Universal recovery from a decrease of quantum relative entropy
The data processing inequality states that the quantum relative entropy between two states $\rho$ and $\sigma$ can never increase by applying the same quantum channel $\mathcal{N$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $\rho$ and the closest recovered state $(\mathcal{R \circ \mathcal{N)(\rho)$, where $\mathcal{R$ is a recovery map with the property that $\sigma = (\mathcal{R \circ \mathcal{N)(\sigma)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $\sigma$ and the quantum channel $\mathcal{N$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
beta.tex

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ExplicitUniversalRecoveryMap.tex
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\begin{document}

\title{Universal recovery from a decrease of quantum relative entropy}

\author[1]{Marius Junge}
\author[2]{Renato Renner}
\author[2]{David Sutter}
\author[3]{Mark M.~Wilde}
\author[4]{Andreas Winter}

 \affil[1]{Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801-2975, USA}
\affil[2]{Institute for Theoretical Physics, ETH Zurich, Switzerland}
\affil[3]{Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA}
\affil[4]{ICREA \&{} F\'{\i}sica Te\`{o}rica: Informaci\'{o} i Fen\`{o}mens
 Qu\`{a}ntics, Universitat Aut\`{o}noma de Barcelona, ES-08193
 Bellaterra (Barcelona), Spain.
}

 
\date{}


\maketitle

\begin{abstract}
The data processing inequality states that the quantum relative entropy between two states $\rho$ and $\sigma$ can never increase by applying the same quantum channel
$\mathcal{N}$
to both states. This inequality can be strengthened with a remainder term in the form of a distance
between $\rho$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(\rho)$, where $\mathcal{R}$ is a recovery map with the property that
$\sigma = (\mathcal{R} \circ \mathcal{N})(\sigma)$. We show the existence of an \emph{explicit} recovery map that is \emph{universal} in the sense that it  depends only on $\sigma$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.
\end{abstract}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Suppose that an experiment, depending on an unknown parameter $\theta \in \Theta$, is described by classical probability distributions $\nu_{\theta}$ on a sample space $(X,\Omega)$. A \emph{statistic}, i.e., a measurable map $\cN:(X,\Omega) \to (Y,\Sigma)$ is called \emph{sufficient} with respect to $\Theta$~\cite{fisher22}, if the conditional probability does not depend on $\theta$, i.e., if there exists a probability distribution $\nu$ on $(X,\Omega)$ such that
\begin{align}
\nu_{\theta}\bigl(X|\,\cN(X) \bigr) = \nu\bigl(X|\,\cN(X)\bigr) \quad \textnormal{for all} \quad \theta \in \Theta \ .
\end{align}
In such a case, $\cN(X)$ contains the same information about the parameter $\theta \in \Theta$ as $X$ does.

This concept has been generalized to the quantum setup~\cite{Pet86,Petz88,mosonyi04,anna06}. For two Hilbert spaces $A$ and $B$, let $\D(A)$ denote the set of density operators on $A$ and let $\TPCP(A,B)$ be the set of trace-preserving completely positive maps from $A$ to $B$. Let $\Q(A)$ denote some subset of $\D(A)$. A quantum channel $\cN \in \TPCP(A,B)$ is called \emph{sufficient} (or \emph{reversible}) \emph{with respect to} $\Q(A)$, if there exists a recovery map $\cR \in \TPCP(B,A)$ such that 
\begin{align}
(\cR \circ \cN)(\rho) = \rho \quad \textnormal{for all} \quad \rho \in \Q(A) \ .
\end{align}

The \emph{data processing inequality} (also known as \emph{monotonicity of the relative entropy}) states that the relative entropy between two states $\rho$ and $\sigma$ --- defined as $D(\rho \| \sigma):= \tr(\rho (\log \rho - \log \sigma))$ if $\supp(\rho) \subseteq \supp(\sigma)$ and $+\infty$ otherwise --- is non-increasing under physical evolutions~\cite{lindblad75,uhlmann77}, i.e., $D(\rho \| \sigma) \geq D(\cN(\rho)\| \cN(\sigma))$, where $\cN$ is a quantum channel. This fundamental entropy inequality has many applications in quantum information theory and is closely related to the sufficiency of $\cN$. 
As shown in~\cite{Pet86,Petz88,anna06,anna206}, for a given $\sigma \geq 0$ a quantum channel $\cN \in \TPCP(A,B)$ is sufficient with respect to $\Q(A)$ if and only if $D(\rho \| \sigma) = D(\cN(\rho) \| \cN(\sigma))$ for all $\rho \in \Q(A)$. (We assume that the set $\Q(A)$ is such that $\supp(\rho) \subseteq \supp(\sigma)$ for all $\rho \in \Q(A)$.) Furthermore, it is known that the data processing inequality holds with equality if and only if there exists a recovery map $\cR \in \TPCP(B,A)$ that simultaneously reverses the action of the physical evolution $\cN$ on both states~\cite{Pet86,Petz88,Pet03}, i.e., $(\cR \circ \cN)(\rho)=\rho$ and $(\cR \circ \cN)(\sigma)=\sigma$.
If the Hilbert spaces $A$ and $B$ are assumed to be separable and $\sigma \in \TC(A)$ is positive semi-definite, then the
recovery map can be taken as the \emph{Petz recovery map} 
$\cP_{\sigma,\cN}$
(also known as the \emph{transpose map}), 
 defined as the unique solution to
\begin{align} \label{eq_EqPetzIntro}
 \langle a_2, \cN^{\dagger}(a_1)  \rangle_{\sigma} = \langle \cP^{\dagger}_{\sigma,\cN}(a_2),a_1  \rangle_{\cN(\sigma)} �\quad \textnormal{for all} \quad a_1, a_2 \in \BB(B) \ ,
\end{align}
where $\BB(B)$ denotes the set of bounded operators on $B$, $\TC(A)$ the set of trace-class operators on $A$, 
$\left \langle a, b \right \rangle_{\omega}:=\tr(a^{\dagger} \omega^{\frac{1}{2}} b \omega^{\frac{1}{2}})$ is a 
 weighted inner product with $\omega$ taken to be positive semi-definite and trace class~\cite{Pet86,Petz88,Pet03},
 and $\cN^{\dagger}$ denotes the adjoint map of $\cN$. The Petz recovery map is completely positive and trace non-increasing (see, e.g., \cite{OP93}). 
 In the case that the Hilbert spaces $A$ and $B$ are finite-dimensional, then, on the support of $\cN(\sigma)$, the Petz recovery map takes the form  
\begin{align} \label{eq_Petzmap}
\cP_{\sigma,\cN} \ : \  X_B \mapsto \sigma^{\frac{1}{2}} \cN^{\dagger}\bigl( \cN(\sigma)^{-\frac{1}{2}} X_B\, \cN(\sigma)^{-\frac{1}{2}} \bigr) \sigma^{\frac{1}{2}} \ .
\end{align}
(Following the standard convention, $\sigma^{-1}$ is defined to be the inverse of $\sigma$, when $\sigma$ is considered as an operator acting only on the support of $\sigma$.)



The concept of sufficient statistics can be made robust.
A quantum channel $\cN \in \TPCP(A,B)$ is \emph{$\eps$-sufficient} with respect to $\Q(A)$ if there exists a recovery map $\cR_{\eps} \in \TPCP(B,A)$ such that~\cite{anna14} 
\begin{align}
\frac{1}{2}\norm{\rho - (\cR_\eps \circ \cN)(\rho)}_1 \leq \eps \quad \textnormal{for all} \quad \rho \in \Q(A) \ ,
\end{align}
for some $\eps \in [0,1]$.
Together with the case $\eps=0$ discussed above, this motivates the question if there exists a refined version of the data processing inequality in terms of recoverability~\cite{WL12}, which would serve as an alternative characterization of approximate sufficient statistics.

An inequality that is closely related to the monotonicity of the relative entropy is the \emph{strong subadditivity} of quantum entropy~\cite{LieRus73_1,LieRus73}, which ensures that for any tripartite state $\rho_{ABC} \in \D(A\otimes B \otimes C)$ the conditional mutual information is non-negative, i.e., $I(A:C|B)_{\rho}:=H(AB)_{\rho}+H(BC)_{\rho}-H(ABC)_{\rho}-H(B)_{\rho} \geq 0$, where $H(A)_{\rho}:=-\tr(\rho_A \log \rho_A)$ denotes the von Neumann entropy. This inequality has been strengthened recently with a remainder term in the form of a distance to the closest recovered state. It was shown in~\cite{FR14}, that for any density operator $\rho_{ABC}$ there exists a trace-preserving completely positive map (the \emph{recovery map}) $\cR_{B \to BC}$ such tha
\begin{align} \label{eq_FR}
I(A:C|B)_{\rho} \geq - 2 \log F(\rho_{ABC},\cR_{B \to BC}(\rho_{AB})) \ ,
\end{align}
where the fidelity of $\rho$ and $\sigma$ is defined by $F(\rho,\sigma):=\norm{\sqrt{\rho}\sqrt{\sigma}}_1$.
If $A$, $B$, and $C$ are finite-dimensional Hilbert spaces, on the support of $\rho_B$, $\cR_{B \to BC}$ can be taken as a rotated Petz recovery map, i.e., a trace-preserving completely positive map of the form
\begin{align} \label{eq_rotPETZ}
X_B \mapsto V_{BC} \rho_{BC}^{\frac{1}{2}} (\rho_B^{-\frac{1}{2}} U_B X_B U_B^{\dagger} \otimes \id_C) \rho_{BC}^{\frac{1}{2}} V_{BC}^{\dagger} \ ,
\end{align}
where $V_{BC}$ and $U_B$ are unitaries on $B\otimes C$ and $B$, respectively.

The result of~\cite{FR14}, whose proof is based on de Finetti type arguments and properties of R\'enyi entropies, has been extended and generalized in various ways. In~\cite{BHOS14}, based on the quantum state redistribution protocol \cite{DY08} and de Finetti type arguments, it was shown that the fidelity term can be replaced by a \emph{measured relative entropy} $\MD$, which is never smaller than the fidelity term, i.e.,
\begin{align} \label{eq_fernando}
I(A:C|B)_{\rho} \geq \MD(\rho_{ABC} \| \cR_{B \to BC}(\rho_{AB})) \geq- 2 \log F(\rho_{ABC},\cR_{B \to BC}(\rho_{AB})) \ .
\end{align}
 The measured relative entropy is defined as the supremum of the relative entropy with measured inputs over all positive operator-valued measures (POVMs) $\cM = \{M_x\}$, i.e., 
\begin{align} \label{eq_defMeasRelEnt}
\MD(\rho \| \sigma) := \sup \Bigl\{ D\bigl( \cM(\rho) \big\| \cM(\sigma) \bigr) : \cM(\rho) = \sum_{x} \tr(\rho M_x) \proj{x} \text{ with } \sum_{x} M_x = \id \Bigr\} \ ,
\end{align}
where $\{\ket{x}\}$ is a finite set of orthonormal vectors.
We note that the tighter bound from \cite{BHOS14} came at the cost of losing all information about the structure of the recovery map. In~\cite{SFR15}, it was shown that there exists a recovery map both satisfying~\eqref{eq_fernando} and possessing a universality property, in the sense that it only depends on the marginal $\rho_{BC}$. Furthermore, for a linearized version of~\eqref{eq_FR} it was shown that the recovery map has the form of a rotated Petz recovery map with commuting unitaries, i.e., a recovery map of the form in \eqref{eq_rotPETZ} where $V_{BC}$ and $U_B$ commute with $\rho_{BC}$ and $\rho_B$, respectively. 

In view of approximate sufficiency of quantum channels discussed above, it would be helpful to have a generalization of~\eqref{eq_FR} in terms of relative entropies. This has been established in~\cite{Wilde15} with a proof technique based on the notion of a R\'enyi generalization of a relative entropy difference \cite{SBW14} and Hadamard's three-line theorem. It was shown that for any two states $\rho$ and $\sigma$ with $\supp(\rho) \subseteq \supp(\sigma)$ and any channel $\cN$ there exists a recovery map $\cR$ such that $(\cR \circ \cN)(\sigma)=\sigma$ and
\begin{align} \label{eq_wilde}
D(\rho \| \sigma) - D\bigl(\cN(\rho) \| \cN(\sigma) \bigr) \geq - 2 \log F\bigl(\rho,(\cN\circ\cR)(\rho)\bigr) \ .
\end{align}
Furthermore, the recovery map was shown to be a rotated Petz recovery map with unitaries $U$ and $V$ in the algebra generated by
 $\sigma$ and $\cN(\sigma)$, respectively.
Very recently, another different proof technique was found~\cite{STH15}, based on the concavity and monotonicity of the operator logarithm, which shows that there exists a recovery map $\cR$ such that
\begin{align} \label{eq_STH}
D(\rho \| \sigma) - D\bigl(\cN(\rho) \| \cN(\sigma) \bigr)
& \geq \MD\bigl(\rho \| (\cR \circ \cN)(\rho)\bigr) \\
& \geq - 2 \log F\bigl(\rho,(\cN\circ\cR)(\rho)\bigr) \
\label{eq_STH_1}.
\end{align}
The recovery map was shown to be a convex combination of rotated Petz recovery maps with unitaries $U$ and $V$ in the algebra generated by $\sigma$ and $\cN(\sigma)$, respectively, and therefore satisfies $(\cR \circ \cN)(\sigma)=\sigma$.

Neither in~\cite{Wilde15} nor in~\cite{STH15} could the recovery map satisfying~\eqref{eq_wilde} and~\eqref{eq_STH}, respectively, be shown to be universal, in the sense that it could be taken independent of $\rho$. In this article, we show that for separable (not necessarily finite-dimensional) Hilbert spaces there exists an \emph{explicit}, universal recovery map $\cR_{\sigma,\cN}$ that fulfills~\eqref{eq_wilde} and depends only on $\sigma$ and $\cN$. In addition, we show that there exists a universal recovery map that satisfies~\eqref{eq_STH} for finite-dimensional Hilbert spaces. We note that by the Fuchs-van de Graaf inequality~\cite{fuchs99} the fidelity can be transferred into a trace distance term such that~\eqref{eq_wilde} and~\eqref{eq_STH_1} provide alternative characterizations for approximate sufficient statistics.

\paragraph{Result.} We show that for any non-negative operator $\sigma$ and for any channel $\cN$ there exists an \emph{explicit} and \emph{universal} recovery map $\cR_{\sigma,\cN}$  such that
\begin{align} \label{eq_res_intro}
D(\rho \| \sigma) - D\bigl(\cN(\rho) \| \cN(\sigma) \bigr) \geq - 2 \log F\bigl(\rho,(\cR_{\sigma,\cN}\circ\cN)(\rho)\bigr)
\end{align}
for all density operators $\rho$ such that $\supp(\rho) \subseteq \supp(\sigma)$. A consequence of the  universality of the recovery map $\cR_{\sigma,\cN}$ is that $(\cR_{\sigma,\cN}\circ\cN)(\sigma)=\sigma$.
 We refer to Theorem~\ref{thm_explicit} for a more precise statement.

