# Noise mitigation in quantum teleportation

Zi-Jian Xu  School of Physical Science and Technology & Lanzhou Center for
Theoretical Physics, Lanzhou University, Lanzhou 730000, China  Key Laboratory
of Quantum Theory and Applications of MoE & Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Jun-Hong
An* —<EMAIL_ADDRESS>School of Physical Science and Technology & Lanzhou
Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China  Key
Laboratory of Quantum Theory and Applications of MoE & Key Laboratory of
Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China

###### Abstract

Permitting the transmission of unknown quantum states over long distances by
using entanglement, quantum teleportation serves as an important building
block for many quantum technologies. However, in the noisy intermediate-scale
quantum era, the practical realization of quantum teleportation is inevitably
challenged by the noise-induced decoherence. We here propose a noise
mitigation mechanism applicable in both the discrete- and continuous-variable
quantum teleportation schemes. Via investigating the non-Markovian decoherence
dynamics of the two types of quantum teleportation schemes, we find that, as
long as a bound state is formed in the energy spectrum of the total system
consisting of the involved subsystems and their respective reservoirs, the
quantum superiority of the fidelity is persistently recovered. Supplying an
insightful understanding on the noise mitigation protocols, our result paves
the way to the practical realization of noise-tolerant quantum teleportation.

##  I Introduction

Quantum teleportation allows the transfer of an arbitrary unknown quantum
state between distant parties by using quantum entanglement [1, 2, 3]. It is
not only a basic protocol of quantum communication, but also a building block
in the realization of various large-scale quantum technologies. It plays
important roles in quantum repeaters [4], which is pivotal for quantum
communication over long distances [5], quantum computation [6, 7, 8, 9],
quantum network [10], and quantum secret sharing [11]. It is also used as a
tool for exploring fundamental physics, such as closed time-like curves [12]
and black-hole evaporation [13]. Following two parallel directions to encode
information in discrete- [1] or continuous-variable [14] system, quantum
teleportation has been realized in various systems [15, 16, 17, 18, 19, 20].
Outstanding performances have been achieved in terms of teleportation distance
in aid of the satellite [21].

Being in the noisy intermediate-scale quantum era [22, 23, 24], we are
generally forced to make a persistent effort to beat the unwanted decoherence
effect induced by different kinds of noises in quantum technologies. An ideal
teleportation process requires a noiseless quantum channel established by
sharing a pure maximally entangled state. The protection of quantum channel
from the decoherence-induced disentanglement is a prerequisite for the
realization and application of quantum teleportation [25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. It has been found that
the quantum advantages of teleportation typically disappears in a noisy
setting, with decoherence degrading quantum to classical. Many strategies, for
example, error correction [38], weak measurement [39], entanglement
purification [25], and noiseless linear amplification [41], have been proposed
to overcome the destructive impacts of the decoherence on quantum
teleportation. However, the analysis and methodologies considered thus far to
address the decoherence effect on quantum teleportation are based on the Born-
Markovian approximation. Given the inherent non-Markovian nature of the
decoherence dynamics [44, 45, 46, 47], it is expected that such treatments are
insufficient. Furthermore, with the rapid development of experimental
techniques, more and more quantum systems have exhibited the non-Markovian
effect in experiments [48, 49, 50, 51, 52, 53, 54, 55]. It was really found
that the non-Markovian effect may play an active role in retrieving the
quantum superiority from the noises in several protocols of quantum
technologies [56, 57, 58, 59, 60, 61, 62]. A natural question is: What is the
non-Markovian effect on the noisy quantum teleportation?

Figure 1: Schematic diagrams of (a) discrete- and (b) continuous-variable
quantum teleportation.

In this work, we propose a mechanism to recover the quantum superiority of
teleportation influenced by the local dissipative noises. Via investigating
the non-Markovian decoherence dynamics of the teleportation schemes for both
the discrete- and continuous-variable systems, we find that the teleportation
performance is sensitively determined by the feature of the energy spectrum of
the total system formed by the involved subsystems of the quantum channels and
their local reservoirs. When a bound state is present in the energy spectrum,
the fidelity for both the teleportation schemes in the long-time steady state
surpasses their corresponding classical limits, which are also the achievable
bounds under the Born-Markovian approximate description to the decoherence.
Our result demonstrates that the non-Markovian effect has a self-healing
ability to rescue the teleportation from the noise-induced degradation.
Refreshing our understanding of the noise effect on quantum teleportation, our
result paves the way for overcoming the destructive influence of the
decoherence on quantum teleportation in practical noises.

This paper is organized as follows. In Sec. II, we begin by presenting the
ideal quantum teleportation schemes for both the discrete- and continuous-
variable systems. In Sec. III, we study the non-Markovian decoherence dynamics
of the two teleportation schemes. The dominate role of the bound state of the
involved subsystems of the quantum channels and their respective local
reservoirs in the teleportation performance is revealed in this section.
Section IV is devoted to the numerical calculation to verify our result. We
conclude the paper after a brief discussion on the experimental relevance in
Sec. V.

