# Enhanced high-dimensional teleportation in correlated amplitude damping
noise by weak measurement and environment-assisted measurement

Xing Xiao  College of Physics and Electronic Information, Gannan Normal
University, Ganzhou 341000, Jiangxi, China  Tian-Xiang Lu
<EMAIL_ADDRESS>College of Physics and Electronic Information, Gannan
Normal University, Ganzhou 341000, Jiangxi, China  Yan-Ling Li
<EMAIL_ADDRESS>School of Information Engineering, Jiangxi University
of Science and Technology, Ganzhou, 341000, Jiangxi, China

###### Abstract

High-dimensional teleportation provides various benefits in quantum networks
and repeaters, but all these advantages rely on the high-quality distribution
of high-dimensional entanglement over a noisy channel. It is essential to
consider correlation effects when two entangled qutrits travel consecutively
through the same channel. In this paper, we present two strategies for
enhancing qutrit teleportation in correlated amplitude damping (CAD) noise by
weak measurement (WM) and environment-assisted measurement (EAM). The fidelity
of both approaches has been dramatically improved due to the probabilistic
nature of WM and EAM. We have observed that the correlation effects of CAD
noise result in an increase in the probability of success. A comparison has
demonstrated that the EAM scheme consistently outperforms the WM scheme in
regard to fidelity. Our research expands the capabilities of WM and EAM as
quantum techniques to combat CAD noise in qutrit teleportation, facilitating
the development of advanced quantum technologies in high-dimensional systems.

##  I Introduction

Quantum teleportation is not only a fascinating protocol in the theory of
quantum information [1], but also a fundamental part of long-distance quantum
communication and quantum networks [2] since it allows the non-local
transmission of an unknown quantum state. Over the last thirty years,
outstanding progress has been achieved in both the theoretical and
experimental study of quantum teleportation [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
In particular, there has been very rapid development in long-distance quantum
teleportation, such as photonic quantum teleportation based on optical fibre
networks and free-space satellites [13, 14, 15, 16, 17, 18].

While the qubit (two-level system) serves as the basic unit of quantum
information, a single particle in the actual world commonly has numerous
degrees of freedom, thereby comprising a high-dimensional quantum system.
Therefore, it is also of substantial interest to understand the functions of
higher-level systems, such as the qutrit (three-level system) or the qudit
(d-level system). In fact, quantum information protocols that rely on qutrits
or qudits possess certain advantages over qubit-based schemes due to the
additional dimensions in Hilbert space [19]. These benefits encompass, but are
not confined to an enhanced channel capacity [20, 21, 22], improved noise
resilience [23, 24], more efficient encoding of quantum information [25], a
more reliable simulation of quantum dynamics [26, 27], and an increased
sensitivity in quantum imaging schemes [28]. These advantages are particularly
intriguing in the upcoming noisy intermediate-scale quantum (NISQ) era since
the “quality” is of critical importance [29]. Recently, there has been
significant progress in high-dimensional quantum teleportation experiments
[30, 31, 32, 33, 34].

It is well known that the quantum superiority of quantum teleportation is
mainly due to the initial shared entanglement between the sender and the
receiver. The initial shared entanglement between the sender and the receiver
is widely acknowledged as the primary reason for the superiority of quantum
teleportation. However, quantum entanglement is very fragile in real-world
environments. Therefore, the noise in the channel is a significant constraint
on the fidelity of quantum teleportation. In the last decades, more attention
has been paid to dealing with the noise in quantum teleportation [35, 36, 37].
However, the majority of previous investigations have concentrated on the
memoryless channel, where the noises are assumed to have different origins and
are treated independently. While this presumption is reasonable in specific
physical scenarios, it may not be justifiable in numerous more practical
circumstances [38, 39, 40, 41, 42]. The noises noises may share a common
origin, resulting in their correlation with one another [43, 44]. For example,
successive use of the same channel may cause the correlated noise between
inputs, as the channel may retain memory between successive transmissions.
Unlike the uncorrelated channel, which can be expressed as a tensor product of
independent and identical completely positive, trace-preserving (CPTP) maps,
the Kraus operators of the correlated channel map cannot be expressed in a
tensor product form. The study of the correlated channel has attracted
considerable attention in the field of quantum information, since memory
effects cannot be avoided during high rate transmission [45, 46, 47, 48, 49,
50, 51, 52, 53, 54].

In this paper, we present two techniques for improving the fidelity of qutrit
teleportation in the presence of correlated amplitude damping (CAD) noise,
through the use of weak measurement (WM) and environment-assisted measurement
(EAM), as displayed in Fig. 1. Within the WM approach, pre-WM and post quantum
measurement reversal (QMR) are performed on the system once before and once
after it passes through the CAD channel. The purpose of pre-WM is to bring the
initially entangled state closer to a ‘lethargic’ state that is more resilient
to the CAD noise, while the post-QMR aims to eliminate the effects of the pre-
WM and the CAD noise, thus restoring the teleported state. The power of WM to
combat AD noise has been demonstrated in many theoretical and experimental
works [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. Nevertheless, the
potential of WM to handle CAD noise, particularly qutrit CAD noise, has yet to
be explored. Although the fact that correlation effects in CAD noise can
improve the fidelity of teleportation has been reported in Ref. [68], we
emphasize that the combination of WM and QMR makes further efforts to improve
fidelity. In the EAM approach, a measurement such as photon counting is
performed on the noisy environment coupled to the Qutrit, followed by a QMR
operation on the system conditioned by the measurement results. We find that
the aid of EAM enables the fidelity of the teleportation to be almost
completely recovered to 1, thereby eliminating the decoherence effect of the
CAD noise. However, it should be acknowledged that the implementation of our
approaches is subject to a certain probability. Fortunately, the correlation
effects of CAD noise can increase the probability of success. It is
interesting to note that the EAM scheme is superior to WM scheme both in terms
of fidelity and probability of success. The reason for this is that the post-
performed EAM gathers information from both the system and the noisy channel.

Figure 1: (color online) Schematic illustrations of WM and EAM enhanced qutrit
teleportation in CAD noise. (a) The circuit of qutrit teleportation with the
assistance of WM and QMR. (b) The circuit of qutrit teleportation with the
assistance of EAM and QMR. $H$, $L_{C}$, $X$ and $Z$ are generalized logic
gates for qutrit. $(m,n)$ denotes the result of the joint measurement
performed by Alice.

The structure of this paper is as follows. In Sec. II, we introduce some
related concepts such as quantum teleportation, CAD noise, WM and QMR for the
qutrit case. In Sec. III, we then show how the fidelity of teleportation could
be enhanced by WM and QMR. In Sec. IV, we propose another scheme to improve
the fidelity with the assistance of EAM and QMR. A comparison of the two
schemes is discussed in detail in Sec. V. Finally, the conclusions are
summarized in Sec. VI.

##  II Basic concepts

###  II.1 CAD noise for qutrits

Amplitude damping noise is a prototype model of a dissipative interaction
between a quantum system and its environment, such as spontaneous emission in
the atomic system or photon loss in the optical system [2]. Assuming that two
qutrits successively pass through an AD channel, the correlation effects
should be considered if the relaxation time of the channel is not too short
compared with the time interval between the two consecutive qutrits entering
the channel. In this case, the channel is a CAD channel which cannot be
factorized as those on each one.

