# Information capacity analysis of fully correlated multi-level amplitude
damping channels

Rajiuddin Sk<EMAIL_ADDRESS>Prasanta K. Panigrahi
<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of
Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India

###### Abstract

The primary objective of quantum Shannon theory is to evaluate the capacity of
quantum channels, which is a challenging task in many instances. Recently,
Multi-level Amplitude Damping channel has been introduced, and the
corresponding quantum capacity of the channel has been analyzed for a quantum
system of dimension d=3 [S. Chessa, V. Giovannetti, Commun. Phys. 4, 22
(2021)]. In this paper, we have investigated the information capacity of
Multi-level Amplitude Damping Channel for dimension d=3 in presence of
correlation between successive applications of the channel. We derive the
single-shot classical capacities and quantum capacities associated with a
different class of maps for the three-level system. Additionally, we compute
the quantum and classical capacities in entanglement-assisted scenarios.

††preprint: APS/123-QED

##  I Introduction

The central objective of quantum information and communication theory is to
comprehend how information can be securely stored, processed, and transmitted
utilizing practical frameworks rooted in quantum mechanics. Quantum
communication protocol can be realized by physical quantum system which
exploits the inherent properties of quantum mechanics to enable novel ways of
information processing. In the real world, however, noise is inevitable in
this process, and each type of noise can be described by a quantum channel.
Hence, the quantity of maximum information that can be efficiently transmitted
via a quantum channel is a topic of importance for practical implementation of
information processing tasks. The term channel capacity is used to quantify
this information, and it is the main focus of our paper [1, 2]. From the
pioneering work of Shanon, one can compute the capacity of classical channels
using the principle and framework of classical information theory [3]. In a
classical scenario, only classical information is transmitted, and the
classical capacity is the only concern. However, in the quantum context,
quantum states can be transmitted as well, and the quantum capacity is
included in addition to the classical capacity. Furthermore, the family of
capacities linked with a quantum channel can be expanded in the presence of
extra resources shared between the communicating parties, such as
entanglement, which is referred to as entanglement-assisted capacity [4, 5, 6,
7].

In quantum operational theory, noisy quantum channel is described by
completely positive trace preserving linear map (CPTP). The capacity of qubit
systems, both classical and quantum, has been explored for different channels
[8, 9, 10]. The detection scheme for quantum capacity [11, 12, 13] has been
experimentally verified by Cuevas et. al., [14], as reported in their study.
In a simplest case quantum channel is memoryless, i.e., the channel acts
independently on successive applications. However, in practical scenarios,
memory effect or correlation may be present between successive applications of
the channels. When the information transmission rate is high, such memory
effects become unavoidable and have been experimentally investigated in
optical fibers [15] and solid-state devices [16, 17]. Over the last decade,
quantum memory channel [18] has gained significant attention from the
researchers. Moreover, it has been shown that the effect of memory between
successive applications of the channel enhances the quantum capacity of a
channel [19, 20]. The primary challenge in computing quantum channel
capacities arises from the theory that the entropic measures associated with
them exhibit super-additivity property, and the calculation involves a
regularization process. Because of this difficulty, there are only a few
memory channels whose capacities has been analyzed completely [19, 20, 21].

Among various noise models, amplitude damping channel is one of the simplest
model, and it has been extensively studied for qubit systems in analyzing
capacities [22, 23, 24], dynamics of quantum correlations [25] etc. The
classical and quantum capacity of two-level (qubit) system are well understood
because they can be characterized by single decay, and this channel exhibits
degradable or anti-degradable properties [26]. The classical and quantum
capacity of a two-level amplitude damping channel, when the channel is fully
correlated, has been studied by D’Arrigo et.al. [27]. However, a general
treatment for the d-dimensional case (qudit) is currently unavailable, and it
is unlikely to be realizable. Consequently, the ADC for $d>2$ must be tackled
individually for each dimension, and the literature regarding the capacities
of multi-level amplitude damping (MAD) channel is still limited [28, 29].
There has been a recent increase in interest among researchers in studying
their capacity analysis. This is because, it has been demonstrated that they
offer potential advantages in terms of computation [30, 31, 32], communication
[33], and error correction [34, 35]. Consequently, there is a need for a
comprehensive characterization of communication performance under all
available noise regimes and general noise models. Qudit systems have been the
focus of considerable attention, with particular emphasis on the three-
dimensional systems, known as qutrits, due to their comparative accessibility
in both experimental and theoretical settings [36, 37, 38, 39, 40].

The focus of this paper is on the fully correlated multi-level amplitude
damping channel (MAD channel), in which two qutrits simultaneously relaxed to
the ground state from the excited states. Making use of Lindblad master
equation and finding out the Kraus operator, we have characterised the fully
correlated MAD channel. We specifically analyze the fully correlated MAD on
the qutrit space and systematically examine the quantum capacity of the
channel in large region of the damping parameter space. Our aim is to
determine the conditions under which these quantities can be determined. We
also compute the single-shot classical capacities in some regions of the
damping parameters space. Finally, the quantum and classical capacities have
been analyzed in entanglement assisted scenario.

The organization of the paper is as follows. In Sec II, we have discussed
about MAD channel and the model corresponding to fully correlated MAD channel.
Sec. III deals with an overview of complementary channel and the degradability
of channel. The covariance property of the channel is also discussed in this
section. In Sec. IV, we have derived explicitly the single-shot classical
capacity in special cases of fully correlated MAD channel. While Sec V
contains the analysis of quantum capacity in different scenarios. In Sec. VI,
we have discussed the entanglement assisted capacity of the channel. Finally,
the concluding remark is given in Sec. VII.

Figure 1: Schematic representation of fully correlated MAD channel for a
three-level system. A and B are two qutrits that undergo these relaxations
when they are fully correlated.

##  II The Model

In this section, we provide a brief review of some of the key concepts
associated with the multi-level amplitude damping (MAD) noise model, including
memoryless and correlated scenarios [1, 2]. We also analyze the fully
correlated MAD channel for three-level system from the Lindblad master
equation approach. A MAD channel is a completely positive trace preserving
(CPTP) mapping [4, 5] described by the following set of Kraus operators [41]:

$\begin{gathered}E_{0}\equiv|0\rangle\langle 0|+\sum_{1\leq l\leq
d-1}\sqrt{1-\zeta_{l}}\ket{k}\bra{k}\\\
E_{kl}\equiv\sqrt{p_{lk}}\ket{k}\bra{l},\\\ \end{gathered}$ (1)

where $\\{\ket{l}\\}$ are the set of orthonormal basis of the Hilbert space
$\mathcal{H}_{S}$ with $0\leq k,l\leq d-1$, $p_{lk}$ are the decay parameter
and $\zeta_{l}=\sum_{0\leq k<l}p_{lk}\leq 1$. In this work, we will limit our
examination to the particular category of MAD channels illustrated in Eq. (1)
that are linked to three-level system or qutrit system. For a single qutrit
system, the evolution of the density is governed by the relation
$\rho_{t}=\sum E_{n}\rho E_{n}^{\dagger}$, where $E_{n}$ is defined in Eq. (1)
and n can take values from 0 to 3.