##  II Ideal quantum teleportation

Quantum teleportation is the transfer of an unknown quantum state over a long
distance by using quantum channel formed by entanglement and classical channel
via classical communication [1]. Depending on the properties of the Hilbert
space of the involved systems, it can be classified into discrete- and
continuous-variable [14, 18, 63, 17] quantum teleportation.

In the case of the discrete-variable teleportation, both the quantum channel
and the unknown state are constructed by two-level systems (TLSs) [16], see
Fig. 1(a). Consider that Alice wants to transfer an unknown state
$\ket{\varphi_{1}}=a\ket{e_{1}}+b\ket{g_{1}}$ carried by the first TLS on her
hands to Bob, where $|e_{l}\rangle$ and $|g_{l}\rangle$ are the excited and
ground states of the $l$th TLS, respectively. She first shares an entangled
pair of TLSs in the state
$\ket{\Phi_{23}}=\frac{1}{\sqrt{2}}(\ket{e_{2}e_{3}}+\ket{g_{2}g_{3}})$ as a
quantum channel with Bob. The quantum channel evolves with time under a
unitary dynamics governed by the Hamiltonian
$\hat{H}_{0}=\omega_{0}\sum_{l=2}^{3}\hat{\sigma}_{l}^{\dagger}\hat{\sigma}_{l}$,
with $\hat{\sigma}_{l}=|g_{l}\rangle\langle e_{l}|$. Thus, the state of the
tripartite TLS dynamically evolves to

$\displaystyle\ket{\Phi_{\text{T}}(t)}$ $\displaystyle=$ $\displaystyle
2^{-1}[|\Phi^{+}_{12}\rangle\hat{I}_{3}+|\Phi^{-}_{12}\rangle\hat{\sigma}^{z}_{3}+|\Psi^{+}_{12}\rangle\hat{\sigma}_{3}^{x}$
(1) $\displaystyle+|\Psi^{-}_{12}\rangle
i\hat{\sigma}_{3}^{y}](ae^{-i2\omega_{0}t}|e_{3}\rangle+|g_{3}\rangle),$

where $|\Phi^{\pm}\rangle=(|ee\rangle\pm|gg\rangle)/\sqrt{2}$ and
$|\Psi^{\pm}\rangle=(|eg\rangle\pm|ge\rangle)/\sqrt{2}$ are the four Bell
states [64]. Then, Alice makes a Bell-state measurement to her two TLSs. If
the $k$th Bell state $|\text{Bell}_{12}^{(k)}\rangle$, with
$|\text{Bell}\rangle\in\\{|\Phi^{\pm}\rangle,|\Psi^{\pm}\rangle\\}$, is
obtained, the state of Bob’s TLS collapses to
$\langle\text{Bell}_{12}^{(k)}|\Phi_{123}(t)\rangle$. Alice tells Bob her
measurement result $k$ via the classical channel. Finally, Bob performs a
local operation $\hat{U}_{3}^{(k)}$ on his TLS according to the result $k$. If
the measurement result is $|\Phi_{12}^{+}\rangle$, $|\Phi_{12}^{-}\rangle$,
$|\Psi_{12}^{+}\rangle$, and $|\Psi_{12}^{-}\rangle$, he chooses the operation
$\hat{I}$, $\hat{\sigma}^{z}$, $\hat{\sigma}^{x}$, and $-i\hat{\sigma}^{y}$,
respectively. Bob’s final state then becomes

$|\varphi_{3}^{(k)}\rangle=P_{k}^{-1/2}\hat{U}_{3}^{(k)}\langle\text{Bell}^{(k)}_{12}|\Phi_{\text{T}}(t)\rangle,$
(2)

where $P_{k}=|\langle\text{Bell}_{12}^{(k)}|\Phi_{\text{T}}(t)\rangle|^{2}$.
Setting $a=\cos(\theta/2)$ and $b=\sin(\theta/2)\mathit{e}^{\mathit{i}\phi}$,
we can evaluate the average fidelity as

$\displaystyle\bar{F}$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\frac{\sin\theta}{4\pi}d\theta\int_{0}^{2\pi}d\phi\sum_{k=1}^{4}P_{k}|\langle\varphi|\varphi_{3}^{(k)}\rangle|^{2}$
(3) $\displaystyle=$ $\displaystyle[2+\cos(2\omega_{0}t)]/3.$

Thus, a maximal average fidelity $\max_{t}\bar{F}(t)=1$ is achieved at times
$t=n\pi/\omega_{0}$, with $n$ being positive integers.