\paragraph{Rotated and twirled Petz recovery map.} We next introduce two classes of recovery maps that take the form of a rotated (or twirled) Petz recovery map. These are the building blocks for an explicit recovery map that satisfies~\eqref{eq_res_intro}.
In~\cite{Wilde15}, for $t \in \R$ the following rotated version of the Petz recovery map (given in~\eqref{eq_Petzmap}) has been introduced
\begin{align} \label{eq_MarkPetzmap}
\cR^t_{\sigma,\cN} \ : X_B \mapsto \sigma^{-\ci t} \cP_{\sigma, \cN}\bigl( \cN(\sigma)^{\ci t} X_B\, \cN(\sigma)^{-\ci t} \bigr) \sigma^{\ci t} \ .
\end{align}
A less specific form of a rotated Petz recovery map is given by
\begin{align}
  \cR_{\sigma,\cN}^{U,V}\ :
   X_B \mapsto  V  \cP_{\sigma, \cN}\bigl(   U  X_{B} U  \bigr)    V \ ,
\end{align}
where $U$ and $V$ denote unitaries that commute with $\cN(\sigma)$ and $\sigma$, respectively. Let $\cU_{\sigma,A}$ be the set of unitaries on $A$ that commute with $\sigma$.
Let $\Pos(A)$ denote the set of non-negative operators on a Hilbert space $A$. (If $A$ is not finite-dimensional, we take the convention that operators in $\Pos(A)$ are trace-class.)
 For any $\sigma \in \Pos(A)$ and $\cN \in \TPCP(A,B)$ we denote the convex hull of rotated Petz recovery maps by 
\begin{align} \label{eq_convexHullRotatedPetz}
\RRR_{\sigma,\cN}:=\conv\Bigl(\cR_{\sigma,\cN}^{U,V} \ : \  U \in \cU_{\cN(\sigma),B} \quad \textnormal{and} \quad V \in \cU_{\sigma,A} \Bigr) \ .
\end{align}
We note that the rotated Petz recovery map $\cR_{\sigma,\cN}^{U,V}$ is trace non-increasing and completely positive. For any $\sigma \in \Pos(A)$, we define 
\begin{align}
\D_{\sigma}(A):=\{\rho \in \D(A) : \supp(\rho) \subseteq \supp(\sigma)\} \ .
\end{align}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}

\begin{theorem}\label{thm_explicit}
Let $A$ and $B$ be separable Hilbert spaces. For any $\sigma \in \Pos(A)$, any $\rho \in \D_{\sigma}(A)$ and any $\cN \in \TPCP(A,B)$ we have
\begin{align} \label{eq_explcitMainRes}
D(\rho \dd \sigma) - D\bigl(\cN(\rho) \dd \cN(\sigma) \bigr) \geq - 2 \log F\bigl(\rho,( \cR_{\sigma,\cN} \circ\cN)(\rho)\bigr) \ ,
\end{align}
where 
\begin{align}
 \cR_{\sigma,\cN}(\cdot):= \int_{\R} \di t  \,\beta_0(t) \,  \cR^{\frac{t}{2}}_{\sigma,\cN}(\cdot)
\end{align}
on the support of $\cN(\sigma)$ with $\cR^t_{\sigma,\cN}  : X_B \mapsto \sigma^{-\ci t} \cP_{\sigma, \cN}\bigl( \cN(\sigma)^{\ci t} X_B\, \cN(\sigma)^{-\ci t} \bigr) \sigma^{\ci t}$ and $\beta_0$ a probability density function on $\R$ defined by 
\begin{align} \label{eq_defBeta}
\beta_0(t):=\frac{\pi}{2}\bigl(\cosh(\pi t) + 1 \bigr)^{-1} \ .
\end{align}
If $A$ and $B$ are separable, $\cP_{\sigma, \cN}$ is defined as the adjoint of the unique linear map
$\cP_{\sigma, \cN}^{\dag}$
satisfying~\eqref{eq_EqPetzIntro} with domain
$\supp(\cN(\sigma))$ and range $\supp(\sigma)$.
If $A$ and $B$ are finite-dimensional, this unique linear map $\cP_{\sigma, \cN}$ is given by~\eqref{eq_Petzmap}. 
\end{theorem}
Figure~\ref{fig_beta} depicts the probability density $\beta_0$ as a function of $t\in \R$. 
We note that the recovery map $\cR_{\sigma,\cN}$ that satisfies~\eqref{eq_explcitMainRes} can be chosen such that it projects everything outside of the support of $\cN(\sigma)$ to zero.
For the case of finite-dimensional Hilbert spaces we can tighten the inequality~\eqref{eq_explcitMainRes} at the cost of losing the explicit form of the recovery map.
\begin{theorem}\label{thm_monoRelEnt}
Let $A$ and $B$ be finite-dimensional Hilbert spaces. For any $\sigma \in \Pos(A)$ and any $\cN \in \TPCP(A,B)$ there exists a recovery map $\widetilde{\cR}_{\sigma,\cN}$ inside the set $\RRR_{\sigma,\cN}$ defined in~\eqref{eq_convexHullRotatedPetz}, such that for all $\rho \in \D_{\sigma}(A)$ 
\begin{align} \label{eq_strictMonoBound}
D(\rho \dd \sigma) - D\bigl(\cN(\rho) \dd \cN(\sigma) \bigr) 
&\geq \MD\bigl(\rho, (\widetilde{\cR}_{\sigma,\cN} \circ \cN)(\rho) \bigr) \\
&\geq - 2 \log F\bigl(\rho,(\widetilde{\cR}_{\sigma,\cN}\circ\cN)(\rho)\bigr)  \ . \label{eq_strictFidBound}
\end{align}
\end{theorem}
We note that the measured relative entropy is a quantity that satisfies several desirable properties (see, e.g.,~\cite[Lemma~B.3]{SFR15}). Therefore the bound in \eqref{eq_strictMonoBound} can be particularly helpful when it is used as a proof tool (see, e.g., Corollaries~\ref{cor_concCondEnt} and~\ref{cor_jointConvRelEnt}). Furthermore, as discussed in~\cite{BHOS14} the right-hand side of~\eqref{eq_strictMonoBound} can be substantially larger than~\eqref{eq_strictFidBound} in some cases.


\begin{remark}[Functoriality properties] \label{rmk_properties}
The recovery maps $\cR_{\sigma,\cN}$ and
$\widetilde{\cR}_{\sigma,\cN}$ stated in~Theorems~\ref{thm_explicit} and \ref{thm_monoRelEnt}, respectively, satisfy apart from~\eqref{eq_explcitMainRes} several desirable ``functoriality'' properties. Some of them have been stated in~\cite{WL12,LiWin14,Wilde15}. Since
both recovery maps satisfy these properties, we abbreviate them both as $\cR_{\sigma,\cN}$ in what immediately follows.
\begin{enumerate}
\item \textbf{Universality.} The recovery map does not depend on $\rho$. This follows directly from Theorem~\ref{thm_explicit}.

\item \textbf{Perfect reconstruction of $\sigma$ from $\cN(\sigma)$.} The recovery map satisfies $( \cR_{\sigma,\cN} \circ \cN)(\sigma)=\sigma$. This is clear from the fact that any rotated Petz map of the form in \eqref{eq_MarkPetzmap} perfectly recovers $\sigma$  \cite{Wilde15}, and thus so does any convex combination of these maps. Alternatively, as the recovery map predicted by Theorem~\ref{thm_explicit} that satisfies~\eqref{eq_explcitMainRes} is universal, the assertion follows by choosing $\rho = \sigma/\tr(\sigma)$.

\item \textbf{Normalization.} In case $\cN = \cI$, where $\cI$ denotes the identity map, we have $ \cR_{\sigma,\cN}(\cdot)=\Pi_{\sigma}(\cdot)\Pi_{\sigma}$. Thus if $\sigma$ has full support the recovery map is equal to the identity channel. 
This follows directly by~\cite[Section~4.1]{Wilde15} and by definition of the recovery map $\cR_{\sigma,\cN}(\cdot)$.

\item \textbf{Stabilization.} For any
$\sigma \in \Pos(A)$, any $\cN \in \TPCP(A,B)$, any reference system $R$, and any full-rank $\tau \in \Pos(R)$, we have $ \cR_{\sigma\otimes \tau_R,\cN\otimes \cI_R}(\cdot) =  \cR_{\sigma,\cN} \otimes \cI_R(\cdot)$. This follows by combining~\cite[Section~4.2]{Wilde15} together with the normalization property discussed above.


\end{enumerate}
We note that by following~\cite[Sections~4.2 and 4.3]{Wilde15} together with the strengthening of Theorem~\ref{thm_explicit} given in~\eqref{eq_explcitMainResStrong} we can obtain remainder terms that fulfill some parallel and serial composition rules.
\end{remark}


The proof of Theorem~\ref{thm_explicit} consists of two parts. We first prove the statement for finite-dimensional Hilbert spaces $A$ and $B$ by employing a strengthened version of Hadamard's three-line theorem that is due to Hirschman \cite{H52}. By an approximation argument we show that the result remains valid for separable Hilbert spaces.
Theorem~\ref{thm_monoRelEnt} is proven differently. Via Sion's minimax theorem~\cite[Corollary~3.3]{Sion58}, we show the existence of a universal recovery map that satisfies~\eqref{eq_strictMonoBound} for any finite set of states $\rho \in \D(A)$ (see Step~\hyperlink{step_1_1}{1}). We then show, by using the concept of an $\eps$-net, how this implies the existence of a universal recovery map that satisfies~\eqref{eq_strictMonoBound} for all $\rho \in \D(A)$. 


\begin{figure}[!htb]
\centering
\input{beta.tex}
\caption{This plot depicts the probability density $\beta_0$ defined in~\eqref{eq_defBeta} as a function of $t \in�\R$. We see that it is peaked around $t=0$ which corresponds to the Petz recovery map, i.e., $\mathcal{R}^{t=0}_{\sigma, \mathcal{N}} = \mathcal{P}_{\sigma, \mathcal{N}}$.}
\label{fig_beta}
\end{figure}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem~\ref{thm_explicit}}
\subsection*{\hypertarget{step_0_1}{Step 1:} Proof for  finite-dimensional Hilbert spaces}
In this step we assume that the Hilbert spaces $A$ and $B$ are finite-dimensional. We note that we will prove a slightly stronger version of Theorem~\ref{thm_explicit} that by concavity of the logarithm and the fidelity immediately implies Theorem~\ref{thm_explicit}.
\begin{theoremStrong}
For any $\sigma \in \Pos(A)$, any $\rho \in \D_{\sigma}(A)$ and any $\cN \in \TPCP(A,B)$ we have
\begin{align} \label{eq_explcitMainResStrong}
D(\rho \dd \sigma) - D\bigl(\cN(\rho) \dd \cN(\sigma) \bigr) \geq - 2 \int_{\R} \di t \, \beta_0(t) \,  \log F\bigl(\rho,( \cR^{\frac{t}{2}}_{\sigma,\cN} \circ\cN)(\rho)\bigr) \ ,
\end{align}
where $\cR^t_{\sigma,\cN}(\cdot)$ is defined in~\eqref{eq_MarkPetzmap} and the probability density function  $\beta_0(t):=\frac{\pi}{2}(\cosh(\pi t) + 1)^{-1}$. 
\end{theoremStrong}

\begin{remark}
Inequality~\eqref{eq_explcitMainResStrong} together with the fact that the mapping $t\mapsto \cR_{\sigma,\cN}^t$ is continuous implies that for any $\sigma \in \Pos(A)$, $\rho \in \D_{\sigma}(A)$ and $\cN \in \TPCP(A,B)$ such that $D(\rho \| \sigma)=D(\cN(\rho) \| \cN(\sigma))$ we have $(\cR^t_{\sigma,\cN}\circ \cN)(\rho)=\rho$ and $(\cR^t_{\sigma,\cN}\circ \cN)(\sigma)=\sigma$ for  all $t \in \R$ with $\cR^t_{\sigma,\cN}(\cdot)$ defined in~\eqref{eq_MarkPetzmap}. This follows because $F(\omega,\tau) \in [0,1]$ and $F(\omega,\tau) = 1$ if and only if $\omega = \tau$ for density operators $\omega$ and $\tau$.
\end{remark}

Our proof of \eqref{eq_explcitMainResStrong} is similar to the approach taken in \cite{Wilde15}. There are two main ingredients: a R\'enyi generalization of a relative entropy difference \cite{SBW14} and Hirschman's improvement of the Hadamard three-line theorem \cite{H52}. We begin by recalling these two ingredients and then proceed to a proof of \eqref{eq_explcitMainResStrong}. 

Let $\mathrm{L}(A)$ denote the space of bounded linear operators acting on a Hilbert space $A$. For any $L \in \mathrm{L}(A)$ the \emph{Schatten $p$-norm} is defined as
\begin{align}
\norm{L}_p := \bigl( \tr(|L|^p) \bigr)^{\frac{1}{p}} \quad \textnormal{for} \quad p\in [1,\infty) \ ,
\end{align}
where $|L|:=\sqrt{L^\dagger L}$. 
%We extend this definition to $p \in (0,1)$, but note that in this case $\norm{L}_p$ is not a norm. \mmw{Do we actually use the norm for $p<1$? If not, maybe remove this statement...}
A R\'{e}nyi generalization of a relative entropy difference is defined as
\cite{SBW14}%
\begin{equation}
\widetilde{\Delta}_{\alpha}(  \rho,\sigma,\mathcal{N})  :=
\frac{2\alpha}{\alpha-1}\log\left\Vert \left(  \left[  \mathcal{N}(
\rho)  \right]  ^{\frac{1-\alpha}{2\alpha}}\left[
\mathcal{N}(  \sigma)  \right]  ^{\frac{\alpha-1}{2\alpha}}\otimes \id_{E}\right)  U_{A\rightarrow BE}\, \sigma^{\frac{
1-\alpha}{2\alpha}}\rho^{\frac12}\right\Vert _{2\alpha}%
,\label{eq:renyi-diff}%
\end{equation}
where $\alpha\in(0,1)\cup(1,\infty)$, and $U_{A\rightarrow BE}$ is an
isometric extension of the channel $\mathcal{N}$. That is, $U_{A\rightarrow
BE}$ is a linear isometry satisfying $\operatorname{tr}_{E}(U_{A\rightarrow
BE}(  \cdot) U_{A\rightarrow BE}^{\dag})=\mathcal{N}\left(
\cdot\right)  $ and $U_{A\rightarrow BE}^{\dag}U_{A\rightarrow BE}=\id_{A}$. All
isometric extensions of a channel are related by an isometry acting on the
environment system~$E$, so that the definition in \eqref{eq:renyi-diff} is
invariant under any such choice. Recall also that the adjoint $\mathcal{N}%
^{\dag}$\ of a channel is given in terms of an isometric extension $U$ as
$\mathcal{N}^{\dag}(  \cdot)  =U^{\dag}(  (
\cdot)  \otimes \id_{E})  U$.

\begin{lemma}
[\cite{SBW14,Wilde15}]The following limit holds for $\rho$, $\sigma$, and
$\mathcal{N}$ as given in the statement of Theorem~\ref{thm_explicit}:%
\begin{equation}
\lim_{\alpha\rightarrow1}\widetilde{\Delta}_{\alpha}(  \rho
,\sigma,\mathcal{N})  =D(\rho\Vert\sigma)-D\bigl(  \mathcal{N}(
\rho) \big\|  \mathcal{N}(  \sigma)  \bigr)
.\label{eq:rel-ent-diff-a-1}%
\end{equation}
\end{lemma}

For $\alpha=\frac12$, observe that%
\begin{align}
\widetilde{\Delta}_{\frac12}(  \rho,\sigma,\mathcal{N})    &
=-2\log\left\Vert \left(  \left[  \mathcal{N}(  \rho)  \right]
^{\frac12}\left[  \mathcal{N}(  \sigma)  \right]  ^{-\frac12}\otimes
\id_{E}\right)  U_{A\rightarrow BE}\,\sigma^{\frac12}\rho^{\frac12}\right\Vert _{1}
 =-2 \log  F\Bigl(  \rho,\mathcal{P}_{\sigma,\mathcal{N}}\bigl(\mathcal{N}(  \rho)  \bigr)  \Bigr)  .
\end{align}

The following lemma is based on Hirschman's improvement of the Hadamard three-line theorem \cite{H52}, and for completeness, we provide a proof in Appendix~\ref{sec:op-hirschman-proof}.

\begin{lemma}
\label{thm:op-hirschman}
Let $S:=\left\{  z\in\mathbb{C}:0\leq\operatorname{Re}\left\{  z\right\}
\leq1\right\} $ and let $G:S\rightarrow
L(\mathcal{H})$ be a bounded map that is holomorphic on the interior of $S$
and continuous on the boundary. Let $\theta\in(0,1)$ and define $p_{\theta}$
by%
\begin{equation}
\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}\ ,
\end{equation}
where $p_{0},p_{1}\in\left[  1,\infty\right]  $. Then the following bound
holds%
\begin{equation}
\log\bigl(  \left\Vert G(\theta)\right\Vert _{p_{\theta}}\bigr)  \leq
\int_{\R}\di t\ \Bigl(  \alpha_{\theta}(t)\log \bigl(  \left\Vert
G(\ci t)\right\Vert _{p_{0}}^{1-\theta}\bigr)  +\beta_{\theta}(t)\log\bigl(
\left\Vert G(1+\ci t)\right\Vert _{p_{1}}^{\theta}\bigr)  \Bigr) \ ,\label{eq:oper-hirschman}%
\end{equation}
where $\alpha_{\theta}(t)$ and $\beta_{\theta}(t)$ are defined by
\begin{align}
\alpha_{\theta}(t)   :=\frac{\sin(\pi\theta)}{2(1-\theta)\bigl(
\cosh(\pi t)-\cos(\pi\theta)\bigr)  } \qquad \textnormal{and} \qquad
\beta_{\theta}(t)  :=\frac{\sin(\pi\theta)}{2\theta\bigl( \cosh(\pi
t)+\cos(\pi\theta)\bigr)  }\ .\label{eq:pt}%
\end{align}
\end{lemma}

\begin{remark}
Fix $\theta\in(0,1)$. Observe that $\alpha_{\theta}(t),\beta_{\theta}(t)\geq0$
for all $t\in\mathbb{R}$ and we have
\begin{equation}
\int_{\R}\di t\ \alpha_{\theta}(t)=\int_{\R}%
\di t\ \beta_{\theta}(t)=1 \ ,
\end{equation}
(see, e.g., \cite[Exercise~1.3.8]{G08}) so that $\alpha_{\theta}(t)$ and
$\beta_{\theta}(t)$ can be interpreted as probability density functions.
Furthermore, the following limit holds
\begin{equation}
\lim_{\theta\searrow0}\beta_{\theta}(t)=\frac{\pi}{2\bigl(  \cosh(\pi
t)+1\bigr) } =\beta_{0}(t)\ ,\label{eq:dist-limit}%
\end{equation}
where $\beta_{0}$ is also a probability density function on $\R$.
\end{remark}%