As we all know, an arbitrary completely positive, trace-preserving (CPTP) map
$\mathcal{E}$ can be written as
$\mathcal{E}(\rho)=\sum_{j}E_{j}\rho_{0}E_{j}^{\dagger}$ in the operator-sum
representation, where $E_{j}$ are the Kraus operators. Note that for the
uncorrelated AD channel, the overall dynamical map can be expressed as a
tensor product of the individual AD maps: $\mathcal{E}_{\rm
AD}^{(n)}(\rho)=\mathcal{E}_{\rm AD}^{\otimes n}(\rho_{0})$. However, such a
tensorial decomposition is not valid for the CAD channel. The map of a CAD
channel is a combination of the uncorrelated AD channel and fully correlated
AD (FCAD) channel:

$\displaystyle\mathcal{E}_{\rm CAD}(\rho)$ $\displaystyle=$
$\displaystyle(1-\mu)\mathcal{E}_{\rm AD}^{\otimes
2}(\rho)+\mu\mathcal{E}_{\rm FCAD}(\rho),$ $\displaystyle=$
$\displaystyle(1-\mu)\sum_{i,j=0}^{2}E_{ij}\rho
E_{ij}^{\dagger}+\mu\sum_{k=0}^{2}A_{kk}\rho A_{kk}^{\dagger},$

where $\mu\in[0,1]$ is the correlation parameter. We can recover the
uncorrelated AD channel by setting $\mu=0$ and obtain the FCAD channel if
$\mu=1$. The explicit expressions of the Kraus operators $E_{ij}=E_{i}\otimes
E_{j}$ and $A_{kk}$ are determined by solving the Lindblad equation (please
see the details in Appendix A) [48, 68]

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\
0&\sqrt{\overline{d}_{1}}&0\\\
0&0&\sqrt{\overline{d}_{2}}\end{array}\right),E_{1}=\left(\begin{array}[]{ccc}0&\sqrt{d_{1}}&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$ (8) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&\sqrt{d_{2}}\\\ 0&0&0\\\
0&0&0\end{array}\right),$ (12)

with $d_{1}=1-\exp(-\gamma_{10}t)$, $d_{2}=1-\exp(-\gamma_{20}t)$ and the
symbol $\overline{d}=1-d$. The Kraus operators of the correlated part are
given as

$\displaystyle A_{00}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}{\rm\bf I}_{4\times 4}&&&\\\
&\sqrt{\overline{d}_{1}}&&\\\ &&{\rm\bf I}_{3\times 3}&\\\
&&&\sqrt{\overline{d}_{2}}\end{array}\right),A_{11}=\left(\begin{array}[]{c|c|c}{\rm\bf
0}_{1\times 4}&\sqrt{d_{1}}&{\rm\bf 0}_{1\times 4}\\\ \hline\cr{\rm\bf
0}_{4\times 4}&{\rm\bf 0}_{4\times 1}&{\rm\bf 0}_{4\times 4}\\\
\hline\cr{\rm\bf 0}_{4\times 4}&{\rm\bf 0}_{4\times 1}&{\rm\bf 0}_{4\times
4}\end{array}\right),$ (20) $\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c|c|c}{\rm\bf O}_{1\times 4}&{\rm\bf
0}_{1\times 4}&\sqrt{d_{2}}\\\ \hline\cr{\rm\bf 0}_{4\times 4}&{\rm\bf
0}_{4\times 4}&{\rm\bf 0}_{4\times 1}\\\ \hline\cr{\rm\bf 0}_{4\times
4}&{\rm\bf 0}_{4\times 4}&{\rm\bf 0}_{4\times 1}\end{array}\right),$ (24)

where ${\rm\bf I}_{n\times n}$ is the $n$-dimensional identity matrix and
${\rm\bf 0}_{n\times m}$ is the zero matrix of $n\times m$.

###  II.2 Standard quantum teleportation for qutrit

Quantum teleportation aims to transfer an unknown quantum state from the
sender Alice to the receiver Bob. In what follows we discuss how to implement
the teleportation of a qutrit state

$|\psi\rangle_{\rm in}=\alpha|0\rangle+\beta|1\rangle+\delta|2\rangle,$ (25)

with $\alpha=\cos\theta_{1}$,
$\beta=\sin\theta_{1}\cos\theta_{2}e^{i\phi_{1}}$,
$\delta=\sin\theta_{1}\sin\theta_{2}e^{i\phi_{2}}$ and
($0<\theta_{1},\theta_{2}\leq\frac{\pi}{2}$, $0<\phi_{1},\phi_{2}\leq 2\pi$).

(i) The first step in teleportation is to establish the entanglement between
Alice and Bob. This could be done in a number of ways. Here, we assume that
the entangled particle pairs (qutrit 2 and qutrit 3) are generated by the
third party Charlie

$|\Phi\rangle_{(23)}=\frac{1}{\sqrt{3}}(|00\rangle+|11\rangle+|22\rangle),$
(26)

where the subscript “(23)” denotes the qutrits 2 and 3, respectively. Charlie
distributes qutrit 2 and qutrit 3 to Alice and Bob through a public AD channel
and then via two separate noiseless private channels. Because of the
successive uses of the public AD channel, the correlation effects should be
taken into account. Therefore, the noisy channel can be modelled as a CAD
channel.

(ii) First, Alice interacts the qutrit 1 containing the teleported information
with her half of the entangled pair (qutrit 2) thorugh the logic gates $L_{C}$
and $H$, where
$H=\frac{1}{\sqrt{3}}\sum_{m,n=0}^{2}\mathrm{e}^{2\pi\mathrm{i}mn/3}|m\rangle\langle
n|$ is the Hadamard gate of the qutrit and
$L_{C}|m\rangle\otimes|n\rangle=|m\rangle\otimes|n\ominus m\rangle$ is the
CNOT left shift gate for two qutrits, where $\ominus$ denotes subtraction
module $3$. Then Alice performs measurements on her particles (qutrit 1 and
qutrit 2). Finally, Alice sends her result $(m,n)$ to Bob through a classical
channel.

(iii) According to the result $(m,n)$, Bob applies a series of recovery
operations $X^{-n}$ and $Z^{3-m}$ on his particle to obtain the teleported
state. $X^{n}|k\rangle=|k\oplus n\rangle$ and
$Z^{m}|k\rangle=\mathrm{e}^{2\pi\mathrm{i}km/3}|k\rangle$ are NOT gate and
phase gate for single qutrit.

In the noiseless case, Bob can obtain the exact state of the form of Eq. (25).
However, the inevitable noise in the procedure will make the state obtained by
Bob different from the teleported state, i.e., the fidelity is less than 1.

###  II.3 WM and QMR for qutrit

The WM that we consider is a positive operator valued measure (POVM), which is
different from the so-called strong measurement (i.e., von Neumann projective
measurement). WM does not completely destroy the measured system, thereby
retaining the measured state reversible. Although the WM for qubit has been
widely explored in many previous literatures [55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67], the specific form of WM for qutrit was first proposed by
Xiao in Ref. [69, 70]. The WM for single-qutrit is written as

$\displaystyle\mathcal{M}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\sqrt{\overline{p}_{1}}&0\\\ 0&0&\sqrt{\overline{p}_{2}}\end{array}\right).$
(30)

The parameters $p_{1},p_{2}\in[0,1]$ are usually known as the strengths of WM.
$\mathcal{M}$ and the other two POVM elements $\mathcal{M}_{1}={\rm
diag}(0,\sqrt{p_{1}},0)$ and $\mathcal{M}_{2}={\rm diag}(0,0,\sqrt{p_{2}})$
satisfy the completeness relation
$\mathcal{M}\mathcal{M}^{\dagger}+\mathcal{M}_{1}\mathcal{M}_{1}^{\dagger}+\mathcal{M}_{2}\mathcal{M}_{2}^{\dagger}=I$.
However, the measurement operators $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are
not reversible. We only focus on the measurement operator $\mathcal{M}$, which
maps the qutrit to a state that can still be recovered by proper operations,
e.g., QMR. Physically, the measurement operator $\mathcal{M}$ could be
realized by adding a detector to monitor the dissipation of the system.
Whenever there is a click, we discard the result. Therefore, this post-
selection removes the outcomes of $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ and
keeps the result of $\mathcal{M}$ (no click).