For two memoryless use of MAD channel the evolution is

$\rho_{t}=\Phi(\rho)=\sum_{ij}E_{i}^{m}\otimes
E_{j}^{m}\rho{E_{i}^{m}}^{\dagger}\otimes{E_{j}^{m}}^{\dagger},$ (2)

where $E^{m}$ refers to the Kraus operator corresponding to uncorrelated
channel or memoryless channel. If the subsequent action of channels has some
correlation, it is not possible to write the Kraus operators simply as the
tensor product of individual single qutrit Kraus operator [42]. The evolution
of the density matrix $\rho$ for two uses of channel with arbitrary degrees of
memory can be written as

$\rho_{t}=(1-\mu)\sum_{ij}E_{i}^{m}\otimes
E_{j}^{m}\rho{E_{i}^{m}}^{\dagger}\otimes{E_{j}^{m}}^{\dagger}+\mu\sum_{k}E_{k}^{c}\rho{E_{k}^{c}}^{\dagger},$
(3)

where $\mu$ is the memory parameter and $E^{c}$ corresponds to Kraus operator
for fully correlated channel. For, $\mu=0$ the channel is said to be
memoryless or uncorrelated channel, whereas for $\mu=1$, the channel is said
to be fully correlated or full memory channel. The explicit form of the Kraus
operators for the fully correlated MAD channel can be obtained from the
solution of the Lindblad master equation, which we will derive below
methodically. The time evolution of three-level system is governed by the
following Lindblad master equation:

$\dot{\rho}=\mathcal{L}\rho=\mathcal{L}_{c}(\rho)+\mathcal{D}(\rho)=-i[H,\rho]+\mathcal{D}(\rho),$
(4)

where $\rho$ is the density matrix of the three-level system and $-i[H,\rho]$
represents the coherent evolution which is unitary in nature, and the
$\mathcal{D}(\rho)$ is the damping part which indicates the non-unitary
evolution.
In case of Markov quantum channel, a stochastic map can be written as

$\rho_{0}\rightarrow\rho_{t}=\Phi(\rho_{0})=e^{\mathcal{L}t}\rho_{0}.$ (5)

The above equation gives the dynamics of the system coupled with the
reservoir. The non-unitary part, which gives the dissipation of the density
matrix is

$\begin{split}\mathcal{D}(\rho)=\frac{\Gamma_{3}}{2}\left(2\sigma_{12}\rho\sigma_{21}-\sigma_{22}\rho-\rho\sigma_{22}\right)+\frac{\Gamma_{2}}{2}(2\sigma_{02}\rho\sigma_{20}-\\\
\sigma_{22}\rho-\rho\sigma_{22})+\frac{\Gamma_{1}}{2}\left(2\sigma_{01}\rho\sigma_{10}-\sigma_{11}\rho-\rho\sigma_{11}\right).\\\
\end{split}$ (6)

In the above equation, $\Gamma_{3}$, $\Gamma_{2}$ and $\Gamma_{1}$ are the
spontaneous decay rate corresponding to the transition of atom from
$\ket{2}\rightarrow\ket{1}$, $\ket{2}\rightarrow\ket{0}$ and
$\ket{1}\rightarrow\ket{0}$ respectively. The transition operator
$\sigma_{kl}$ indicates the transition of atom from
$\ket{k}\rightarrow\ket{l}$, i.e., $\sigma_{kl}=\ket{k}\bra{l}$. These
transitions are governed by the interaction between the system (S) and
environment ($\mathcal{E}$), which has the form,
$(\sigma_{S}\otimes\sigma^{\dagger}_{\mathcal{E}}+\sigma^{\dagger}_{S}\otimes\sigma_{\mathcal{E}})$
($\sigma$ and $\sigma^{\dagger}$ are respectively the lowering and raising
operator)

We now extend the Lindblad equation for the case of two three-level atoms,
where the action of the amplitude damping is fully correlated:

$\begin{split}\mathcal{D}^{c}(\rho)=\frac{\Gamma_{3}}{2}\left(2\mathcal{S}_{12}\rho\mathcal{S}_{21}-\mathcal{S}_{22}\rho-\rho\mathcal{S}_{22}\right)+\frac{\Gamma_{2}}{2}(2\mathcal{S}_{02}\rho\mathcal{S}_{20}-\\\
\mathcal{S}_{22}\rho-\rho\mathcal{S}_{22})+\frac{\Gamma_{1}}{2}\left(2\mathcal{S}_{01}\rho\mathcal{S}_{10}-\mathcal{S}_{11}\rho-\rho\mathcal{S}_{11}\right),\\\
\end{split}$ (7)

where $\mathcal{S}_{kl}=\sigma_{kl}\otimes\sigma_{kl}$. They decay process
associated with two qutrits, A and B, is displayed in Fig. (1). There are
several methods available to solve the master equation of the form shown in
Eq. (7). We adopt the method proposed by Daffer et al. [43], wherein they have
used left,$\\{\mathbb{L}_{i}\\}$ and right, $\\{\mathbb{R}_{i}\\}$, damping
basis with damping eigenvalue $\lambda_{i}$ for a Lindblad superoperator that
yields the image of a trace preserving, completely positive map:

$\rho_{t}=\phi(\rho_{0})=\sum_{i}\operatorname{tr}(\mathbb{L}_{i}\rho_{0})e^{\lambda_{i}t}\mathbb{R}_{i}.$
(8)

Here, the left and right eigenoperators, $\\{\mathbb{L}_{i}\\}$ and
$\\{\mathbb{R}_{i}\\}$ have the same eigenvalue $\lambda_{i}$ and they satisfy
the eigenvalue equations, $\mathbb{L}\mathcal{D}=\lambda\mathbb{L}$ and
$\mathcal{D}\mathbb{R}=\lambda\mathbb{R}$ respectively. They also obey the
duality relation
$\operatorname{tr}\\{\mathbb{L}_{i}\mathbb{R}_{j}\\}=\delta_{ij}$.