We can now readily establish the desired result in \eqref{eq_explcitMainResStrong}.
In what follows, we abbreviate the isometric extension
$U_{A\rightarrow BE}$ of the channel $\mathcal{N}$
as $U$. Pick%
\begin{equation}
G(  z)  :=\left(  \left[  \mathcal{N}(  \rho)
\right]  ^{\frac{z}{2}}\left[  \mathcal{N}(  \sigma)  \right]
^{-\frac{z}{2}}\otimes \id_{E}\right)  U\, \sigma^{\frac{z}{2}}\rho^{\frac12}\ ,
\end{equation}
$p_{0}=2$, $p_{1}=1$, and $\theta\in\left(  0,1\right)  $, which fixes
$p_{\theta}=\frac{2}{1+\theta}$. The operator valued-function $G(
z)  $ satisfies the conditions needed to apply
Lemma~\ref{thm:op-hirschman}. For the choices above, we find that%
\begin{align}
\left\Vert G(  \theta)  \right\Vert _{\frac{2}{1+\theta}  }
&  =\left\Vert \left(  \left[  \mathcal{N}(  \rho)  \right]
^{\frac{\theta}{2}}\left[  \mathcal{N}(  \sigma)  \right]  ^{-\frac{\theta}{2}}\otimes \id_{E}\right)  U\, \sigma^{\frac{\theta}{2}}\rho^{\frac12}\right\Vert _{\frac{2}{
1+\theta}  }\ ,
\end{align}
and
\begin{align}
\left\Vert G(  \ci t)  \right\Vert _{2} &  =\left\Vert \left(  \left[
\mathcal{N}(  \rho)  \right]  ^{\frac{\ci t}{2}}\left[  \mathcal{N}(
\sigma)  \right]  ^{-\frac{\ci t}{2}}\otimes \id_{E}\right)  U\sigma^{\ci t}\rho
^{\frac12}\right\Vert _{2}\leq\left\Vert \rho^{\frac12}\right\Vert _{2}%
=1 \ ,\label{eq:M0} 
\end{align}
as well as
\begin{align}
\left\Vert G(  1+\ci t)  \right\Vert _{1} &  =\left\Vert \left(
\left[  \mathcal{N}(  \rho)  \right]^{\frac{1+\ci t}{2}}\left[  \mathcal{N}(  \sigma)  \right]  ^{-\frac{1+\ci t}{2}}\otimes \id_{E}\right)  U \,\sigma^{\frac{1+\ci t}{2}}\rho^{\frac12}%
\right\Vert _{1}\nonumber\\
& =\left\Vert \left(  \left[  \mathcal{N}(  \rho)  \right]
^{\frac{\ci t}{2}}\left[  \mathcal{N}(  \rho)  \right]^{\frac12}\left[
\mathcal{N}(  \sigma)  \right]  ^{-\frac{\ci t}{2}}\left[  \mathcal{N}(
\sigma)  \right]^{-\frac12}\otimes \id_{E}\right)  U\sigma^{\frac12}\sigma
^{\frac{\ci t}{2}}\rho^{\frac12}\right\Vert _{1}\nonumber\\
&  =\left\Vert \left(  \left[  \mathcal{N}(  \rho)  \right]
^{\frac12}\left[  \mathcal{N}(  \sigma)  \right]  ^{-\frac{\ci t}{2}}\left[
\mathcal{N}(  \sigma)  \right]  ^{-\frac12}\otimes \id_{E}\right)
U\sigma^{\frac12}\sigma^{\frac{\ci t}{2}}\rho^{\frac12}\right\Vert _{1}\nonumber\\
&  = F\bigl(  \rho,(\mathcal{R}_{\sigma,\mathcal{N}}^{\frac{t}{2}} \circ
\mathcal{N})(  \rho)    \bigr) \ .\label{eq:M1}%
\end{align}
Then we can apply the fact that $\left\Vert G(  \ci t)  \right\Vert
_{2}\leq1$ and \eqref{eq:oper-hirschman} to conclude that the following bound
holds for all $\theta\in(0,1)$
\begin{align}
\log \left\Vert \left(  \left[  \mathcal{N(}\rho)\right]  ^{\frac{\theta}{2}}\left[  \mathcal{N(}\sigma)\right]^{-\frac{\theta}{2}}\otimes \id_{E}\right) U\, \sigma^{\frac{\theta}{2}}\rho^{\frac12}\right\Vert _{\frac{2}{1+\theta} }
\leq \int_{\R}\di t\ \beta_{\theta}(t)\log \left( F\bigl(  \rho,(\mathcal{R}_{\sigma,\mathcal{N}}^{\frac{t}{2}} \circ
\mathcal{N})(  \rho)  \bigr)    ^{\theta} \right) \ ,
\end{align}
which implies%
\begin{align}
-\frac{2}{\theta}\log  \left\Vert \left(  \left[  \mathcal{N}(\rho)\right]  ^{\frac{\theta}{2}}\left[  \mathcal{N(}\sigma)\right]  ^{-\frac{\theta}{2}}\otimes \id_{E}\right)  U\sigma^{\frac{\theta}{2}}\rho^{\frac12}\right\Vert _{\frac{2}{1+\theta} }  \label{eq:critical-inequality}
\geq-2\int_{\R}\di t\ \beta_{\theta}(t)\log  F\bigl(  \rho,(\mathcal{R}_{\sigma,\mathcal{N}}^{\frac{t}{2}} \circ
\mathcal{N})(  \rho)   \bigr) \  .
\end{align}
Letting $\theta=\frac{ 1-\alpha}{\alpha}$, we see that this is the
same as%
\begin{equation}
\widetilde{\Delta}_{\alpha}(  \rho,\sigma,\mathcal{N})  \geq
- 2 \int_{\R}\di t\ \beta_{\frac{ 1-\alpha}{\alpha}}%
(t)\log F\bigl(  \rho,(\mathcal{R}_{\sigma,\mathcal{N}}^{\frac{t}{2}} \circ
\mathcal{N})(  \rho)    \bigr)
.\label{eq:alpha-bound}%
\end{equation}
Since the inequality in \eqref{eq:critical-inequality} holds for all
$\theta\in\left(  0,1\right)  $ and thus \eqref{eq:alpha-bound} holds for all
$\alpha\in\left(  \frac12,1\right)  $, we can take the limit as $\alpha\nearrow1$
and apply \eqref{eq:rel-ent-diff-a-1}, \eqref{eq:dist-limit}, and the dominated convergence theorem to conclude
that \eqref{eq_explcitMainResStrong} holds.


\begin{remark} \label{rmk_subnormalized}
If $A$ and $B$ are finite-dimensional Hilbert spaces the statement in Theorem~\ref{thm_explicit} can be slightly generalized. For any $\sigma \geq 0$ and any trace non-increasing completely positive map $\cN$ with Kraus operators $\{N_i\}$ such that $0 \neq \sum_i N_i^\dag N_i \leq \id$ the recovery map $ \cR_{\sigma,\cN}$ defined in Theorem~\ref{thm_explicit} satisfies \eqref{eq_explcitMainRes} for all subnormalized density operators $\rho \geq 0$ with $0 < \tr(\rho)\leq 1$ such that $\supp(\rho) \subseteq \supp(\sigma)$. This follows by the same argument given in Step~\hyperlink{step_1_1}{1} by using $U_{A\rightarrow BE} = \sum_i N_i \otimes \ket{i}_E$ as an isometric extension of the trace non-increasing and completely positive map $\mathcal{N}$.
\end{remark}

%%%
\subsection*{\hypertarget{step_0_1}{Step 2:} Extension to infinite dimensions}

In this step the Hilbert spaces $A$ and $B$ are assumed to be separable (not necessarily finite-dimensional). We will show how to lift the stronger version of Theorem~\ref{thm_explicit} for finite-dimensional Hilbert spaces, such that it can apply to states $\rho$ and $\sigma$ and a channel $\cN$ associated with separable Hilbert spaces. This is accomplished via a limiting argument. Let $\{\Pi^a_A\}_{a \in \N}$ and $\{\Pi^b_B \}_{b \in \N}$ be sequences of finite-rank projectors on $A$ and $B$, respectively, that converge to $\id_A$ and $\id_B$, respectively, with respect to the weak operator topology, meaning that
\begin{equation}
\lim_{a\to \infty} \langle \psi \vert
\Pi^a_A \vert \phi \rangle = \langle \psi \vert
 \phi \rangle
\end{equation}
for all vectors 
$\vert \phi \rangle, \vert \psi \rangle \in A$ (similarly for $\Pi^b_B \to \id_B$). For $\sigma \in \Pos(A)$ and $\rho \in \D_{\sigma}(A)$ we consider projected versions
\begin{align} \label{eq_projRhoA}
\sigma^a :=\Pi_A^a \, \sigma \, \Pi_A^a \quad \textnormal{and} \quad \rho^a :=\Pi_A^a \, \rho \,  \Pi_A^a \ .
\end{align}
We note that the sequences $\{\rho^a \}_{a \in \N}$ and $\{\sigma^a�\}_{a \in \N}$ converge to $\rho$ and $\sigma$, respectively, in the trace norm (see, e.g., Corollary 2 of~\cite{FAR11} or Lemma~11.1 of \cite{holevo_book}).
Let $\cS^a$ be the set of non-negative operators that is generated by~\eqref{eq_projRhoA} for all $\rho \in \cS$. For any $\cN \in \TPCP(A,B)$ we define its analogue with a projection at the input and output as
\begin{align}
\cN^{a,b}(\cdot) := \Pi_B^b \cN\bigl( \Pi_A^a ( \cdot) \Pi_A^a\bigr) \Pi_B^b \ .
\end{align}
Note that $\cN^{a,b}$ converges to $\cN$ in the weak operator topology, in the sense that
\begin{equation}
\lim_{a,b\to \infty}
\langle \phi' \vert \cN^{a,b}(\vert \phi \rangle \langle \psi \vert) \vert \psi' \rangle
  = \langle \phi' \vert \cN(\vert \phi \rangle \langle \psi \vert) \vert \psi' \rangle ,
  \label{eq:channel-convergence-weak-op}
\end{equation}  
  for all vectors 
$\vert \phi \rangle, \vert \psi \rangle \in A$ and
$\vert \phi' \rangle, \vert \psi' \rangle \in B$ (cf., the discussions in \cite{SH08}). 


We start by proving two lemmas that show how the difference of relative entropies and the fidelity, respectively, change when considering projected states.
\begin{lemma} \label{lem_one}
For any $\sigma \in \Pos(A)$, any $\rho \in \D_{\sigma}(A)$, and any $\cN \in \TPCP(A,B)$, we have
\begin{align}
\limsup_{a \to \infty} \limsup_{b \to \infty}  D(\rho^a \| \sigma^a) - D\bigl(\cN^{a,b}(\rho^a) \big\| \, \cN^{a,b}(\sigma^a) \bigr) \leq D(\rho \| \sigma) - D\bigl(\cN(\rho) \big\| \, \cN(\sigma) \bigr) \ .
\end{align}
\end{lemma}
\begin{proof}
By Lemma~3 in~\cite{lindblad74} and since the relative entropy is lower semicontinuous~\cite[Exercise~7.22]{holevo_book}, we obtain
\begin{equation}
D(\rho^a \| \sigma^a) \leq D(\rho \| \sigma) \leq \liminf_{a \to \infty} D(\rho^a \| \sigma^a) \ ,
\end{equation}
which implies that
\begin{align} \label{eq_s1}
\lim_{a \to \infty} D(\rho^a \| \sigma^a) = D(\rho \| \sigma) \ .
\end{align}
The lower semicontinuity of the relative entropy implies that
\begin{align}
 \liminf_{a \to \infty} \liminf_{b \to \infty} D\bigl(\cN^{a,b}(\rho^a) \| \cN^{a,b}(\sigma^a) \bigr) 
& \geq \liminf_{a \to \infty} D\bigl(\cN(\rho^a) \| \cN(\sigma^a) \bigr) \\
& \geq D\bigl(\cN(\rho) \| \cN(\sigma) \bigr) \ .
\end{align}
This proves the assertion.
\end{proof}
%%%%%%%

\begin{lemma} \label{lem_PetzWellDefined}
The Petz recovery map $\cP_{\sigma^a,\cN^{a,b}}$ defined in~\eqref{eq_Petzmap} satisfies $\lim_{a\to \infty} \lim_{b \to \infty} \cP_{\sigma^a,\cN^{a,b}} = \cP_{\sigma,\cN}$ (in the sense of \eqref{eq:channel-convergence-weak-op}), where $\cP_{\sigma,\cN}$ is a recovery map that is defined as the (unique) linear map with domain
$\operatorname{supp}(\cN(\sigma))$ and range
$\operatorname{supp}(\sigma)$
satisfying
\begin{align} \label{eq_EqPetz}
\left \langle a_2, \cN^{\dagger}(a_1) \right \rangle_{\sigma} = \left \langle \cP^{\dagger}_{\sigma,\cN}(a_2),a_1 \right \rangle_{\cN(\sigma)} �\quad \textnormal{for all} \quad a_1 \in \BB(B), \, a_2 \in \BB(A)\ ,
\end{align}
with the weighted inner product $\left \langle a, b \right \rangle_{\omega}:=\tr(a^{\dagger} \omega^{\frac{1}{2}} b \omega^{\frac{1}{2}})$ and $\omega$ positive semi-definite and trace class. 
\end{lemma}
\begin{proof}
We begin by outlining the proof.
As shown by Petz~\cite{Pet86,Petz88,Pet03} (see also \cite[Chapter~8]{OP93}), the map $\cP_{\sigma^a,\cN^{a,b}}$ defined in~\eqref{eq_Petzmap} is the unique linear map
with domain
$\operatorname{supp}(\cN^{a,b}(\sigma^a))$ and range
$\operatorname{supp}(\sigma^a)$
 satisfying
\begin{align} \label{eq_setEq}
\left \langle a_2, (\cN^{a,b})^{\dagger}(a_1) \right \rangle_{\sigma^a} = \left \langle \cP^{\dagger}_{\sigma^a,\cN^{a,b}}(a_2),a_1 \right \rangle_{\cN^{a,b}(\sigma^a)} �\quad \textnormal{for all} \quad a_1 \in \BB(B), \, a_2 \in \BB(A) \ ,
\end{align}
and the map $\cP_{\sigma,\cN}$ is the unique linear map
with domain
$\operatorname{supp}(\cN(\sigma))$ and range
$\operatorname{supp}(\sigma)$
 satisfying
\begin{align} \label{eq_setEq-inf-dim}
\left \langle a_2, (\cN)^{\dagger}(a_1) \right \rangle_{\sigma} = \left \langle \cP^{\dagger}_{\sigma,\cN}(a_2),a_1 \right \rangle_{\cN(\sigma)} �\quad \textnormal{for all} \quad a_1 \in \BB(B), \, a_2 \in \BB(A) \ .
\end{align}
We will first show that
\begin{align} \label{eq_s1_converge}
\lim_{a \to \infty} \lim_{b \to \infty} \left \langle a_2, (\cN^{a,b})^{\dagger}(a_1) \right \rangle_{\sigma^a}  
=\left \langle a_2, \cN^{\dagger}(a_1) \right \rangle_{\sigma} ,
\end{align}
for all $a_1 \in \BB(B),  a_2 \in \BB(A)$,
which by \eqref{eq_setEq} and \eqref{eq_setEq-inf-dim} implies that
\begin{align}
\lim_{a \to \infty} \lim_{b \to \infty} \left \langle \cP^{\dagger}_{\sigma^a,\cN^{a,b}}(a_2),a_1 \right \rangle_{\cN^{a,b}(\sigma^a)} = \left \langle \cP^{\dagger}_{\sigma,\cN}(a_2),a_1 \right \rangle_{\cN(\sigma)}
\label{eq:petz-converge-almost-final-1}
\end{align}
for all $a_1 \in \BB(B),  a_2 \in \BB(A)$.
After showing that
\begin{align}
\lim_{a \to \infty} \lim_{b \to \infty} \left \langle \cP^{\dagger}_{\sigma^a,\cN^{a,b}}(a_2),a_1 \right \rangle_{\cN^{a,b}(\sigma^a)} = 
\lim_{a \to \infty} \lim_{b \to \infty} \left \langle \cP^{\dagger}_{\sigma^a,\cN^{a,b}}(a_2),a_1 \right \rangle_{\cN(\sigma)} ,\label{eq:petz-converge-almost-final}
\end{align}
we can conclude from \eqref{eq:petz-converge-almost-final-1} and \eqref{eq:petz-converge-almost-final} that
\begin{align} 
\lim_{a \to \infty} \lim_{b \to \infty} \left \langle \cP^{\dagger}_{\sigma^a,\cN^{a,b}}(a_2),a_1 \right \rangle_{\cN(\sigma)} =
\left \langle \cP^{\dagger}_{\sigma,\cN}(a_2),a_1 \right \rangle_{\cN(\sigma)}. \label{eq:petz-converge-almost-final-2}
\end{align}
From there, we argue that this implies the convergence
$\lim_{a\to \infty} \lim_{b \to \infty} \cP_{\sigma^a,\cN^{a,b}} = \cP_{\sigma,\cN}$,
 in the sense of \eqref{eq:channel-convergence-weak-op}.

