\begin{document}

\begin{CJK*}{UTF8}{gbsn}
\title{On the relation between quantum Darwinism and approximate  quantum Markovianity}

\author{Xiao-Kan Guo~(郭肖侃)}
 \email{kankuohsiao@whu.edu.cn}
\affiliation{College of Physical Science \& Technology, Bohai University,
Jinzhou 121013, China}
\author{Zhiqiang Huang (黄志强)}
\email{hzq@wipm.ac.cn}
\affiliation{State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China}
\date{\today}

\begin{abstract} 
There are strong evidences in the literature that quantum non-Markovianity would hinder the presence of Quantum Darwinism. 
 In this note, we study  the relation between quantum Darwinism and approximate quantum Markovianity for open quantum systems by exploiting the properties of quantum conditional mutual information. We first show that for approximately Markovian quantum processes the conditional mutual information still has the scaling property for Quantum Darwinism. Then two general bounds on the backflow of information  are obtained, with which we can show that the presence of Quantum Darwinism restricts the information backflow and the quantum non-Markovianity must be small.
\end{abstract}

\maketitle
\end{CJK*}

\section{Introduction}
Quantum decoherence gains importance  both in the foundational studies such as quantum measurements and in the practical studies such as the open quantum systems that are applicable to modern quantum technologies. An important recent development in quantum decoherence, known as Quantum Darwinism \cite{Z03,OPZ1,OPZ2,BZ05,BZ06,BZ08,QD,ZQZ09,ZQZ10,RZ10,RZ11,RZZ12,ZRZ14,ZRZ16,ZZ17,ULZZJ19}, explains how  classical objectivity emerges from the quantum formalism. Basically, in Quantum Darwinism, one treats the information transmission from the system $S$ to a fragment $F$ of the environment $E$ as a quantum channel, so that the accessible classical information as bounded by the Holevo information for different fragments $F_i$ of the environment $E$ can be compared. If these fragments agree upon the accessible information  to some 
reasonable extent, then we say the classical information thus obtained is objective. In this sense, Quantum Darwinism tells us how the environment witnesses the system in the process of quantum decoherence as an effective measurement channel.

As the Holevo bound involves an optimization procedure, it is difficult to calculate it. To circumvent this computational difficulty,  an alternative formulation of Quantum Darwinism is proposed in the recent works \cite{22a,22b,22c}, where one makes measurements on the fragments $F_i$ of $E$ and asks for the maximal information about $S$  that one can extract from these measurements. In contrast, in the original framework of Quantum Darwinism one makes measurements on the systems $S$ and considers the Holevo information defined by the conditional entropy conditioned on $S$. Such a change of measurements is reasonable in the sense that the two settings lead to the same scaling behavior of the Holevo information.
 It turns out that such a change of measurements not only makes the computations easier, but also reveals deep relations between the correlations in the environment $E$ and the information extracted from measuring $F_i$. For example, the Koashi-Winter monogamy relation can be invoked in this new framework to show the trade-off relation for the quantum correlations between $S$ and $F_i$ and the classical information of the $F_i$'s.

Interestingly, in this new framework of Quantum Darwinism, one can consider also the scaling of the conditional mutual information $I(S:F_l|F_k)$ \cite{22b}. This form of conditional mutual information also makes appearance in our recent work on quantum non-Markovianity \cite{HG21,HG22}. 
We therefore expect to study the connections between Quantum Darwinism and quantum (non-)Markovianity from a new perspective, in light of these new results. Previous studies on the connection between Quantum Darwinism and quantum (non-)Markovianity relies on specific models. In these model studies \cite{DM1,DM2,DM3,DM4,DM5,MS21}, there are strong evidences showing that  the presence of non-Markovianity would suppress Quantum Darwinism. However, there are  some models  \cite{DM7, OdD19,DM9} in which the  non-Markovian nature of  environment does not  prevent  the appearance of Quantum Darwinism. That the presence of non-Markovianity would hinder Quantum Darwinism is expected from the general understanding that quantum non-Markovianity is related to the information backflow from environment to system, but the absence of relations in several models calls upon further study in this direction.

In this note, we consider the relations between Quantum Darwinism and approximate   quantum Markovianity in a general setting. This is  operationally meaningful since the Holevo information decreases under  quantum operations and hence the quantum operations can only be operated approximately \cite{Cai03}.