Let us consider the initial density matrix of two three-level atoms in the
absence of any interaction with the environment as

$\rho=\left(\begin{array}[]{ccccc}\rho_{00}&\cdots&\rho_{04}&\cdots&\rho_{08}\\\
\vdots&&\vdots&&\vdots\\\ \rho_{40}&\cdots&\rho_{44}&\cdots&\rho_{48}\\\
\vdots&&\vdots&&\vdots\\\
\rho_{80}&\cdots&\rho_{84}&\cdots&\rho_{88}\end{array}\right).$ (9)

We solve the Eq. (7) by converting the initial density matrix in Hilbert state
into Fock-Liouville space and finding the corresponding Lindblad super-
operator. After that, we find the left and right eigenbasis and corresponding
eigenvalues, which will give the output density matrix according to Eq. (8).
In our two qutrit system the Lindblad super-operator is $81\times 81$ matrix
with the eigen values ($\lambda_{i}$) are $(0)_{49}$,
$-(\Gamma_{2}+\Gamma_{3})_{1}$,$-((\Gamma_{2}+\Gamma_{3})/2)_{14}$,
$(\Gamma)_{1}$, $(-\Gamma_{1}/2)_{14}$ and
$((\Gamma_{1}+\Gamma_{2}+\Gamma_{3})/2)_{2}$ are respectively. Note that the
subscripts attached to eigenvalues correspond to the frequency of occurrence
of each specific eigenvalue.

Evidently, the dynamical evolution of the input density matrix can be written
as

ρ_t=
(~ρ00ρ01ρ02ρ03e-Γ1t/2ρ04ρ05ρ06ρ07e-Γ2+Γ32tρ08ρ10ρ11ρ12ρ13e-Γ1t/2ρ14ρ15ρ16ρ17e-(Γ2+Γ3)2tρ18ρ20ρ21ρ22ρ23e-Γ1t/2ρ24ρ25ρ26ρ27e-(Γ2+Γ3)2tρ28ρ30ρ31ρ32ρ33e-Γ1t/2ρ34ρ35ρ36ρ37e-Γ2+Γ32tρ38e-Γ1t/2ρ40e-Γ1t/2ρ41e-Γ1t/2ρ42e-Γ1t/2ρ43~ρ44e-Γ1t/2ρ45e-Γ1t/2ρ46e-Γ1t/2ρ47e-Γ1+Γ2+Γ32tρ48ρ50ρ51ρ52ρ53e-Γ1t/2ρ54ρ55ρ56ρ57e-Γ2+Γ32tρ58ρ60ρ61ρ62ρ63e-Γ1t/2ρ64ρ65ρ66ρ67e-Γ2+Γ32tρ68ρ70ρ71ρ72ρ73e-Γ1t/2ρ74ρ75ρ76ρ77e-Γ2+Γ32tρ78e-Γ2+Γ32tρ80e-Γ2+Γ32tρ81e-Γ2+Γ32tρ82e-Γ2+Γ32tρ83e-Γ1+Γ2+Γ32tρ84e-Γ2+Γ32tρ85e-Γ2+Γ32tρ86e-Γ2+Γ32tρ87e-(Γ2+Γ3)tρ88).

(10)

In the above density matrix,

$\displaystyle\tilde{\rho}_{00}=$
$\displaystyle\rho_{00}+\rho_{44}(1-e^{-\Gamma_{1}t})+\rho_{88}(1-e^{-(\Gamma_{2}+\Gamma_{3})t}$
$\displaystyle-\Theta(\Gamma)e^{-\Gamma_{1}t}+\Theta(\Gamma)e^{-(\Gamma_{2}+\Gamma_{3})t})$

and

$\tilde{\rho}_{44}=e^{-\Gamma_{1}t}\rho_{44}+\Theta(\Gamma)(e^{-\Gamma_{1}t}-e^{-(\Gamma_{2}+\Gamma_{3})t}),$

where $\Theta(\Gamma)=\Gamma_{3}/(\Gamma_{3}+\Gamma_{2}-\Gamma_{1})$. As we
have discussed earlier, the dynamical evolution of a two-qutrit state $\rho$
subject to a multi-level amplitude damping channel with full Markovian memory
can be written in terms of Kraus representation as $\rho_{t}=\sum_{n}E_{n}\rho
E^{\dagger}_{n}$. It is evident that the Kraus operator $E_{n}$, can be
computed from the relation
$\sum_{i}\operatorname{tr}(\mathbb{L}_{i}\rho_{0})e^{\lambda_{i}t}\mathbb{R}_{i}=\sum_{n}E_{n}\rho
E^{\dagger}_{n}$. The explicit form of Kraus operators is determined from the
solution of correlated Lindblad equation, which is given as

$\displaystyle E_{00}=$
$\displaystyle\ket{00}\bra{00}+\sqrt{1-p_{1}}\ket{11}\bra{11}$
$\displaystyle+\sqrt{(1-p_{2})(1-p_{3})}\ket{22}\bra{22}+\sum_{\begin{subarray}{c}i,j=0\\\
i\neq j\end{subarray}}^{2}\ket{ij}\bra{ij}$ $\displaystyle E_{11}=$
$\displaystyle\sqrt{p_{1}}\ket{00}\bra{11}$ $\displaystyle E_{22}=$
$\displaystyle\sqrt{p_{1}+(1-\Theta(\Gamma))p_{123}}\ket{00}\bra{22}$
$\displaystyle E_{33}=$
$\displaystyle\sqrt{\Theta(\Gamma)p_{123}}\ket{11}\bra{22},$ (11)

where $\Theta(\Gamma)=\Gamma_{3}/(\Gamma_{3}+\Gamma_{2}-\Gamma_{1})$,
$p_{123}=e^{-\Gamma_{1}t}-e^{-(\Gamma_{2}+\Gamma_{3})t}=(1-p_{1})-(1-p_{2})(1-p_{3})$
and $\Gamma_{2}+\Gamma_{3}>\Gamma_{1}$. We have written the Kraus operators in
matrix form in the supplementary material to make its structure easier to
understand. The CPTP condition of the transformation is satisfied when
$(1-p_{1})\geq(1-p_{2})(1-p_{3})$, which gives accessible values of $p_{1}$,
$p_{2}$ and $p_{3}$, shown in colored region of Fig. (2). The Kraus operators
mentioned above fulfil the completeness relation $\sum
E_{n}^{\dagger}E_{n}=I$. Nevertheless, the fact that $\sum
E_{n}E_{n}^{\dagger}\neq I$ implies that the channel is non-unital.

Figure 2: The accessible domain of the damping parameters. The coloured region
gives the values of the damping parameters for which the transformation is
CPTP.

##  III Channel properties

In the calculation of the quantum capacity of a channel, the optimization of
the von Neuman entropy of both quantum channel and its complementary channel
is a necessary step. In this section, we will provide a brief overview of
complementary channels, including the degradability and anti-degradability
characteristics of quantum channel. We also explore the covariance property of
the channel and its complement.