In the case of the continuous-variable teleportation, the information is
encoded in quantized optical fields. When Alice wants to transfer an unknown
coherent state $|\varphi_{1}\rangle=\hat{D}_{1}(\alpha)|0_{1}\rangle$, with
$\hat{D}_{1}(\alpha)=e^{\alpha\hat{a}_{1}^{\dagger}-\alpha^{*}\hat{a}_{1}^{\dagger}}$,
to Bob, she first shares a two-mode squeezed vacuum state
$|\Phi_{23}\rangle=\exp[r(\hat{a}_{2}\hat{a}_{3}-\hat{a}^{\dagger}_{2}\hat{a}_{3}^{\dagger})]|0_{2}0_{3}\rangle$
as a quantum channel with Bob [65], see Fig. 1(b). Considering the unitary
dynamics governed by
$\hat{H}_{0}=\omega_{0}\sum_{l=2}^{3}\hat{a}_{l}^{\dagger}\hat{a}_{l}$ of the
quantum channel, the state of the total system reads

$\displaystyle|\Phi_{\text{T}}(t)\rangle=\cosh^{-1}re^{\frac{-|\alpha|^{2}}{2}-\alpha\hat{a}_{1}^{\dagger}-\tanh
re^{-2i\omega_{0}t}\hat{a}_{2}^{\dagger}\hat{a}_{3}^{\dagger}}|0_{1}0_{2}0_{3}\rangle.$
(4)

Then, Alice makes a homodyne measurement to the two optical fields on her
hands. It consists of the following two steps. The first step Alice needs to
do is to couple the optical fields $\hat{a}_{1}$ and $\hat{a}_{2}$ by a 50:50
beam splitter, which can be described by
$\hat{V}=\exp[\frac{\pi}{4}(\hat{a}_{1}^{\dagger}\hat{a}_{2}-\hat{a}_{2}^{\dagger}\hat{a}_{1})]$.
The second one is to measure two commuting quadrature operators
$\hat{X}_{1}=(\hat{a}_{1}+\hat{a}_{1}^{\dagger})/2$ and
$\hat{P}_{2}=(\hat{a}_{2}-\hat{a}_{2}^{\dagger})/(2i)$. The homodyne
measurement, with the results $x_{1}$ and $p_{2}$, collapses the state of
Bob’s optical field into

$\displaystyle{\langle
x_{1}p_{2}|\hat{V}|\Phi_{\text{T}}(t)\rangle\over\sqrt{P_{z}}}$
$\displaystyle=$ $\displaystyle
c(z)e^{(\alpha-\sqrt{2}z)e^{-i2\omega_{0}t}\tanh
r\hat{a}^{\dagger}_{3}}|0_{3}\rangle,~{}$ (5)

where $z=x_{1}-ip_{2}$, $c(z)={e^{-{|\alpha|^{2}\over
2}-|z|^{2}+\sqrt{2}\alpha z^{*}}\over\sqrt{2^{-1}\pi P_{z}}\cosh r}$,
$\hat{X}_{1}|x_{1}\rangle=x_{1}|x_{1}\rangle$,
$\hat{P}_{2}|p_{2}\rangle=p_{2}|p_{2}\rangle$, and $P_{z}=|\langle
x_{1}p_{2}|\hat{V}|\Phi_{\text{T}}(t)\rangle|^{2}$. In the derivation of Eq.
(5), $\langle
p_{1}x_{2}|0_{1}0_{2}\rangle=({2\over\pi})^{1/2}e^{-x^{2}-p^{2}}$ has been
used. Alice tells Bob her measurement result $z$ via the classical channel.
Finally, Bob makes a displacement operation $\hat{D}_{3}(\sqrt{2}z)$ on his
optical field to convert Eq. (5) into

$\displaystyle|\varphi^{(z)}_{3}\rangle=c(z)\big{|}[\sqrt{2}z+(\alpha-\sqrt{2}z)e^{-i2\omega_{0}t}\tanh
r]_{3}\big{\rangle}.$ (6)

If a series of teleported states are in sequence given to Alice and the
homodyne detector is able to respond to all of eigenvalues of the quadrature
operators in the teleportation, then the average fidelity over all the
measurement results is [66]

$\displaystyle\bar{F}(t)=\int
dx_{1}dp_{2}P_{z}|\langle\alpha|\varphi^{(z)}_{3}\rangle|^{2}={1/2\cosh^{-2}r\over
1-\tanh r\cos(2\omega_{0}t)}.$ (7)

A maximal average fidelity $\max_{t}\bar{F}(t)=(1+e^{-2r})^{-1}$ is achieved
when $t=n\pi/\omega_{0}$. It tends to one when $r$ approaches infinity.