We first use straightforward arguments from the approximately complete positive  maps \cite{BDW16} to explain in \S\ref{S3} that for  approximately Markovian processes the scaling property of Quantum Darwinism  still holds, meaning that for the quantum non-Markovianity to hinder Quantum Darwinism a large non-Markovianity quantifier should be required. In \S\ref{S4}, we study the backflow of correlation as quantified by the conditional mutual information and find such  backflow of correlation is upper bounded by both the classical information and the quantum discord. If the suppressions as in \cite{22b} hold, then the backflow of information, which is one way of characterizing quantum non-Markovianity, is also greatly suppressed, thereby corroborating the approximately Markovian results discussed in \S\ref{S3}.

\section{Quantum Darwinism: Decohering the system or eavesdropping the environment}\label{S2}
We start   by reviewing briefly both the standard formulation of Quantum Darwinism \cite{Z03,OPZ1,OPZ2,BZ05,BZ06,BZ08,QD,ZQZ09,ZQZ10,RZ10,RZ11,RZZ12,ZRZ14,ZRZ16,ZZ17} and the recent formulation given in \cite{22a,22b,22c}. 

In Quantum Darwinism one considers the setting  of an open quantum system  $S$ interacting with an environment $E$ which consists of $N$ fragments (or subenvironments) $F_i$. The interaction between $S$ and $E$, or the quantum decoherence of $S$ by $E$, is considered as a quantum communication channel $\Lambda_{S\rightarrow F_i}$ such that information is transmitted from $S$ to $F_i$. Namely, correlation is created between $S$ and $F_i$ as the quantum  mutual information
\begin{equation}
I(S:F_i)=H_S+H_{F_i}-H_{SF_i}\neq0
\end{equation}
where $H_S=-\text{tr}\rho_S\log_2\rho_S$ is the von Neumann entropy.

To extract classical information about $S$ from $F_i$, one considers the quantum decoherence of   $S$ by projecting it to the pointer basis, $\{\ket{\hat{s}}\}$ \cite{Z03,PBasis}, and hence one obtains the Holevo information
\begin{equation}\label{2}
\chi(\hat{\Pi}_S:F_i)=H\Bigl(\sum_{\hat{s}}p_{\hat{s}}\rho_{F_i|\hat{s}}\Bigr)-\sum_{\hat{s}}p_{\hat{s}}H(\rho_{F_i|\hat{s}}),
\end{equation}
where $\hat{\Pi}_S=\sum_{\hat{s}}\pi_{\hat{s}}\ket{\hat{s}}\bra{\hat{s}}$ and $\rho_{F_i|\hat{s}}=\braket{\hat{s}|\rho_{SF_i}|\hat{s}}/p_{\hat{s}}$ is the fragment state conditioned on system's pointer state with the $p_{\hat{s}}$ being the probabilities for projecting to the pointer basis. The Holevo information is an upper bound on the classical  information transmittable in the quantum channel $\Lambda_{S\rightarrow F_i}$. For the successful extraction of information about $S$, one sets the condition
\begin{equation}\label{3}
\braket{\chi(\hat{\Pi}_S:F_i)}_{\# F_\delta}\approx(1-\delta) H_S
\end{equation}
where $\delta$ is a small number called the information deficit, and $\braket{}_{\# F_\delta}$  denotes the average over the fragments of size ${\# F_\delta}$.
Here, $H_S=H(\hat{\Pi}_S)$ quantifies the missing information about $S$ computed in the pointer basis, i.e. the classical information content of the pointer state when the quantum state is decohered. 
The condition \eqref{3} says that an observer only needs a number ${\# F_\delta}$ of fragments $F_i$ to retrieve $(1-\delta) H_S$ classical bits of information about $S$, and therefore the rest of the environment is redundant. Let $\# E=N$ be the size of the environment $E$, then we define the redundancy as
\begin{equation}
R_\delta=\frac{\# E}{\# F_\delta}.
\end{equation}
When the fragments $F_i$ are of the same size, we have $R_\delta=1/f_\delta$ with $f_\delta$ the fraction of the relevant fragments.

