined as ``adjusted subadditivity of relative entropy''. As explained by the author himself, this definition was already present in an earlier draft of our present article, which we had shared with him (see also the recently published thesis \cite{thesisangela}). Furthermore, the techniques that we introduce here are different from his, and more in line with the classical literature on the subject.}
	\begin{align}\tag{$\operatorname{AT(c,d)}$}
D(\rho\|E^{\mathcal{M}}_*(\rho))\le c\,\big(D(\rho\|E_{1*}(\rho))+D(\rho\|E_{2*}(\rho))\big)+d\,.
\end{align}
As opposed to the classical setting, conditional expectations arising from dissipative evolutions on quantum lattice spin systems do not satisfy the commuting square condition in general, even at infinite temperature. This difference is exclusively due to the non-commutativity of the underlying algebras, which explains the introduction of the \textit{weak} constant $d$. Here, we estimate both constants $c$ and $d$ in terms of the interactions appearing in the Hamiltonian of the system. Unlike previous work on the subject, our inequality exactly reduces to that of \cite{cesi2001quasi} for commuting algebras.

As mentioned previously, our main application of these inequalities is in the context of mixing times of quantum lattice spin systems - although we expect these inequalities and their proof techniques to find other applications in quantum information theory. In \cite{cesi2001quasi}, Cesi used his inequality in order to show the exponential convergence in relative entropy of classical Glauber dynamics on lattice systems towards equilibrium, independently of the lattice size. In a forthcoming paper, we will make use of the approximate tensorization inequality to show similar convergences for dissipative quantum Gibbs samplers.

\paragraph{Outline of the paper}
In \Cref{sec2}, we review basic mathematical concepts used in this paper, and more particularly the notion of a noncommutative conditional expectation. We derive theoretical expressions on the \textit{strong} ($c$) and \textit{weak} ($d$) constants for general von Neumann algebras in \Cref{sec:strong-quasi-tensorization}, where our main result is stated in \Cref{theo_AT_pinching}, and subsequently apply them to obtain strengthenings of uncertainty relations. In \Cref{sec:condexp}, we review the conditional expectations arising from Petz recovery maps and from Davies generators and show in  \Cref{coro_equal_cond}, states that both conditional expectations coincide.  Finally, in \Cref{sec:dabies}, we derive explicit bounds on the constants $c$ and $d$ for conditional expectations associated to Gibbs samplers on lattice spin systems in terms of the interactions of the corresponding Hamiltonian.

\section{Notations and definitions}\label{sec2}
In this section, we fix the basic notation used in the paper, and introduce the necessary definitions.

\subsection{Basic notations}

Let $({\cal H},\langle .|.\rangle)$ be a finite dimensional Hilbert space of dimension $d_{\cal H}$. We denote by ${\cal B}({\cal H})$ the Banach space of bounded operators on ${\cal H}$,  by ${\cal B}_{\operatorname{sa}}({\cal H})$ the subspace of self-adjoint operators on ${\cal H}$, i.e. ${\cal B}_{\operatorname{sa}}({\cal H})=\left\{X={\cal B}({\cal H});\ X=X^*\right\}$, where the adjoint of an operator $Y$ is written as $Y^*$, and by ${\cal B}_+({\cal H})$ the cone of positive semidefinite operators on ${\cal H}$. We will also use the same notations ${\cal N}_{\operatorname{sa}}$ and ${\cal N}_+$ in the case of a von Neumann subalgebra ${\cal N}$ of ${\cal B}({\cal H})$. The identity operator on ${\cal N}$ is denoted by ${\mathds{1}}_{\cal N}$, dropping the index ${\cal N}$ when it is unnecessary. In the case of ${\cal B}(\mathbb{C}^\ell)$, $\ell\in\mathbb{N}$, we will also use the notation ${\mathds{1}}$ for ${\mathds{1}}_{\mathbb{C}^\ell}$. Similarly, given a map $\Phi:{\cal B}({\cal H})\to{\cal B}({\cal H})$, we denote its dual with respect to the Hilbert-Schmidt inner product as $\Phi_*$. We also denote by ${\rm{id}}_{{\cal B}({\cal H})}$, or simply ${\rm{id}}$, resp. ${\rm{id}}_\ell$, the identity superoperator on ${\cal B}({\cal H})$, resp. ${\cal B}(\mathbb{C}^\ell)$. We denote by $\mathcal{D}({\cal H})$ the set of positive semidefinite, trace-one operators on ${\cal H}$, also called \emph{density operators},  by ${\cal D}_+({\cal H})$ the subset of full-rank density operators, and by ${\cal D}_{\leq}({\cal H})$ the set of subnormalized states. In the following, we will often identify a density matrix $\rho\in\mathcal{D}({\cal H})$ and the \emph{state} it defines, that is the positive linear functional ${\cal B}({\cal H})\ni X\mapsto\mathop{\rm Tr}\nolimits(\rho \,X)$.