###  III.1 Complementary channel and degradibility

Let $\mathcal{O}(\mathcal{H})$ represent the space of positive linear
operators on a Hilbert space $\mathcal{H}$. A quantum channel $\Phi$, map the
input system S into the corresponding output system $S^{\prime}$:
$\mathcal{O}(\mathcal{H_{S}})\rightarrow\mathcal{O}(\mathcal{H_{S^{\prime}}})$.
If $\mathcal{E}$ and $\mathcal{E}^{\prime}$ represent the corresponding
environment of the input system and output system, then from the Stinespring
representation [44, 45], we can write

$\Phi(\rho_{S})=\operatorname{tr}_{\mathcal{E}^{\prime}}(V\rho_{S}V^{\dagger}),$
(12)

where V represents an isometry:
$\mathcal{H}_{S}\rightarrow\mathcal{H}_{S^{\prime}}\otimes\mathcal{H}_{\mathcal{E}^{\prime}}$.
In this configuration, the complementary map $\tilde{\Phi}$, which maps the
input system to the output environment:
$\mathcal{O}(\mathcal{H_{S}})\rightarrow\mathcal{O}(\mathcal{H_{\mathcal{E}^{\prime}}})$,
is defined as

$\tilde{\Phi}(\rho_{S})=\operatorname{tr}_{S^{\prime}}(V\rho_{S}V^{\dagger}).$
(13)

If $E_{k}$ are the Kraus operators which generate the map $\Phi$, and
$\ket{k}_{\mathcal{E}}$ are the basis corresponding to the environment, then
one can express the operator V in the following way:

$V=\sum_{k}E_{k}\otimes\ket{k}_{\mathcal{E}}.$ (14)

We can express Eq. (13), equivalently as follows:

$\tilde{\phi}(\rho_{S})=\sum_{k,l}\operatorname{tr}[E_{k}\rho_{S}E_{k}^{\dagger}]\ket{k}\bra{l}_{\mathcal{E}}.$
(15)

Let us now revisit the property of degradable and anti-degradable channels
[46]. The quantum channel denoted by $\Phi$, is called degradable, if there
exist a map $\Phi_{D}$:
$\mathcal{O}(\mathcal{H_{S^{\prime}}})\rightarrow\mathcal{O}(\mathcal{H_{\mathcal{E}^{\prime}}})$
such that

$\tilde{\Phi}=\Phi_{D}\circ\Phi.$ (16)

The channel concatenation is represented by the symbol ”$\circ$”. On the
contrary, the map is said to be anti-degradable if there exists a map
$\Phi_{AD}$:
$\mathcal{O}(\mathcal{H_{\mathcal{E}^{\prime}}})\rightarrow\mathcal{O}(\mathcal{H_{S^{\prime}}})$
such that

$\Phi=\Phi_{AD}\circ\tilde{\Phi}.$ (17)

If the mapping $\Phi$ is invertible, then simply we can make the inversion of
it to construct the super-operators $\tilde{\Phi}\circ\Phi^{-1}$ or
$\Phi\circ\tilde{\Phi}^{-1}$ and checking the CPTP of these super-operators we
can conclude whether the channel is degradable or anti degradable. The
complete positivity of the super-operators is determined by examining the
positivity of its Choi matrices [47]. One can represent quantum channels as a
matrix in the vector space since it connects the vector space of linear
operators. This can be done by vectorization of density matrices:

$\displaystyle\rho_{S}=$ $\displaystyle\sum_{kl}\rho_{kl}|k\rangle_{S}\langle
l|\longrightarrow|\rho\rangle\rangle=\sum_{kl}\rho_{kl}|k\rangle_{S}\otimes|l\rangle_{S}$
(18)
$\displaystyle\Phi\left(\rho_{S}\right)\longrightarrow\mathcal{M}_{\Phi}|\rho\rangle\rangle.$

In the above equation, $\mathcal{M}_{\Phi}$ is a $d^{2}_{S^{\prime}}\times
d^{2}_{S}$ dimension matrix, connects $\mathcal{H}_{S^{\prime}}^{\otimes 2}$
and $\mathcal{H}_{S}^{\otimes 2}$. So, starting from Eq. (16), one can write
the following identity:

$\mathcal{M}_{\tilde{\Phi}}=\mathcal{M}_{\Phi_{D}}\circ\mathcal{M}_{\Phi}.$
(19)

Using this equality one can represent the superoperator
$\tilde{\Phi}\circ\Phi^{-1}$ as
$\mathcal{M}_{\tilde{\Phi}}\mathcal{M}^{-1}_{\Phi}$ provided that
$\mathcal{M}_{\Phi}$ is invertible.

###  III.2 Covariance of the channel

This section explores the covariance properties of the channel $\Phi$, which
enables us to inspect the properties of a general quantum channel with respect
to certain unitary transformations. First, we define three unitary matrices

$\mathtt{V}_{1}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{pmatrix},\mathtt{V}_{2}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&1\\\
\end{pmatrix},\mathtt{V}_{3}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{pmatrix}.$

Now, we define sixteen unitary operations
($\mathbf{U}_{0},\dots,\mathbf{U}_{15}$) using above three matrices as
follows:

$\mathbf{U}_{i}=\mathtt{V}_{m}\otimes\mathtt{V}_{n}\quad\forall
m,n\;s.t\;0\leq m\;\geq 3\;and\;0\leq n\;\geq 3,$

where m and n takes values 0,1,2,3 and note that
$\mathtt{V}_{0}=\mathbb{I}_{3\times 3}$.

All the unitary operations $\mathbf{U}_{i}$ either commute or anti-commute
with the Kraus operator given in Eq. (II). For a particular $\mathbf{U}_{i}$,
for instance $\mathbf{U}_{1}=\mathtt{V}_{0}\otimes\mathtt{V}_{1}$, the Kraus
operator $E_{00}$ and $E_{11}$ commute with the $\mathbf{U}_{1}$, whereas,
$E_{22}$ and $E_{33}$ anti-commute with $\mathbf{U}_{1}$:
$E_{00}\mathbf{U}_{1}=\mathbf{U}_{1}E_{00}$,
$E_{11}\mathbf{U}_{1}=\mathbf{U}_{1}E_{11}$,
$E_{22}\mathbf{U}_{1}=-\mathbf{U}_{1}E_{22}$, and
$E_{33}\mathbf{U}_{1}=-\mathbf{U}_{1}E_{33}$.