By using the complementary relation for the Holevo information and the quantum discord \cite{ZZ13}, we can rewrite the condition \eqref{3} as
\begin{equation}\label{5}
I(S:F_i)\approx(1-\delta) H_S.
\end{equation}
This is because the Holevo information is the upper bound on the transmittable classical information in the pointer state,
\begin{equation}\label{6}
\chi(\hat{\Pi}_S:F_i)=\max_{{s}}I({\Pi}_S:F_i)=J(\hat{\Pi}_S:F_i)
\end{equation}
where the maximum is taken over all possible eigenbases, and
$J(\hat{\Pi}_S:F_i)$ is the classical correlation used in the definition of quantum discord. When the quantum correlation is small, we obtain \eqref{5}, with the understanding that it has been averaged, i.e. $I(S:F_i)=\braket{I(S:F_i)}_{\# F_\delta}$.

The most important property of $I(S:F_i)$ for Quantum Darwinism is  its {\it scaling property} under the increase of $\# F_\delta$. The study of various models confirms the general behavior of $I(S:F_i)$: At the initial stage, $I(S:F_i)$ follows a steep rise  with increasing $\# F_\delta$ as the classical correlation increases; then it saturates a long and flat plateau meaning that at this stage the added fragments are redundant, i.e. when 
\begin{equation}\label{7}
I(S:F_i)\geqslant(1-\delta) H_S;\end{equation}
finally, $I(S:F_i)$ follows again a steep rise to reach its maximum as the quantum correlation increases.

Now from \eqref{6} we see that it involves an optimization, which is generically hard. Also, since the Holevo information is only an upper bound, the above conditions for information retrieval can in fact be relaxed. To improve upon these limitations, the recent work \cite{22a} considers instead the measurements on $F_i$. Let us denote the part being measured by a check, e.g. the Holevo information in \eqref{2} becomes the asymmetric mutual information $J(\check{S}:F_i)$. When  $F_i$ is measured,  one has similarly 
\begin{equation}
J(S:\check{F}_i)=\chi(S:\check{F}_i),
\end{equation}
which is the maximal information about $S$ one can extract by measuring $F_i$. In fact, this $J(S:\check{F}_i)$ also upper bounds the classical accessible information \cite{22c}.
Numerical results  show that the scaling property for Quantum Darwinism still holds for $J(S:\check{F}_i)$. In particular, the plateau condition can be written as
\begin{equation}\label{9}
J(S:\check{F}_i)\geqslant(1-\delta) H_S.
\end{equation}

Using the symmetric mutual information $I(S:F_i)$ and the asymmetric mutual information $J(S:\check{F}_i)$, one can define the fragmentary discord, $D(S:\check{F}_i)=I(S:F_i)-J(S:\check{F}_i)$. If we consider each fragment $F_i$ has its own information deficit $\delta_i$, and consider the averaged discord $\bar{D}(S:\check{F}_i)=\frac{1}{N}\sum_{i=1}^ND(S:\check{F}_i)$, then the following bound holds \cite{22b},
\begin{equation}\label{10}
\bar{D}(S:\check{F}_i)\leqslant\delta H_S
\end{equation}
where $\delta=\sum_{i=1}^N\delta_i/N$. Similarly, for the $\# F_\delta\leqslant\frac{N}{2}$ fragments, the bound can be rewritten as
\begin{equation}\label{11}
\bar{D}(S:\check{F}_i)\leqslant\bigl[1-(1-\delta)\frac{R_\delta}{N}\bigr] H_S.
\end{equation}
Clearly, since the information deficit is chosen to be small, the quantum correlation as quantified by the quantum discord is greatly suppressed. Importantly, the plateau condition \eqref{9} implies 
\begin{equation}\label{12}
I(S:F_l|F_k)\leqslant2\delta H_S,
\end{equation}
for $k\geqslant\#F_\delta,~k+l\leqslant N-\# F_\delta$. With \eqref{12}, one can directly test Quantum Darwinism by inspecting the scaling of the conditional mutual information $I(S:F_l|F_k)$, without facing the problem of optimization.

\section{Relation to approximately quantum Markovian processes}\label{S3}
We consider in this section the case in which the quantum channels involved deviate a little  from complete positivity. The complete positivity of the open quantum system dynamics is related to the quantum data processing inequality  \cite{Bus14}, and the deviation from the   quantum data processing inequality  indicates quantum non-Markovianity. A small deviation from the   quantum data processing inequality can give rise to an approximately complete positive map, and the converse is also true \cite{BDW16}.