##  III Noisy effects

In practice, the quantum channel is inevitably influenced by the noise-induced
decoherence. It causes the degradation of the entanglement and deteriorates
the performance of quantum teleportation. Depending on whether the system in
the channel has energy exchange with the noise or not, the decoherence is
classified into dissipation and dephasing. The description of decoherence is
based on the idea of the open system. Starting from the unitary dynamics of
the total system formed the system and its noise and tracing over the degrees
of freedom the noise, the dynamics of the open system is achieved. A widely
used approximation during this procedure is the Born-Markovian approximation
[67]. We here will go beyond this approximation [44, 45, 46] and exactly
evaluate the effects of local dissipative noises on the two types of schemes
of quantum teleportation.

The dissipative noise can be described by a bosonic reservoir. Consider that
each subsystem of the quantum channel feels a local reservoir. The Hamiltonian
for both the discrete- and continuous-variable quantum channels can be
universally written as $\hat{H}=\sum_{l=2}^{3}\hat{H}_{l}$, with

$\hat{H}_{l}=\omega_{0}\hat{o}_{l}^{\dagger}\hat{o}_{l}+\sum_{k}[\omega_{k}\hat{b}_{l,k}^{\dagger}\hat{b}_{l,k}+(g_{k}\hat{b}_{l,k}^{\dagger}\hat{o}_{l}+\text{h.c.})],$
(8)

where $\hat{b}_{l,k}$ is the annihilation operator of the $k$th mode with
frequency $\omega_{k}$ of the reservoir felt by the $l$th subsystem of the
quantum channel, $g_{k}$ is the coupling strength, $\hat{o}=\hat{\sigma}$ for
the discrete-variable teleportation, and $\hat{o}=\hat{a}$ for the continuous
one. Commonly, the coupling strength is further characterized by the spectral
density $J(\omega)=\sum_{k}\left|g_{k}\right|^{2}\delta(\omega-\omega_{k})$.
It may have the form
$J(\omega)=\eta\omega^{s}\omega_{c}^{1-s}e^{-\omega/\omega_{c}}$, where $\eta$
is a dimensionless coupling constant, $\omega_{c}$ is a cut-off frequency to
avoid infrared catastrophe, and $s$ is the Ohmicity parameter. The reservoir
is classified into sub-Ohmic for $0<s<1$, Ohmic for $s=1$, and super-Ohmic for
$s>1$ [68]. Under the condition that the reservoirs are initially in the
vacuum state, one can trace out the degrees of freedom of the reservoirs
exactly. The dynamics governed by Eq. (8) for the discrete-variable system is
exactly solvable because only the Hilbert subspaces with the total excitation
numbers $N=0$ and $1$ are involved [69]. The exact dynamics for the
continuous-variable system is obtainable by the Feynman-Vernon’s influence-
functional theory in the coherent-state representation [70, 71]. We obtain the
non-Markovian master equation as

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle\sum_{l=2,3}\\{-i\Omega(t)[\hat{o}^{\dagger}_{l}\hat{o}_{l},\rho(t)]+\Gamma(t)\check{\mathcal{L}}_{l}\rho(t)]\\},$
(9)

where $\rho(t)$ is the reduced density matrix of the quantum channel formed by
the subsystems $\hat{o}_{2}$ and $\hat{o}_{3}$,
$\check{\mathcal{L}}_{l}\rho(t)=2\hat{o}_{l}\rho(t)\hat{o}^{\dagger}_{l}-[\hat{o}^{\dagger}_{l}\hat{o}_{l},\rho(t)]_{+}$
is the Lindblad superoperator, $\Gamma(t)\equiv-\text{Re}[\dot{u}(t)/u(t)]$ is
the decay rate, and $\Omega(t)\equiv-\text{Im}[\dot{u}(t)/u(t)]$ is the
renormalized frequency. The time-dependent function $u(t)$ is determined by

$\dot{u}(t)+\mathit{i}\omega_{0}u(t)+\int_{0}^{t}d\tau\mu(t-\tau)u(\tau)=0,$
(10)

where $\mu(x)\equiv\int_{0}^{\infty}d\omega J(\omega)e^{-\mathit{i}\omega x}$
is the noise correlation function and the initial condition is $u(0)=1$.

In the special case when the system-reservoir coupling is weak and the
characteristic time scale of the correlation function of the reservoirs is
much smaller than that of the system, we can apply the Born-Markovian
approximation to Eq. (10) by replacing $u(\tau)$ by $u(t)$ and extending the
upper limit of the integration from $t$ to infinity. Then using the identity
$\lim_{t\rightarrow\infty}\int_{0}^{t}d\tau
e^{-i(\omega-\omega_{0})\tau}=\pi\delta(\omega-\omega_{0})+i{\mathcal{P}\over\omega_{0}-\omega}$,
with $\mathcal{P}$ being the Cauchy principal value, the Born-Markovian
approximate solution of Eq. (10) reads

$u_{\text{BMA}}(t)\backsimeq
e^{-[\kappa+i(\omega_{0}+\Delta_{\omega_{0}})]t},$ (11)

where $\kappa=\pi J(\omega_{0})$ and
$\Delta_{\omega_{0}}=\mathcal{P}\int_{0}^{\infty}\frac{J(\omega)}{\omega_{0}-\omega}d\omega$.