\subsection{Entropic quantities and $\mathbb{L}_p$ spaces} Throughout this paper, we will use various distance measures between states and between observables. We collect them here for sake of clarity: given a state $\rho\in{\cal D}({\cal N})$, its \textit{von Neuman entropy} is defined by
\begin{align*}
S(\rho):=-\mathop{\rm Tr}\nolimits\big[ \rho\,\ln\rho \big]\,.
\end{align*}
Next, when $\rho\equiv \rho_{AB}\in {\cal D}({\cal H}_A\otimes {\cal H}_B)$ is the state of a bipartite quantum system, its \textit{conditional entropy} is defined by
\begin{align*}
S(A|B)_\rho:=S(\rho_{AB})-S(\rho_B)\,,
\end{align*}
where $\rho_B:=\mathop{\rm Tr}\nolimits_A(\rho)$ corresponds to the marginal of $\rho$ over the subsystem ${\cal H}_B$. More generally, given two positive semidefinite operators $\rho,\sigma\in {\cal B}_+({\cal H})$, the \textit{relative entropy} between $\rho$ and $\sigma$ is defined as follows \cite{Umegaki-RelativeEntropy-1962}:
\begin{align*}
D(\rho\|\sigma):=\left\{\begin{aligned}
&\mathop{\rm Tr}\nolimits[\rho\,(\ln\rho-\ln\sigma)]\,\,\,\,\mathop{\rm supp}\nolimits(\rho)\subset\mathop{\rm supp}\nolimits(\sigma)\\
&+\infty\,\,\qquad\qquad\qquad\text{else}
\end{aligned}\right.
\end{align*}
Moreover, given (possibly subnormalized) positive semidefinite operators $\rho\ge 0$ and $\sigma>0$, their \textit{max-relative entropy} is defined as \cite{datta2009minmaxrelativeentropies}:
\begin{align*}
D_{\max}(\rho\|\sigma):= \inf\{\lambda|\,\rho\le \mathrm{e}^{\lambda}\sigma    \}  \equiv\ln\,(\|\sigma^{-\frac{1}{2}}\,\rho\,\sigma^{-\frac{1}{2}}\|_\infty)\,.
\end{align*}
From the max-relative entropy, we can define the \textit{max-information} of a (possibly subnormalized) bipartite state $\rho_{AB}\in{\cal B}_+({\cal H}_A\otimes {\cal H}_B)$ as follows \cite{berta2011quantum}:
\begin{align*}
I_{\max}(A:B)_{\rho}\equiv I_{\max}({\cal H}_A:{\cal H}_B)_{\rho} :=\inf_{\tau_{B}\in{\cal D}({\cal H})}\,D_{\max}(\rho_{AB}\|\rho_A\otimes \tau_B)\,.
\end{align*}
Given a subalgebra ${\cal N}$ of ${\cal B}({\cal H})$ and $\sigma\in{\cal D}_+({\cal N})$, we define the modular maps $\Gamma_\sigma:{\cal N}\to {\cal B}({\cal H})$ and $\Delta_\sigma:{\cal N}\to{\cal N}$ as follows
\begin{align*}
	\Gamma_\sigma(X):=\sigma^{1/2}\,X\,\sigma^{1/2}\,\qquad\Delta_\sigma(X)=\sigma\,X\,\sigma^{-1}\,.
	\end{align*}
Then for any $p\ge 1$ and $X\in{\cal N}$, its non-commutative weighted $\mathbb{L}_p(\sigma)$-norm is defined as \cite{Kosaki-noncommLp-1984}:
\begin{align*}
	\|X\|_{\mathbb{L}_p(\sigma)}:=\mathop{\rm Tr}\nolimits\big[ |\Gamma_\sigma^{\frac{1}{p}}(X)|^p  \big]^{\frac{1}{p}}\,
	\end{align*}
	and $\|X\|_{\mathbb{L}_{\infty}(\sigma)}=\|X\|_\infty$, the operator norm of $X$, which we will often denote by $\| X \|$ too, dropping the subindex. We call the space ${\cal B}({\cal H})$ endowed with the norm $\|.\|_{\mathbb{L}_p(\sigma)}$ the \textit{quantum $\mathbb{L}_p(\sigma)$ space}. In the case $p=2$, we have a Hilbert space, with corresponding $\sigma$-KMS scalar product
\begin{align}\label{KMSinner}
	\langle X,\,Y\rangle_\sigma:=\mathop{\rm Tr}\nolimits[\sigma^{1/2}X^*\sigma^{1/2}Y]\,.
	\end{align}
Weighted $\mathbb{L}_p$ norms enjoy the following useful properties:
\begin{itemize}
	\item[-] H\"{o}lder's inequality: for any $p,\hat{p}\ge 1$ such that $p^{-1}+\hat{p}^{-1}=1$, and any $X,Y\in{\cal N}$:
	\begin{align*}
		\langle X,\,Y\rangle_\sigma\le \|X\|_{\mathbb{L}_p(\sigma)}\,\|Y\|_{\mathbb{L}_{\hat{p}}(\sigma)}\,.
		\end{align*}
		Here, $\hat{p}$ is the \textit{H\"{o}lder conjugate} of $p$.
	\item[-] Duality of norms: for any $p\ge 1$ of H\"{o}lder conjugate $\hat{p}$, and any $X\in{\cal N}$:
	\begin{align*}
		\|X\|_{\mathbb{L}_p(\sigma)}=\sup_{\|Y\|_{\mathbb{L}_{\hat{p}}(\sigma)}\le 1}\,\langle Y,\,X\rangle_\sigma\,.
		\end{align*}
	\item[-] For any completely positive, unital linear map $\Phi:{\cal N}\to {\cal N}$ such that $\Phi_*(\sigma)=\sigma$, any $p\ge 1$ and any $X\in{\cal N}$:
	\begin{align}\label{Phiinvariantnorm}
		\|\Phi(X)\|_{\mathbb{L}_p(\sigma)}\le \|X\|_{\mathbb{L}_p(\sigma)}\,.
		\end{align}
	\end{itemize}

\subsection{Conditional expectations} Here we introduce the main object studied in this paper:

\begin{definition}[Conditional expectations~\cite{OhyaPetz-Entropy-1993}]
 Let ${\mathcal{M}}\subset {\cal N}$ be a von Neumann subalgebra of ${\cal N}$. Given a state $\sigma\in{\cal D}_+({\mathcal{M}})$, a linear map $E:{\cal N}\to{\mathcal{M}}$ is called a \textit{conditional expectation} with respect to $\sigma$ of ${\cal N}$ onto ${\mathcal{M}}$ if the following conditions are satisfied:
\begin{itemize}
	\item[-] For all $X\in{\cal N}$, $\|E[X]\|\le \|X\|$;
	\item[-] For all $X\in {\mathcal{M}}$, $E[X]=X$;
	\item[-] For all $X\in{\cal N}$, $\mathop{\rm Tr}\nolimits[\sigma E[X]]=\mathop{\rm Tr}\nolimits[\sigma X]$.
\end{itemize}
\end{definition}

A conditional expectation satisfies the following useful properties (see \cite{Aspects2003} for proofs and more details):

\begin{proposition}\label{propositioncondexp}
	Conditional expectations generically satisfy the following properties:
	\begin{itemize}
		\item[(i)] The map $E$ is completely positive and unital.
		\item[(ii)] For any $X\in{\cal N}$ and any $Y,Z\in{\mathcal{M}}$, $E[YXZ]=Y E[X]Z$.
		\item[(iii)] $E$ is self-adjoint with respect to the scalar product $\langle .,\,.\rangle_\sigma$. In other words:
		\begin{align*}
			\Gamma_\sigma\circ E=E_*\circ\Gamma_\sigma   \,,
		\end{align*}
		where $E_*$ denotes the adjoint of $E$ with respect to the Hilbert-Schmidt inner product.
		\item[(iv)] $E$ commutes with the modular automorphism group of $\sigma$: for any $s\in\mathbb{R}$,
		\begin{align}
			\Delta_\sigma^{is}\circ E=E\circ \Delta^{is}_\sigma\,.
		\end{align}
		\item[(v)] Uniqueness: given a von Neumann subalgebra ${\mathcal{M}}\subset {\cal N}$ and a faithful state $\sigma$, the existence of a conditional expectation $E$ is equivalent to the invariance of ${\mathcal{M}}$ under the modular automorphism group $(\Delta_\sigma^{is})_{s\in\mathbb{R}}$.  In this case, $E$ is uniquely determined by $\sigma$.
	\end{itemize}
\end{proposition}

With a slight abuse of notations, given a finite-dimensional von Neumann subalgebra ${\cal N}=E[{\cal B}({\cal H})]$ of ${\cal B}({\cal H})$, we denote by ${\cal D}({\cal N}):= E_{*}({\cal D}({\cal H}))$ its corresponding set of states that are invariant by $E$, so that ${\cal D}({\cal H})\equiv {\cal D}({\cal B}({\cal H}))$. Similarly, the set of subnormalized states on the algebra ${\cal N}$ is defined as ${\cal D}_{\le}({\cal N}) $. We also introduce the concept of a conditional covariance: given a von Neumann-subalgebra ${\mathcal{M}}\subset {\cal N}$, a conditional expectation $E^{\mathcal{M}}$ from ${\cal N}$ onto ${\mathcal{M}}$ and a quantum state $\sigma\in{\cal D}_+({\mathcal{M}})$, where ${\cal D}({\mathcal{M}})$ is defined with respect to $E^{\mathcal{M}}$, we define the \textit{conditional covariance} functional as follows: for any two $X,Y\in{\cal N}$,
\begin{align}\label{condcov}
	\operatorname{Cov}_{{\mathcal{M}},\sigma}(X,Y):=\langle X-E^{\mathcal{M}}[X],\,Y-E^{\mathcal{M}}[Y]\rangle_\sigma\,.
\end{align}