To apply these deviation results to Quantum Darwinism, we consider the following novel scenario: Instead of focusing on the system $S$, we take the measurement made on the fragment $F_k$ as a quantum channel $\Lambda_{F_k\rightarrow F_k'}$. After the system $S$ has decohered and the information about $S$ has been transmitted to $F_k$, we assume that the fragment $F_k$ no longer interacts with $S$ but the interactions between different  $F_i$'s are still possible. With this assumption,
the quantum channel $\Lambda_{F_k\rightarrow F_k'}$ can be considered as an $F_k$-$F_l$ open system where $F_k$ is the new open ``system'' of an observer's concern and another fragment $F_l$ with $l\neq k$ is the new environment for $F_k$. On the other hand,  the system $S$ becomes now a {\it steerable } extension of the $F_k$-$F_l$ open system, because the states $\rho_{F_kF_l}$ can be engineered according to the pointer basis of $S$. This way, we have adapted the new setting of Quantum Darwinism to that of \cite{Bus14}.

Now suppose that the   quantum channel $\Lambda_{F_k\rightarrow F_k'}$  is approximately CPTP, in the sense that 
\begin{equation}
\frac{1}{2}\norm{\sigma_{SF'_k}-\Lambda_{F_k\rightarrow F_k'}(\rho_{SF_k})}_1\leqslant\epsilon
\end{equation}
where $\epsilon\in[0,1]$ and $\sigma_{SF_kF_l}=U_{F_kF_l}(\rho_{SF_kF_l})U^\dag_{F_kF_l}$ is the partially evolved state. 
Then by the Alicki-Fannes-Winter inequality \cite{AF04,W16}, we have 
\begin{equation}
I(S:F'_k)_\sigma\leqslant I(S:F_k)+2\epsilon\log\abs{S}+(1+\epsilon)h[\frac{\epsilon}{1+\epsilon}]
\end{equation}
where $h[x]=-x\log x-(1-x)\log(1-x)$ is the binary entropy. Furthermore, by 
Theorem 8 of \cite{BDW16}, we also have the  bound on the  conditional mutual information
\begin{equation}\label{15}
I(S:F_l|F_k)\leqslant2\epsilon\log\abs{S}+(1+\epsilon)h[\frac{\epsilon}{1+\epsilon}].
\end{equation}

The bound in \eqref{15} has the same qualitative structure as \eqref{12}: two bounds both depends on a small parameter and  the ``entropy'' of $S$.\footnote{In \eqref{15}, the term $\log\abs{S}$ can be roughly understood as the statistical-mechanical entropy in the Boltzmann sense.} 
We therefore see that, with a small deviation from complete positivity, the scaling behavior \eqref{12}  of conditional mutual information for Quantum Darwinism {\it  can still hold}, if we choose proper parameters $\epsilon,\delta$, and
 assume that the fragment $F_k$ no longer interacts with the system $S$. In other words, the  approximately Markovian quantum processes will not hinder the presence of Quantum Darwinism. This understanding is consistent with the strict  bounds on the deviation from the quantum Markovianity  \cite{FR15,FMP19}.
 