In the general non-Markovian case, the exact evaluation of the average
fidelity $\bar{F}(t)$ need the numerical solving of Eq. (10). However, via
analyzing the long-time behavior of $u(t)$, we can obtain an analytical form
of $\bar{F}(t)$ in the steady-state limit of the quantum channel. This is
helpful for us to build up a clear physical picture on the performance of our
quantum-teleportation schemes under the impact of the local reservoirs. A
Laplace transform to Eq. (10) results in
$\tilde{u}(p)=[p+i\omega_{0}+\int_{0}^{\infty}\frac{J(\omega)}{p+i\omega}d\omega]^{-1}$.
$u(t)$ is obtained by the inverse Laplace transform to $\tilde{u}(p)$, which
can be done by finding the poles from

$Y(E)\equiv\omega_{0}-\int_{0}^{\infty}\frac{J(\omega)}{\omega-E}d\omega=E,~{}(E=ip).$
(12)

It is interesting to find that the roots of Eq. (12) are just the
eigenenergies of each subsystem and its local reservoir in the single-
excitation subspace. To be specific, we expand the eigenstate of $\hat{H}_{l}$
as
$\ket{\Phi}=(x\hat{o}_{l}^{\dagger}+\sum_{k}y_{k}\hat{b}_{k}^{\dagger})|\varnothing_{l},\\{0_{k}\\}\rangle$,
where $|\varnothing\rangle=|g\rangle$ and $|0\rangle$ for the discrete- and
continuous-variable ones, respectively. Substituting it into
$\hat{H}_{l}\ket{\Phi}=E\ket{\Phi}$ with $E$ being the eigenenergy, we can
readily find that the roots $E$ satisfy the same equation as Eq. (12). This
result implies that the decoherence dynamics of both of the quantum channels
is essentially determined by the energy-spectrum characteristic of each local
subsystem-reservoir system. It gives us an insightful message to control the
decoherence dynamics of the quantum channels by artificially engineering the
structure of the energy spectrum. Because $Y(E)$ is a monotonically decreasing
function in the regime $E<0$, Eq. (12) has one and only one isolated root
$E_{b}$ provided $Y(0)<0$. We call the eigenstate of $E_{b}$ the bound state.
On the other hand, since $Y(E)$ is not well analytic in the regime $E>0$ due
to the divergence of its integration, Eq. (12) has infinite roots in this
regime. They form a continuous energy band. Then, after applying the inverse
Laplace transform using the contour integration and the residue theorem, we
obtain [72]

$u(t)=Ze^{-iE_{b}t}+\int_{0}^{\infty}\frac{J(E)e^{-iEt}dE}{(E-\omega_{0}-\Delta_{E})^{2}+[\pi
J(E)]^{2}},$ (13)

where the first term with
$Z\equiv[1+\int_{0}^{\infty}\frac{J(\omega)}{(E_{b}-\omega)^{2}}d\omega]^{-1}$
is contributed from the potentially formed bound state and the second term is
from the band energies. Contributed by the branch cut of the contour, the
second term approaches zero in the long-time regime due to the out-of-phase
interference of the continuously changing oscillation frequency $E$. Thus, if
the bound state is absent, we have $u(\infty)=0$ characterizing a complete
decoherence; while if the bound state energy is formed, we have
$u(\infty)\simeq Ze^{-iE_{b}t}$ implying a decoherence suppression. The
condition $Y(0)<0$, under which the bound state is formed, can be evaluated
for the Ohmic-family spectral density as
$\omega_{0}-2\eta\omega_{c}\gamma(s)\leq 0$, where $\gamma(s)$ is the Euler’s
gamma function. Thus, via efficiently engineering the system frequency
$\omega_{0}$ and the parameters in the spectral density to form the bound
state, we can suppress the decoherence in quantum teleportation. To verify
this expectation, we investigate the decoherence effects on the two types of
quantum teleportation schemes in the following.

###  III.1 Discrete-variable case

Solving the master equation (9) with $\hat{o}=\hat{\sigma}$ under the initial
state $\rho_{23}(0)=\ket{\Phi_{23}}\bra{\Phi_{23}}$, we obtain

$\displaystyle\rho_{23}(t)$ $\displaystyle=$
$\displaystyle\\{[P_{t}|e\rangle\langle e|+(1-P_{t})|g\rangle\langle
g|]^{\otimes 2}+|g\rangle\langle g|^{\otimes 2}$ (14)
$\displaystyle+[u^{2}(t)|e\rangle\langle g|^{\otimes 2}+\text{h.c.}]\\}/2,$

where $P_{t}=|u(t)|^{2}$. Repeating the similar procedure as the ideal case,
we get the average fidelity as