\subsection{Quantum Markov semigroups}
The basic model for the evolution of an open system in the Markovian regime is given by a quantum Markov semigroup (or QMS) $(\mathcal{P}_t)_{t\ge0}$ acting on ${\cal B}({\cal H})$. Such a semigroup is characterised by its generator, called the Lindbladian $\mathcal{L}$, which is defined on ${\cal B}({\cal H})$ by ${\cal L}(X)={\lim}_{t\to 0}\,\frac{1}{t}\,(\mathcal{P}_t(X)-X)$ for all $X\in{\cal B}({\cal H})$. Recall that by the GKLS Theorem \cite{Lind,[GKS76]}, ${\cal L}$ takes the following form: for all $X\in{\cal B}({\cal H})$,
\begin{equation}\label{eqlindblad}
{\cal L}(X)=i[H,X]+\frac{1}{2}\sum_{k=1}^l{\left[2\,L_k^*XL_k-\left(L_k^*L_k\,X+X\,L_k^*L_k\right)\right]}
\end{equation}
where $H\in{\cal B}_{\operatorname{sa}}({\cal H})$, the sum runs over a finite number of \textit{Lindblad operators} $L_k\in{\cal B}({\cal H})$, and $[\cdot,\cdot]$ denotes the commutator defined as $[X,Y]:=XY-YX$, $\forall X,Y\in{\cal B}({\cal H})$. The QMS is said to be \textit{faithful} if it admits a full-rank invariant state $\sigma$. When the state $\sigma$ is the unique invariant state, the semigroup is called \textit{primitive}. Further assuming the self-adjointness of the generator ${\cal L}$ with respect to the inner product (\ref{KMSinner}), there exists a conditional expectation $E\equiv E_{\mathcal{F}}$ onto the fixed-point subalgebra ${\mathcal{F}}({\cal L}):=\{X\in{\cal B}({\cal H}):\,{\cal L}(X)=0\}$ such that
\begin{align*}
\mathcal{P}_t(X)\underset{t\to\infty}{\to}E[X]\,.
\end{align*}

\section{Lattice spin systems with commuting Hamiltonians}\label{sec:dabies}

In this section, we aim at estimating the strong and weak constants appearing in  \Cref{theo_AT_pinching} in the context of lattice spin systems, and compare them with previous conditions in the classical and quantum literature. Given a finite lattice $\Lambda \subset\subset \mathbb{Z}^d$, we define the tensor product Hilbert space ${\cal H}:={\cal H}_\Lambda\equiv\bigotimes_{k\in\Lambda}{\cal H}_k$, where for each $k\in\Lambda$, ${\cal H}_k\simeq \mathbb{C}^\ell$, $\ell\in\mathbb{N}$. Then, let $\Phi:\Lambda\to {\cal B}_{\operatorname{sa}}({\cal H}_{\Lambda}) $ be an $r$-local potential, i.e. for any $j\in \Lambda$, $\Phi(j)$ is self-adjoint and supported on a ball of radius $r$ around site $j$. We assume further that $\| \Phi(j) \|\le K$ for some constant $K<\infty$. The potential $\Phi$ is said to be a \textit{commuting potential} if for any $i,j\in\Lambda$, $[\Phi(i),\Phi(j)]=0$.  Given such a local, commuting potential, the Hamiltonian on a subregion $A\subseteq \Lambda$ is defined as
\begin{align}
H_A=\sum_{j\in A}\,\Phi(j)\,.
\end{align}
Next, the corresponding Gibbs state corresponding to the region $A$ and at inverse temperature $\beta$ is defined as
\begin{align}
\sigma_A=\frac{\mathrm{e}^{-\beta H_A}}{\mathop{\rm Tr}\nolimits[\mathrm{e}^{-\beta H_A}]}\,.
\end{align}
Note that this is in general not equal to the state $\mathop{\rm Tr}\nolimits_B[\sigma_\Lambda]$.

\subsection{Heat-bath generators on lattice spin systems}

Let  $\Lambda \subset\subset \mathbb{Z}^d$ be a finite lattice and $\Phi:\Lambda\to {\cal B}_{\operatorname{sa}}({\cal H}_{\Lambda}) $ an $r$-local commuting potential.  Denote by $\sigma$ the associated Gibbs state. Then,  the \textit{heat-bath generator} is defined as
\begin{equation}
{\cal L}_\Lambda^{H} (X) := \underset{k \in \Lambda}{\sum} \left(  {\cal A}_{k, \sigma} (X) -X   \right)
\end{equation}
for every $X \in {\cal B}({\cal H}_\Lambda)$, where we are writing ${\cal A}_{k, \sigma}(X)=(\mathop{\rm Tr}\nolimits_k[\sigma])^{-1/2}\mathop{\rm Tr}\nolimits_k[\sigma^{1/2} X \sigma^{1/2}] (\mathop{\rm Tr}\nolimits_k[\sigma])^{-1/2}$ (see Section \ref{sec:condexp}). Note that the dual map of the first term of each summand above is a Petz recovery map for the partial trace. Similarly, for any $A \subseteq \Lambda$, we define ${\cal L}_A^{H}$ the generator in $A$, in which the summation is over elements $k \in A$.

In \cite{BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}, we addressed the problem proving the positivity of the MLSI constant associated to the heat-bath dynamics for quantum spin chains. We showed results of positivity of this constant, under the assumption of some conditions of clustering of correlations on the Gibbs state, via results of quasi-factorization of the relative entropy, i.e. results of strong approximate tensorization in which the term in the left-hand side has no conditional expectation (and thus it is not a ‘conditional relative entropy').

The new notion of approximate tensorization AT$(c,d)$ introduced in this paper allows to take an orthogonal approach in that problem, since now we are considering the conditional expectation $E_\sigma$ associated to the heat-bath generator, as opposed to \cite{BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019}, where we focused on the dual of ${\cal A}_{\sigma}$, i.e. the Petz recovery map, which is not a conditional expectation. Hence, now we can prove new results on AT$(c,d)$ that might be of use to take a further step in the study of the positivity of the MLSI.