Using these relation it is straightforward to prove that

$\displaystyle\Phi(\mathbf{U}_{1}\rho\mathbf{U}_{1})=$ (20) $\displaystyle
E_{00}\mathbf{U}_{1}\rho\mathbf{U}_{1}E_{00}^{\dagger}+E_{11}\mathbf{U}_{1}\rho\mathbf{U}_{1}E_{11}^{\dagger}$
$\displaystyle+E_{22}\mathbf{U}_{1}\rho\mathbf{U}_{1}E_{22}^{\dagger}+E_{33}\mathbf{U}_{1}\rho\mathbf{U}_{1}E_{33}^{\dagger}$
$\displaystyle=\mathbf{U}_{1}E_{00}\rho
E_{00}^{\dagger}\mathbf{U}_{1}+\mathbf{U}_{1}E_{11}\rho
E_{11}^{\dagger}\mathbf{U}_{1}$
$\displaystyle+(-\mathbf{U}_{1}E_{22})\rho(-E_{22}^{\dagger}\mathbf{U}_{1})+(-\mathbf{U}_{1}E_{33})\rho(-E_{33}^{\dagger}\mathbf{U}_{1})$
$\displaystyle=\mathbf{U}_{1}\Phi(\rho)\mathbf{U}_{1}.$

In the same way, we can prove the covariance under others $\mathbf{U}_{i}$.
Now, we will create some unitary matrices that will swap the diagonal elements
of the density matrix with each other. These unitary matrices have the
following form:

$\displaystyle\mathbf{V}_{1}=$
$\displaystyle\ket{00}\bra{00}+\ket{01}\bra{02}+\ket{10}\bra{12}+\ket{02}\bra{01}+\ket{20}\bra{21}$
(21)
$\displaystyle+\ket{11}\bra{11}+\ket{12}\bra{10}+\ket{21}\bra{20}+\ket{22}\bra{22}$
$\displaystyle\mathbf{V}_{2}=$
$\displaystyle\ket{00}\bra{00}+\ket{01}\bra{21}+\ket{10}\bra{02}+\ket{02}\bra{10}+\ket{20}\bra{12}$
$\displaystyle+\ket{11}\bra{11}+\ket{12}\bra{20}+\ket{21}\bra{01}+\ket{22}\bra{22}$
$\displaystyle\mathbf{V}_{3}=$
$\displaystyle\ket{00}\bra{00}+\ket{01}\bra{10}+\ket{10}\bra{01}+\ket{02}\bra{20}+\ket{20}\bra{02}$
$\displaystyle+\ket{11}\bra{11}+\ket{12}\bra{21}+\ket{21}\bra{12}+\ket{22}\bra{22}$
$\displaystyle\mathbf{V}_{4}=$
$\displaystyle\ket{00}\bra{00}+\ket{01}\bra{12}+\ket{10}\bra{20}+\ket{02}\bra{21}+\ket{20}\bra{10}$
$\displaystyle+\ket{11}\bra{11}+\ket{12}\bra{01}+\ket{21}\bra{02}+\ket{22}\bra{22}$
$\displaystyle\mathbf{V}_{5}=$
$\displaystyle\ket{00}\bra{00}+\ket{01}\bra{20}+\ket{10}\bra{21}+\ket{02}\bra{12}+\ket{20}\bra{01}$
$\displaystyle+\ket{11}\bra{11}+\ket{12}\bra{02}+\ket{21}\bra{10}+\ket{22}\bra{22}.$

The action of the unitaries defined above is to swap the position of the
diagonal without affecting the states $\ket{00}$, $\ket{11}$ and $\ket{22}$.
It is straightforward to confirm that $\mathbf{V}_{i}$ commutes with the Kraus
operators. So, the channel $\Phi$ is covariant under the operation of the
unitaries $\mathbf{V}_{i}$, i.e.,

$\Phi(\mathbf{V}_{i}\rho\mathbf{V}_{i})=\mathbf{V}_{i}\Phi(\rho)\mathbf{V}_{i}.$

We are operating these swap unitaries to make the optimization procedure
easier, which will be clear in the next section.

##  IV Classical capacity

The classical capacity of a quantum channel, $\mathcal{C}$, represents its
capability to transmit classical information reliably. It is determined by the
maximum quantity of classical information that can be successfully transmitted
per use of the channel. In order to calculate the classical capacity, it is
typically necessary to perform optimization over multiple uses of the channel:

$\mathcal{C}(\Phi)=\lim_{n\longrightarrow\infty}\frac{1}{n}\bar{\chi}\left(\Phi^{\otimes
n}\right),$ (22)

where
$\bar{\chi}(\Phi)=\max_{\xi_{j},\rho_{j}}\chi\left\\{\Phi,\left(\xi_{j},\rho_{j}\right)\right\\}$
and $\chi(\Phi,\left\\{\xi_{j},\rho_{j}\right\\})$ is the Holevo quantity over
single uses of the channel. Generally, the Holevo quantity obeys the super-
additive condition. So, the regularization in Eq. (22) is necessary to find
out the classical capacity.

In this section, we mainly focus on single-shot classical capacity
$\mathcal{C}_{1}$ of a fully correlated three-level amplitude damping channel.
The single-shot classical capacity $\mathcal{C}_{1}$ is determined by
optimizing the Holevo quantity $\chi$ over single uses of the channel $\Phi$
and over possible ensembles $\\{\xi_{j},\rho_{j}\\}$, which is

$\displaystyle\mathcal{C}_{1}=$
$\displaystyle\max_{\xi_{k}\rho_{j}\in\mathcal{H}}\chi(\Phi,\\{\xi_{j},\rho_{j}\\})$
(23) $\displaystyle=$
$\displaystyle\max_{\xi_{j}\rho_{j}\in\mathcal{H}}\\{S\left(\Phi(\rho)\right)-\sum_{j}\xi_{j}S\left(\Phi\left(\rho_{j}\right)\right)\\}$

with $\\{\xi_{j}\\}$ probabilities and $\rho=\sum\xi_{j}\rho_{j}$ the average
transmitted message. $S(\rho)=-\operatorname{tr}[\rho\log_{2}\rho]$ is the von
Neuman entropy of the state $\rho$. The first term of $\chi$ corresponds to
the entropy of the channel output for the input quantum state $\rho$, while
the second term gives the average entropy of the channel output. It is
possible to discover an ensemble of pure input states for any ensemble of
mixed input states, such that the resulting output states have a value of
$\chi$ that is equal to or greater than the original ensemble [48]. So, we
define an ensemble of pure state $\\{\xi_{j},\ket{\psi_{j}}\\}$ with the
single-shot classical capacity,

$\displaystyle\mathcal{C}_{1}=\max_{\xi_{j}\rho_{j}\in\mathcal{H}}\\{S\left(\Phi(\rho)\right)-\sum_{j}\xi_{j}S\left(\Phi\left(\ket{\psi_{j}}\bra{\psi_{j}}\right)\right)\\},$
(24)

where $\rho=\sum_{j}\xi_{j}\ket{\psi_{j}}\bra{\psi_{j}}$. Now, our aim is to
search for the ensemble which will maximize $\chi$. In the following section,
we give a comprehensive description of the method we used.