The QMR $\mathcal{R}$ for single-qutrit is

$\displaystyle\mathcal{R}=\left(\begin{array}[]{ccc}\sqrt{\overline{q}_{1}\overline{q}_{2}}&0&0\\\
0&\sqrt{\overline{q}_{2}}&0\\\
0&0&\sqrt{\overline{q}_{1}}\end{array}\right)=\mathcal{F}\mathcal{M}\mathcal{F}\mathcal{M}\mathcal{F},$
(34)

where $\mathcal{F}$ is the trit-flip operation $\mathcal{F}=|0\rangle\langle
2|+|1\rangle\langle 0|+|2\rangle\langle 1|$. Notice that the second equality
of Eq. (34) suggests that the QMR can be realized by the following five
sequential operations on the qutrit: trit-flip ($\mathcal{F}$), qutrit WM
($\mathcal{M}$), trit-flip ($\mathcal{F}$), another qutrit WM ($\mathcal{M}$),
and trit-flip ($\mathcal{F}$). The trit-flip operation $\mathcal{F}$ can be
realized by a $\pi$ pulse applied on the transition
$|1\rangle\leftrightarrow|2\rangle$ and followed by another $\pi$ pulse to
interchange the populations between $|0\rangle$ and $|1\rangle$, i.e., by the
series of two $\pi$ pulses
$\pi^{|1\rangle\leftrightarrow|2\rangle}\pi^{|0\rangle\leftrightarrow|1\rangle}$
[71].

##  III Enhancing the fidelity of teleportation by WM and QMR

We are now in a position to demonstrate the basic idea of our WM scheme. In
order to enhance the fidelity of teleportation, the pre-WMs and post-QMRs are
carried out before and after the CAD noise, as illustrated in Fig. 1(a).
Charlie performs WMs on the entangled pairs immediately after preparing the
initial entangled state. These pre-WMs are designed to project the entangled
state (26) onto a state that is insensitive to CAD noise. Then he distributes
them through the CAD channel. When Alice and Bob receive their qutrits, they
perform QMRs on their own qutrit. The QMRs aim to recover the initial
entanglement between qutrits 2 and 3. After these three steps, Alice and Bob
will share the entangled state (unnormalized)

$\rho_{(23)}^{\rm WM}=\mathcal{R}_{(23)}\mathcal{E}_{\rm
CAD}\Big{[}\mathcal{M}_{(23)}\big{(}|\Phi\rangle_{(23)}\langle\Phi|\big{)}\mathcal{M}_{(23)}^{\dagger}\Big{]}\mathcal{R}_{(23)}^{\dagger}.$
(35)

$\mathcal{M}_{(23)}$ and $\mathcal{R}_{(23)}$ are defined as
$\mathcal{M}_{(23)}=\mathcal{M}_{(2)}\otimes\mathcal{M}_{(3)}$ and
$\mathcal{R}_{(23)}=\mathcal{R}_{(2)}\otimes\mathcal{R}_{(3)}$.
$\rho_{(23)}^{\rm WM}$ is a $9\times 9$ matrix which has following non-zero
elements in the basis $\\{|j,k\rangle=|3j+k+1\rangle\\}$:

$\displaystyle\rho_{11}=$
$\displaystyle\frac{1}{3}\\{\overline{q}_{1}^{2}\overline{q}_{2}^{2}[1+(\overline{\mu}d_{1}^{2}+\mu
d_{1})\overline{p}_{1}^{2}+(\overline{\mu}d_{2}^{2}+\mu
d_{2})\overline{p}_{2}^{2}]\\},$ $\displaystyle\rho_{22}=$
$\displaystyle\rho_{44}=\frac{1}{3}\overline{\mu}d_{1}\overline{d}_{1}\overline{p}_{1}^{2}\overline{q}_{1}\overline{q}_{2}^{2},$
$\displaystyle\rho_{33}=$
$\displaystyle\rho_{77}=\frac{1}{3}\overline{\mu}d_{2}\overline{d}_{2}\overline{p}_{2}^{2}\overline{q}_{1}^{2}\overline{q}_{2},$
$\displaystyle\rho_{55}=$
$\displaystyle\frac{1}{3}(\overline{\mu}\overline{d}_{1}^{2}+\mu\overline{d}_{1})\overline{p}_{1}^{2}\overline{q}_{2}^{2},$
$\displaystyle\rho_{99}=$
$\displaystyle\frac{1}{3}(\overline{\mu}\overline{d}_{2}^{2}+\mu\overline{d}_{2})\overline{p}_{2}^{2}\overline{q}_{1}^{2},$
(36) $\displaystyle\rho_{15}=$
$\displaystyle\rho_{51}^{*}=\frac{1}{3}(\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}})\overline{p}_{1}\overline{q}_{1}\overline{q}_{2}^{2},$
$\displaystyle\rho_{19}=$
$\displaystyle\rho_{91}^{*}=\frac{1}{3}(\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}})\overline{p}_{2}\overline{q}_{1}^{2}\overline{q}_{2},$
$\displaystyle\rho_{59}=$
$\displaystyle\rho_{95}^{*}=\frac{1}{3}(\overline{\mu}\overline{d}_{1}\overline{d}_{2}+\mu\sqrt{\overline{d}_{1}\overline{d}_{2}})\overline{p}_{1}\overline{p}_{2}\overline{q}_{1}\overline{q}_{2}$

Through the procedure of quantum teleportation, Bob obtains the state

$\displaystyle\rho_{\rm out}^{\rm WM}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\left[\begin{array}[]{ccc}\epsilon_{00}&\epsilon_{01}&\epsilon_{02}\\\
\epsilon_{10}&\epsilon_{11}&\epsilon_{12}\\\
\epsilon_{20}&\epsilon_{21}&\epsilon_{22}\end{array}\right],$ (40)

where
$\epsilon_{00}=(\rho_{11}+\rho_{55}+\rho_{99})\alpha^{2}+(\rho_{33}+\rho_{44})|\beta|^{2}+(\rho_{22}+\rho_{77})|\delta|^{2}$,
$\epsilon_{11}=(\rho_{22}+\rho_{77})\alpha^{2}+(\rho_{11}+\rho_{55}+\rho_{99})|\beta|^{2}+(\rho_{33}+\rho_{44})|\delta|^{2}$,
$\epsilon_{22}=(\rho_{33}+\rho_{44})\alpha^{2}+(\rho_{22}+\rho_{77})|\beta|^{2}+(\rho_{11}+\rho_{55}+\rho_{99})|\delta|^{2}$,
$\epsilon_{01}=\epsilon_{10}^{*}=\alpha\beta^{*}(\rho_{15}+\rho_{59}+\rho_{91})$,
$\epsilon_{02}=\epsilon_{20}^{*}=\alpha\delta^{*}(\rho_{19}+\rho_{51}+\rho_{95})$,
$\epsilon_{12}=\epsilon_{21}^{*}=\beta\delta^{*}(\rho_{15}+\rho_{59}+\rho_{91})$,
and $N=\rho_{11}+\rho_{22}+\rho_{33}+\rho_{44}+\rho_{55}+\rho_{77}+\rho_{99}$
is the normalization factor.