Before that, we need to prove the following result concerning the kernel of the generator.

\begin{proposition}\label{prop:ker_gen_heatbath}
For every $A \subseteq \Lambda$, the following holds:
\begin{equation}
\ker({\cal L}_A^{H}) = \{X \, : \, E_{A,\sigma}(X) = X\} \, ,
\end{equation}
where $E_{A,\sigma}(X) = \underset{n \rightarrow \infty}{\operatorname{lim}} \mathcal{A}^n_{A,\sigma}(X) $, for ${\cal A}_{A, \sigma}(X)=(\mathop{\rm Tr}\nolimits_A[\sigma])^{-1/2}\mathop{\rm Tr}\nolimits_A[\sigma^{1/2} X \sigma^{1/2}] (\mathop{\rm Tr}\nolimits_A[\sigma])^{-1/2}$.
\end{proposition}

\begin{proof}
By virtue of \cite[Theorem 6]{BardetCapelLuciaPerezGarciaRouze-HeatBath1DMLSI-2019},  we know that for $X \in {\cal B}({\cal H})$,
\begin{equation}
X= \mathcal{A}_{A, \sigma} (X) \, \,  \Leftrightarrow \, \, X= \mathcal{A}_{k, \sigma} (X)  \, \; \forall k \in A \,.
\end{equation}
We conclude by the equivalence (2) $\Leftrightarrow$ (3) in Proposition \ref{prop_condexp}.
\end{proof}

\subsection{Davies generators on lattice spin systems}

Consider the Hamiltonian $H_\Lambda:=H^{\Sigma}_\Lambda$ of the system on the lattice $\Lambda$. Introduce also the Hamiltonian $H^{\operatorname{HB}}$ of the heat bath, as well as a set of system-bath interactions $\{ S_{\alpha,k}\otimes B_{\alpha,k} \}$, where $\alpha$ label all the operators $S_{\alpha,k}$ and $B_{\alpha,k}$ associated to the site $k\in\Lambda$. Here, we assume that the operators $S_{\alpha,k}$ form an orthonormal basis of self-adjoint operators in ${\cal B}_{\operatorname{sa}}({\cal H})$ with respect to the Hilbert-Schmidt inner product (think of the qubit Pauli matrices). The Hamiltonian of the universe composed of the system and its heat-bath is given by
\begin{align}
H=H_\Lambda+H^{\operatorname{HB}}+\sum_{\alpha,k\in\Lambda}S_{\alpha,k}\otimes B_{\alpha,k}\,.
\end{align}

Assuming that the bath is in a Gibbs state, by a standard argument (e.g. weak coupling limit, see \cite{[SL78]}), the evolution on the system can be approximated by a quantum Markov semigroup whose generator is of the following form:
\begin{align}\label{totalgenerator}
{\cal L}^{\operatorname{D},\beta}_\Lambda(X)=i[H_\Lambda,X]+\sum_{k\in\Lambda}\,{\cal L}^{\operatorname{D},\beta}_k(X)\,,
\end{align}
where
\begin{align}\label{lindblad}
{\cal L}^{\operatorname{D},\beta}_k(X)=\sum_{\omega,\alpha}\,\chi^{\beta}_{\alpha,k}(\omega)\,\Big(  S_{\alpha,k}^*(\omega)XS_{\alpha,k}(\omega)-\frac{1}{2}\,\big\{  S_{\alpha,k}^*(\omega)S_{\alpha,k}(\omega),X \big\}   \Big)\,.
\end{align}

Similarly, define the generator ${\cal L}^\beta_A$ by restricting the sum in \Cref{totalgenerator} to the sublattice $A$:
\begin{align}\label{localgenerator}
{\cal L}^{\operatorname{D},\beta}_A(X)=i[H_A,X]+\sum_{k\in A}\,{\cal L}^{\operatorname{D},\beta}_k(X)\,.
\end{align}
Note that ${\cal L}^{\operatorname{D},\beta}_A$ acts non-trivially on $A_\partial:=\{k\in\Lambda:\,d(k,A)\le r\}$. Then, for any region $A\subset \Lambda$, we define the conditional expectation onto the algebra ${\cal N}_A$ of fixed points of ${\cal L}_A$ with respect to the Gibbs state $\sigma=\sigma_\Lambda$ as follows \cite{Kastoryano2014}: given an adequate decomposition ${\cal H}_\Lambda:=\bigoplus_{i\in I_{{\cal N}_A}}\,{\cal H}_i\otimes {\cal K}_i$ of the total Hilbert space ${\cal H}_\Lambda$ of the lattice spin system,
\begin{align}\label{Daviescond}
E^{\operatorname{D},\beta}_A[X]:=\lim_{t\to \infty}\mathrm{e}^{t{\cal L}^{\operatorname{D},\beta}_A}(X)\equiv \sum_{i\in I_{{\cal N}_A}}\, \mathop{\rm Tr}\nolimits_{{\cal K}^A_i}(P^{A}_i ({\mathds{1}}_{{\cal H}_i^A}\otimes \sigma_i^A)\,X  P^{A}_i) \otimes {\mathds{1}}_{{\cal K}^A_i}\,,
\end{align}
for some fixed full-rank states $\sigma_i^A$ on ${\cal K}_i^A$. In was shown in Lemma 11 of \cite{Kastoryano2014} that the generator of the Davies semigroups corresponding to a local commuting potential is \textit{frustration-free}. This means that the state $\sigma$ is invariant with respect to any ${\cal L}_A^{\operatorname{D},\beta}$, $A\subseteq \Lambda$. Therefore, the conditional expectations $E^{\operatorname{D},\beta}_A$ are all defined with respect to $\sigma$.

\subsection{Preliminary results}\label{subsec:preliminaries}

In Theorem \ref{coro_equal_cond}, we have proven that the conditional expectations arising from the Petz recovery map and Davies semigroups coincide and, as a consequence, the two examples of dissipative dynamics for quantum spin systems mentioned in the previous two subsections have the same associated conditional expectation. Indeed, note that the fixed points of the conditional expectation arising from the Petz recovery map coincides with the kernel of the conditional heat-bath generator by Proposition \ref{prop:ker_gen_heatbath}.  Hence, for the rest of the paper, all the results presented concern this conditional expectation.

In Proposition \ref{prop_classical_inftyT}, we will show that approximate tensorization AT($1,0$) holds at infinite temperature for classical Hamiltonians. However,  it is not clear (and we strongly believe the opposite) that this remains true for non-classical commuting Gibbs states, which is the reason behind the introduction of our notion of weak approximate tensorization. Nevertheless, we can still prove in general two interesting results.

\begin{proposition}\label{prop_cond_ABdisjoint}
 Let $A,B\subset\Lambda$ be two regions separated by at least a distance $2r$, that is such that $A\partial\cap B\partial=\emptyset$. Then ${\cal N}_A$ and ${\cal N}_B$ form a commuting square, that is,
 \begin{equation}\label{eq_prop_cond_ABdisjoint1}
  E_A^{\beta}\circ E_B^{\beta}=E_B^{\beta}\circ E_A^{\beta} = E_{A\cup B}^{\beta}\,.
 \end{equation}
 Consequently, for all $\rho\in{\cal D}({\cal H}_\Lambda)$,
 \begin{equation}\label{eq_prop_cond_ABdisjoint2}
  D\big(\rho\|E_{A\cup B*}^{\beta}(\rho)\big)\leq D\big(\rho\|E_{A*}^{\beta}(\rho)\big)+D\big(\rho\|E_{B*}^{\beta}(\rho)\big)\,.
 \end{equation}
\end{proposition}

\begin{proof}
Remark that by definition of the map $A_{\sigma_{A}}$, it only acts non-trivially on $A_\partial$ and as identity on $(A_\partial)^c$. Consequently, as $E_A=\lim_{n\to\infty} A_{\sigma_A}^n$, this property carries over to the conditional expectation and we have $E_A=E_A\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal H}_{A_\partial^c}}$ by slight abuse of notations. Similarly, $E_B=E_B\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal H}_{B_\partial^c}}$. This shows the result since $A_\partial\cap B_\partial=\emptyset$.
\end{proof}