The channel covariance properties discussed in the earlier section are
employed in this section to find the form of the ensembles
$\\{\xi_{j},\ket{\psi_{j}}\\}$ that solve the maximization problems (24). In
order to make the optimization simpler, we first find another ensemble
$\\{\xi^{\prime}_{j},\ket{\psi^{\prime}_{j}}\\}$ by replacing each state
$\ket{\psi_{j}}$ of the ensemble $\\{\xi_{j},\ket{\psi_{j}}\\}$ with the set
$\\{\mathbf{U}_{0}\ket{\psi_{j}},\dots,\mathbf{U}_{15}\ket{\psi_{j}}\\}$,
where each state occurs with probability $\xi^{\prime}_{j}=\xi_{j}/16$ [49].
The corresponding density operator of the new ensemble,
$\rho^{\prime}=\sum_{j}\xi^{\prime}_{j}\ket{\psi^{\prime}_{j}}\bra{\psi^{\prime}_{j}}$
can be written as

$\displaystyle\rho^{\prime}$
$\displaystyle=\sum_{j}\frac{\xi_{j}}{16}\left(\ket{\psi_{j}}\bra{\psi_{j}}+\sum_{i=1}^{15}\mathbf{U}_{i}\ket{\psi_{j}}\bra{\psi_{j}}\mathbf{U}_{i}\right)$
(25)
$\displaystyle=\frac{1}{16}\left(\rho+\sum_{i=1}^{15}\mathbf{U}_{i}\rho\mathbf{U}_{i}\right).$

The density matrix $\rho^{\prime}$ has the same diagonal elements as $\rho$,
while its off-diagonal elements completely vanish. Now, our aim is to show
that

$\chi(\Phi,\\{\xi^{\prime}_{j},\ket{\psi^{\prime}}_{j}\\})\geqslant\chi\left(\Phi,\left\\{\xi_{j},\ket{\psi_{j}}\right\\}\right).$
(26)

Given that von Neumann entropy remains unchanged under unitary operations, one
may write $S(\phi(\rho^{\prime}_{j}))=S(\phi(\rho_{j}))$. Hence, the second
term of the Holevo quantity become

$\displaystyle\sum_{j}\xi^{\prime}_{j}S\left(\Phi\left(\ket{\psi^{\prime}_{j}}\bra{\psi^{\prime}_{j}}\right)\right)$
$\displaystyle=16\sum_{j}\frac{\xi_{j}}{16}S\left(\Phi\left(\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\right)\right)$
(27)
$\displaystyle=\sum_{j}\xi_{j}S\left(\Phi\left(\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\right)\right).$

Using the concavity property of von Neuman entropy we find the output entropy
corresponding to the $\rho^{\prime}$ as

$\displaystyle S\left(\Phi({\rho^{\prime}})\right)$
$\displaystyle=S\left(\Phi\left(\frac{1}{16}\rho+\frac{1}{16}\sum_{i=1}^{15}\mathbf{U}_{i}\rho\mathbf{U}_{i}\right)\right)$
(28)
$\displaystyle\geqslant\frac{1}{16}S\left(\Phi(\rho)\right)+\frac{1}{16}\sum_{i=1}^{15}S\left(\Phi\left(\mathbf{U}_{i}\rho\mathbf{U}_{i}\right)\right)$
$\displaystyle=S\left(\Phi(\rho)\right).$

So, the validity of the Eq. (26) is proved from the Eq. (27) and (28). We can
conclude from the above proof that one can construct an ensemble with the same
diagonal element as any given ensemble of pure states, such that the density
matrix of the new ensemble has zero off-diagonal entries and the Holevo
quantity of this new ensemble is at least as large as that of the original
ensemble.

We introduce a generic input state $\\{\xi_{j},\ket{\psi_{j}}\\}$:

$\displaystyle\ket{\psi}_{j}$
$\displaystyle=a_{j}\ket{00}+b_{j}\ket{01}+c_{j}\ket{02}+d_{j}\ket{10}+e_{j}\ket{11}$
(29) $\displaystyle+f_{j}\ket{12}+g_{j}\ket{20}+h_{j}\ket{21}+k_{j}\ket{22},$

where
$a_{j},\;b_{j},\;c_{j},\;d_{j},\;e_{j},\;f_{j},\;g_{j},\;h_{j}\;\text{and}\;k_{j}\in\mathbb{C}$
and satisfy the normalization condition. we can write the corresponding
density matrix $\rho=\sum_{j}\xi_{j}\ket{\psi_{j}}\bra{\psi_{j}}$.

Eventually, the corresponding density matrix $\rho^{\prime}$, which gives the
upper bound of Holevo quantity, become the diagonal matrix:

$\rho^{\prime}=\begin{pmatrix}\alpha&&&&&&&&\\\ &\beta_{1}&&&&&&&\\\
&&\beta_{2}&&&&&&\\\ &&&\beta_{3}&&&&&\\\ &&&&\gamma&&&&\\\
&&&&&\beta_{4}&&&\\\ &&&&&&\beta_{5}&&\\\ &&&&&&&\beta_{6}&\\\
&&&&&&&&\delta\\\ \end{pmatrix},$ (30)

where we have assumed

$\displaystyle\alpha=\sum_{j}\xi_{j}|a_{j}|^{2},\;\beta_{1}=\sum_{j}\xi_{j}|b_{j}|^{2},\;\beta_{2}=\sum_{j}\xi_{j}|c_{j}|^{2},$
$\displaystyle\beta_{3}=\sum_{j}\xi_{j}|d_{j}|^{2},\;\gamma=\sum_{j}\xi_{j}|e_{j}|^{2},\;\beta_{4}=\sum_{j}\xi_{j}|f_{j}|^{2},$
$\displaystyle\beta_{5}=\sum_{j}\xi_{j}|g_{j}|^{2},\;\beta_{6}=\sum_{j}\xi_{j}|h_{j}|^{2},\;\delta=\sum_{j}\xi_{j}|k_{j}|^{2}\;.$