Thus, we need to establish \eqref{eq_s1_converge} and
\eqref{eq:petz-converge-almost-final}, and we begin by proving
\eqref{eq_s1_converge}. Consider that%
\begin{equation}
\left \langle a_{2},(\mathcal{N}^{a,b})^{\dag}(a_{1})\right\rangle_{\sigma}=\text{tr}%
\left(a_{2}^{\dag}(\sigma^{a})^{\frac{1}{2}}(\mathcal{N}^{a,b})^{\dag}%
(a_{1})(\sigma^{a})^{\frac{1}{2}}\right)\, .
\end{equation}
Then to prove \eqref{eq_s1_converge}, we need to show that the following limit
holds for all bounded $a_{1}$ and $a_{2}$:%
\begin{equation}
\lim_{a\rightarrow\infty}\lim_{b\rightarrow\infty}\left\vert \text{tr}%
(a_{2}^{\dag}(\sigma^{a})^{\frac{1}{2}}(\mathcal{N}^{a,b})^{\dag}%
(a_{1})(\sigma^{a})^{\frac{1}{2}})-\text{tr}(a_{2}^{\dag}\sigma^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}})\right\vert
=0 \, .\label{eq:first-limit-petz}%
\end{equation}
From the fact that any bounded square operator can be written as a sum of a
real part and an imaginary part, each which of in turn can be written as a sum
of a positive and negative part \cite[Chapter~2]{KR83}, it suffices to prove the limit in
\eqref{eq:first-limit-petz} for positive semi-definite $a_{1}$ and $a_{2}$. So
we take $a_{1}$ and $a_{2}$ positive semi-definite in what follows. Consider
that%
\begin{multline}
\left\vert \text{tr}\left(a_{2}(\sigma^{a})^{\frac{1}{2}}(\mathcal{N}^{a,b})^{\dag
}(a_{1})(\sigma^{a})^{\frac{1}{2}}\right)-\text{tr}\left(a_{2}\sigma^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right)\right\vert
\label{eq:main-ineq-convergence}\\
\leq\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}%
}(\mathcal{N}^{a,b})^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-(\sigma
^{a})^{\frac{1}{2}}\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}\right]
\right)  \right\vert \\
+\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}}%
\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-\sigma^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert \, .
\end{multline}
We begin with the first term on the right of \eqref{eq:main-ineq-convergence}:%
\begin{align}
&  \left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}%
}(\mathcal{N}^{a,b})^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-(\sigma
^{a})^{\frac{1}{2}}\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}\right]
\right)  \right\vert \nonumber\\
& \hspace{40mm}  =\left\vert \text{tr}\left(  \left(  \mathcal{N}^{a,b}\left[  (\sigma
^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  -\mathcal{N}\left[
(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  \right)
a_{1}\right)  \right\vert \\
&  \hspace{40mm} \leq\left\Vert \mathcal{N}^{a,b}\left[  (\sigma^{a})^{\frac{1}{2}}%
a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  -\mathcal{N}\left[  (\sigma
^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  \right\Vert
_{1}\left\Vert a_{1}\right\Vert _{\infty}\, ,
\end{align}
where we applied the H\"older inequality in the last line.
So now we focus on bounding the trace norm above. Consider that $(\sigma
^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}$ is a trace-class operator
because%
\begin{align}
\left\Vert (\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}%
}\right\Vert _{1} &  \leq\left\Vert (\sigma^{a})^{\frac{1}{2}}\right\Vert
_{2}\left\Vert a_{2}\right\Vert _{\infty}\left\Vert (\sigma^{a})^{\frac{1}{2}%
}\right\Vert _{2}\\
&  =\left\Vert \sigma^{a}\right\Vert _{1}\left\Vert a_{2}\right\Vert _{\infty
}\\
&  \leq\left\Vert \sigma\right\Vert _{1}\left\Vert a_{2}\right\Vert _{\infty}\, .
\end{align}
So then%
\begin{align}
&  \left\Vert \mathcal{N}^{a,b}\left[  (\sigma^{a})^{\frac{1}{2}}a_{2}%
(\sigma^{a})^{\frac{1}{2}}\right]  -\mathcal{N}\left[  (\sigma^{a})^{\frac
{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  \right\Vert _{1} \nonumber \\
& \hspace{30mm}  =\left\Vert \Pi_{B}^{b}\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac
{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \Pi_{B}%
^{b}-\mathcal{N}\left[  (\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac
{1}{2}}\right]  \right\Vert _{1}\\
& \hspace{30mm} \leq\left\Vert \Pi_{B}^{b}\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a}%
)^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \Pi_{B}%
^{b}-\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma
^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right\Vert _{1} \nonumber \\
&\hspace{40mm}  +\left\Vert \mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a}%
)^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]
-\mathcal{N}\left[  (\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}%
}\right]  \right\Vert _{1} \, .
\end{align}
We now bound the last two terms. Let%
\begin{equation}
\omega_{B}^{a}:=\frac{\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}%
}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  }{\text{tr}\left(
\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma
^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right)  } \,\, .
\end{equation}
Then the first term is bounded as%
\begin{align}
&  \left\Vert \Pi_{B}^{b}\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac
{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \Pi_{B}%
^{b}-\mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma
^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right\Vert _{1} \nonumber \\
&\hspace{30mm}  =\text{tr}\left(  \mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}%
}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right)  \left\Vert
\Pi_{B}^{b}\omega^{a}\Pi_{B}^{b}-\omega^{a}\right\Vert _{1}\\
&\hspace{30mm}  \leq\text{tr}\left(  \mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac
{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right)  \left[
2\sqrt{1-\text{tr}(\Pi_{B}^{b}\omega^{a})}\right] \, ,
\end{align}
where the last line follows from a standard result (see, e.g., \cite[Lemma~A.2]{SFR15}).
Taking the limit as $b\rightarrow\infty$ and as $a\rightarrow\infty$ then
makes this term arbitrarily small. For the other term, consider that%
\begin{align}
  \left\Vert \mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}}%
a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  -\mathcal{N}\left[
(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma^{a})^{\frac{1}{2}}\right]  \right\Vert
_{1} \nonumber 
&  \leq\left\Vert \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma
^{a})^{\frac{1}{2}}\Pi_{A}^{a}-(\sigma^{a})^{\frac{1}{2}}a_{2}(\sigma
^{a})^{\frac{1}{2}}\right\Vert _{1} \\
&  = 0 ,
\end{align}
where the last equality follows because $\Pi^a_A (\sigma^a)^{\frac{1}{2}} = (\sigma^a)^{\frac{1}{2}}$.

Now we handle the second term on the right in
\eqref{eq:main-ineq-convergence}:%
\begin{multline}
\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}}%
\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-\sigma^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert
\label{eq:main-ineq-convergence-2}\\
\leq\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-(\sigma^{a})^{\frac{1}%
{2}}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert
\\
+\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}-\sigma^{\frac{1}{2}}%
\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert .
\end{multline}
The first term on the right in \eqref{eq:main-ineq-convergence-2}\ is bounded as follows:
\begin{align}
&  \left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}%
}\mathcal{N}^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}}-(\sigma^{a})^{\frac{1}%
{2}}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert
\nonumber\\
& \hspace{40mm} =\left\vert \text{tr}\left(  a_{2}(\sigma^{a})^{\frac{1}{2}}\mathcal{N}%
^{\dag}(a_{1})\left[  (\sigma^{a})^{\frac{1}{2}}-\sigma^{\frac{1}{2}}\right]
\right)  \right\vert \\
&\hspace{40mm}  \leq\left[  \text{tr}\left(  a_{2}^{2}\sigma^{a}\right)  \text{tr}\left(
\left[  \mathcal{N}^{\dag}(a_{1})\right]  ^{2}\left[  (\sigma^{a})^{\frac
{1}{2}}-\sigma^{\frac{1}{2}}\right]  ^{2}\right)  \right]  ^{\frac{1}{2}%
}\\
&\hspace{40mm}  \leq\left[  \left\Vert a_{2}\right\Vert _{\infty}^{2}\left\Vert
a_{1}\right\Vert _{\infty}^{2}\text{tr}\left(  \sigma^{a}\right)
\text{tr}\left(  \left[  (\sigma^{a})^{\frac{1}{2}}-\sigma^{\frac{1}{2}%
}\right]  ^{2}\right)  \right]  ^{\frac{1}{2}}\\
&\hspace{40mm}  \leq\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert
_{\infty}\left\Vert \sigma^{a}\right\Vert _{1}^{\frac{1}{2}}\left\Vert (\sigma
^{a})^{\frac{1}{2}}-\sigma^{\frac{1}{2}}\right\Vert _{2}\\
&\hspace{40mm}  \leq\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert
_{\infty}\left\Vert \sigma\right\Vert _{1}^{\frac{1}{2}}\left[  \left\Vert \sigma
^{a}-\sigma\right\Vert _{1}\right]  ^{\frac{1}{2}}\, .
\end{align}
The first inequality follows from the Cauchy-Schwarz inequality. The last
inequality is a consequence of the Powers-Stormer inequality. By essentially
the same reasoning, we obtain the following bound on the other term on the
right in \eqref{eq:main-ineq-convergence-2}:%
\begin{equation}
\left\vert \text{tr}\left(  a_{2}\left[  (\sigma^{a})^{\frac{1}{2}}%
\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}-\sigma^{\frac{1}{2}}%
\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}}\right]  \right)  \right\vert
\leq\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert _{\infty
}\left\Vert \sigma\right\Vert _{1}^{\frac{1}{2}}\left[  \left\Vert \sigma^{a}%
-\sigma\right\Vert _{1}\right]  ^{\frac{1}{2}} \, .
\end{equation}
Putting everything together, we find that
\begin{multline}
\left\vert \text{tr}(a_{2}^{\dag}(\sigma^{a})^{\frac{1}{2}}(\mathcal{N}%
^{a,b})^{\dag}(a_{1})(\sigma^{a})^{\frac{1}{2}})-\text{tr}(a_{2}^{\dag}%
\sigma^{\frac{1}{2}}\mathcal{N}^{\dag}(a_{1})\sigma^{\frac{1}{2}})\right\vert
\\
\leq\text{tr}\left(  \mathcal{N}\left[  \Pi_{A}^{a}(\sigma^{a})^{\frac{1}{2}%
}a_{2}(\sigma^{a})^{\frac{1}{2}}\Pi_{A}^{a}\right]  \right)  \left[
2\sqrt{1-\text{tr}(\Pi_{B}^{b}\omega^{a})}\right] 
+2\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert _{\infty
}\left\Vert \sigma\right\Vert _{1}^{1/2}\left[  \left\Vert \sigma^{a}%
-\sigma\right\Vert _{1}\right]  ^{\frac{1}{2}} ,
\end{multline}
from which we can conclude \eqref{eq_s1_converge}.

We now prove \eqref{eq:petz-converge-almost-final}, by a reasoning that is
very similar to the above. Consider that%
\begin{align}
&  \left\vert \langle\mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag}%
(a_{2}),a_{1}\rangle_{\mathcal{N}^{a,b}(\sigma^{a})}-\langle\mathcal{P}%
_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag}(a_{2}),a_{1}\rangle_{\mathcal{N}%
(\sigma)}\right\vert \nonumber\\
& \hspace{30mm} =\left\vert \text{tr}\left(  \mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}%
}^{\dag}(a_{2})\left[  \mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}%
a_{1}\mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}-\mathcal{N}(\sigma)^{\frac
{1}{2}}a_{1}\mathcal{N}(\sigma)^{\frac{1}{2}}\right]  \right)  \right\vert
\\
&\hspace{30mm}  \leq\left\vert \text{tr}\left(  \mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}%
}^{\dag}(a_{2})\mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}a_{1}\left[
\mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}-\mathcal{N}(\sigma)^{\frac{1}{2}%
}\right]  \right)  \right\vert \nonumber \\
& \hspace{50mm} +\left\vert \text{tr}\left(  \mathcal{P}_{\sigma
^{a},\mathcal{N}^{a,b}}^{\dag}(a_{2})\left[  \mathcal{N}^{a,b}(\sigma
^{a})^{\frac{1}{2}}-\mathcal{N}(\sigma)^{\frac{1}{2}}\right]  a_{1}%
\mathcal{N}(\sigma)^{\frac{1}{2}}\right)  \right\vert \, .
\end{align}
By the same methods as given above, we find that%
\begin{multline}
\left\vert \text{tr}\left(  \mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag
}(a_{2})\mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}a_{1}\left[  \mathcal{N}%
^{a,b}(\sigma^{a})^{\frac{1}{2}}-\mathcal{N}(\sigma)^{\frac{1}{2}}\right]
\right)  \right\vert \\
\leq\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert _{\infty
}\left\Vert \sigma\right\Vert _{1}^{\frac{1}{2}}\left[  \left\Vert \mathcal{N}^{a,b}(\sigma^{a})-\mathcal{N}(\sigma
)\right\Vert _{1}\right]  ^{\frac{1}{2}} \, ,
\end{multline}
and%
\begin{multline}
\left\vert \text{tr}\left(  \mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag
}(a_{2})\left[  \mathcal{N}^{a,b}(\sigma^{a})^{\frac{1}{2}}-\mathcal{N}%
(\sigma)^{\frac{1}{2}}\right]  a_{1}\mathcal{N}(\sigma)^{\frac{1}{2}}\right)
\right\vert \\
\leq\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert _{\infty
}\left\Vert \sigma\right\Vert _{1}^{\frac{1}{2}}\left[  \left\Vert \mathcal{N}^{a,b}(\sigma^{a})-\mathcal{N}(\sigma
)\right\Vert _{1}\right]  ^{\frac{1}{2}}\, .
\end{multline}
Finally, consider that%
\begin{align}
\left\Vert \mathcal{N}^{a,b}(\sigma^{a})-\mathcal{N}(\sigma)\right\Vert _{1}
&  \leq\left\Vert \mathcal{N}^{a,b}(\sigma^{a})-\mathcal{N}^{a,b}%
(\sigma)\right\Vert _{1}+\left\Vert \mathcal{N}^{a,b}(\sigma)-\mathcal{N}%
(\sigma)\right\Vert _{1}\\
&  \leq\left\Vert \sigma^{a}-\sigma\right\Vert _{1}+\left\Vert \Pi
^{b}\mathcal{N}(\Pi^{a}\sigma\Pi^{a})\Pi^{b}-\mathcal{N}(\Pi^{a}\sigma\Pi
^{a})\right\Vert _{1}+\left\Vert \mathcal{N}(\Pi^{a}\sigma\Pi^{a}%
)-\mathcal{N}(\sigma)\right\Vert _{1}\\
&  \leq\left\Vert \sigma^{a}-\sigma\right\Vert _{1}+\left\Vert \Pi
^{b}\mathcal{N}(\Pi^{a}\sigma\Pi^{a})\Pi^{b}-\mathcal{N}(\Pi^{a}\sigma\Pi
^{a})\right\Vert _{1}+\left\Vert \Pi^{a}\sigma\Pi^{a}-\sigma\right\Vert _{1},
\end{align}
which converges also in the limit as $b\rightarrow\infty$ and as
$a\rightarrow\infty$. Putting
everything together, we find that%
\begin{multline}
\left\vert \langle\mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag}%
(a_{2}),a_{1}\rangle_{\mathcal{N}^{a,b}(\sigma^{a})}-\langle\mathcal{P}%
_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag}(a_{2}),a_{1}\rangle_{\mathcal{N}%
(\sigma)}\right\vert \\
\leq2\left\Vert a_{2}\right\Vert _{\infty}\left\Vert a_{1}\right\Vert
_{\infty}\left\Vert \sigma\right\Vert _{1}^{\frac{1}{2}}\left[  2 \left\Vert \sigma^{a}-\sigma\right\Vert _{1}+\left\Vert
\Pi^{b}\mathcal{N}(\Pi^{a}\sigma\Pi^{a})\Pi^{b}-\mathcal{N}(\Pi^{a}\sigma
\Pi^{a})\right\Vert _{1}\right]  ^{\frac{1}{2}}\, ,
\end{multline}
which allows us to conclude \eqref{eq:petz-converge-almost-final}, and in turn, \eqref{eq:petz-converge-almost-final-2}.