The fidelity of this teleportation is now given by

$\displaystyle F^{\rm WM}$ $\displaystyle=_{\rm in}\langle\psi|\rho_{\rm
out}^{\rm WM}|\psi\rangle_{\rm in}$
$\displaystyle=A_{1}+(\alpha^{2}|\beta|^{2}+\alpha^{2}|\delta|^{2}+|\beta|^{2}|\delta|^{2})(A_{2}+A_{3}-2A_{1}),$

where $A_{1}=(\rho_{11}+\rho_{55}+\rho_{99})/N$,
$A_{2}=(\rho_{22}+\rho_{33}+\rho_{44}+\rho_{77})/N$ and
$A_{3}=(\rho_{15}+\rho_{51}+\rho_{19}+\rho_{91}+\rho_{59}+\rho_{95})/N$.

Considering that the teleported state $|\psi\rangle_{\rm in}$ is unknown, one
should calculate the average fidelity. Since Eq. (III) only contains the terms
$|\beta|^{2}$ and $|\delta|^{2}$, hence the average fidelity could be written
as

$\displaystyle\langle F\rangle^{\rm WM}$
$\displaystyle=\frac{2}{\pi^{2}}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}F^{\rm
WM}d\Omega,$ (42) $\displaystyle=\frac{1}{4}(1+A_{1}+A_{3}).$

with
$d\Omega=\sin^{3}\theta_{1}\cos\theta_{1}d\theta_{1}d\phi_{1}\sin\theta_{2}\cos\theta_{2}d\theta_{2}d\phi_{2}$.
It is obvious that the fidelity $\langle F\rangle^{\rm WM}$ is highly
dependent on the strength of the WM and QMR. In oder to fight the CAD noise
and obtain the maximum average fidelity, it is crucial to choose the optimal
strength of the post-QMRs.

It is generally believed that entanglement is the key resource of quantum
teleportation and that entanglement degradation is directly responsible for
the fidelity loss. Therefore, the most intuitive way to improve the average
fidelity is to choose a post-QMR with an appropriate strength that keeps the
entangled state $\rho_{(23)}^{\rm WM}$ as close as possible to the initial
state $|\Phi\rangle_{(23)}$. The decays of both uncorrelated the AD noise and
the FCAD noise could be regarded as quantum transitions that lead to the
excited states jumping to the only ground state with certain probabilities.
Thus, one can technically use the trick of ‘unraveling’ the excitation into
‘jump’ and ‘no-jump’ scenarios and work with pure states [71]. It can be shown
that in the CAD channel the optimal strength of the QMRs yield to (please see
the details in Appendix B)

$\displaystyle\overline{q}_{\rm 1,opt}^{\rm WM}$ $\displaystyle=$
$\displaystyle\overline{p}_{1}\Big{(}\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}}\Big{)},$
(43) $\displaystyle\overline{q}_{\rm 2,opt}^{\rm WM}$ $\displaystyle=$
$\displaystyle\overline{p}_{2}\Big{(}\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}}\Big{)}.$

For the sake of simplicity, we assume $d_{1}=d_{2}=d$, $p_{1}=p_{2}=p$ and
consequently $q_{1}=q_{2}=q$. Thus the optimal strength of post-QMR is

$q_{\rm opt}^{\rm
WM}=1-\overline{p}\Big{(}\overline{\mu}\overline{d}+\mu\sqrt{\overline{d}}\Big{)}.$
(44)

The average fidelity of Eq. (42) will be written as

$\displaystyle\langle F\rangle^{\rm WM}_{\rm
opt}=\frac{1}{4}+\frac{\big{[}2(\overline{\mu}d+\mu)d\overline{p}^{2}+5\big{]}(\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}+4(\overline{\mu}\overline{d}+\mu)}{4\Big{\\{}\big{[}2(\overline{\mu}d+\mu)d\overline{p}^{2}+1\big{]}(\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}+2(\overline{\mu}\overline{d}+\mu)+4\overline{\mu}d(\overline{\mu}\overline{d}+\mu\sqrt{\overline{d}})\Big{\\}}}.$
(45)

Since the WMs and QMRs are not unitary, the improvement of fidelity is not
deterministic but probabilistic. Under the condition of Eq. (44), the
probability of success is

$\displaystyle P_{\rm opt}^{\rm WM}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\overline{p}^{4}\overline{d}^{2}(\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}\Big{\\{}\big{[}2(\overline{\mu}d+\mu)d\overline{p}^{2}+1\big{]}(\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}$
(46) $\displaystyle+$ $\displaystyle
2(\overline{\mu}\overline{d}+\mu)+4\overline{\mu}d(\overline{\mu}\overline{d}+\mu\sqrt{\overline{d}})\Big{\\}}.$

Figure 2: (color online) (a) The average fidelity $\langle F\rangle_{\rm
opt}^{\rm WM}$ and (b) the probability of success $P_{\rm opt}^{\rm WM}$ as a
function of the noise strength $d$ for different values of $p$ with $\mu=0.8$.

In Fig. 2, we plot the average fidelity $\langle F\rangle_{\rm opt}^{\rm WM}$
and the probability of success $P_{\rm opt}^{\rm WM}$ as a function of the
noise strength $d$ for different values of $p$. From Fig. 2, we can summarize
the following conclusions. (i) In the absence of WM and QMR, i.e., $p=q=0$,
the average fidelity decreases as the noise strength increases, which directly
indicates the degradation of the entanglement initially shared between Alice
and Bob, since the fidelity depends on the entanglement. Note that the
fidelity bound of classical teleportation in a 3-dimensional single-qutrit
case is 1/2 [72]. The quantum advantage is always preserved in the CAD noise,
meaning that quantum teleportation is more robust to noise in higher-
dimensional systems. (ii) The fidelity can be partially improved by the post-
QMR even without the pre-WM ($p=0$). It could be considered as a simplified
version of error correction based on prior knowledge of the CAD noise. (iii)
The combination of pre-WM and post-QMR is indeed able to further enhance the
fidelity of teleportation in the CAD noise. (iv) The probability of success
decreases with increasing strength of pre-WM, which means that high fidelity
teleportation is realized at the expense of low probability of success, as
shown in Fig. 2(b).

Figure 3: (color online) (a) The average fidelity $\langle F\rangle_{\rm
opt}^{\rm WM}$ and (b) the probability of success $P_{\rm opt}^{\rm WM}$ as a
function of the correlated factor $\mu$ and WM strength $p$ with $d=0.6$.