$\displaystyle\bar{F}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}{\sin\theta\over
4\pi}d\theta\int_{0}^{2\pi}d\phi\sum_{k=1}^{4}\langle\varphi|\hat{U}_{3}^{(k)}\langle\text{Bell}^{(k)}_{12}|\varphi_{1}\rangle\langle\varphi_{1}|$
(15)
$\displaystyle\otimes\rho_{23}(t)|\text{Bell}^{(k)}_{12}\rangle\hat{U}_{3}^{(k)\dagger}|\varphi\rangle$
$\displaystyle=$
$\displaystyle\\{2+|u(t)|^{2}(|u(t)|^{2}-1)+\text{Re}[u(t)^{2}]\\}/3.$

It is readily checked that $u(t)=e^{-i\omega_{0}t}$ and thus $\bar{F}(t)$
tends to the ideal result of Eq. (3) in the noiseless limit. Substituting the
Born-Markovian approximation result (11) into Eq. (15), we also obtain

$\bar{F}_{\text{BMA}}(t)=\\{2+e^{-2\kappa t}[e^{-2\kappa
t}-2\sin^{2}(\omega_{0}t)]\\}/3,$ (16)

where the frequency shift $\Delta_{\omega_{0}}$ has been renormalized into the
bare frequency $\omega_{0}$. $\bar{F}_{\text{BMA}}(t)$ tends to $2/3$ with
time, which is just the classical-communication limit [73]. It means that the
quantum advantage of teleportation is completely destroyed by the Markovian
dissipative noises. A similar result was also reported in many previous works.

In the non-Markovian case, it is natural to expect that $\bar{F}(t)$ tends to
$2/3$ with time too when no bound state is formed in the energy spectrum of
each TLS and its local reservoir. We focus on the situation in the presence of
the bound state. Substituting $\lim_{t\rightarrow\infty}u(t)=Ze^{-iE_{b}t}$ in
the situation into Eq. (15), we obtain

$\lim_{t\rightarrow\infty}\bar{F}(t)=\\{2+Z^{2}[Z^{2}-2\sin^{2}(E_{b}t)]\\}/3.$
(17)

It is remarkable to find that
$\max\lim_{t\rightarrow\infty}\bar{F}(t)=(2+Z^{4})/3$ is always larger than
the classical fidelity $2/3$. Therefore, the quantum superiority of the
teleportation is retrieved in the non-Markvoain dynamics as long as the bound
state is formed.

###  III.2 Continuous-variable case

Within the framework of the path-integral influence-function method, the
solution of the master equation (9) for the continuous-variable quantum
channel is [70, 71]

$\displaystyle\rho(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t)$
$\displaystyle=$ $\displaystyle\int
d\mu(\boldsymbol{\alpha}_{i})d\mu(\boldsymbol{\alpha}_{i}^{\prime})\mathcal{J}(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t|\boldsymbol{\alpha}_{i},\bar{\boldsymbol{\alpha}}_{i}^{\prime};0)$
(18)
$\displaystyle\times\rho(\boldsymbol{\bar{\alpha}}_{i},\boldsymbol{\alpha}_{i}^{\prime};0),$

where
$\rho(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t)=\langle\boldsymbol{\alpha}_{f}|\rho_{23}(t)|\boldsymbol{\alpha}_{f}^{\prime}\rangle$
is the reduced density matrix expressed in coherent-state representation and
$\mathcal{J}(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t|\boldsymbol{\bar{\alpha}}_{i},\boldsymbol{\alpha}_{i}^{\prime};0)$
is the propagating function. We have used the coherent-state representation
$|\boldsymbol{\alpha}\rangle=\prod_{l=2}^{3}|\alpha_{l}\rangle$, with
$|\alpha_{l}\rangle=\exp(\alpha_{l}\hat{a}_{l}^{\dagger})|0_{l}\rangle$, which
obeys the resolution of identity $\int
d\mu\left(\boldsymbol{\alpha}\right)|\boldsymbol{\alpha}\rangle\langle\boldsymbol{\alpha}|=1$
with the integration measures
$d\mu\left(\boldsymbol{\alpha}\right)\equiv\prod_{l}e^{-\bar{\alpha}_{l}\alpha_{l}}\frac{d^{2}\alpha_{l}}{\pi}$.
$\bar{\boldsymbol{\alpha}}$ denotes the complex conjugate of
$\boldsymbol{\alpha}$. The propagating function reads

$\displaystyle\mathcal{J}(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t|\boldsymbol{\alpha}_{i},\bar{\boldsymbol{\alpha}}_{i}^{\prime};0)=\exp\\{\sum_{l=2,3}\big{[}u(t)\bar{\alpha}_{lf}\alpha_{li}$
$\displaystyle~{}~{}~{}~{}~{}+\bar{u}(t)\bar{\alpha}_{li}^{\prime}\alpha_{lf}^{\prime}+[1-\left|u(t)\right|^{2}]\bar{\alpha}_{li}^{\prime}\alpha_{li}\big{]}\big{\\}}.$
(19)