It remains to argue for the convergence $\lim_{a\rightarrow\infty}%
\lim_{b\rightarrow\infty}\mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}%
}=\mathcal{P}_{\sigma,\mathcal{N}}$, in the sense of \eqref{eq:channel-convergence-weak-op}. Consider that
\eqref{eq:petz-converge-almost-final-2} implies that%
\begin{equation}
\lim_{a\rightarrow\infty}\lim_{b\rightarrow\infty}\langle\phi^{\prime
}|\mathcal{N}(\sigma)^{\frac{1}{2}}\mathcal{P}_{\sigma^{a},\mathcal{N}^{a,b}%
}^{\dag}(|\phi\rangle\langle\psi|)\mathcal{N}(\sigma)^{\frac{1}{2}}%
|\psi^{\prime}\rangle=\langle\phi^{\prime}|\mathcal{N}(\sigma)^{\frac{1}{2}%
}\mathcal{P}_{\sigma,\mathcal{N}}^{\dag}(|\phi\rangle\langle\psi
|)\mathcal{N}(\sigma)^{\frac{1}{2}}|\psi^{\prime}\rangle \, ,
\end{equation}
for all $|\phi\rangle,|\psi\rangle\in A$ and $|\phi^{\prime}\rangle
,|\psi^{\prime}\rangle\in B$. This establishes convergence of $\mathcal{P}%
_{\sigma^{a},\mathcal{N}^{a,b}}^{\dag}$ to $\mathcal{P}_{\sigma,\mathcal{N}%
}^{\dag}$ for all $|\phi\rangle,|\psi\rangle\in A$ and $|\phi^{\prime}%
\rangle,|\psi^{\prime}\rangle\in\operatorname{supp}(\mathcal{N}(\sigma))$,
which in turn allows us to conclude convergence of $\mathcal{P}_{\sigma
^{a},\mathcal{N}^{a,b}}$ to $\mathcal{P}_{\sigma,\mathcal{N}}$.
\end{proof}


Before stating the following lemma, we introduce the shorthand
\begin{equation}
\cU_{\omega,t} : X \mapsto \omega^{\ci t} X \omega^{-\ci t} \ ,
\end{equation}
where $\omega$ is a positive semi-definite operator.
%%%%%%







\begin{lemma} \label{lem_two}
For any $\sigma \in \Pos(A)$, any $\rho \in \D_{\sigma}(A)$, any $\cN \in \TPCP(A,B)$, and all $t\in \mathbb{R}$, the following limit holds
\begin{align}
\lim_{a \to \infty} \lim_{b \to \infty} F\bigl(\rho^a, (\cR^t_{\sigma^a,\cN^{a,b}} \circ \cN^{a,b})(\rho^a) \bigr) = F\bigl(\rho, (\cR^t_{\sigma,\cN} \circ \cN)(\rho) \bigr)\, .
\end{align}
\end{lemma}
\begin{proof}
%We note that for any $t \in \R$ the mapping $\cR^t_{\sigma^a,\cN^{a,b}}$ is trace non-increasing.
%Thus for any $a,b \in \N$ we find
%\begin{align}
%\norm{\cR^t_{\sigma^a,\cN^{a,b}}\bigl( \cN^{a,b}(\rho^a) \bigr)}_1 \leq \norm{ \cN^{a,b}(\rho^a)}_1 \leq \norm{\rho^a}_1 \leq 1 \ .
%\end{align}
Consider that we have the following convergences:
\begin{align}
\lim_{a \to \infty} \lim_{b \to \infty}
\cU_{\cN^{a,b}(\sigma^a),t}  = \cU_{\cN(\sigma),t}  \qquad \textnormal{and} \qquad
\lim_{a \to \infty} 
\cU_{\sigma^a,t}  = \cU_{\sigma,t} \, ,
\end{align}
each understood to be in the weak sense in \eqref{eq:channel-convergence-weak-op}.
The serial concatenation of weakly converging channels converges to the serial concatenation of the limit, so that $\lim_{a \to \infty} \lim_{b \to \infty} \cR^t_{\sigma^a,\cN^{a,b}} \circ \cN^{a,b} =  \cR^t_{\sigma,\cN} \circ \cN
$, where we used the above and Lemma~\ref{lem_PetzWellDefined}. Then we can conclude that the fidelity converges because  it is continuous in its inputs (see, e.g.,~\cite[Lemma~B.9]{FR14}), and for our case considered here, convergence in the weak sense implies convergence in the trace norm \cite[Chapter~11]{holevo_book}. This proves the assertion.


\end{proof}



By invoking the stronger version of Theorem~\ref{thm_explicit} for finite-dimensional Hilbert spaces (that has been proven in Step~\hyperlink{step_0_1}{1})\footnote{Recall that by Remark~\ref{rmk_subnormalized}, Theorem~\ref{thm_explicit} remains valid for subnormalized states and a trace non-increasing completely positive map.} together with Lemmas~\ref{lem_one} and~\ref{lem_two},
the fact that any separable Hilbert space is isomorphic to $\ell^2$~\cite[Theorem~II.7]{simon_book}, and the dominated convergence theorem,
we find for any $\rho \in \D_{\sigma}(A)$
\begin{align}
0 &\leq \limsup_{a \to \infty} \limsup_{b \to \infty}  \Bigg[ D(\rho^ a \| \sigma^a) - D\bigl(\cN^{a,b}(\rho^a) \| \cN^{a,b}(\sigma^a) \bigr) \nonumber \\
& \hspace{50mm} + 2 \int_{\R} \ \di t
\ \beta_0(t)  \ 
\log F\bigl(\rho^a , (\cR^{\frac{t}{2}}_{\sigma^a,\cN^{a,b}} \circ \cN^{a,b})(\rho^a) \bigr) \Bigg]  \\
&\leq D(\rho \| \sigma) - D\bigl(\cN(\rho) \| \cN(\sigma) \bigr)  + 2 
\int_{\R} \ \di t \ \beta_0(t)  \ 
\log F\bigl(\rho, (\cR_{\sigma,\cN}^{\frac{t}{2}} \circ \cN)(\rho) \bigr) \ .
\end{align}
This establishes the stronger version of Theorem~\ref{thm_explicit} in \eqref{eq_explcitMainResStrong}.
We can then apply concavity of the logarithm and the fidelity to conclude Theorem~\ref{thm_explicit}.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem~\ref{thm_monoRelEnt}} \label{sec_pf_finiteDim}
For this proof we first introduce some notation.
For any $H \in \Pos(A)$ where $\Pos(A)$ denotes the set of non-negative operators on $A$ and where $A$ is a finite-dimensional Hilbert space let $H = \sum_{k\in [d]} \lambda_k P_k$ be an eigenvalue decomposition of $H$, where $d \leq \dim(A)$ and $\{P_k \}_{k \in [d]}$ are mutually orthogonal projectors. For $\phi =(\phi_1,\ldots,\phi_d) \in [0,2\pi]^{\times d}$ we define the unitary
\begin{align} \label{eq_defUnitary}
U_H^{\phi} := \sum_{k \in [d]} \exp(\ci \phi_k) P_k \ .
\end{align}
With the help of these unitaries we define yet another rotated Petz recovery map
\begin{align} \label{eq_rotPetzMapSpecial}
  \cT_{\sigma,\cN}^{\varphi,\vartheta}\ :
   X_B \mapsto  U_{\sigma}^{\vartheta}  \cP_{\sigma, \cN}\bigl(   U_{\mathcal{N}(\sigma)}^{\varphi} X_{B} U_{\mathcal{N}(\sigma)}^{\varphi\dag}  \bigr)    U_{\sigma}^{\vartheta\dag} \ .
\end{align}
 For any $\sigma \in \Pos(A)$ and $\cN \in \TPCP(A,B)$ we denote the convex hull of rotated Petz recovery maps by 
\begin{align} \label{eq_convexHullRotatedPetzSpecial}
\RR_{\sigma,\cN}:=\conv\Bigl(\cT_{\sigma,\cN}^{\varphi,\vartheta} \ : \  \vartheta \in [0,2\pi]^{\times d_1}, \varphi \in[0,2\pi]^{\times d_2} \Bigr) \ ,
\end{align}
where $d_1 \leq \dim(A)$ and $d_2 \leq \dim(B)$.
We note that the rotated Petz recovery map $\cT_{\sigma,\cN}^{\varphi,\vartheta}$ is trace non-increasing and completely positive.
We further note that clearly $\RRR_{\sigma,\cN} \subseteq \RR_{\sigma,\cN}$ with $\RRR_{\sigma,\cN}$ defined in~\eqref{eq_convexHullRotatedPetz}.
%%%%%%%%
\subsection*{\hypertarget{step_1_1}{Step 1:}  Proof for a finite set of states}
For any $\sigma \in \Pos(A)$ consider a finite set of density operators $\{\rho^x\}_{x \in \cX}$ with $\rho^x \in \D_{\sigma}(A)$ for all $x \in \cX$ and $|\cX| < \infty$. Furthermore let $\nu$ be a probability measure on $\cX$. For any channel $\mathcal{N} \in \TPCP(A,B)$ we define the following non-negative operators
\begin{align} \label{eq_setStates}
\rho_{XA}  := \sum_{x \in \cX} \nu(x) \proj{x}_X \otimes \rho^x  \quad \textnormal{and} \quad \sigma_{XA}  := \sum_{x \in \cX} \nu(x) \proj{x}_X \otimes\sigma  = \rho_X \otimes \sigma \ .
\end{align}
This then gives
\begin{align} \label{eq_ssssd}
[(\cI_{X}\otimes\mathcal{N}) (  \sigma_{XA} )]^{-\frac{1}{2}}  =\bigl[\bigl(\rho_{X}\otimes\mathcal{N}(\sigma)\bigr)\bigr]^{-\frac{1}{2}}=\rho_{X}^{-\frac{1}{2}}\otimes\mathcal{N}(\sigma)^{-\frac{1}{2}} \ .
\end{align}
Since $(\cI_{X}\otimes\mathcal{N})^{\dag} =\cI_{X}\otimes\mathcal{N}^{\dag}$ and $\sigma_{XA}^{\frac{1}{2}}=(\rho_{X}\otimes\sigma)^{\frac{1}{2}} =\rho_{X}^{\frac{1}{2}}\otimes\sigma^{\frac{1}{2}}$, the Petz recovery map for $\sigma_{XA}$ and $\cI_X \otimes \cN$ can be written as
\begin{align} \label{eq_PetzDecay}
Y_{XB}     \mapsto \cT_{\sigma_{XA},\cI_{X}\otimes\mathcal{N}}\left( Y_{XB}\right)  
&=\sigma_{XA}^{\frac{1}{2}}  (\cI_{X}\otimes\mathcal{N})^{\dag}\left(  [(\cI_{X}\otimes\mathcal{N})  ( \sigma_{XA})]^{-\frac{1}{2}} \, Y_{XB} \,  [( \cI_{X} \otimes\mathcal{N}) ( \sigma_{XA})]^{-\frac{1}{2}}\right)    \sigma_{XA}^{\frac{1}{2}} \nonumber \\
& =( \rho_{X}^{\frac{1}{2}}\otimes\sigma^{\frac{1}{2}})  (\cI_{X}\otimes\mathcal{N}^{\dag})  \bigl(  (\rho_{X}^{-\frac{1}{2}}\otimes\mathcal{N} (\sigma)^{-\frac{1}{2}}) Y_{XB}  (  \rho_{X}^{-\frac{1}{2}}\otimes\mathcal{N}( \sigma)^{-\frac{1}{2}})  \bigr)(\rho_{X}^{\frac{1}{2}}\otimes\sigma^{\frac{1}{2}}) \nonumber \\
& =(\Pi_{\supp(\nu)}\otimes\cT_{\sigma,\mathcal{N}})(Y_{XB}) \ ,
\end{align}
where $\Pi_{\supp(\nu)}$ denotes a projector onto the support of $\nu$. 
Furthermore, we have that for any $\vartheta \in [0,2\pi]^{\times d_1}$ and for any $\varphi \in [0,2\pi]^{\times d_2}$ there exist $\tilde \vartheta, \bar \vartheta \in  [0,2\pi]^{\times d_1}$ and $\tilde \varphi, \bar \varphi \in  [0,2\pi]^{\times d_2}$ such that
\begin{align} \label{eq_unitaryDec}
U_{\sigma_{XA}}^{\vartheta} = U_{\rho_X}^{\tilde \vartheta} \otimes U_{\sigma}^{\bar \vartheta } \quad \textnormal{and} \quad
U_{(\Pi_{\supp(\nu)} \otimes \cN)(\sigma_{XA})}^{\varphi} = U_{\rho_X}^{\tilde \varphi} \otimes U_{\cN(\sigma)}^{\bar \varphi} \ .
\end{align}
Theorem~3.3 of~\cite{STH15} ensures that
\begin{multline} \label{eq_STHH}
D(\rho_{XA}\dd \sigma_{XA} ) - D \bigl( (\Pi_{\supp(\nu)}\otimes\mathcal{N} )  (\rho_{XA})  \dd (\Pi_{\supp(\nu)}\otimes\mathcal{N})(\sigma_{XA})  \bigr)\\
-  \min \bigl\{ \MD\bigl(\rho_{XA}\big\| (\cT_{\sigma_{XA},\cI_X\otimes\cN} \circ \Pi_{\supp(\nu)}\otimes\mathcal{N})  (\rho_{XA} ) \bigr) : \cT_{\sigma_{XA},\Pi_{\supp(\nu)} \otimes \cN} \in \RR_{\sigma_{XA},\Pi_{\supp(\nu)} \otimes \cN} \bigr \} \geq 0 \ .
\end{multline}
Combining~\eqref{eq_setStates},~\eqref{eq_ssssd},~\eqref{eq_PetzDecay}, and~\eqref{eq_unitaryDec} shows that for any $\vartheta \in [0,2\pi]^{\times d_1}$ and for any $\varphi \in [0,2\pi]^{\times d_2}$ there exist $\bar \vartheta \in [0,2\pi]^{\times d_1}$ and $\bar \varphi \in [0,2\pi]^{\times d_2}$ such that
\begin{align}
\cT^{\varphi,\vartheta}_{\sigma_{XA},\Pi_{\supp(\nu)} \otimes \cN}(\rho_{XA}) = \sum_{x \in \cX} \nu(x) \proj{x}_X \otimes \cT^{\bar \varphi,\bar \vartheta}_{\sigma, \cN}(\rho^x) \ .
\end{align}
This, together with a simple property of the relative and the measured relative entropy ensuring that they decompose for orthogonal states (see Lemmas~B.2 and B.3 in~\cite{SFR15}), implies that~\eqref{eq_STHH} can be rewritten as
\begin{align} \label{eq_midddstep}
 \max_{\cT_{\sigma, \cN} \in \RR_{\sigma,\cN}}  \sum_{x \in \cX} \nu(x) \Big(  D(\rho^{x}\dd \sigma) - D\bigl(\mathcal{N}(\rho^{x})\big\| \mathcal{N} (\sigma)\bigr)    - \MD\bigl(\rho^x\big\| (\cT_{\sigma,\cN} \circ\cN)(\rho^x) \bigr)  \Bigr) \geq 0 \ .
\end{align}
We note that the set $\RR_{\sigma,\cN}$ (defined in Equation~\eqref{eq_convexHullRotatedPetz}) is compact since $\TPCP(B,A)$ is compact (see~\cite[Remark~C.3]{SFR15}) and since the intersection of a compact set with a closed set is compact. Furthermore, the mapping $\RR_{\sigma,\cN} \ni \cT_{\sigma,\cN} \mapsto \MD(\rho^{x}\|(\cT_{\sigma,\cN}\circ \mathcal{N})(\rho^{x})) \in \R_+$ is lower semicontinuous (see proof of Lemma~\ref{lem_DeltaContInR}). By the extreme value theorem this implies that the maximum in~\eqref{eq_midddstep} is attained.