Now, we utilize the covariance property of the channel with respect to the
unitary swap operation defined in Eq. (21). We start with the ensemble
$\\{\xi^{\prime}_{j},\ket{\psi^{\prime}_{j}}\\}$ defined in Eq. (30), we
create another ensemble by replacing each state $\ket{\psi^{\prime}_{j}}$ with
the set of states
$\\{\ket{\psi^{\prime}_{j}},\mathbf{V}_{i}\ket{\psi^{\prime}_{j}}\\}$, with
each occurs with probability $\xi^{\prime}_{j}/6$. The new ensemble denoted by
$\\{\bar{\xi}_{j},\ket{\bar{\psi_{j}}}\bra{\bar{\psi_{j}}}\\}$. The new
density operator:

$\displaystyle\bar{\rho}$
$\displaystyle=\sum_{j}\frac{\xi^{\prime}_{j}}{6}\left(\ket{\psi^{\prime}_{j}}\bra{\psi^{\prime}_{j}}+\sum_{i=1}^{5}\mathbf{V}_{i}\ket{\psi^{\prime}_{j}}\bra{\psi^{\prime}_{j}}\mathbf{V}_{i}\right)$
(31)
$\displaystyle=\frac{1}{6}\left(\rho^{\prime}+\sum_{i=1}^{5}\mathbf{V}_{i}\rho^{\prime}\mathbf{V}_{i}\right).$

We will prove that $\\{\bar{\xi}_{j},\ket{\bar{\psi}_{j}}\\}$ has the Holevo
quantity $\chi$, which is greater equal to that of the ensemble
$\\{\xi^{\prime}_{j},\ket{\psi^{\prime}_{j}}\\}$. The first term of the Holevo
quantity $\chi$ takes the following form:

$\displaystyle S\left(\Phi({\bar{\rho}})\right)$
$\displaystyle=S\left(\Phi\left(\frac{1}{6}\rho^{\prime}+\frac{1}{6}\sum_{i=1}^{5}\mathbf{V}_{i}\rho^{\prime}\mathbf{V}_{i}\right)\right)$
(32)
$\displaystyle\geqslant\frac{1}{6}S\left(\Phi(\rho^{\prime})\right)+\frac{1}{6}\sum_{i=1}^{5}S\left(\Phi\left(\mathbf{V}_{i}\rho^{\prime}\mathbf{V}_{i}\right)\right)$
$\displaystyle=S\left(\Phi(\rho^{\prime})\right).$

Now, we will prove that the second term of $\chi$ will remain invariant under
swap unitaries:

$\displaystyle\sum_{j}\bar{\xi}_{j}S\left(\Phi\left(\ket{\bar{\psi}_{j}}\bra{\bar{\psi}_{j}}\right)\right)$
$\displaystyle=6\sum_{j}\frac{\xi^{\prime}_{j}}{6}S\left(\Phi\left(\left|\psi^{\prime}_{j}\right\rangle\left\langle\psi^{\prime}_{j}\right|\right)\right)$
(33)
$\displaystyle=\sum_{j}\xi^{\prime}_{j}S\left(\Phi\left(\left|\psi^{\prime}_{j}\right\rangle\left\langle\psi^{\prime}_{j}\right|\right)\right).$

The above two equations (32) and (33) prove that the Holevo quantity of the
ensemble $\\{\bar{\xi}_{j},\ket{\bar{\psi}_{j}}\\}$ yields the upper bound of
the ensemble $\\{\xi^{\prime}_{j},\ket{\psi^{\prime}_{j}}\\}$. Hence, we can
say that the Holevo quantity of the ensemble
$\\{\bar{\xi}_{j},\ket{\bar{\psi}_{j}}\\}$ is at least as large as that of the
original ensemble $\\{\xi_{j},\ket{\psi_{j}}\\}$.

The subsequent section examines the single-shot classical capacity of two
specific channel types: the single decay channel and the V-type channel [28].
These channels are obtained by imposing constraints on the decay parameters,
and we have computed their classical capacity using some algebraic inequality
and the convex property of binary Shannon entropy. Due to the complex
structure of eigenvalues of the output state in the case of $\Lambda$-type
channel and three decay rate channel, we could not calculate the classical
capacity.

####  IV.0.1 V-type channel

The lowermost energy level in this damping channel only interacts with the two
higher energy levels, and the transition from $\ket{2}\rightarrow\ket{1}$ is
not permitted. The Kraus operators that represent the V-type channel can be
derived by setting $p_{3}$ to zero in Equation (II). The resulting Kraus
operators are provided below:

$\displaystyle E_{00}=$
$\displaystyle\ket{00}\bra{00}+\sqrt{1-p_{1}}\ket{11}\bra{11}+\sqrt{1-p_{2}}\ket{22}\bra{22},$
$\displaystyle+\sum_{\begin{subarray}{c}i,j=0\\\ i\neq
j\end{subarray}}^{2}\ket{ij}\bra{ij}$ $\displaystyle E_{11}=$
$\displaystyle\sqrt{p_{1}}\ket{00}\bra{11},\,E_{22}=\sqrt{p_{2}}\ket{00}\bra{22}.$

These are the same Kraus operators used for V-type transition in a three-level
system in Ref. [50]. If the quantum channel $\phi_{(p_{1},p_{2},0)}$ acts on
the generic state given by Eq. (29), the density matrix become

$\displaystyle\rho^{\prime\prime}_{j}$
$\displaystyle=E_{00}(\ket{\psi_{j}}\bra{\psi_{j}})E_{00}^{\dagger}+E_{11}(\ket{\psi_{j}}\bra{\psi_{j}})E_{11}^{\dagger}$
(35) $\displaystyle+E_{22}(\ket{\psi_{j}}\bra{\psi_{j}})E_{22}^{\dagger}.$

The matrix form of the density matrix $\rho^{\prime\prime}_{j}$ can be
obtained from the Eq. (17) of supplementary material by putting
$\rho=\ket{\psi_{j}}\bra{\psi_{j}}$. The density matrix has seven dimensional
noiseless subspace span
$\\{\ket{00},\ket{01},\ket{10},\ket{02},\ket{20},\ket{12},\ket{21}\\}$. The
two nonzero eigenvalues are

$\eta^{\pm}_{j}=\frac{1}{2}[1\pm\sqrt{1-l_{j}^{2}}],$

where
$l_{j}^{2}=4(1-|a|_{j}^{2}-p_{1}|e|_{j}^{2}-p_{2}|k|_{j}^{2})(p_{1}|e|_{j}^{2}+p_{2}|k|_{j}^{2})$
From the expression of eigenvalues we can see that $\eta_{j}$ depends only on
the absolute value of the coefficients and is independent of the phase. So, we
can assume the state parameters are real.

The average entropy of the output state is found as

$\sum_{j}\xi_{j}S(\Phi(\ket{\psi_{j}}\bra{\psi_{j}}))=\sum_{j}\xi_{j}H_{2}(\eta_{j}),$
(36)

where $H_{2}(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ is commonly known as
Shanon’s binary entropy.