Next, let us consider the Holevo information of the ensemble of states $\mathcal{S}=\{p_{\hat{s}},\rho_{\hat{s}}\}$,
\begin{equation}
\chi(\mathcal{S})=H\Bigl(\sum_{\hat{s}}p_{\hat{s}}\rho_{\hat{s}}\Bigr)-\sum_{\hat{s}}p_{\hat{s}}H(\rho_{\hat{s}}).
\end{equation}
Then the Holevo information \eqref{2} of the quantum channel $\Lambda_{S\rightarrow F_i}$ is the Holevo information $\chi(\mathcal{F})$ of the output ensemble $\mathcal{F}=\{p_{\hat{s}},\rho_{F_i|\hat{s}}\}$. The monotonicity of the Holevo information under quantum channels give $\chi(\mathcal{S})\geqslant\chi(\mathcal{F})$, and when the equality holds there exists a (Petz) recovery channel $\Phi_{F_i\rightarrow S}$ such that $\Phi_{F_i\rightarrow S}\circ\Lambda_{S\rightarrow F_i}(\rho_{\hat{s}})=\rho_{\hat{s}}$ \cite{HJPW04}. The existence of recovery channel is equivalent to the  backflow of {\it all} information from $F_i$ to $S$, and we see that this requires the Holevo information of the ensemble of states is intact under the quantum channel $\Lambda_{S\rightarrow F_i}$, i.e. 
\begin{equation}
\chi(\mathcal{S})=\chi(\mathcal{F})=\chi(\hat{\Pi}_S:F_i).\end{equation}
A small deviation from this equality is allowable, if the following bound holds (cf. Theorem 6 of \cite{BDW16})
\begin{equation}\label{18}
\chi(\mathcal{S})-\chi(\mathcal{F})\geqslant-2\log\sum_{\hat{s}}p_{\hat{s}}\sqrt{F}(\rho_{\hat{s}},\Phi\circ\Lambda(\rho_{\hat{s}}))
\end{equation}
where $\sqrt{F}(\rho,\sigma)=\norm{\sqrt{\rho}\sqrt{\sigma}}_1$ is the root of the fidelity. The inequality \eqref{18} is the constraint for the existence of the approximate recovery channel or approximately full information backflow. 

As far as the quantum decoherence or emergence of classicality is concerned, we expect a large violation of the condition \eqref{18} to reach an irreversible information transmission. However, when strong non-Markovianity is present, it is still  possible for such an approximate recovery channel to exist.

\section{Relation to backflow of correlation}\label{S4}
\begin{CJK*}{UTF8}{gbsn}
It is well-known by now that  the quantum non-Markovianity  of open quantum dynamics can be quantified by the information backflow  from environment to system, or the increase in distinguishability of system's states \cite{BLP09}. Recently, we have shown in \cite{HG21} that the backflow of quantum mutual information can be equivalently reformulated as the backflow of conditional mutual information in a system-environment-ancilla setting. Here, we shall consider the backflow of conditional mutual information in the new setting of Quantum Darwinism.

We consider first the pure state $\rho_{SE}$
for the system-environment total system. Since for pure states both the quantum discord and the classical correlation takes the maximal value of $H_S$, we have 
\begin{equation}
I(S:E)=D(S:\check{E})+J(S:\check{E})=2H_S.
\end{equation}
In the system-environment-ancilla setting of \cite{HG21}, we require the ancilla-system initial state is maximally entangled, i.e. $I(A:S)_{\rho_i}=2H_A=2\ln d_A$. In other words, initially the ancillary $A$ possesses all the information about $S$ (and vice versa). After the information of $S$ is spread into $E$, we have in general
$I(A:S)\neq2H_A$ and $I(A:SE)=2H_A$. 
The memory effect is reflected by the non-monotonicity of the  conditional mutual information $I(A:E|S)=I(A:SE)-I(A:S)$. 

Notice that  the ancillary $A$ can be considered either as a fragment $F_i$ of the environment or  a special observer who makes measurements on $S$. 
We first consider the former interpretation of $A$ as fragment of $E$.
To avoid confusions, let us label the   $(S^*,E^*,A^*)$ in the system-environment-ancilla setting with a $*$.
We can take the $A^*$ as the system $S$ in Quantum Darwinism setting and take $S^*$ as a marked fragment $F_1$ in Quantum Darwinism setting. Let us consider the following
\begin{equation}
I(A^*:E^*_{sub}|S^*)\equiv I(S:{F}_l|{F}_1),
\end{equation}
where the RHS is understood as the information transmission from $S$ to $F_1$ in the setting of Quantum Darwinism, which can be alternatively understood as in the LHS as information flow from $A^*$ to $S^*$. 
If we adopt the interpretation of $A^*$ as a fragment of environment, this information transmission is the backflow of information. 
So we have
    \begin{align}
        &I(A^*:E^*_{sub}|S^*)= I(S:{F}_l|{F}_1)\notag \\
        =&I(S:F_{l+1})-I(S:F_1)\notag \\
        =&I(S:F_{\#F_\delta})-I(S:F_1)+I(S:F_{l+1})-I(S:F_{\#F_\delta})\notag \\
        =&I(S:{F}_{\#F_\delta-1}|\mathcal{F}_1)+I(S:{F}_{l-\#F_\delta+1}|{F}_{\#F_\delta})\notag \\
        \leqslant&I(S:F_{\#F_\delta})-I(S:F_1)+2\delta H_S \label{20}\\
        \approx&J(S;F_{\#F_\delta})+D(S;F_{\#F_\delta})-(1-2\delta-\delta')H(S)\label{21}
    \end{align}
where in \eqref{20} we have used the scaling \eqref{12}, and in \eqref{21} we have used \eqref{5} and $\delta'$ is the information deficit for this particular case, $I(S:F_1)\approx(1-\delta')H_S$. By expressing \eqref{5} in terms of $J$,  we obtain
\begin{equation}
I(A^*:E^*_{sub}|S^*)\lessapprox(\delta'+\delta)H_S +D(S;F_{\#F_\delta})\label{22}.
\end{equation}
This result \eqref{22} shows that the conditional mutual information $I(A^*:E^*_{sub}|S^*)$ is  bounded both by the classical information  and the quantum correlation. This clearly shows that  the backflow of information contains both classical and quantum correlations. Now, in the  parameter range $ l\leqslant N-2\#F_\delta$, by further using the bound \eqref{11}, we see that the conditional mutual information, which is to be backflowed from $A^*$ to $S^*$, is greatly suppressed. 