The belief that, compared to classical Hamiltonians, general commuting Gibbs states do not satisfy strong AT at infinite temperature leads us to consider weak AT instead, as suggested by the result in \cite{gao2017strong}. As already mentioned in the general setting of \Cref{corollaruweak}, they obtain the following weak AT for $\beta=0$:
\begin{equation}\label{eq_weakAT_beta0}
   D\big(\rho\|E_{A\cup B*}^{\beta=0}(\rho)\big)\leq D\big(\rho\|E_{A*}^{\beta=0}(\rho)\big)+D\big(\rho\|E_{B*}^{\beta=0}(\rho)\big)+d\,,
\end{equation}
where $d=\sup_{\rho}\,\inf_{\eta} D_{\max}(E_{A*}^{\beta=0}\circ E_{B*}^{\beta=0}(\rho)\|E_{A\cup B*}^{\beta=0}(\eta))$. One can then wonder if the only difference with the classical case comes from this defect at infinite temperature. The next proposition goes in this direction. To state it we need the following notation. Given a subset $A\subset\Lambda$ and using that $\sigma=e^{-\beta\,H_\Lambda}$ is an invariant state, we decompose the total hamiltonian according to the fixed-point algebra of $E_A^\beta$:
\[H_\Lambda=\sum_{i\in I_{{\cal N}_A}}\,H_i^A\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i^A}+1\hspace{-0.27em}\mathrm{l}_{{\cal H}_i^A}\otimes H_i^{\Lambda\backslash A}\,.\]
We then denote by $H^{A\partial}$ the part of the Hamiltonian acting only on $A$ and its boundary:
\[H^{A\partial}=\sum_{i\in I_{{\cal N}_A}}\,H_i^A\otimes 1\hspace{-0.27em}\mathrm{l}_{{\cal K}_i^A}\,.\]
\begin{proposition}\label{prop_AT_HS}
Define $d$ as above. Then for all $\rho\in{\cal D}({\cal H}_\Lambda)$,
 \begin{equation}\label{eq_prop_cond_ABdisjoint}
  D\big(\rho\|E_{A\cup B*}^{\beta}(\rho)\big)\leq \,e^{4\beta\|H^{A\partial}\|}\,\Big(D\big(\rho\|E_{A*}^{\beta}(\rho)\big)+D\big(\rho\|E_{B*}^{\beta}(\rho)\big)\Big)+e^{2\beta\|H^{A\partial}\|}\,d\,.
 \end{equation}
\end{proposition}

\begin{proof}
 The proof of this result follows the same steps as \Cref{HSAT}, but replacing  $\lambda_{\max}(\sigma)$ and $1/(\lambda_{\min}(\sigma)$ by $e^{2\beta\|H^{A\partial}\|}$, thanks to Proposition 4.2 in \cite{junge2019stability}.
\end{proof}

\begin{remark}
This argument is similar to that of Lemma 2.2 of \cite{caputo2015approximate}. For the same reason as in the classical setting therein, the strong constant can however be very large, since it depends on the norm $\|H^{A\partial}\|$, which generally depends linearly on $|A|$. For this reason, we will rather consider applying \Cref{theo_AT_pinching} in the next section, which will provide us with better bounds under the assumption of strong clustering of correlations.
\end{remark}

\subsection{Clustering of correlations on lattice spin systems}\label{subsubsec:L1-clustering-correlations}
We recall the expression entering the definition of the constant $c_1$ appearing in \Cref{theo_AT_pinching}: given ${\cal N}_{A\cup B}\equiv E_{A\cup B}[{\cal N}_\Lambda]:=\bigoplus_{i\in I_{{\cal N}_{A\cup B}}}\,{\cal B}({\cal H}_i)\otimes {\mathds{1}}_{{\cal K}_i} $,
\begin{align}\label{C2cond}
c(A,B):=\max_{i\in I_{{\cal N}_{A\cup B}}}\|E_{A}^{(i)}\circ E_{B}^{(i)}-E_{A\cup B}^{(i)}:\,\mathbb{L}_1(\tau_i)\to\mathbb{L}_\infty({\cal N}_\Lambda)\|\,.
\end{align}
In the case of the classical Glauber dynamics that will be introduced in \Cref{classicalglauber}, and when the algebra ${\cal N}$ corresponds to the linear span of the rank-one projections onto classical product basis $|\eta\rangle$, $\eta\in\Omega_\Lambda$, the Hilbert spaces ${\cal H}_i$ appearing in the decomposition of the algebra ${\mathcal{M}}$ all are one dimensional. Moreover, the indices $i\in I_{{\cal N}_{A\cup B}}$ correspond to the set of boundary conditions $\omega\in\Omega_{\partial{A\cup B}}$ and the states $\tau_\omega$ correspond to the conditional Gibbs measures $\mu^\omega_{A\cup B} $ (see Section \ref{classicalglauber} for the notation). Hence $c(A,B)$ reduces to
\begin{align*}
c(A,B)^{\operatorname{cl}}:=\max_{\omega\in\Omega_{ (A\cup B)^c}} \,\|E_A^\omega\circ E_B^\omega-E_{A\cup B}^\omega: \,\mathbb{L}_1(\mu^\omega_{A\cup B})\to\mathbb{L}_\infty(\mu^\omega_{A\cup B})\| \,.
\end{align*}
In \cite{cesi2001quasi}, such norms were estimated: under the condition of \textit{complete analyticity} of Dobrushin and Shlosman \cite{[DS85],Dobrushin1985,[DS87]}, which characterizes the absence of a phase transition and hence is satisfied for one dimensional systems, and for all dimensions above a critical temperature, Cesi showed that for any two subsets $A,B\subset \Lambda$, and any two positive constants $\kappa,\xi$ such that $|\partial B\cap (A\cup B)|\,\kappa\mathrm{e}^{-\xi\,d(b\backslash A,A\backslash B)}\le 1$, the following holds:
\begin{align*}
c(A,B)\le \kappa\mathrm{e}^{-\xi\, d(B\backslash A,A\backslash B)+1}\,.
\end{align*}
By H\"{o}lder's inequality, this bound in particular implies the so-called \textit{(conditional) clustering of correlations} (see Section \ref{sec:clustering}): for all $\omega\in \Omega_{(A\cup B)^c}$ and any $f\in \mathbb{L}_\infty(\Omega_{A\cup B})$,
\begin{align*}
\mu^\omega_{A\cup B}\big(E_A^\omega[f],\,E_B^\omega[f]   \big)\le \kappa\,\mathrm{e}^{-\xi \,d(B\backslash A,A\backslash B)}\,\|f\|^2_{\mathbb{L}_1(\mu^\omega_{A\cup B})}\,,
\end{align*}
where $\mu^\omega_{A\cup B}\big(E_A^\omega[f],\,E_B^\omega[f]   \big):= \mu^\omega_{A\cup B}\big((E_A^\omega[f]-E_{A\cup B}^\omega[f])(\,E_B^\omega[f] -E_{A\cup B}^\omega[f] )\big)$ denotes the correlation function. Motivated by this classical setting, we introduce the notion of a conditional quantum clustering of correlations:

\begin{definition}\label{def:expcondl1clust}
	Let $\Lambda \subset \subset \mathbb{Z}^d$ be a finite lattice and ${\cal N}_{\Lambda}\subset {\cal B}({\cal H}_\Lambda)$ a subalgebra of ${\cal B}({\cal H}_\Lambda)$. Let $\mathcal{E}^{\Lambda}:=\{E_A;\,A\subset \Lambda\}$ a family of conditional expectations on ${\cal N}_{\Lambda}$. We say that $\mathcal{E}^{\Lambda}$ satisfies a \textit{conditional $\mathbb{L}_1$-clustering of correlations} if there exist constants $\kappa,\xi>0$ such that, for any $A,B\subset \Lambda$, with $A\cap B\ne \emptyset$,
	\begin{align*}
	c(A,B)\le \kappa\,\mathrm{e}^{-\xi d(A\backslash B,B\backslash A)}\,.
	\end{align*}
\end{definition}

\paragraph{Block diagonal states in the energy basis}

Recall that for a classical Hamiltonian (i.e. a Hamiltonian that is diagonalizable in the classical product basis $\{|\eta\rangle\}_{\eta\in \Omega_{\Lambda}}$), and classical states, the inequality (\ref{eqgeneral}) reduces to that found in \cite{cesi2001quasi}, with $d=0$ and $\eta=\mathcal{P}_{\mathcal{M}}(\rho)$. More generally, one can find approximate tensorization for states that are block diagonal in the energy eigenbasis of the Hamiltonian $H$. This is the content of next proposition. Once again, we introduce some notations before stating it.

Given the spectral decomposition $H_\Lambda:=\sum_{\varepsilon\in{\rm sp}(H_\Lambda)}\,\varepsilon P_\varepsilon$ of the Hamiltonian $H_\Lambda$, let $${\cal N}_{{\rm sp}(H_\Lambda)}:=\{ X\in{\cal B}({\cal H}_\Lambda):\,P_\varepsilon X P_{\varepsilon'}=\delta_{\varepsilon,\varepsilon'}P_\varepsilon X P_\varepsilon ~\forall \varepsilon,\varepsilon'\in {\rm sp}(H_\Lambda)\}$$ be the algebra of operators $X$ that are block diagonal in the energy eigenbasis.
\begin{proposition}\label{stabilizer}
 Given any sublattice $A\subset \Lambda$, $E_A[{\cal N}_{{\rm sp}(H_\Lambda)}]\subset {\cal N}_{{\rm sp}(H_\Lambda)}$.

 With a slight abuse of notations, we denote by $E_A^{(i)}:=E_A[P_i(\cdot)P_i]$ the restriction onto the block $i\in I_{{\cal N}_A}$ of $E_A\equiv E_A|_{{\cal N}_{{\rm sp}(H_\Lambda)}}:{\cal N}_{{\rm sp}(H_\Lambda)}\mapsto {\cal N}_A\cap {\cal N}_{{\rm sp}(H_\Lambda)}$.

 Moreover, the following approximate tensorization holds for any two sublattices $A,B\subset \Lambda$, $A\cap B\ne \emptyset$:
\begin{align*}
D(\rho\|E_{A\cup B*}(\rho))\le \frac{1}{1-2 c} \left( D(\rho\|E_{A*}(\rho))+D(\rho\|E_{B*}(\rho))+3\max_{\varepsilon\in{\rm sp}(H_\Lambda)}\,\ln(\dim(P_\varepsilon{\cal H}_\Lambda))+4\,c \right) \,,
\end{align*}
for $$c:=\max_{i\in I_{{\cal N}_{A\cup B}}}\,\|E_A^{(i)}\circ E_B^{(i)}-E_{A\cup B}^{(i)}:\,\mathbb{L}_1(\sigma^{A\cup B}_i)\to \mathbb{L}_\infty({\cal N}_{{\rm sp}(H_\Lambda)})\|\,,$$
where $\sigma_i^A$ was defined in (\ref{Daviescond}).
	\end{proposition}
\begin{proof}
	First of all, we prove that the algebra ${\cal N}_{{\rm sp}(H_\Lambda)}$  is preserved under the action of the conditional expectations $E_A$ onto an arbitrary sublattice $A\subset \Lambda$. To show this, it is enough to prove that ${\cal L}^{\operatorname{D},\beta}_{A}({\cal N}_{{\rm sp}(H_\Lambda)})\subset {\cal N}_{{\rm sp}(H_\Lambda)}$. This is done using the expression (\ref{lindblad}) together with \Cref{eq!}: for any $\omega,\alpha$, we have
	\begin{align*}
 S_{\alpha,k}^*(\omega)P_\varepsilon XP_\varepsilon S_{\alpha,k}(\omega)  =\sum_{\substack{\varepsilon_1-\varepsilon_1'=\omega\\\varepsilon_2-\varepsilon_2'=\omega}}\,P_{\varepsilon_1'}S_{\alpha,k}^*P_{\varepsilon_1}P_\varepsilon XP_\varepsilon P_{\varepsilon_2} S_{\alpha,k}P_{\varepsilon_2'}=\sum_{\varepsilon'=\varepsilon-\omega}\,P_{\varepsilon'}S_{\alpha,k}^* P_\varepsilon X P_\varepsilon S_{\alpha,k}P_{\varepsilon'}\in{\cal N}_{{\rm sp}(H_\Lambda)}\,.
	\end{align*}
	Similarly
	\begin{align*}
\big\{  S_{\alpha,k}^*(\omega)S_{\alpha,k}(\omega),X \big\}=\sum_{\substack{\varepsilon_1-\varepsilon_1'=\omega\\\varepsilon_2-\varepsilon_2'=\omega}}\big\{	P_{\varepsilon_1'}\,S_{\alpha,k}^*P_{\varepsilon_1}P_{\varepsilon_2}S_{\alpha,k}P_{\varepsilon_2'},\,P_\varepsilon X P_\varepsilon\big\}=\sum_{\varepsilon'=\omega+\varepsilon} \big\{  P_\varepsilon S_{\alpha,k}^*P_{\varepsilon'}S_{\alpha,k},P_\varepsilon XP_\varepsilon\big\}\in{\cal N}_{{\rm sp}(H_\Lambda)}\,.
	\end{align*}
	Thus, ${\cal L}^{\operatorname{D},\beta}_A({\cal N}_{{\rm sp}(H_\Lambda)})\subset {\cal N}_{{\rm sp}(H_\Lambda)}$, and hence $E_A[ {\cal N}_{{\rm sp}(H_\Lambda)}]\subset  {\cal N}_{{\rm sp}(H_\Lambda)}$  Therefore, the restriction $E_A:=E_A|_{{\cal N}_{{\rm sp}(H_\Lambda)}}$ defines a conditional expectation onto the algebra ${\cal N}_{{\rm sp}(H_\Lambda)}\cap {\cal N}_A$. Moreover, from the bounds found in \Cref{propboundsd1d2}, we directly have the following crude estimates
	\begin{align*}
	&d_1\le \max_{\varepsilon\in{\rm sp}(H_\Lambda)}\,\ln(\operatorname{dim}(P_\varepsilon{\cal H}_\Lambda))\\
	&d_2\le 2\max_{\varepsilon\in {\rm sp}(H_\Lambda)}\,\ln(\operatorname{dim}(P_\varepsilon{\cal H}_\Lambda))\,.
	\end{align*}
	We conclude from the joint use of \Cref{theo_AT_pinching} and \Cref{propboundsd1d2}.
	\end{proof}
\begin{remark}
The motivation behind the statement of \Cref{stabilizer} comes from the problem of thermal stability of quantum error correcting codes \cite{Temme2014,temme2015fast,KITAEV20032}. There, the initial state $\rho$ is typically assumed to be supported in the so-called code space, that is the ground space of some stabilizer Hamiltonian $H_\Lambda$. For translational invariant error correcting codes, every eigenspace has dimension independent of the lattice size (cf. the $2D$ toric code has four dimensional eigenspaces). Therefore, upon showing conditional $\mathbb{L}_1$ clustering of correlations, we find that the
 weak constant in the approximate tensorization is independent of $|\Lambda|$. In that case, for any two fixed sublattices $A,B\subset\subset \mathbb{Z}^d$ we find the following exact tensorization of the normalized relative entropy: given any sequence $\rho_\Lambda\in {\cal N}_{\operatorname{gs}(H_\Lambda)}$ supported in the ground space of $H_\Lambda$:
 \begin{align*}
 \limsup_{\Lambda\nearrow \mathbb{Z}^d}\,\frac{D(\rho_\Lambda\|E_{A\cup B*}(\rho_\Lambda))}{|\Lambda|}\le  \limsup_{\Lambda\nearrow \mathbb{Z}^d}\,\frac{D(\rho_\Lambda\|E_{A*}(\rho_\Lambda))}{|\Lambda|}+ \limsup_{\Lambda\nearrow \mathbb{Z}^d}\,\frac{D(\rho_\Lambda\|E_{ B*}(\rho_\Lambda))}{|\Lambda|}\,.
 \end{align*}
 Such bounds will prove crucial in showing fast thermalization of dissipative Gibbs samplers in a forthcoming paper.
\end{remark}