The coherent-state representation of the two-mode squeezed state as the
initial state is

$\displaystyle\rho(\boldsymbol{\bar{\alpha}}_{i},\boldsymbol{\alpha}^{\prime}_{i};0)=\cosh^{-2}r\exp[-\tanh
r(\bar{\alpha}_{1}\bar{\alpha}_{2}+\alpha_{1}^{\prime}\alpha_{2}^{\prime})].$
(20)

Substituting Eqs. (20) and (19) into Eq. (18) and performing the Gaussian
integration, we obtain

$\rho(\boldsymbol{\bar{\alpha}}_{f},\boldsymbol{\alpha}_{f}^{\prime};t)=a\exp\normalsize[\sum_{l\neq
l^{\prime}}(\frac{b}{2}\bar{\alpha}_{lf}\bar{\alpha}_{l^{\prime}f}+c\bar{\alpha}_{lf}\alpha_{lf}^{\prime}+\frac{b^{\ast}}{2}\alpha_{lf}^{\prime}\alpha_{l^{\prime}f}^{\prime})],$
(21)

with $a=x\cosh^{-2}r$, $b=-xu(t)^{2}\tanh r$,
$c=x|u(t)|^{2}(1-|u(t)|^{2})\tanh^{2}r$, and
$x=[1-(1-|u(t)|^{2})^{2}\tanh^{2}r]^{-1}$. A similar procedure as the ideal
case leads to the average fidelity as

$\displaystyle\bar{F}(t)$ $\displaystyle=$ $\displaystyle\int
dx_{1}dp_{2}\langle\varphi|\hat{D}_{3}(\sqrt{2}z)\langle
x_{1}p_{2}|\hat{V}|\varphi_{1}\rangle\langle\varphi_{1}|\rho_{23}(t)\hat{V}^{\dagger}$
(22)
$\displaystyle\otimes|x_{1}p_{2}\rangle\hat{D}^{\dagger}_{3}(\sqrt{2}z)|\varphi\rangle={a\over
2}[1+\text{Re}(b)-c]^{-1}.$

In the noiseless limit, $u(t)=e^{-i\omega_{0}t}$ and thus $a=\cosh^{-2}r$,
$b=-e^{-2i\omega_{0}t}\tanh r$, and $c=0$, which readily reduces $\bar{F}(t)$
to Eq. (7). Considering the Born-Markovian approximation solution (11), we
obtain

$\bar{F}_{\text{BMA}}(t)=\\{2+\sinh(2r)e^{-2\kappa t}[\tanh
r-\cos(2\omega_{0}t)]\\}^{-1}.$ (23)

Equation (23) tends to $1/2$ in the long-time condition, which is the
classical (i.e., no entanglement) bound for teleporting coherent states [74,
75]. Therefore, the quantum advantage of teleportation again is destroyed by
the Markovian noises.

Figure 2: (a) Evolution of $|u(t)|$ obtained via numerically solving Eq. (10)
and $Z$ (blue line) in different $\omega_{c}$. (b) Energy spectrum of the
total system formed by the system and the environment in different
$\omega_{c}$ obtained via numerically solving Eq. (12). We use $\eta=0.2$ and
$s=1$, where the bound state is formed when $\omega_{c}>5\omega_{0}$.

In the general non-Markovian dynamics, substituting
$\lim_{t\rightarrow\infty}u(t)=Ze^{-iE_{b}t}$ in the presence of the bound
state into Eq. (22), we have

$\lim_{t\rightarrow\infty}\bar{F}(t)=\\{2+\sinh(2r)Z^{2}[\tanh
r-\cos(2E_{b}t)]\\}^{-1}.$ (24)

Its maximum
$\max\lim_{t\rightarrow\infty}\bar{F}(t)=[2-Z^{2}(1-e^{-2r})]^{-1}$ is
achieved when $t=n\pi/(2E_{b})$ with $n$ being odd numbers. It is always
greater than the Born-Markovian approximate result, i.e., $1/2$. Therefore,
being similar to the discrete-variable case, we can retrieve the quantum
superiority of the noisy teleportation in the non-Markovian dynamics by
engineering the formation of the bound state.

Figure 3: Evolution of the average fidelity $\bar{F}(t)$ of Eq. (15) (the
lower surface), its transient maxima in Eq. (17) (the upper surface), and its
steady-state maxima $\max\lim_{t\rightarrow\infty}\bar{F}(t)=(2+Z^{4})/3$ (red
line) in different $\omega_{c}$ in the discrete-variable teleportation case.
The parameters are the same as Fig. 2.