Since~\eqref{eq_midddstep} is valid for any probability measure $\nu$ on $\cX$ we obtain
\begin{align} \label{eq_minmaxFirst}
\min_{\nu \in \PPP(\cX)}  \max_{\cT_{\sigma, \cN} \in \RR_{\sigma,\cN}}  \sum_{x \in \cX} \nu(x) \Big(  D(\rho^{x}\dd \sigma) - D\bigl(\mathcal{N}(\rho^{x})\big\| \mathcal{N} (\sigma)\bigr)    - \MD\bigl(\rho^x\big\| (\cT_{\sigma,\cN} \circ\cN)(\rho^x) \bigr)  \Bigr) \geq 0 \ ,
\end{align}
where $\PPP(\cX)$ denotes the set of probability measures on $\cX$.
We note that the minimum is attained. To see this, we first remark that the set $\cX$ is clearly compact and as a result, the set $\PPP(\cX)$ is weak* compact~\cite[Theorem~15.11]{aliprantis07}. Furthermore, the function $\PPP(\cX) \ni \mu \mapsto  \sum_{x \in \cX} \nu(x) [ D(\rho^{x}\dd \sigma) - D(\mathcal{N}(\rho^{x})\| \mathcal{N} (\sigma))    - \MD(\rho^x \| (\cT_{\sigma,\cN} \circ\cN)(\rho^x)) ] \in  \R$ is continuous with respect to the weak* topology, since expectation values are continuous in the weak* topology. The extreme value theorem thus ensures that the minimum is attained. 

To simplify notation, let 
\begin{align}
f(\nu,\cT_{\sigma,\cN}):= \sum_{x \in \cX} \nu(x) \Big(  D(\rho^{x}\dd \sigma) - D\bigl(\mathcal{N}(\rho^{x})\big\| \mathcal{N} (\sigma)\bigr)    - \MD\bigl(\rho^x\big\| (\cT_{\sigma,\cN} \circ\cN)(\rho^x) \bigr)  \Bigr) \ .
\end{align}
For any $\cT_{\sigma,\cN} \in \RR_{\sigma,\cN}$, the function $\PPP(\cX) \ni \nu \mapsto f(\nu,\cT_{\sigma,\cN}) \in \R$ is linear and bounded (since $\rho^x \in \D_{\sigma}(A)$ for all $x \in \cX$) and hence continuous. Furthermore, for any $\nu \in \PPP(\cX)$ the function $\RR_{\sigma,\cN} \ni \cT_{\sigma,\cN} \mapsto f(\nu,\cT_{\sigma,\cN})\in \R$ is concave~\cite[Lemma~B.4]{SFR15} and upper semicontinuous (as shown in the proof of Lemma~\ref{lem_DeltaContInR}). The sets $\PPP(\cX)$ and $\RR_{\sigma,\cN}$ are both convex and as discussed above $\RR_{\sigma,\cN}$ is compact. As a result, Sion's minimax theorem~\cite[Corollary~3.3]{Sion58} implies that
\begin{equation}
\max_{\cT_{\sigma, \cN} \in \RR_{\sigma,\cN}} \min_{\nu \in \PPP(\cX)}    \sum_{x \in \cX} \nu(x) \Big(  D(\rho^{x}\dd \sigma) - D\bigl(\mathcal{N}(\rho^{x})\big\| \mathcal{N} (\sigma)\bigr)    - \MD\bigl(\rho^x\big\| (\cT_{\sigma,\cN} \circ\cN)(\rho^x) \bigr)  \Bigr)\geq 0 \ .
\end{equation}
This shows that there exists a recovery map $\cT_{\sigma,\cN} \in \RR_{\sigma,\cN}$ that satisfies
\begin{align}
D(\rho^{x}\dd \sigma) - D\bigl(\mathcal{N} (\rho^{x})\dd \mathcal{N}(\sigma)\bigr)  \geq \MD\bigl(\rho^x\big\| (\cT_{\sigma,\cN} \circ\cN)(\rho^x) \bigr) \quad \textnormal{for all}\quad x \in \cX\ .
\end{align}
Since $\RRR_{\sigma,\cN} \subseteq \RR_{\sigma,\cN}$, this proves the statement of Theorem~\ref{thm_monoRelEnt} for a finite set of states $\{\rho^x\}_{x \in \cX}$.


%%%%%%%%
\subsection*{\hypertarget{step_1_2}{Step 2:}  Extensions to an infinite set of states}
We now show how to extend the statement proven in Step~\hyperlink{step_1_1}{1} for a finite set of states to the set of all states.
Let $\cS := \D(A)$ denote the set of density operators on $A$. Furthermore, for $\cT \in \TPCP(B,A)$, $\sigma \in \Pos(A)$, and $\cN \in \TPCP(A,B)$ we define the function family
\begin{align} \label{eq_DeltaFooDef}
\Delta_{\cT,\sigma,\cN} \ : \ \ \ &\D_{\sigma}(A) \to \R \cup \{-\infty \}  \nonumber \\
&\rho \mapsto D(\rho \dd \sigma) - D\bigl(\cN(\rho) \dd \cN(\sigma) \bigr) - \MD\bigl(\rho\big\| (\cT \circ\cN)(\rho) \bigr) \ .
\end{align}

We start by proving two regularity properties of the $\Delta$-function family defined above.
\begin{lemma} \label{lem_DeltaContInRho}
For any $\cT \in \TPCP(B,A)$, any $\sigma \in \Pos(A)$, and any $\cN \in \TPCP(A,B)$, the function $\D_{\sigma}(A) \ni \rho \mapsto \Delta_{\cT,\sigma,\cN}(\rho) \in \R$ defined in~\eqref{eq_DeltaFooDef} is upper semicontinuous.
\end{lemma}
\begin{proof}
For any fixed $\sigma \in \Pos(A)$, the function $\D_{\sigma}(A) \ni \rho \mapsto D(\rho \dd \sigma) = \tr(\rho(\log \rho - \log \sigma) = - H(\rho) - \tr(\rho \log \sigma)$ is continuous. This follows by Fannes' inequality~\cite{fannes73} that shows that $\D(A) \ni \rho \mapsto H(\rho) \in \R$ is continuous. Furthermore the function $\D_{\sigma}(A) \ni \rho \mapsto - \tr(\rho \log \sigma) \in \R$ is bounded and linear and thus continuous. Since $\D(A) \ni \rho \mapsto \cN(\rho)$ is linear and bounded it is continuous and thus the function $\D_{\sigma}(A) \ni \rho \mapsto D(\cN(\rho) \dd \cN(\sigma))$ is also continuous. In addition, the function $\D_{\sigma}(A) \ni \rho \mapsto \MD (\rho\big\| (\cT \circ\cN)(\rho))$ is lower semicontinuous~\cite[Lemma~3.2]{SFR15} which then proves the assertion.
\end{proof}

\begin{lemma} \label{lem_DeltaContInR}
For any $\sigma \in \D(A)$, any $\rho \in \D_{\sigma}(A)$, and any $\cN \in \TPCP(A,B)$, the function $\TPCP(B,A) \ni\cT \mapsto \Delta_{\cT,\sigma,\cN}(\rho) \in \R$ defined in~\eqref{eq_DeltaFooDef} is upper semicontinuous. 
\end{lemma}
\begin{proof}
The assertion follows from the fact that the mapping $\RR_{\sigma,\cN} \ni \cT_{\sigma,\cN} \mapsto {\MD(\rho\|(\cT_{\sigma,\cN}\circ \mathcal{N})(\rho))} \in \R_+$ is lower semicontinuous. To see this, we first note that by definition we have ${\MD(\rho \| (\cT_{\sigma,\cN} \circ \cN)(\rho))} = \sup_{\cM \in \mathbb{M}}{D(\cM(\rho)\| \cM((\cT_{\sigma,\cN} \circ \cN)(\rho)))} $. Since $\cT_{\sigma,\cN}$, $\cN$, and $\cM$ are linear bounded operators and hence continuous we find that $\cT_{\sigma,\cN} \mapsto D(\cM(\rho^x)\| \cM((\cT_{\sigma,\cN} \circ \cN)(\rho^x)))$ is continuous as the logarithm $\R^+ \ni x \mapsto \log x \in \R$ is continuous. As the supremum of continuous functions is lower semicontinuous~\cite[Chapter~IV, Section~6.2, Theorem~4]{bourbaki_book}, the assertion follows.
\end{proof}




Since $\cS$ is compact (see, e.g.,~\cite[Lemma~C.1]{SFR15}), it follows that for any $\eps >0$ there exists a finite set $\cS^\eps$ of density operators on $A$ such that any $\rho \in \cS$ is $\eps$-close to an element of $\cS^\eps$. We assume without loss of generality that $\cS^{\eps'} \subset \cS^\eps$ for $\eps' \geq \eps$. Let $\cT^\eps \in \RR_{\sigma,\cN}$ be such that $\inf_{\rho \in \cS^\eps} \Delta_{\cR^\eps,\sigma,\cN}(\rho) \geq 0$ whose existence has been established in Step~\hyperlink{step_1_1}{1}. The set
$\RR_{\sigma,\cN}$ is compact because
$\TPCP(B,A)$ is compact~\cite[Remark~C.3]{SFR15} and the intersection of a compact set with a closed set is compact.
Since $\RR_{\sigma,\cN}$ is compact, there exists a decreasing sequence $\{ \eps_n�\}_{n \in \N}$ and $\tilde \cT \in \RR_{\sigma,\cN}$ such that
\begin{align}
\lim_{n \to \infty} \eps_n = 0 \quad \textnormal{and} \quad \tilde \cT = \lim_{n \to \infty} \cT^{\eps_n} \ .
\end{align}
Thus for all $n \in \N$ we find by Lemma~\ref{lem_DeltaContInR}
\begin{multline}
\inf_{\rho \in \cS^{\eps_n}} \Delta_{\tilde \cT,\sigma,\cN}(\rho) 
\geq \inf_{\rho \in \cS^{\eps_n}} \limsup_{m \to \infty} \Delta_{\cT^{\eps_m},\sigma,\cN}(\rho) 
\geq \limsup_{m \to \infty} \inf_{\rho \in \cS^{\eps_n}} \Delta_{\cT^{\eps_m},\sigma,\cN}(\rho)\\
\geq \limsup_{m \to \infty} \inf_{\rho \in \cS^{\eps_m}} \Delta_{\cT^{\eps_m},\sigma,\cN}(\rho)
\geq 0 \ , \label{eq_midstep}
\end{multline}
where the final step follows by Step~~\hyperlink{step_1_1}{1}. For any $\rho \in \cS$ let $\rho_n \in \cS^{\eps_n}$ be such that $\lim_{n \to \infty} \rho_n = \rho$. (By the definition of $\cS^{\eps_n}$, it follows that such a sequence $\{ \rho_n \}_{n \in \N}$ with $\rho_n \in \cS^{\eps_n}$ always exists.) By Lemma~\ref{lem_DeltaContInRho} and~\eqref{eq_midstep} we find for any $\rho \in \cS$
\begin{align}
\Delta_{\tilde \cT, \sigma,\cN}(\rho) 
= \Delta_{\tilde \cT,\sigma,\cN}(\lim_{n \to \infty} \rho_n) 
\geq \limsup_{n \to \infty} \Delta_{\tilde \cT, \sigma,\cN}(\rho_n) 
\geq  \limsup_{n \to \infty}\inf_{\rho \in \cS^{\eps_n}} \Delta_{\tilde \cT, \sigma,\cN}(\rho) 
 \geq 0 \ .
\end{align}
Since $\RRR_{\sigma,\cN} \subseteq \RR_{\sigma,\cN}$, this proves the first inequality in~\eqref{eq_strictMonoBound}. 

The second inequality in~\eqref{eq_strictMonoBound} is a consequence of the monotonicity of the R\'enyi divergence in the order parameter~\cite{csiszar95} and of the fact that for any two states there exists an optimal measurement that does not increase their fidelity~\cite[Section~3.3]{Fuc96}.
This thus concludes the proof of Theorem~\ref{thm_monoRelEnt} for finite-dimensional Hilbert spaces.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Universal refinements of other entropy inequalities}
It is well known (see e.g.~\cite{ruskai02}) that the monotonicity of the relative entropy under trace-preserving completely positive maps is closely related to (i) strong subadditivity, (ii) concavity of the conditional entropy, and (iii) joint convexity of the relative entropy. Based on this relation, Theorem~\ref{thm_monoRelEnt} can be used to derive universal remainder terms for the statements (i)-(iii). We note that universal means that the recovery map that occurs in the remainder term does not depend on all possible parameters. Within this section we assume that $A$, $B$, and $C$ are finite-dimensional Hilbert spaces. For $n \in \N$, the $n$-simplex is denoted by $\Delta_{n}:=\{  x\in\R^{n} : \ x\geq 0, \ \sum_{i=1}^{n} x_{i}=1\}$. The following three corollaries consist of two statements each. Statement 1 is derived from Theorem~\ref{thm_explicit} which has the advantage that the remainder term is explicit (and hence also universal). However, since the lower bound of Theorem~\ref{thm_explicit} is in form of a logarithm of a fidelity we cannot fully simplify the remainder terms (e.g., the convex sum appears inside the logarithm). This is circumvented in Statement 2 (which is derived from Theorem~\ref{thm_monoRelEnt})\footnote{Recall that the lower bound in Theorem~\ref{thm_monoRelEnt} is in the form of a measured relative entropy.%
%--- a quantity that has several nice properties.
} at the cost that the remainder term is not explicit, but still universal.

Applying Theorems~\ref{thm_explicit} and \ref{thm_monoRelEnt} for $\sigma=\id_A \otimes \rho_{BC} \in \Pos(A\otimes B \otimes C)$, $\rho=\rho_{ABC} \in \D_{\sigma}(A\otimes B \otimes C)$, and $\cN_{ABC \to AB}=\cI_{AB}\otimes\tr_C$ implies a strengthened version of the result that has been established in~\cite[Theorem~2.1, Corollary~2.4, and Remark~2.5]{SFR15}.
\begin{corollary}[Strong subadditivity] \label{cor_SSA}\
\begin{enumerate}
  \item For any density operator $\rho_{ABC} \in \D(A\otimes B \otimes C)$ we have
  \begin{align}
  I(A:C|B)_{\rho} \geq -2 \log F\bigl(\rho_{A B C}, \cR_{B \to BC}(\rho_{A B})\bigr) \ ,
  \end{align}
where $\cR_{B \to BC}(\cdot)= \int_{\R} \di t \beta_0(t) \cR^t_{\rho_{BC},\tr_C}(\cdot)$ as defined in Theorem~\ref{thm_explicit}.
\item  For any density operator $\rho_{B C} \in \D(B\otimes C)$ there exists a recovery map $\cR_{B \to B C}$ inside the set $\RRR_{\rho_{BC},\tr_C}$ defined in~\eqref{eq_convexHullRotatedPetz}, such that for any extension $\rho_{A B C}$ on $A \otimes B \otimes C$ we have
  \begin{align} \label{eq_main}
    I(A:C|B)_{\rho} \geq  \MD\left(\rho_{ABC} \big\| \cR_{B \to BC}(\rho_{AB})\right) \geq -2 \log F\bigl(\rho_{A B C}, \cR_{B \to BC}(\rho_{A B})\bigr) \ .
  \end{align}
\end{enumerate}
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 From Corollary~\ref{cor_SSA} we can deduce a universal remainder term for the concavity of the conditional entropy. We note that a similar remainder term has been conjectured in~\cite{BLW14}. Furthermore, a remainder term that however is neither universal nor explicit has been proven in~\cite[Corollary~12]{Wilde15}.

\begin{corollary}[Concavity of conditional entropy]\label{cor_concCondEnt} \
\begin{enumerate}
\item For any $\rho_{AB} \in \D(A\otimes B)$ we have
\begin{align}
H(A|B)_{\rho} - \sum_{x \in \cX} \nu(x) H(A|B)_{\rho^x}  
\geq - 2 \log \sum_{x\in \cX} \nu(x)  F\bigl(\rho_{AB}^x,\cR_{B \to AB}(\rho_B^x)\bigr)  \ ,
\end{align}
for all ensembles $\{\nu(x),\rho_{AB}^x \}_{x\in \cX}$ with $\nu \in \Delta_{\cX}$ and $\rho_{AB}^x \in \D(A \otimes B)$ such that $\rho_{AB}=\sum_{x \in \cX}\nu(x) \rho_{AB}^x$ where $\cR_{B \to AB}(\cdot)=\int_{\R} \di t \beta_0(t) \cR^{t}_{\rho_{AB},\tr_A}(\cdot)$ as defined in Theorem~\ref{thm_explicit}.