\subsection{Classical Hamiltonian over quantum systems}\label{classicalglauber}

In this section, we investigate the case of a quantum lattice spin system undergoing a classical Glauber dynamics, whose the framework was already studied in \cite{Cubitt2015}. These semigroups correspond to Davies generators whose Hamiltonian is classical, that is, diagonal in a product basis of ${\cal H}_\Lambda$. In order to make the connexion with the classical Glauber dynamics over a classical system (i.e. initially diagonal in the product basis), We introduce the generator more explicitly: consider a lattice spin system over $\Gamma=\mathbb{Z}^d$ with classical configuration space $S=\{+1,-1\}$, and, for each $\Lambda\subset \Gamma$, denote by $\Omega_\Lambda=S^\Lambda$ the space of configurations over $\Lambda$. Next, given a classical finite-range, translationally invariant potential $\{J_A\}_{A\in\Gamma}$ and a boundary condition $\tau\in\Omega_{\Lambda^c}$, define the Hamiltonian over $\Lambda$ as

\begin{align*}
H_\Lambda^\tau(\sigma)=-\sum_{A\cap\Lambda\ne 0}\,J_A(\sigma\times\tau),~~~~~~\forall\sigma\in\Omega_\Lambda\,.
\end{align*}
The classical Gibbs state corresponding to such Hamiltonian is then given by
\begin{align*}
\mu_\Lambda^\tau(\sigma)=(Z_{\Lambda}^\tau)^{-1}\,\exp\big( -H_{\Lambda}^\tau(\sigma)\big)\,,
\end{align*}
Next, define the Glauber dynamics for a potential $J$ as the Markov process on $\Omega_\Lambda$ with the generator
\begin{align*}
(L_\Lambda f)(\sigma)=\sum_{x\in\Lambda}\,c_{J}(x,\sigma)\nabla_xf(\sigma)\,,
\end{align*}
where $\nabla_xf(\sigma)=f(\sigma^x)-f(\sigma)$ and $\sigma^x$ is the configuration obtained by flipping the spin at position $x$. The numbers $c_J(x,\sigma)$ are called transition rates and must satisfy the following assumptions:
\begin{itemize}
	\item[1.] There exist $c_m,c_M$ such that $0<c_m\le c_J(x,\sigma)\le c_M<\infty$ for all $x,\sigma$.
	\item[2.] $c_J(x,.)$ depends only on spin values in $b_r(x)$.
	\item[3.] For all $k\in\Gamma$, $c_J(x,\sigma')=c_J(x+k,\sigma)$ id $\sigma'(y)=\sigma(y+k)$ for all $y$.
	\item[4.] Detailed balance: for all $x\in\Gamma$, and all $\sigma$
	\begin{align*}
	\exp\left(-\sum_{A\ni x}J_A(\sigma)\right)c_J(x,\sigma)=c_J(x,\sigma^x)\exp\left(   -\sum_{A\ni x}J_A(\sigma^x)\right)\,.
	\end{align*}
\end{itemize}
These assumptions constitute sufficient conditions for the corresponding Markov process to have the Gibbs states over $\Lambda$ as stationary points. Next, we introduce the notion of a quantum embedding of the aforementioned classical Glauber dynamics. This is the Lindbladian of corresponding Lindblad operators given by
\begin{align}\label{Lindbladops}
L_{x,\eta}:=\sqrt{c_J(x,\eta)}\,|\eta^x\rangle\langle \eta|\otimes {\mathds{1}}\,,~~~\forall x\in\Lambda,\,\eta\in\Omega_{b_x(r)}\,.
\end{align}
It was shown in \cite{Cubitt2015} that such a dynamics is KMS-symmetric with respect to the state $\mu_\Lambda^\tau$ as embedded into the computational basis. Moreover, the set of fixed points in the Schr\"{o}dinger picture corresponds to the convex hull of the set of Gibbs states over $\Lambda$, $\{\mu_\Lambda^\tau|\tau\in\Omega_{\Lambda^c}\}$. In the Heisenberg picture, this implies that the fixed-point algebras ${\mathcal{F}}({\cal L}_A)$ are expressed as
\begin{align}\label{fixedpoints}
{\mathcal{F}}({\cal L}_A):=\bigoplus_{\omega\in\Omega_{\partial A}}\,|\omega\rangle\langle\omega|_{\partial A}\otimes {\mathds{1}}_{A}\otimes {\cal B}({\cal H}_{A_\partial^c})\,.
\end{align}
Equivalently,
\begin{align}\label{classcond}
E_{A*}(\rho)=\sum_{\omega\in\Omega_{\partial A}}\,|\omega\rangle\langle\omega|_{\partial A}\otimes \sigma^\omega_{A}\otimes \mathop{\rm Tr}\nolimits_A(\langle \omega|\rho|\omega\rangle)\,,
\end{align}
where $\sigma^\omega_A$ denotes the Gibbs state $\mu^\omega_A$ embedded into the computational basis.