##  IV Numerical results

To verify our noise mitigation mechanism in quantum teleportation, we first
plot in Fig. 2(a) the non-Markovian evolution of $|u(t)|$ in different
$\omega_{c}$ by choosing $s=1$ and $\eta=0.2$. Two typical behaviors are
observed. For $\omega_{c}<5\omega_{0}$, $|u(t)|$ decays exclusively to zero.
When $\omega_{c}>5\omega_{0}$, $|u(t)|$ halts to decay and tends to a finite
value. It is interesting to see that this finite value $|u(\infty)|$ matches
exactly with $Z$ evaluated from the bound state, see the blue line in Fig.
2(a). Completely different from the Markovian result, such decoherence
suppression is a distinctive feature of our non-Markovian dynamics. The energy
spectrum of the total system formed by the each subsystem of the quantum
channel and its local reservoir in Fig. 2(b) reveals that a bound state is
formed when $\omega_{c}>5\omega_{0}$, which coincided exactly with the regime
where $|u(\infty)|$ tends to a finite value in Fig. 2(a). The result
demonstrates that the decisive role of the bound state in suppressing the non-
Markovian decoherence of the quantum channel. This gives us an insightful
inspiration to retrieve the superiority of quantum teleportation in the
presence of the practical noise by engineering the formation of the bound
state.

Figure 3 shows the evolution of the average fidelity $\bar{F}(t)$ in Eq. (15)
of the discrete-variable quantum teleportation. It clearly indicates that
$\bar{F}(t)$ tends to $2/3$ in the absence of the bound state when
$\omega_{c}<5\omega_{0}$. As long as the bound state is present, the transient
maxima of $\bar{F}$, which matches with the analytical result in Eq. (17)
evaluated from the bound state, approaches
$\max\lim_{t\rightarrow\infty}\bar{F}(t)=(2+Z^{4})/3$ in the long-time limit,
see the red line in Fig. 3. It fully reveals the recovery of the quantum
advantage of the teleportation due to the formation of the bound state in the
non-Markovian decoherence dynamics. The result in the continuous-variable case
in Fig. 4 also confirms our expectation. It shows that when
$\omega_{c}<5\omega_{0}$, $\bar{F}(t)$ approaches 1/2, which shows no
difference from the Markovian approximate result in Eq. (23). When
$\omega_{c}>5\omega_{0}$, the transient maxima of $\bar{F}(t)$ tend to the
values evaluated from the analytical Eq. (24). Its long-time maxima fit very
well with $\max\lim_{t\rightarrow\infty}\bar{F}(t)=[2-Z^{2}(1-e^{-2r})]^{-1}$,
which surpasses 1/2 and partially retrieves the quantum advantage from the
destruction of noise.

Figure 4: Evolution of the average fidelity $\bar{F}(t)$ of Eq. (22) (the
lower surface), its transient maxima in Eq. (24) (the upper surface), and its
steady-state maxima
$\max\lim_{t\rightarrow\infty}\bar{F}(t)=[2-Z^{2}(1-e^{-2r})]^{-1}$ (red line)
in different $\omega_{c}$ in the continuous-variable teleportation case. $r=2$
and other parameters are the same as Fig. 2.

These results demonstrates the important role of the bound state and non-
Markovian dynamics in recovering quantum advantage of the discrete- and
continuous-variable teleportation in the noisy situation. Such amazing result
is caused by the suppressed decoherence induced by the bound state in the non-
Markovian dynamics. Therefore, we can mitigate the loss of quantum advantage
of teleportation [26, 27, 28, 29, 30] via engineering the formation of the
bound state. It should be noted that the distinguished roles played by the
bound state in other quantum protocols has been reported [56, 60, 61].

##  V Discussion and conclusion

It should be emphasized that the bound-state-favored superiority in our
quantum teleportation of both discrete- and continuous-variable systems is
independent of the explicit form of the spectral density. Although only the
Ohmic-family spectral density is displayed in this paper, our result can be
generalized to other cases without difficulty. The effects of the non-
Markovian dynamics and the bound state have been observed in recent
experiments [51, 52, 53, 54, 55], which provides a strong support in the
experimental realization of our noisy mitigation scheme.

In summary, we have proposed a noise mitigation mechanism for quantum
teleportation being universal for both the discrete- and continuous-variable
quantum channels, under which the the quantum superiority in the teleportation
fidelity is retrieved. This is in sharp contrast to the result under the Born-
Markovian approximation, where the fidelity exponentially decays to or even
worse than its classical limit. Our analysis reveals that it is due to the
constructive interplay between the non-Markovian effect and the bound state of
the total system consisting of the involved systems of the quantum channel and
their reservoirs: The bound state supplies the intrinsic ability and the non-
Markovian effect supplies the dynamical way to achieve the good performance.
Efficiently overcoming the decoherence obstacle in realizing quantum
teleportation, our result provides an experimentally feasible strategy to
realize the high-fidelity teleportation in practice.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grants No. 12275109 and No. 12247101) and the Supercomputing Center of
Lanzhou University.

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