Figure 3(a) displays the behavior of the fidelity $\langle F\rangle_{\rm
opt}^{\rm WM}$ as a function of $\mu$ and $p$ at a fixed value of the noisy
strength $d=0.6$. It is clear that the correlation effects have a positive
effect on the behavior of the fidelity, which is in consistent with that
obtained in Ref. [68]. On the other hand, the combined action of pre-WM and
post-QMR can further enhance fidelity based on correlation effects. The higher
the strength of the WM, the higher the fidelity. In the limit $p\rightarrow
1$, the fidelity is close to 1 because the entangled state
$|\Phi\rangle_{(23)}$ is completely recovered by the optimal QMR. Although the
optimal condition in Eq. (44) requires prior knowledge of the noise, it can
almost completely eliminate the effect of the CAD noise without knowing the
information of the teleported state. However, it should be noted that the
higher fidelity comes at the cost of a lower probability of success. Generally
speaking, there is a trade-oﬀ between fidelity improvement and probability of
success in these probabilistic schemes, and it should be carefully balanced in
realistic situations. For some tasks it may be preferable to have as high a
fidelity as possible regardless of the low probability of success, whereas for
other tasks the probability of success is more important if the fidelity is
high enough. Fortunately, the correlation effects can increase the probability
of success to some extent, as shown in Fig. 3(b).

##  IV Enhancing the fidelity of teleportation by EAM and QMR

Unlike WM, which is applied to the quantum system, EAM is applied to the
environment coupled to the system of interest [73]. Assuming that the
environment is in a vacuum state $|0\rangle_{\rm env}$, then the evolution of
the total system (i.e., qutrit plus environment) is determined by $\rho_{\rm
tot}=U(\rho_{\rm s}(0)\otimes|0\rangle_{\rm env}\langle 0|)U^{\dagger}$, where
$U=\exp(-i\hat{H}_{\rm tot}t)$ is the time-dependent evolution operator and
$\rho_{\rm s}(0)$ is the initial state of the system. The reduced density
matrix of the system $\rho_{\rm s}(t)$ can be obtained by tracing over the
environmental degrees of freedom, which yields to

$\displaystyle\rho_{\rm s}(t)$ $\displaystyle=$ $\displaystyle\sum_{n}{{}_{\rm
env}}\langle n|U|0\rangle_{\rm env}\rho_{\rm s}(0)_{\rm env}\langle
0|U^{\dagger}|n\rangle_{\rm env}$ (47) $\displaystyle=$
$\displaystyle\sum_{n}E_{n}\rho_{\rm s}(0)E_{n}^{\dagger},$

where $E_{n}={{}_{\rm env}}\langle n|U|0\rangle_{\rm env}$ is the Kraus
operator of the environmental noise. $\\{|n\rangle_{\rm env}\\}$ are the
complete basis of the environment.

A photon counting measurement on the excitation of the environment will
collapse the state of the environment into the $k^{\rm th}$ eigenstate if $k$
photons are collected. Subsequently, the system will be projected into a state
that corresponds to the $k^{\rm th}$ outcome [73, 74]:

$\rho^{k}_{\rm s}=E_{k}\rho_{\rm s}(0)E_{k}^{\dagger}.$ (48)

If the operator $E_{k}$ is reversible, then one can perform the reversal
operations to recover the initial state of the system. Obviously, only
$E_{00}$ and $A_{00}$ in Eq. (II.1) satisfy the reversibility condition. Thus,
we only focus on the measurement results of $k=0$ since they are reversible,
while discarding the results of $k\neq 0$. This post-selection allows the
system to collapse into a specific state that can be recovered by QMR.

According to the procedure illustrated in Fig. 1(b), after the sequences of
CAD, EAM and QMRs, Alice and Bob will share the entangled state (unnormalized)

$\rho_{(23)}^{\rm
EAM}=\mathcal{R}_{(23)}\Big{[}(1-\mu)E_{00}|\Phi\rangle_{(23)}\langle\Phi|E_{00}^{\dagger}+\mu
A_{00}|\Phi\rangle_{(23)}\langle\Phi|A_{00}^{\dagger}\Big{]}\mathcal{R}_{(23)}^{\dagger},$
(49)

which has the following non-zero elements:

$\displaystyle\rho^{\prime}_{11}=$
$\displaystyle\frac{1}{3}\\{\overline{q}_{1}^{2}\overline{q}_{2}^{2}(\mu
M_{1}+\overline{\mu}M_{2})\\},$ $\displaystyle\rho^{\prime}_{55}=$
$\displaystyle\frac{1}{3}(\mu\overline{d}_{1}M_{1}+\overline{\mu}\overline{d}_{1}^{2}M_{2})\overline{q}_{2}^{2},$
$\displaystyle\rho^{\prime}_{99}=$
$\displaystyle\frac{1}{3}(\mu\overline{d}_{2}M_{1}+\overline{\mu}\overline{d}_{2}^{2}M_{2})\overline{q}_{1}^{2},$
(50) $\displaystyle\rho^{\prime}_{15}=$
$\displaystyle\rho_{51}^{{}^{\prime}*}=\frac{1}{3}(\mu\sqrt{\overline{d}_{1}}M_{1}+\overline{\mu}\overline{d}_{1}M_{2})\overline{q}_{1}\overline{q}_{2}^{2},$
$\displaystyle\rho^{\prime}_{19}=$
$\displaystyle\rho_{91}^{{}^{\prime}*}=\frac{1}{3}(\mu\sqrt{\overline{d}_{2}}M_{1}+\overline{\mu}\overline{d}_{2}M_{2})\overline{q}_{1}^{2}\overline{q}_{2},$
$\displaystyle\rho^{\prime}_{59}=$
$\displaystyle\rho_{95}^{{}^{\prime}*}=\frac{1}{3}(\mu\sqrt{\overline{d}_{1}\overline{d}_{2}}M_{1}+\overline{\mu}\overline{d}_{1}\overline{d}_{2}M_{2})\overline{q}_{1}\overline{q}_{2},$

where $M_{1}=(1+\overline{d}_{1}^{2}+\overline{d}_{2}^{2})/3$ and
$M_{2}=(1+\overline{d}_{1}+\overline{d}_{2})/3$.

Through the procedure of quantum teleportation, Bob obtains the state

$\displaystyle\rho_{\rm out}^{\rm EAM}$ $\displaystyle=$
$\displaystyle\frac{1}{M}\left[\begin{array}[]{ccc}\epsilon^{\prime}_{00}&\epsilon^{\prime}_{01}&\epsilon^{\prime}_{02}\\\
\epsilon^{\prime}_{10}&\epsilon^{\prime}_{11}&\epsilon^{\prime}_{12}\\\
\epsilon^{\prime}_{20}&\epsilon^{\prime}_{21}&\epsilon^{\prime}_{22}\end{array}\right],$
(54)

where
$\epsilon^{\prime}_{00}=(\rho^{\prime}_{11}+\rho^{\prime}_{55}+\rho^{\prime}_{99})\alpha^{2}$,
$\epsilon^{\prime}_{11}=(\rho^{\prime}_{11}+\rho^{\prime}_{55}+\rho^{\prime}_{99})|\beta|^{2}$,
$\epsilon^{\prime}_{22}=(\rho^{\prime}_{11}+\rho^{\prime}_{55}+\rho^{\prime}_{99})|\delta|^{2}$,
$\epsilon^{\prime}_{01}=\epsilon_{10}^{{}^{\prime}*}=\alpha\beta^{*}(\rho^{\prime}_{15}+\rho^{\prime}_{59}+\rho^{\prime}_{91})$,
$\epsilon^{\prime}_{02}=\epsilon_{20}^{{}^{\prime}*}=\alpha\delta^{*}(\rho^{\prime}_{19}+\rho^{\prime}_{51}+\rho^{\prime}_{95})$,
$\epsilon^{\prime}_{12}=\epsilon_{21}^{{}^{\prime}*}=\beta\delta^{*}(\rho^{\prime}_{15}+\rho^{\prime}_{59}+\rho^{\prime}_{91})$,
and $M=\rho^{\prime}_{11}+\rho^{\prime}_{55}+\rho^{\prime}_{99}$ is the
normalization factor.