\item For any $\rho_{AB} \in \D(A\otimes B)$ there exists recovery map $\cR_{B \to AB} \in \RRR_{\rho_{AB},\tr_A}$ with $\RRR_{\rho_{AB},\tr_A}$ defined in~\eqref{eq_convexHullRotatedPetz}, such that
\begin{multline} \label{eq_concavityCEnt}
H(A|B)_{\rho} - \sum_{x \in \cX} \nu(x) H(A|B)_{\rho^x} 
\geq \sum_{x \in \cX} \nu(x) \MD\bigl(\rho_{AB}^x\big\|\cR_{B \to AB}(\rho_B^x)\bigr) \\
\geq - 2\sum_{x \in \cX} \nu(x) \log F\bigl(\rho_{AB}^x,\cR_{B \to AB}(\rho_B^x)\bigr) \ ,
\end{multline}
for all ensembles $\{\nu(x),\rho_{AB}^x \}_{x\in \cX}$ with $\nu \in \Delta_{\cX}$ and $\rho_{AB}^x \in \D(A \otimes B)$ such that $\rho_{AB}=\sum_{x \in \cX}\nu(x) \rho_{AB}^x$.
\end{enumerate}
\end{corollary}
\begin{proof}
We will provide a proof for Statement 2. Statement 1 follows by exactly the same reasoning.
Let us consider the following classical-quantum state
\begin{align}
\phi_{XAB} = \sum_{x \in \cX} \nu(x) \proj{x}_X \otimes \rho_{AB}^x \ .
\end{align}
By Corollary~\ref{cor_SSA} and a simple fact ensuring that the measured relative entropy for orthogonal input states decomposes (see Lemma~B.3 in~\cite{SFR15}), we find
\begin{multline}
H(A|B)_{\rho} - \sum_{x \in \cX} \nu(x) H(A|B)_{\rho^x}  = H(A|B)_{\phi} - H(A|BX)_{\phi} = I(X:A|B)_{\phi}  \\
 \geq \MD\bigl(\phi_{XAB} \| \cR_{B \to AB}(\phi_{XB}) \bigr) = \sum_{x \in \cX} \nu(x) \MD\bigl(\rho_{AB}^x \big| \cR_{B \to AB}(\rho_B^x) \bigr) \ .
\end{multline}
We note that by Corollary~\ref{cor_SSA} the recovery map $\cR_{B \to AB}$ is universal in the sense that it only depends on $\rho_{AB}$. 
The second inequality in~\eqref{eq_concavityCEnt} is a consequence of the monotonicity of the R\'enyi divergence in the order parameter~\cite{csiszar95} and of the fact that for any two states there exists an optimal measurement that does not increase their fidelity~\cite[Section~3.3]{Fuc96}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Theorems~\ref{thm_explicit} and \ref{thm_monoRelEnt} imply a universal remainder term for the joint convexity of the relative entropy. We note that a similar remainder term has been conjectured in~\cite{SBW14}. Corollary~13 in~\cite{Wilde15} proves such a remainder term that however is neither universal nor explicit.
\begin{corollary}[Joint convexity of relative entropy] \label{cor_jointConvRelEnt} \
\begin{enumerate}
\item For any $\nu \in \Delta_{\cX}$, $\{ \sigma^x \}_{x \in \cX}$ with $\sigma^x \in \Pos(A)$ for all $x \in \cX$, $\sigma_{XA}=\sum_{x \in \cX} \nu(x) {\proj{x}_X \otimes \sigma^x}$, any $\rho_x \in \D_{\sigma^x}(A)$ and $\rho=\sum_{x \in \cX} \nu(x) \rho^x$ we have
\begin{align}
\sum_{x \in \cX} \nu(x) D(\rho^x \dd \sigma^x) - D(\rho \dd \sigma) 
\geq -2 \log \sum_{x\in \cS} \nu(x) F\bigl(\rho^x, \cR_{\sigma_{XA},\tr_X}(\rho) \bigr)  \ ,
\end{align}
where $\cR_{\sigma_{XA},\tr_X}(\cdot) = \int_{\R} \di t \beta_0(t) \cR^t_{\sigma_{XA},\tr_X}(\cdot)$ as defined in Theorem~\ref{thm_explicit}.

\item For any $\nu \in \Delta_{\cX}$, $\{ \sigma^x \}_{x \in \cX}$ with $\sigma^x \in \Pos(A)$ for all $x \in \cX$, and $\sigma_{XA}=\sum_{x \in \cX} \nu(x) {\proj{x}_X \otimes \sigma^x}$ there exists a recovery map $\cR \in \RRR_{\sigma_{XA},\tr_X}$ such that
\begin{align} \label{eq_convexD}
\sum_{x \in \cX} \nu(x) D(\rho^x \dd \sigma^x) - D(\rho \dd \sigma) 
\geq \sum_{x \in \cX} \nu(x) \MD\bigl( \rho^x \| \cR(\rho)\bigr)
\geq -2 \sum_{x \in \cX} \nu(x) \log F\bigl(\rho^x, \cR(\rho) \bigr)
\end{align}
for all $\{ \rho^x \}_{x \in \cX}$ with $\rho_x \in \D_{\sigma^x}(A)$ and $\rho=\sum_{x \in \cX} \nu(x) \rho^x$.
\end{enumerate}
\end{corollary}
\begin{proof}
We will provide a proof for Statement 2. Statement 1 follows by exactly the same reasoning.
Let us consider the states
\begin{align}
 \rho_{XA}:= \sum_{x \in \cX} \nu(x) \proj{x}_X \otimes \rho^x \quad \textnormal{and} \quad \rho = \rho_A = \sum_{x \in \cX} \nu(x) \rho^x  \ .
\end{align}
Since the relative entropy decomposes for orthogonal input states (see Lemma~B.2 in~\cite{SFR15}), Theorem~\ref{thm_monoRelEnt} implies that there exists a recovery map $\cR \in \RRR_{\sigma_{XA},\tr_X}$ such that
\begin{multline}
\sum_{x \in \cX} \nu(x) D(\rho^x \dd \sigma^x) - D(\rho \dd \sigma)  = D(\rho_{XA} \dd \sigma_{XA}) - D(\rho_A \dd \sigma_A) \\
 \geq \MD\bigl(\rho_{XA} \| \cR(\rho) \bigr)
 =\sum_{x \in \cX} \nu(x) \MD\bigl(\rho^x \| \cR(\rho) \bigr) \ .
\end{multline}
The second inequality in~\eqref{eq_convexD} is a consequence of the monotonicity of quantum R\'enyi divergence in the order parameter~\cite{MLDSFT13} and of the fact that for any two states there exists an optimal measurement that does not increase their fidelity~\cite[Section~3.3]{Fuc96}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Approximate quantum error correction}
Theorem~\ref{thm_explicit} allows for establishing an alternate, information-theoretic
set of conditions for approximate quantum error correction. Recall that in
quantum error correction the main goal is to protect a set of states from the
action of a noisy quantum channel, by encoding this set into a subspace of a
given Hilbert space. The quantum error correction conditions given in the seminal works \cite{BDSW96,KL97} are necessary and
sufficient for perfect quantum error correction. Although
these works were essential for gaining an understanding of quantum error
correction, the conditions given represent a strong idealization, since we can never hope in
practice to have perfect quantum error correction. So this realization
motivated some follow-up works on approximate quantum error correction (see
\cite{BK02,Mandayam2012} and references therein), in which one seeks to
determine conditions under which approximate recovery from the action of a
given noisy channel is possible. We mention that some of these works made
essential use of the Petz recovery map.

In \cite{HMPB11}, information-theoretic conditions for perfect error
correction were given in terms of the quantum relative entropy. We recall
their result:\ Let $A$ be a Hilbert space, let $C$ be a subspace of $A$
(called the \textquotedblleft codespace\textquotedblright), and let $\Pi$
denote the projection onto $C$. Then%
\begin{equation}
\{\forall\, \rho\in \D(C)\ \textnormal{ : } \ D(\rho\Vert\Pi)=D(\mathcal{N}(\rho)\Vert\mathcal{N}%
(\Pi)) \}
\iff \{\forall \, \rho\in \D(C) \ \textnormal{ : } \ \rho=(\mathcal{P}_{\Pi,\mathcal{N}%
}\circ\mathcal{N})(\rho) \} \ ,\label{eq:rel-ent-cond}%
\end{equation}
where $\cP_{\Pi,\cN}(\cdot)$ is the Petz recovery map defined in~\eqref{eq_Petzmap}.
\textit{So perfect error correction is possible if and only if for all }%
$\rho\in \D(C)$ \textit{the pairwise distinguishability of }$\rho$\textit{ with
}$\Pi$\textit{ does not decrease under the action of the noisy channel
}$\mathcal{N}$.

Given the statement in \eqref{eq:rel-ent-cond} and the prior work on
approximate quantum error correction, it is natural to wonder whether a robust
version of \eqref{eq:rel-ent-cond} exists. Indeed, Theorem~\ref{thm_explicit} implies
such a statement, establishing necessary and sufficient information-theoretic
conditions for approximate quantum error correction. Loosely speaking, we can
now say that \textit{approximate error correction is possible if any only if
the pairwise distinguishability of }$\rho$\textit{ with }$\Pi$\textit{ does
not decrease by too much under the action of the noisy channel }$\mathcal{N}$ \emph{for all $\rho \in \D(C)$}.
The corollary below is a consequence of Theorem~\ref{thm_explicit} and some other
known facts.

\begin{corollary} \label{cor_QEC}
Let $\varepsilon\in\left[  0,1\right]$, $A$ and $B$ be a finite-dimensional Hilbert spaces, $C$ be a subspace of $A$ (called the
\textquotedblleft codespace\textquotedblright), $\cN \in \TPCP(A,B)$ a quantum channel, and let $\Pi$ denote the
projection onto $C$. If for all $\rho\in \D(C)$ we have 
\begin{equation}
 D(\rho\Vert\Pi)-D\bigl(\mathcal{N}(\rho)\Vert\mathcal{N}(\Pi)\bigr)\leq\varepsilon \ ,
\end{equation}
then it is possible to approximately recover every state $\rho \in \D(C)$, in the sense that we have for all $\rho\in \D(C)$
\begin{equation}
 F\bigl(\rho,(\mathcal{R}_{\Pi,\mathcal{N}}\circ\mathcal{N})(\rho)\bigr)\geq1-\frac{1}{2} \varepsilon\ .\label{eq:fid-approx-QEC}%
\end{equation}
Conversely, if \eqref{eq:fid-approx-QEC} holds for $\varepsilon\in\left[  0,1 \right]  $, then we have for all $\rho\in \D(C)$
\begin{equation}
 D(\rho\Vert\Pi)-D(\mathcal{N}(\rho)\Vert\mathcal{N}(\Pi
))\leq\sqrt{\varepsilon}\log(\dim C)+h_{2}(\sqrt{\varepsilon}) \ ,
\end{equation}
where $h_{2}(p):=-p\log p-(1-p)\log(1-p)$ is the binary entropy, with the
property that $\lim_{p\searrow0}h_{2}(p)=0$.
\end{corollary}

\begin{proof}
The first statement is a direct consequence of Theorem~\ref{thm_explicit}, found by
setting $\rho=\rho$, $\sigma=\Pi$, and $\mathcal{N}=\mathcal{N}$ and then applying the inequality $-\log(x)\geq1-x$,
which holds for $x\in\left[  0,1\right]  $.

To prove the second statement, suppose that \eqref{eq:fid-approx-QEC} holds.
By the Fuchs-van-de-Graaf inequality~\cite{fuchs99}, \eqref{eq:fid-approx-QEC} implies that for all $\rho \in \D(C)$ we have
\begin{equation}
\frac{1}{2}\left\Vert \rho-(\mathcal{R}_{\Pi,\mathcal{N}%
}\circ\mathcal{N})(\rho)\right\Vert _{1}\leq\sqrt{\varepsilon} \ .
\end{equation}
Then consider the following chain of inequalities, which holds for all $\rho\in \D(C)$:
\begin{align}
 D(\rho\Vert\Pi)-D(\mathcal{N}(\rho)\Vert\mathcal{N}(\Pi)) 
& \leq D(\rho\Vert\Pi)-D\bigl((\mathcal{R}_{\Pi,\mathcal{N}}\circ\mathcal{N})(\rho)\Vert(\mathcal{R}_{\Pi,\mathcal{N}}\circ\mathcal{N})(\Pi)\bigr)  \\
& =D(\rho\Vert\Pi)-D\bigl((\mathcal{R}_{\Pi,\mathcal{N}}\circ\mathcal{N})(\rho)\Vert\Pi\bigr) \\
& =-H(\rho)+H((\mathcal{R}_{\Pi,\mathcal{N}}\circ\mathcal{N})(\rho
))+\tr\bigl(\left[  (\mathcal{R}_{\Pi,\mathcal{N}}\circ
\mathcal{N})(\rho)-\rho\right]  \log\Pi\bigr) \\
& \leq\sqrt{\varepsilon}\log\dim(C)+h_{2}(\sqrt{\varepsilon}).
\end{align}
The first inequality is a consequence of monotonicity of quantum relative
entropy with respect to the recovery map $\mathcal{R}_{\Pi,\mathcal{N}}$. The
first equality follows because $(\mathcal{R}_{\Pi,\mathcal{N}}\circ
\mathcal{N})(\Pi))=\Pi$. The second equality is a rewriting, and the last
inequality follows from the Fannes-Audenaert inequality~\cite{audenaert07} (see also~\cite[Lemma~1]{winter15}) and the facts that $\operatorname{supp}(\rho
)\subseteq\operatorname{supp}(\Pi)$, $\operatorname{supp}((\mathcal{R}%
_{\Pi,\mathcal{N}}\circ\mathcal{N})(\rho))\subseteq\operatorname{supp}(\Pi)$,
and $\log\Pi=0$ on the support of $\Pi$.
\end{proof}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
In this work, we showed that for any non-negative operator $\sigma$ and for any channel $\cN$ there exists an \emph{explicit} and \emph{universal} recovery map $\cR_{\sigma,\cN}$ and
\begin{align} \label{eq_res_conc}
D(\rho \| \sigma) - D\bigl(\cN(\rho) \| \cN(\sigma) \bigr) \geq - 2 \log F\bigl(\rho,(\cR_{\sigma,\cN}\circ\cN)(\rho)\bigr)
\end{align}
for all density operators $\rho$ such that $\supp(\rho) \subseteq \supp(\sigma)$. A consequence of its universality is that the recovery map $\cR_{\sigma,\cN}$ satisfies $(\cR_{\sigma,\cN}\circ\cN)(\sigma)=\sigma$. The present work thus constitutes an improvement over~\cite{Wilde15} since the recovery map is constructed explicitly and is independent of $\rho$.
Knowing the explicit structure of the recovery map that satisfies \eqref{eq_res_conc} can be helpful in various scenarios. For example if this is to be implemented by experimentalists or in an actual quantum computer it is helpful to have an explicit recovery map. 
 We showed that the inequality~\eqref{eq_res_conc} implies universal and explicit remainder terms for other entropy inequalities such as strong subadditivity, concavity of the entropy, and the joint convexity of the relative entropy. Furthermore, it can be useful in the context of quantum sufficient statistics and approximate quantum error correction. 

Two questions that remain unanswered are, first, if~\eqref{eq_res_conc} can be tightened in the sense that its right-hand side can be replaced by a relative entropy term of the form $D(\rho \| \cR_{\sigma,\cN}\circ\cN)(\rho))$. Secondly, it is also unknown if a bound in terms of a measured relative entropy (in the form of~\eqref{eq_STH}) remains valid for an $\emph{explicit}$ recovery map --- recall that Theorem~\ref{thm_monoRelEnt} proves the existence of a universal recovery map that satisfies~\eqref{eq_STH} but this recovery map is not explicit. Theorem~\ref{thm_explicit}, which shows  an explicit recovery map that satisfies~\eqref{eq_res_conc}, could therefore provide a new approach to attack these two open questions.