Notice that the backflow of information from $A^*$ to $S^*$ is understood by the standard setting of Quantum Darwinism as the channel from $S$ to $F_i$. Although a little bit strange, this is allowable by the general results on the emergence of Quantum Darwinism in a general many-body quantum system \cite{BPH15,KTPA18,CLAT21,QR21,RZZ16,Rie17,Oll22}.  In a sense, we have modeled the emergence of Quantum Darwinism from $A^*$ to $\{E_{sub}^*\}_{i}\cup\{S^*\}$.

Next, we consider an alternative setting by taking $A$ as an observer who measures $S$. Suppose the initial system-environment state  is
    \begin{equation}\label{SODAW0}
        \ket{\psi_{SE}(0)}=\Bigl(\sum_{{s}} \ket{{s}}_S\Bigr)\otimes \ket{\phi}_E
    \end{equation}
where the $\ket{s}$ is the eigenbasis of $S$ and the normalization is hidden. 
After $A$ measured $S$ by projections $\ket{\hat{s}}\bra{\hat{s}}$ to the pointer basis $\ket{\hat{s}}$, the  joint state is then
    \begin{equation}\label{SOH0}
        \ket{\psi'_{ASE}(0)}=\Bigl(\sum_{\hat{s}} \sqrt{P_{\hat{s}}}\ket{\hat{s}}_A\otimes\ket{\hat{s}}_S\Bigr)\otimes \ket{\phi}_E
    \end{equation}
 where $\sqrt{P_{\hat{s}}}=\braket{\hat{s}|s}$. In this case, $AS$-state is  entangled, and the observer $A$ keeps all the classical information about $S$. We have in fact  returned to the system-environment-ancilla setting of \cite{HG21}.
After the time evolution of $SE$ by $U^{SE}$ (or under the quantum channel $\Lambda_{S\rightarrow E}$), we have 
 \begin{equation}\label{SOH}
    \ket{\psi'_{ASE}}=U^{SE}  \ket{\psi'_{ASE}(0)}=\sum_{\hat{s}} \sqrt{P_{\hat{s}}}\ket{\hat{s}}_A\otimes\ket{\hat{s}}_S\otimes \ket{\phi_{\hat{s}}}_E.
\end{equation}
The $SE$-part is now a branching state
\begin{equation}\label{SOH}
    \ket{\psi_{SE}}=U^{SE}  \ket{\psi_{SE}(0)}=\sum_{\hat{s}} \sqrt{P_{\hat{s}}}\ket{\hat{s}}_S\otimes \ket{\phi_{\hat{s}}}_E.
\end{equation}