Figure 4: (color online) (a) The average fidelities $\langle F\rangle_{\rm
CAD}$, $\langle F\rangle_{\rm opt}^{\rm EAM}$ and (b) the probability of
success $P_{\rm opt}^{\rm EAM}$ as a function of the correlated factor $\mu$
and noise strength $d$.

The average fidelity of EAM scheme could be written as

$\langle F\rangle^{\rm EAM}=\frac{1}{4}(2+B_{1}),$ (55)

where
$B_{1}=(\rho^{\prime}_{15}+\rho^{\prime}_{19}+\rho^{\prime}_{59}+\rho^{\prime}_{51}+\rho^{\prime}_{91}+\rho^{\prime}_{95})/M$.
Again, we have to find the optimal strength of the post-QMRs. The details are
given in Appendix C and the optimal strength of QMR is

$q_{\rm opt}^{\rm
EAM}=1-\Big{(}\overline{\mu}\overline{d}+\mu\sqrt{\overline{d}}\Big{)},$ (56)

Substituting Eq. (56) into Eq. (55), the optimal average fidelity of the EAM
scheme is found to be

$\langle F\rangle_{\rm opt}^{\rm
EAM}=\frac{1}{2}+\frac{2[\overline{\mu}\sqrt{\overline{d}}(1+2\overline{d})+\mu(1+2\overline{d}^{2})](\overline{\mu}\sqrt{\overline{d}}+\mu)+\overline{\mu}\overline{d}+\mu+2\overline{d}^{2}}{2\Big{\\{}[2(\mu\overline{d}+\overline{\mu})\overline{d}+1](\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}+2(\overline{\mu}\overline{d}+\mu)+4\overline{d}^{2}\Big{\\}}}.$
(57)

The corresponding probability of success is

$P_{\rm opt}^{\rm
EAM}=\frac{1}{9}\overline{d}^{2}(\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}\Big{\\{}[2(\mu\overline{d}+\overline{\mu})\overline{d}+1](\overline{\mu}\sqrt{\overline{d}}+\mu)^{2}+2(\overline{\mu}\overline{d}+\mu)+4\overline{d}^{2}\Big{\\}}.$
(58)

In Fig. 4, we analyze the behaviors of $\langle F\rangle_{\rm opt}^{\rm EAM}$
and $P_{\rm opt}^{\rm EAM}$ as a function of correlated factor $\mu$ and noise
strength $d$. We also show the average fidelity $\langle F\rangle_{\rm CAD}$
in the case of CAD noise in Fig. 4(a). We observe that the proposed EAM scheme
significantly improves the fidelity compared to the original unprotected
teleportation. However, we also find that the $\langle F\rangle_{\rm opt}^{\rm
EAM}$ does not increase monotonically with the correlated factor $\mu$ for a
fixed value of $d$ (e.g., $d\rightarrow 1$). This can be understood as
follows: During the process of EAM, we have kept the measurement results of
$k=0$, which correspond to the evolution of both $E_{00}$ and $A_{00}$, QMR
cannot distinguish them exactly. Therefore, the fidelity can be restored to 1
by the operation of QMR only if AD noise ($\mu=0$) or FCAD noise ($\mu=1$) is
involved. In other cases ($0<\mu<1$), the fidelity can be dramatically
improved, but not fully recovered. Figure 4(b) shows that the probability of
success decreases with increasing $d$. The correlation effects also play a
positive role in increasing the probability of success, which is completely
similar to the WM situation shown in Fig. 2(b).

##  V Discussions

We have shown that both the WM and EAM schemes can significantly improve the
fidelity. It seems necessary to make some comparisons between them. Fig. 5(a)
shows the average fidelities $\langle F\rangle_{\rm CAD}$, $\langle
F\rangle_{\rm opt}^{\rm WM}$, $\langle F\rangle_{\rm opt}^{\rm EAM}$ as a
function of the correlated factor $\mu$ and the WM strength $p$ with $d=0.6$.
We find that $\langle F\rangle_{\rm opt}^{\rm WM}$ is always larger than
$\langle F\rangle_{\rm CAD}$, no matter what values $p$ and $\mu$ take. On the
other hand, $\langle F\rangle_{\rm opt}^{\rm EAM}$ is always greater than or
equal to $\langle F\rangle_{\rm opt}^{\rm WM}$ even if the strength of WM is
close to 1. The above conclusion is also valid for other values of $d$.

Another aspect to note is that both the WM and EAM schemes are probabilistic.
As we discussed earlier, the greater improvement in fidelity comes at the cost
of a lower probability of success in both schemes. Therefore, to make a fair
comparison, we need to consider the effect of the probability of success at
the same time. For this purpose, we introduce a quantity called balanced
fidelity improvement

$\langle F\rangle_{\rm imp}=\langle F\rangle_{\rm opt}^{\rm EAM}\times P_{\rm
opt}^{\rm EAM}-\langle F\rangle_{\rm opt}^{\rm WM}\times P_{\rm opt}^{\rm
WM}.$ (59)

The numerical simulation of the balanced fidelity improvement $F_{\rm imp}$ as
a function of the correlated factor $\mu$ and noise strength $d$ is shown in
Fig. 5(b). To obtain greater fidelity, we selected $p=0.9$ as the strength of
WM due to the limited fidelity improvement when $p$ is small. Notably, the
balanced fidelity improvement $\langle F\rangle_{\rm imp}$ is always positive,
which means that the EAM protocol is always better than the WM protocol even
when we consider the probability of success. The reason can be solely
attributed to the differences between WM and EAM. The EAM scheme is
distinguished from WM by the fact that it is performed after the CAD noise,
whereas WM is carried out before it. Essentially, WM only obtains information
about the system, while EAM gathers information about both the system and the
CAD noise. There can be no question that EAM collects more information than
WM. Therefore, it can be deduced that the EAM approach will exceed the WM
approach in enhancing the fidelity of teleportation.

Figure 5: (color online) (a) The average fidelities $\langle F\rangle_{\rm
CAD}$, $\langle F\rangle_{\rm opt}^{\rm WM}$, $\langle F\rangle_{\rm opt}^{\rm
EAM}$ as a function of the correlated factor $\mu$ and WM strength $p$ with
$d=0.6$. (b) The contour plot of the balanced fidelity improvement $\langle
F\rangle_{\rm imp}$ as a function of the correlated factor $\mu$ and noise
strength $d$ with $p=0.9$.

##  VI Conclusions

In summary, we propose two schemes to enhance the fidelity of qutrit
teleportation in the CAD noise. The crucial ingredient for the WM scheme is
the intentional inclusion of pre-WMs. This results in the initial entangled
state collapsing partially towards the $|00\rangle$ state due to the state’s
inactivity in CAD noise. Due to the reversibility of WM, the fidelity of
teleportation could be improved with the help of post-QMRs. The EAM scheme is
based on the idea of feedback control, where an EAM is performed on the
environment to detect the information and thereby restoring the fidelity by
QMRs with a certain probability. Although they are probabilistic, the
correlation effects also contribute positively to improving the probability of
success in both schemes. Moreover, a comprehensive comparison indicates that
the EAM scheme consistently surpasses the WM approach in terms of fidelity.
Our findings indicate that the methods of WM and EAM possess the ability to
handle not only AD noise but also effectively reduce CAD noise. Such a
capability is crucial for various tasks of quantum information processing,
especially when the transmission speed of information is too high to neglect
the correlation effects.