With this expression at hand we can prove that the conditional expectations at infinite temperature satisfy commuting square for any subset $A$ and $B$ of $\Lambda$.

\begin{proposition}\label{prop_classical_inftyT}
 Let $A,B\subset\Lambda$. Then, at $\beta=0$, ${\cal N}_A$ and ${\cal N}_B$ form a commuting square, that is,
 \begin{equation}\label{eq_prop_classical_inftyT}
  E_A\circ E_B=E_B\circ E_A = E_{A\cup B}\,
 \end{equation}
  and consequently, for all $\rho\in{\cal D}({\cal H}_\Lambda)$,
 \begin{equation}\label{eq_prop_classical_inftyT2}
  D\big(\rho\|E_{A\cup B*}^{\beta=0}(\rho)\big)\leq D\big(\rho\|E_{A*}^{\beta=0}(\rho)\big)+D\big(\rho\|E_{B*}^{\beta=0}(\rho)\big)\,.
 \end{equation}
 At finite temperature $\beta>0$, we have
 \begin{equation}\label{eq_prop_classical_inftyT3}
  D_{\max}(E_{A*}\circ E_{B*}(\rho)\| E_{A\cup B*}(\rho))\leq\max_{\substack{\omega_{\partial AB}\\\omega_{\partial B\cap A} }}\,D_{\max}\big(  \sum_{\omega_{\partial A\cap B}}P_{\omega_{\partial A\cap B}}\otimes  \sigma_{A}^{\omega_{\partial A}}\otimes \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},\omega_{\partial B}} \big\| \sigma_{AB}^{\omega_{\partial AB}}  \big)\,,
 \end{equation}
where $P_\omega:=|\omega\rangle\langle\omega|$ and $ \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},{\omega_{\partial B}}}:=\mathop{\rm Tr}\nolimits_{A\cap B}
	\big(\langle \omega_{\partial A\cap B}| \sigma_{B}^{\omega_{\partial B}}|\omega_{\partial A\cap B}\rangle\big)$.

	In particular, \eqref{eq_prop_classical_inftyT3} provides a bound on the weak constants appearing in \Cref{corollaruweak} and \Cref{theo_AT_pinching}.
\end{proposition}
Remark that the first part of this last proposition (i.e. when $\beta=0$) does not depend on the relative positions of $A$ and $B$. Moreover, the upper bound in \Cref{eq_prop_classical_inftyT3} has the advantage of being independent of $\rho$. Moreover, it is equal to $0$ when $\beta=0$. Thus we retrieve the exact tensorization of \Cref{prop_classical_inftyT} at infinite temperature.
\begin{proof}
	Letting $\eta:=E_{A\cup B*}(\rho)$, it is enough to estimate, for any $\rho\in {\cal D}({\cal N})$, $D_{\max}(E_{A*}\circ E_{B*}(\rho)\|E_{A\cup B*}(\rho))$. In order to do so, we first provide an expression for $E_{A*}\circ E_{B*}(\rho)$ which will allow us to compare it more easily to $E_{A\cup B*}(\rho)$. Define
	\begin{align*}
&	 \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},{\omega_{\partial B}}}:=\mathop{\rm Tr}\nolimits_{A\cap B}
	\big(\langle \omega_{\partial A\cap B}| \sigma_{B}^{\omega_{\partial B}}|\omega_{\partial A\cap B}\rangle\big)\\
&	   \rho^{\omega_{\partial B\cap A},\omega_{\partial AB}}_{(A\cup B\partial)^c}:= \langle \omega_{\partial B\cap A}|\mathop{\rm Tr}\nolimits_{A\cup B\backslash \partial B}\big( \langle \omega_{\partial AB}|\rho |\omega_{\partial AB}\rangle  \big)|\omega_{\partial B\cap A}\rangle
\end{align*}
	A lengthy yet easy calculation yields:
	\begin{align*}
	E_{A*}\circ E_{B*}(\rho)=\sum_{\substack{ \omega_{\partial AB}\in \Omega_{\partial AB} \\ \omega_{\partial A\cap B}\in \Omega_{\partial A\cap B}\\\omega_{\partial B\cap A} \in \Omega_{\partial B\cap A}}}\,	P_{\omega_{\partial AB}}\otimes P_{\omega_{\partial A\cap B}}
	\otimes
	\sigma_A^{\omega_{\partial A}}\otimes   \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},{\omega_{\partial B}}} \otimes    \rho^{\omega_{\partial B\cap A},\omega_{\partial AB}}_{(A\cup B\partial)^c}\,,
	\end{align*}
	where $P_\omega:=|\omega\rangle\langle\omega|$. Similarly,
	\begin{align*}
	E_{A\cup B*}(\rho)=\sum_{\substack{\omega_{\partial AB}\in \Omega_{\partial AB}\\\omega_{\partial B\cap A} }}\,P_{\omega_{\partial AB}}\otimes \sigma^{\omega_{\partial AB}}_{AB}\otimes  \rho^{\omega_{\partial B\cap A},\omega_{\partial AB}}_{(A\cup B\partial)^c} \,.
	\end{align*}
	From these two expressions, we see directly that $E_{A\cup B*}=E_{A*}\circ E_{B*}=E_{B*}\circ E_{A*}$ at infinite temperature. In the general situation,
	\begin{align*}
	&D_{\max}(E_{A*}\circ E_{B*}(\rho)\| E_{A\cup B*}(\rho))\\
	&\qquad =\max_{\omega_{\partial AB}}\,
	D_{\max}\Big( \sum_{\substack{\omega_{\partial A\cap B}\\\omega_{\partial B\cap A}}}\,P_{\omega_{\partial A\cap B}}\otimes 	\sigma_A^{\omega_{\partial A}}\otimes   \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},\omega_{\partial B}} \otimes    \rho^{\omega_{\partial B\cap A},\omega_{\partial AB}}_{(A\cup B\partial)^c} \Big\| \sum_{\omega_{\partial B\cap A}} \rho^{\omega_{\partial B\cap A},\omega_{\partial AB}}_{(A\cup B\partial)^c}
	\otimes
	\sigma_{AB}^{\omega_{\partial AB}}    \Big)\\
	&\qquad \le \max_{\substack{\omega_{\partial AB}\\\omega_{\partial B\cap A} }}\,D_{\max}\big(  \sum_{\omega_{\partial A\cap B}}P_{\omega_{\partial A\cap B}}\otimes  \sigma_{A}^{\omega_{\partial A}}\otimes \sigma_{B\backslash A\partial}^{\omega_{\partial A\cup B},\omega_{\partial B}} \big\| \sigma_{AB}^{\omega_{\partial AB}} \big)\,.
	\end{align*}
\end{proof}