To proceed, we observe that joint state $\ket{\psi'_{ASE}}$ can be obtained from the state $\ket{0}_A\otimes\ket{\psi_{SE}}$, with $\ket{0}_A$ a pure state, by a unitary $U^{AS}: \ket{\hat{s}0}\to \ket{\hat{s}\hat{s}}$,
\begin{equation}
    \ket{\psi'_{ASE}}=U^{AS} \ket{0}_A\otimes \ket{\psi_{SE}}.
\end{equation}
Since the pure state has zero von Neumann entropy and the unitary does not increase von Neumann entropy, we have in this case,
\begin{equation}\label{DCMI1}
    I(AS:E_{sub})_{\ket{\psi'_{ASE}}}=  I(S:E_{sub})_{ \ket{\psi_{SE}}}.
\end{equation}
Let us take partial trace of \eqref{SOH}
\begin{equation}\label{27}
    \text{Tr}_{A\overline{E}_{sub}} \Pi_{\psi'}^{ASE}=\sum_{\hat{s}} P_{\hat{s}} \Pi_{\hat{s}}^S \otimes  ( \text{Tr}_{\overline{E}_{sub}} \Pi_{\phi_{\hat{s}}}^{E}).
\end{equation}
where $\overline{E}_{sub}E_{sub}=E$.
Now we can assume the condition of  good decoherence \cite{ZQZ10}  that the
off-diagonal terms of $\rho_{S}$  are negligible, so that  \eqref{27} is a classical-quantum state with zero quantum discord, i.e.
    \begin{equation}\label{DCMI2}
        I(S:E_{sub})_{\ket{\psi'_{ASE}}}=  \chi(\check{S}:E_{sub})_{ \ket{\psi_{SE}}}.
    \end{equation}
Finally, combining \eqref{DCMI1} and \eqref{DCMI2}, we obtain
    \begin{align}
        I(A:E_{sub}|S)_{\ket{\psi'}}&=I(AS:E_{sub})_{\ket{\psi'}}-I(S:E_{sub})_{\ket{\psi'}}\notag \\
       & =I(S:\mathcal{F}_l)- \chi(\check{S}:\mathcal{F}_l)=D(\check{S}:\mathcal{F}_l).\label{29}
    \end{align}
Since we have assumed the good decoherence, we have indeed $D(\check{S}:\mathcal{F}_l)\approx0$ \cite{22a}.

In the present case, the $I(A:E_{sub}|S)_{\ket{\psi'}}$ contains the part that can be possibly  backflowed, and any backflow of conditional mutual information is also bounded as above. Note that, in comparison to \eqref{22}, the classical information is not present in the bound \eqref{29}; this is because the interpretations of $A$ differ in two cases. Here, $A$ contains all the classical information (i.e. the pointer observables) of $S$, while for \eqref{22} this classical information needs to be transmitted.

\end{CJK*}

\section{Conclusion}\label{IV}
We have discussed the relation between Quantum Darwinism and approximate quantum non-Markovianity. In \S\ref{S3}, we have showed that given an approximately Markovian quantum process with an approximate CPTP map, the conditional mutual information still satisfies the scaling property for Quantum Darwinism as is recently given in \cite{22b}, while in \S\ref{S4} we have proved the converse that the existence of Quantum Darwinism greatly suppresses the backflow of correlation. In this sense, we have actually proved, under some physically reasonable assumptions, the following statement:
\begin{quote}
{\it Quantum Darwinism is present in an open quantum system if and only if the open system is Markovian or approximately Markovian. }
\end{quote}

This result is consistent with the numerical works \cite{DM1,DM2,DM3,DM4,DM5,MS21} where
the highest redundancy is reached when the dynamics is fully Markovian,  the redundancy tends to
zero when the dynamics is non-Markovian. (Particularly, \cite{MS21} also considers the information flow from the environment-point-of-view.)
With the explicit bounds obtained in this paper, we can further understand the boundary between these two regimes. 
But the numerical results supporting the irrelevance between Quantum Darwinism and non-Markovianity \cite{DM7, OdD19,DM9} cannot be understood easily from the point of view of present work. These  irrelevance results involve other aspects of classical objectivity such as the spectrum broadcast structures \cite{SBS}, which is worthy of further investigations.

\begin{acknowledgments}
We would like to thank Diogo O. Soares-Pinto for helpful comments. 
ZH  is supported by the National Natural Science Foundation of China under Grant Nos. 12047556, 11725524 and the Hubei Provincial Natural Science Foundation of China under Grant No. 2019CFA003.
\end{acknowledgments}

\end{document}