###### Acknowledgements.

This work was supported by the Funds of the National Natural Science
Foundation of China under Grant Nos. 12265004, 12365003, 12205054 and Jiangxi
Provincial Natural Science Foundation, China under Grant Nos. 20212ACB211004.

##  Appendix A Kraus operators of qutrit CAD noise

The AD noise models the dissipative interaction between a quantum system and
with zero-temperature bath. For the case of $V$-type qutrit system with levels
$|0\rangle$, $|1\rangle$ and $|2\rangle$, the time evolution is determined by
the master equation

$\dot{\rho}=-i[\hat{H},\rho]+\mathcal{L}\rho,$ (60)

where $H$ is the Hamiltonian and $\mathcal{L}$ is the Lindblad superoperator.
Under the Born-Markov approximation, the Lindblad superoperator is

$\displaystyle\mathcal{L}\rho$ $\displaystyle=$
$\displaystyle\frac{\gamma_{10}}{2}(2\sigma_{01}\rho\sigma_{10}-\sigma_{11}\rho-\rho\sigma_{11})$
(61) $\displaystyle+$
$\displaystyle\frac{\gamma_{20}}{2}(2\sigma_{02}\rho\sigma_{20}-\sigma_{22}\rho-\rho\sigma_{22}),$

where $\gamma_{i0}$ are the spontaneous rates of the $i$th level.
$\sigma_{ij}=|i\rangle\langle j|$ is a transition operator between $|i\rangle$
and $|j\rangle$. Note that we have considered the forbidden transition between
two excited states $|1\rangle$ and $|2\rangle$. According to the Kraus
decomposition $\rho(t)=\sum_{i}E_{i}\rho E_{i}^{\dagger}$, the Kraus operators
of AD noise can be obtained as

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\
0&\sqrt{\overline{d}_{1}}&0\\\
0&0&\sqrt{\overline{d}_{2}}\end{array}\right),E_{1}=\left(\begin{array}[]{ccc}0&\sqrt{d_{1}}&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$ (68) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&\sqrt{d_{2}}\\\ 0&0&0\\\
0&0&0\end{array}\right),$ (72)

with $d_{1}=1-\exp(-\gamma_{10}t)$, $d_{2}=1-\exp(-\gamma_{20}t)$. The Kraus
operators of two-qutrit uncorrelated AD channel are tensor products of the
single-qutrit Kraus operators $E_{ij}=E_{i}\otimes E_{j}$.

For characterizing the correlation effects of two qutrits successively passes
through the same AD noise, it is necessary to calculate the correlated
Lindblad equation of the two-qutrit state $\tilde{\rho}$. Note that
$\tilde{\rho}\in C^{9\times 9}$ is the total state of two qutrits. The
correlated version of the Lindblad superoperator $\tilde{\mathcal{L}}$ can be
expressed as [75]

$\displaystyle\tilde{\mathcal{L}}\tilde{\rho}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{10}}{2}\big{(}2\sigma_{01}^{\otimes
2}\tilde{\rho}\sigma_{10}^{\otimes 2}-\sigma_{11}^{\otimes
2}\tilde{\rho}-\tilde{\rho}\sigma_{11}^{\otimes 2}\big{)}$ (73)
$\displaystyle+$
$\displaystyle\frac{\gamma_{20}}{2}\big{(}2\sigma_{02}^{\otimes
2}\tilde{\rho}\sigma_{20}^{\otimes 2}-\sigma_{22}^{\otimes
2}\tilde{\rho}-\tilde{\rho}\sigma_{22}^{\otimes 2}\big{)},$

where $\sigma_{ij}^{\otimes 2}=\sigma_{ij}\otimes\sigma_{ij}$. The above
equation describes the physical process for which the decay emerges
synchronously for both qutrits. The Kraus operators of the correlated part of
the evolution of two-qutrit state are given as [48, 76]

$\displaystyle A_{00}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}{\rm I}_{4\times 4}&&&\\\
&\sqrt{\overline{d}_{1}}&&\\\ &&{\rm I}_{3\times 3}&\\\
&&&\sqrt{\overline{d}_{2}}\end{array}\right),A_{11}=\left(\begin{array}[]{c|c|c}{\rm
O}_{1\times 4}&\sqrt{d_{1}}&{\rm O}_{1\times 4}\\\ \hline\cr{\rm O}_{4\times
4}&{\rm O}_{4\times 1}&{\rm O}_{4\times 4}\\\ \hline\cr{\rm O}_{4\times
4}&{\rm O}_{4\times 1}&{\rm O}_{4\times 4}\end{array}\right),$ (81)
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c|c|c}{\rm O}_{1\times 4}&{\rm O}_{1\times
4}&\sqrt{d_{2}}\\\ \hline\cr{\rm O}_{4\times 4}&{\rm O}_{4\times 4}&{\rm
O}_{4\times 1}\\\ \hline\cr{\rm O}_{4\times 4}&{\rm O}_{4\times 4}&{\rm
O}_{4\times 1}\end{array}\right)$ (85)

##  Appendix B Derivation of Eq. (43)

The most critical factor in preserving fidelity is to protect the initial
shared entanglement between Alice and Bob from the CAD noise. An intuitive
method for protecting entanglement is to ensure the finally entangled state as
close as possible to the initially entangled state. From the perspective of
pure state, one can find a optimal QMR by using the trick of ‘unraveling’ the
excitation into ‘jump’ and ‘no jump’. We assume the initially entangled state
has the form as $(a|00\rangle+b|11\rangle+c|22\rangle)_{\rm s}$ and the
environment is in a vacuum state $|\bf{0}\rangle_{\rm env}$. The procedures of
WM, CAD and QMR can be expressed as:

$\displaystyle\Big{(}a|00\rangle+b|11\rangle+c|22\rangle\Big{)}_{\rm
s}|\bf{0}\rangle_{\rm env}$ (86) $\displaystyle\xrightarrow{\rm WM}$
$\displaystyle\Big{(}a|00\rangle+b\overline{p}_{1}|11\rangle+c\overline{p}_{2}|22\rangle\Big{)}_{\rm
s}|\bf{0}\rangle_{\rm env}$ (87) $\displaystyle\xrightarrow{\rm CAD}$
$\displaystyle\overline{\mu}\Big{(}a|00\rangle+b\overline{p}_{1}\overline{d}_{1}|11\rangle+c\overline{p}_{2}\overline{d}_{2}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm
env}+\overline{\mu}b\overline{p}_{1}\sqrt{d_{1}\overline{d}_{1}}\Big{(}|01\rangle+|10\rangle\Big{)}_{\rm
s}|\bf{1}\rangle_{\rm env}$
$\displaystyle+\overline{\mu}c\overline{p}_{2}\sqrt{d_{2}\overline{d}_{2}}\Big{(}|02\rangle+|20\rangle\Big{)}_{\rm
s}|\textbf{1}\rangle_{\rm
env}+\overline{\mu}\Big{(}b\overline{p}_{1}d_{1}|00\rangle+c\overline{p}_{2}d_{2}|00\rangle\Big{)}_{\rm
s}|\bf 2\rangle_{\rm env}$
$\displaystyle+\mu\Big{(}a|00\rangle+b\overline{p}_{1}\sqrt{\overline{d}_{1}}|11\rangle+c\overline{p}_{2}\sqrt{\overline{d}_{2}}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm
env}+\mu\Big{(}b\overline{p}_{1}\sqrt{d_{1}}+c\overline{p}_{2}\sqrt{d_{2}}\Big{)}|00\rangle_{\rm
s}|\bf 2\rangle_{\rm env}$ $\displaystyle\xrightarrow{\rm QMR}$
$\displaystyle\Big{[}a\overline{q}_{1}\overline{q}_{2}|00\rangle+b\overline{p}_{1}\overline{q}_{2}\big{(}\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}}\big{)}|11\rangle+c\overline{p}_{2}\overline{q}_{1}\big{(}\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}}\big{)}|22\rangle\Big{]}_{\rm
s}|\bf{0}\rangle_{\rm env}$
$\displaystyle+\overline{\mu}b\overline{p}_{1}\sqrt{d_{1}\overline{d}_{1}}\overline{q}_{2}\sqrt{\overline{q}_{1}}\Big{(}|01\rangle+|10\rangle\Big{)}_{\rm
s}|\textbf{1}\rangle_{\rm
env}+\overline{\mu}c\overline{p}_{2}\sqrt{d_{2}\overline{d}_{2}}\overline{q}_{1}\sqrt{\overline{q}_{2}}\Big{(}|02\rangle+|20\rangle\Big{)}_{\rm
s}|\bf{1}\rangle_{\rm env}$
$\displaystyle+\Big{[}b\overline{p}_{1}\overline{q}_{1}\overline{q}_{2}\big{(}\overline{\mu}d_{1}+\mu\sqrt{d_{1}}\big{)}+c\overline{p}_{2}\overline{q}_{1}\overline{q}_{2}\big{(}\overline{\mu}d_{2}+\mu\sqrt{d_{2}}\big{)}\Big{]}|00\rangle_{\rm
s}|\bf{2}\rangle_{\rm env},$

where $|\textbf{n}\rangle_{\rm env}$ denotes $n$ excitations in the
environment. In order to ensure Eq. (B) is close to Eq. (86), one can choose

$\displaystyle\overline{q}_{\rm 1,opt}^{\rm WM}$ $\displaystyle=$
$\displaystyle\overline{p}_{1}\Big{(}\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}}\Big{)},$
(90) $\displaystyle\overline{q}_{\rm 2,opt}^{\rm WM}$ $\displaystyle=$
$\displaystyle\overline{p}_{2}\Big{(}\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}}\Big{)}.$

Then Eq. (B) reduces to (up to a normalized factor $1/\sqrt{\overline{q}_{\rm
1,opt}^{\rm WM}\overline{q}_{\rm 2,opt}^{\rm WM}}$)

$\displaystyle\Big{(}a|00\rangle+b|11\rangle+c|22\rangle\Big{)}_{\rm
s}|{\bf{0}}\rangle_{\rm
env}+\overline{\mu}b\overline{p}_{1}\sqrt{\frac{d_{1}\overline{d}_{1}}{\overline{q}_{\rm
1,opt}^{\rm WM}}}\Big{(}|01\rangle+|10\rangle\Big{)}_{\rm
s}|\textbf{1}\rangle_{\rm
env}+\overline{\mu}c\overline{p}_{2}\sqrt{\frac{d_{2}\overline{d}_{2}}{\overline{q}_{\rm
2,opt}^{\rm WM}}}\Big{(}|02\rangle+|20\rangle\Big{)}_{\rm
s}|\bf{1}\rangle_{\rm env}$
$\displaystyle+\Big{[}b\overline{p}_{1}\big{(}\overline{\mu}d_{1}+\mu\sqrt{d_{1}}\big{)}+c\overline{p}_{2}\big{(}\overline{\mu}d_{2}+\mu\sqrt{d_{2}}\big{)}\Big{]}|00\rangle_{\rm
s}|\bf{2}\rangle_{\rm env}.$ (91)

It is interesting to note that the state of (B) becomes closer to (86) with
the increasing strength of WMs. Particularly, when $p_{1},p_{2}\rightarrow 1$,
the last three terms in Eq. (B) can be neglected and the initial state (86) is
exactly restored.

##  Appendix C Derivation of Eq. (56)

We start with the same initial state, but the sequence of the operations is
CAD, EAM and QMR.

$\displaystyle\Big{(}a|00\rangle+b|11\rangle+c|22\rangle\Big{)}_{\rm
s}|\bf{0}\rangle_{\rm env}$ (92) $\displaystyle\xrightarrow{\rm CAD}$
$\displaystyle\overline{\mu}\Big{(}a|00\rangle+b\overline{d}_{1}|11\rangle+c\overline{d}_{2}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm
env}+\overline{\mu}b\sqrt{d_{1}\overline{d}_{1}}\Big{(}|01\rangle+|10\rangle\Big{)}_{\rm
s}|\bf{1}\rangle_{\rm env}$
$\displaystyle+\overline{\mu}c\sqrt{d_{2}\overline{d}_{2}}\Big{(}|02\rangle+|20\rangle\Big{)}_{\rm
s}|\textbf{1}\rangle_{\rm
env}+\overline{\mu}\Big{(}bd_{1}|00\rangle+cd_{2}|00\rangle\Big{)}_{\rm s}|\bf
2\rangle_{\rm env}$
$\displaystyle+\mu\Big{(}a|00\rangle+b\sqrt{\overline{d}_{1}}|11\rangle+c\sqrt{\overline{d}_{2}}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm
env}+\mu\Big{(}b\sqrt{d_{1}}+c\sqrt{d_{2}}\Big{)}|00\rangle_{\rm s}|\bf
2\rangle_{\rm env}$ $\displaystyle\xrightarrow{\rm EAM}$
$\displaystyle\overline{\mu}\Big{(}a|00\rangle+b\overline{d}_{1}|11\rangle+c\overline{d}_{2}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm
env}+\mu\Big{(}a|00\rangle+b\sqrt{\overline{d}_{1}}|11\rangle+c\sqrt{\overline{d}_{2}}|22\rangle\Big{)}_{\rm
s}|\textbf{0}\rangle_{\rm env}$ (94) $\displaystyle\xrightarrow{\rm QMR}$
$\displaystyle\Big{[}a\overline{q}_{1}\overline{q}_{2}|00\rangle+b\overline{q}_{2}\big{(}\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}}\big{)}|11\rangle+c\overline{q}_{1}\big{(}\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}}\big{)}|22\rangle\Big{]}_{\rm
s}|\bf{0}\rangle_{\rm env}.$ (95)

It is easy to find that the choose of $\overline{q}_{1,\rm opt}^{\rm
EAM}=\overline{\mu}\overline{d}_{1}+\mu\sqrt{\overline{d}_{1}}$ and
$\overline{q}_{2,\rm opt}^{\rm
EAM}=\overline{\mu}\overline{d}_{2}+\mu\sqrt{\overline{d}_{2}}$ can recover
the initial state. If we assume $d_{1}=d_{2}=d$, then the optimal QMR of EAM
scheme is obtained as

$q_{\rm opt}^{\rm
EAM}=1-\Big{(}\overline{\mu}\overline{d}+\mu\sqrt{\overline{d}}\Big{)}.$ (96)

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