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%%%%




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%%%%


\section*{Appendix}


\appendix

\section{Proof of Lemma \ref{thm:op-hirschman}}

\label{sec:op-hirschman-proof}

We begin by recalling Hirschman's strengthening \cite{H52} of the Hadamard
three-line theorem (see, e.g., \cite[Lemma~1.3.8]{G08}):

\begin{lemma}
\label{thm:hirschman}Let $S:=\left\{  z\in\mathbb{C}:0\leq\operatorname{Re}\left\{  z\right\} \leq1\right\}$ and let $g(z)$ be holomorphic on the interior of $S$ and continuous on the
boundary, such that%
\begin{equation}
\sup_{z\in S}\exp\left\{  -a\left\vert \operatorname{Im}z\right\vert \right\}
\log\left\vert g(z)\right\vert \leq A<\infty,
\end{equation}
for some fixed $A$ and $a<\pi$. Then for $\theta\in(0,1)$, the following bound
holds%
\begin{equation}
\log\left\vert g(\theta)\right\vert \leq\int_{\R}\di t\ \Bigl(
\alpha_{\theta}(t)\log\bigl(  \left\vert g(\ci t)\right\vert ^{1-\theta}\bigr)
+\beta_{\theta}(t)\log\bigl(  \left\vert g(1+\ci t)\right\vert ^{\theta}\bigr)
\Bigr)  ,
\end{equation}
where $\alpha_{\theta}(t)$ and $\beta_{\theta}(t)$ are defined in \eqref{eq:pt}.
\end{lemma}


\noindent We can now prove Lemma~\ref{thm:op-hirschman}.

\begin{proof}[Proof of Lemma~\ref{thm:op-hirschman}]
The proof of this theorem is well known, but we provide it for completeness. For fixed $\theta\in(0,1)$, let $q_{\theta}$ be the H\"{o}lder
conjugate of $p_{\theta}$, defined by%
\begin{equation}
\frac{1}{p_{\theta}}+\frac{1}{q_{\theta}}=1 \ .
\end{equation}
Similarly, let $q_{0}$ and $q_{1}$ be H\"{o}lder conjugates of $p_{0}$ and
$p_{1}$, respectively. We can find an operator $X$ such that%
\begin{align}
\left\Vert X\right\Vert _{q_{\theta}}   =1 \quad \textnormal{and} \quad
\operatorname{tr}(  XG(\theta))     =\left\Vert G(\theta)\right\Vert
_{p_{\theta}} \ .
\end{align}
We can write the singular value decomposition for $X$ in the form
$X=UD^{1/q_{\theta}}V$ (implying tr$(D)=1$). For $z\in S$, define%
\begin{equation}
X(z):= UD^{\frac{1-z}{q_{0}}+\frac{z}{q_{1}}}V \ .
\end{equation}
As a consequence, $X(z)$ is holomorphic on the interior of $S$ and continuous on the boundary. Also, observe that $X(\theta)=X$. Then the
following function satisfies the requirements needed to apply
Lemma~\ref{thm:hirschman}:%
\begin{equation}
g(z):=\operatorname{tr}\bigl(  X(z)G(z) \bigr) \   .
\end{equation}
Indeed, we have that%
\begin{equation}
\log\  \left\Vert G(\theta)\right\Vert _{p_{\theta}}
=\log\left\vert g(\theta)\right\vert \leq\int_{\R}\di t\ \left(
\alpha_{\theta}(t)\log\bigl(  \left\vert g(\ci t)\right\vert ^{1-\theta}\bigr)
+\beta_{\theta}(t)\log\bigl(  \left\vert g(1+\ci t)\right\vert ^{\theta}\bigr)
\right)  .\label{eq:app-hirschman}%
\end{equation}
Now, from applying H\"{o}lder's inequality and the facts that $\left\Vert
X(it)\right\Vert _{q_{0}}=1=\left\Vert X(1+\ci t)\right\Vert _{q_{1}}$, we find
that%
\begin{align}
\left\vert g(\ci t)\right\vert  & =\left\vert \operatorname{tr}\bigl(
X(\ci t)G(\ci t)\bigr)  \right\vert \leq\left\Vert X(\ci t)\right\Vert _{q_{0}%
}\left\Vert G(\ci t)\right\Vert _{p_{0}}=\left\Vert G(\ci t)\right\Vert _{p_{0}}
\end{align}
and
\begin{align}
\left\vert g(1+\ci t)\right\vert  & =\left\vert \operatorname{tr}\bigr(
X(1+\ci t)G(1+\ci t)\bigr)  \right\vert \leq\left\Vert X(1+\ci t)\right\Vert _{q_{1}%
}\left\Vert G(1+\ci t)\right\Vert _{p_{1}}=\left\Vert G(1+\ci t)\right\Vert _{p_{1}%
}.
\end{align}
Bounding \eqref{eq:app-hirschman} from above using these inequalities then
gives \eqref{eq:oper-hirschman}.
\end{proof}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\section*{Acknowledgments}

We thank Omar Fawzi, Rupert Frank, Elliott Lieb, Volkher Scholz and Marco Tomamichel for helpful discussions. 
MJ's work was supported by NSF DMS 1501103 and NSF-DMS 1201886. 
RR and DS acknowledge support by the European Research Council (ERC) via grant No.~258932, by the Swiss National Science Foundation (SNSF) via the National Centre of Competence in Research ``QSIT'', and by the European Commission via the project ``RAQUEL''. 
MMW acknowledges support from the NSF under Award No.~CCF-1350397 and the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. He is also grateful to co-authors MJ, RR, and AW for hospitality and support during research visits in summer 2015.
AW�s work was supported by the EU (STREP �RAQUEL�), the ERC (AdG �IRQUAT�), the Spanish MINECO (grant FIS2013-40627-P) with the support of FEDER funds, as well as by the Generalitat de Catalunya CIRIT, project 2014-SGR-966.

\vspace{-3mm}

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ExplicitUniversalRecoveryMap.bbl
\begin{thebibliography}{10}

\bibitem{aliprantis07}
C.~D. Aliprantis and K.~C. Border.
\newblock {\em Infinite Dimensional Analysis}.
\newblock Springer, 2007.

\bibitem{audenaert07}
K.~M.~R. Audenaert.
\newblock A sharp continuity estimate for the von {N}eumann entropy.
\newblock {\em Journal of Physics A: Mathematical and Theoretical},
  40(28):8127, 2007.

\bibitem{BK02}
H.~Barnum and E.~Knill.
\newblock Reversing quantum dynamics with near-optimal quantum and classical
  fidelity.
\newblock {\em Journal of Mathematical Physics}, 43(5):2097--2106, May 2002.
\newblock arXiv:quant-ph/0004088.

\bibitem{BDSW96}
C.~H. Bennett, D.~P. DiVincenzo, J.~A. Smolin, and W.~K. Wootters.
\newblock Mixed-state entanglement and quantum error correction.
\newblock {\em Physical Review A}, 54(5):3824--3851, November 1996.

\bibitem{BLW14}
M.~Berta, M.~Lemm, and M.~M. Wilde.
\newblock Monotonicity of quantum relative entropy and recoverability.
\newblock {\em Quantum Information and Computation}, 15(15 \& 16):1333--1354,
  November 2015.

\bibitem{bourbaki_book}
N.~Bourbaki.
\newblock {\em Elements of Mathematics: General Topology}.
\newblock Hermann, \'{E}diteures des {S}ciences et des {A}rts, 1966.

\bibitem{BHOS14}
F.~G.~S.~L. Brand\~ao, A.~W. Harrow, J.~Oppenheim, and S.~Strelchuk.
\newblock Quantum conditional mutual information, reconstructed states, and
  state redistribution.
\newblock {\em Physical Review Letters}, 115(5):050501, July 2015.

\bibitem{csiszar95}
I.~Csisz\'{a}r.
\newblock Generalized cutoff rates and {R}\'enyi's information measures.
\newblock {\em IEEE Transactions on Information Theory}, 41(1):26--34, Jan
  1995.

\bibitem{DY08}
I.~Devetak and J.~Yard.
\newblock Exact cost of redistributing multipartite quantum states.
\newblock {\em Physical Review Letters}, 100(23):230501, June 2008.

\bibitem{fannes73}
M.~Fannes.
\newblock A continuity property of the entropy density for spin lattice
  systems.
\newblock {\em Communications in Mathematical Physics}, 31(4):291--294, 1973.

\bibitem{FR14}
O.~Fawzi and R.~Renner.
\newblock Quantum conditional mutual information and approximate {M}arkov
  chains.
\newblock {\em Communications in Mathematical Physics}, 340(2):575--611, 2015.

\bibitem{fisher22}
R.~A. Fisher.
\newblock On the mathematical foundations of theoretical statistics.
\newblock {\em Philosophical Transactions of the Royal Society of London Series
  A}, 222:309--368, 1922.

\bibitem{fuchs99}
C.~Fuchs and J.~van~de Graaf.
\newblock Cryptographic distinguishability measures for quantum-mechanical
  states.
\newblock {\em IEEE Transactions on Information Theory}, 45(4):1216--1227, May
  1999.

\bibitem{Fuc96}
C.~A. Fuchs.
\newblock Distinguishability and accessible information in quantum theory.
\newblock {\em PhD Thesis, University of New Mexico}, 1996.
\newblock \href{http://arxiv.org/abs/quant-ph/9601020}{arXiv:quant-ph/9601020}.

\bibitem{FAR11}
F.~Furrer, J.~{\AA}berg, and R.~Renner.
\newblock Min- and max-entropy in infinite dimensions.
\newblock {\em Communications in Mathematical Physics}, 306(1):165--186, 2011.

\bibitem{G08}
L.~Grafakos.
\newblock {\em Classical Fourier Analysis}.
\newblock Springer, second edition, 2008.

\bibitem{HMPB11}
F.~Hiai, M.~Mosonyi, D.~Petz, and C.~Beny.
\newblock Quantum $f$-divergences and error correction.
\newblock {\em Reviews in Mathematical Physics}, 23(7):691--747, August 2011.
\newblock arXiv:1008.2529.

\bibitem{H52}
I.~I. Hirschman.
\newblock A convexity theorem for certain groups of transformations.
\newblock {\em Journal d'Analyse Math\'ematique}, 2(2):209--218, December 1952.

\bibitem{holevo_book}
A.~S. Holevo.
\newblock {\em Quantum Systems, Channels, Information}.
\newblock De Gruyter Studies in Mathematical Physics 16, 2012.

\bibitem{anna14}
A.~Jenc\v{o}v\`a.
\newblock Randomization theorems for quantum channels, 2015.
\newblock \href{http://arxiv.org/abs/1404.3900}{arXiv:1404.3900}.

\bibitem{anna06}
A.~Jenc\v{o}v\`a and D.~Petz.
\newblock Sufficiency in quantum statistical inference.
\newblock {\em Communications in Mathematical Physics}, 263(1):259--276, 2006.

\bibitem{anna206}
A.~Jenc\v{o}v\`a and D.~Petz.
\newblock Sufficiency in quantum statistical inference: A survey with examples.
\newblock {\em Infinite Dimensional Analysis, Quantum Probability and Related
  Topics}, 09(03):331--351, 2006.

\bibitem{KR83}
R.~V. Kadison and J.~R. Ringrose.
\newblock {\em Fundamentals of the Theory of Operator Algebras}, volume I:
  Elementary Theory.
\newblock Academic Press, 1983.

\bibitem{KL97}
E.~Knill and R.~Laflamme.
\newblock Theory of quantum error-correcting codes.
\newblock {\em Physical Review A}, 55(2):900--911, February 1997.

\bibitem{LiWin14}
K.~Li and A.~Winter.
\newblock Squashed entanglement, k-extendibility, quantum {M}arkov chains, and
  recovery maps, 2014.
\newblock \href{http://arxiv.org/abs/1410.4184}{arXiv:1410.4184}.

\bibitem{LieRus73_1}
E.~H. Lieb and M.~B. Ruskai.
\newblock A fundamental property of quantum-mechanical entropy.
\newblock {\em Phys. Rev. Lett.}, 30:434--436, Mar 1973.

\bibitem{LieRus73}
E.~H. Lieb and M.~B. Ruskai.
\newblock Proof of the strong subadditivity of quantum-mechanical entropy.
\newblock {\em Journal of Mathematical Physics}, 14(12):1938--1941, 1973.

\bibitem{lindblad74}
G.~Lindblad.
\newblock Expectations and entropy inequalities for finite quantum systems.
\newblock {\em Communications in Mathematical Physics}, 39(2):111--119, 1974.

\bibitem{lindblad75}
G.~Lindblad.
\newblock Completely positive maps and entropy inequalities.
\newblock {\em Communications in Mathematical Physics}, 40(2):147--151, 1975.

\bibitem{Mandayam2012}
P.~Mandayam and H.~K. Ng.
\newblock Towards a unified framework for approximate quantum error correction.
\newblock {\em Physical Review A}, 86(1):012335, July 2012.
\newblock arXiv:1202.5139.

\bibitem{mosonyi04}
M.~Mosonyi and D.~Petz.
\newblock Structure of sufficient quantum coarse-grainings.
\newblock {\em Letters in Mathematical Physics}, 68(1):19--30, 2004.

\bibitem{MLDSFT13}
M.~M\"{u}ller-Lennert, F.~Dupuis, O.~Szehr, S.~Fehr, and M.~Tomamichel.
\newblock On quantum {R}\'enyi entropies: A new generalization and some
  properties.
\newblock {\em Journal of Mathematical Physics}, 54(12):122203, 2013.

\bibitem{OP93}
M.~Ohya and D.~Petz.
\newblock {\em Quantum Entropy and Its Use}.
\newblock Springer-Verlag, 1993.

\bibitem{Pet86}
D.~Petz.
\newblock Sufficient subalgebras and the relative entropy of states of a von
  {N}eumann algebra.
\newblock {\em Communications in Mathematical Physics}, 105(1):123--131, 1986.

\bibitem{Petz88}
D.~Petz.
\newblock Sufficiency of channels over von {N}eumann algebras.
\newblock {\em The Quarterly Journal of Mathematics}, 39(1):97--108, 1988.

\bibitem{Pet03}
D.~Petz.
\newblock Monotonicity of quantum relative entropy revisited.
\newblock {\em Reviews in Mathematical Physics}, 15(01):79--91, 2003.

\bibitem{simon_book}
M.~Reed and B.~Simon.
\newblock {\em Functional Analysis}.
\newblock Elsevier, Academic Press, 1980.

\bibitem{ruskai02}
M.~B. Ruskai.
\newblock Inequalities for quantum entropy: A review with conditions for
  equality.
\newblock {\em Journal of Mathematical Physics}, 43(9):4358--4375, 2002.

\bibitem{SBW14}
K.~P. Seshadreesan, M.~Berta, and M.~M. Wilde.
\newblock R\'enyi squashed entanglement, discord, and relative entropy
  differences.
\newblock {\em Journal of Physics A: Mathematical and Theoretical},
  48(39):395303, 2015.

\bibitem{SH08}
M.~E. Shirokov and A.~S. Holevo.
\newblock On approximation of infinite-dimensional quantum channels.
\newblock {\em Problems of Information Transmission}, 44(2):3--22, 2008.
\newblock arXiv:0711.2245.

\bibitem{Sion58}
M.~Sion.
\newblock On general minimax theorems.
\newblock {\em Pacific Journal of Mathematics}, 8:171, 1958.

\bibitem{SFR15}
D.~Sutter, O.~Fawzi, and R.~Renner.
\newblock Universal recovery map for approximate {M}arkov chains, 2015.
\newblock accepted for publication at Proceedings of the Royal Society of
  London A; available at
  \href{http://arxiv.org/abs/1504.07251}{arXiv:1504.07251}.

\bibitem{STH15}
D.~Sutter, M.~Tomamichel, and A.~W. Harrow.
\newblock Monotonicity of relative entropy via pinched {P}etz recovery map,
  2015.
\newblock \href{http://arxiv.org/abs/1507.00303}{arXiv:1507.00303}.

\bibitem{uhlmann77}
A.~Uhlmann.
\newblock Relative entropy and the {W}igner-{Y}anase-{D}yson-{L}ieb concavity
  in an interpolation theory.
\newblock {\em Communications in Mathematical Physics}, 54(1):21--32, 1977.

\bibitem{Wilde15}
M.~M. Wilde.
\newblock Recoverability in quantum information theory.
\newblock {\em Proceedings of the Royal Society of London A: Mathematical,
  Physical and Engineering Sciences}, 471(2182):20150338, 2015.

\bibitem{winter15}
A.~Winter.
\newblock Tight uniform continuity bounds for quantum entropies: conditional
  entropy, relative entropy distance and energy constraints, 2015.
\newblock \href{http://arxiv.org/abs/1507.07775}{arXiv:1507.07775}.

\bibitem{WL12}
A.~Winter and K.~Li.
\newblock A stronger subadditivity relation? {W}ith applications to squashed
  entanglement, sharability and separability, 2012.
\newblock
  \href{http://www.maths.bris.ac.uk/~csajw/stronger_subadditivity.pdf}{\url{http://www.maths.bris.ac.uk/~csajw/stronger_subadditivity.pdf}}.

\end{thebibliography}