Finally, we can find another ensemble
$\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\}$ by replacing
$|b_{j}|^{2}=|c_{j}|^{2}=|d_{j}|^{2}=|f_{j}|^{2}=|g_{j}|^{2}=|h_{j}|^{2}=(|b_{j}|^{2}+|c_{j}|^{2}+|d_{j}|^{2}+|f_{j}|^{2}+|g_{j}|^{2}+|h_{j}|^{2})/6$,
which will corresponds to the same density matrix $\bar{\rho}$. And it can be
easily checked that the Holevo quantity will also be unchanged, which is clear
from the expression of eigenvalues $\eta^{\pm}_{j}$.

The series of relations established so far demonstrate that we can find an
ensemble $\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\}$, which enables us to
determine the upper bound of the Holevo quantity of any arbitrary ensemble
$\\{\xi_{j},\ket{\psi_{j}}\\}$. This is because
$\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\}$ is a subset of the original
ensemble $\\{\xi_{j},\ket{\psi_{j}}\\}$. Consequently, maximizing the Holevo
quantity for $\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\}$ will also result
in the maximum for the entire set $\\{\xi_{j},\ket{\psi_{j}}\\}$.

In summary, we have to investigate the classical capacity of the ensemble
$\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\}$, where the states have the
following form:

$\displaystyle\ket{\tilde{\psi}}_{j}$
$\displaystyle=a_{j}\ket{00}+b_{j}\ket{01}\pm b_{j}\ket{02}\pm
b_{j}\ket{10}+e_{j}\ket{11}$ (37) $\displaystyle\pm b_{j}\ket{12}\pm
b_{j}\ket{20}\pm b_{j}\ket{21}+k_{j}\ket{22}.$

And the corresponding density matrix $\bar{\rho}$, which gives the maximum
Holevo bound, is a diagonal matrix with the elements
{$\alpha,\beta,\beta,\beta,\gamma,\beta,\beta,\beta,\delta$}, where

$\displaystyle\alpha=\sum_{j}\tilde{\xi}_{j}|a_{j}|^{2},\;\beta=\sum_{j}\tilde{\xi}_{j}|b_{j}|^{2},\;\gamma=\sum_{j}\tilde{\xi}_{j}|c_{j}|^{2},$
(38) $\displaystyle\delta=\sum_{j}\tilde{\xi}_{j}|d_{j}|^{2}$

and they satisfy normalization relation
$|a_{j}|^{2}+6|b_{j}|^{2}+|c_{j}|^{2}+|d_{j}|^{2}=1$. The entropy of the
output density matrix corresponding to the mapping $\Phi_{(p_{1},p_{2},0)}$
for the input state $\bar{\rho}$ is

$\displaystyle S(\Phi(\bar{\rho}))=$
$\displaystyle-(\alpha+p_{1}\gamma+p_{2}\delta)\log_{2}{(\alpha+p_{1}\gamma+p_{2}\delta)}$
(39)
$\displaystyle-6\beta\log_{2}{\beta}-\gamma(1-p_{1})\log_{2}{((1-p_{1})\gamma)}$
$\displaystyle-\delta(1-p_{2})\log_{2}{((1-p_{2})\delta)}.$

Now, we utilize the following inequality to calculate the lower bound of the
second term of Holevo quantity:

$\displaystyle\sum_{j}\xi_{j}H_{2}\left\\{\frac{1+\sqrt{1-[(6|b_{j}|^{2}+|e_{j}|^{2}+|k_{j}|^{2})^{2}-((6|b_{j}|^{2}+(1-2p_{1})|e_{j}|^{2}+(1-2p_{2})|k_{j}|^{2})^{2}]^{2}}}{2}\right\\}$
(40)
$\displaystyle\geq\sum_{j}\xi_{j}H_{2}\left\\{\frac{1+\sqrt{1-[2p_{1}|e_{j}|^{2}+2p_{2}|k_{j}|^{2}]^{2}}}{2}\right\\}\geq
H_{2}\left\\{\frac{1+\sqrt{1-[2p_{1}\sum_{j}\xi_{j}|e_{j}|^{2}+2p_{2}\sum_{j}\xi_{j}|k_{j}|^{2}]}}{2}\right\\}$
$\displaystyle=H_{2}\left\\{\frac{1+\sqrt{1-[2p_{1}\gamma+2p_{2}\delta]^{2}}}{2}\right\\}.$

The first inequality we obtain using the fact that $X^{2}-Y^{2}\geq(X-Y)^{2}$,
when $X>Y$ and the second inequality we can write from the convexity property
of the binary entropy function $H_{2}(\frac{1+\sqrt{1-x^{2}}}{2})$. The
complete expression of Holevo quantity for the V-type channel is obtained by
maximizing it over all possible values of
$\alpha,\;\beta,\;\gamma\;\text{and}\;\delta$, which is

$\displaystyle\chi(\Phi,\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\})$
$\displaystyle=\max_{\alpha,\beta,\gamma,\delta}\Big{(}(\alpha+p_{1}\gamma+p_{2}\delta)\log_{2}{(\alpha+p_{1}\gamma+p_{2}\delta)}-6\beta\log_{2}{\beta}$
(41)
$\displaystyle-\gamma(1-p_{1})\log_{2}{((1-p_{1})\gamma)}-\delta(1-p_{2})\log_{2}{((1-p_{2})\delta)}$
$\displaystyle+H_{2}\left\\{\frac{1+\sqrt{1-[2p_{1}\gamma+2p_{2}\delta]^{2}}}{2}\right\\}\Big{)}.$

Therefore we can conclude that Eq. (41) gives the single-shot classical
capacity $\mathcal{C}_{1}$ of the V-type channel. In Fig. (3a), we have shown
the behaviour of $\mathcal{C}_{1}$ with respect to the decoherence parameters
$p_{1}$ and $p_{2}$. In the case of complete damping ($p_{1}=1$) of the energy
level $\ket{11}$, the output density matrix becomes eight-dimensional,
indicating the maximum value of the capacity $\log_{2}8$ at $p_{2}=0$. The
figure (3b) illustrates the decay of $\mathcal{C}_{1}$ as a function of
$p_{2}$, under the condition that the energy level $\ket{1}$ is completely
damped.

(a)

(b)

Figure 3: (Color online) (a) The single-shot classical capacity
($\mathcal{C}_{1}$) of channel $\Phi_{(p_{1},p_{2},0)}$ varies according to
the damping parameter $p_{1}$ and$p_{2}$. The value of $\mathcal{C}_{1}$ is
obtained through numerical optimization. (b) The dynamics of $\mathcal{C}_{1}$
with respect to $p_{2}$ when first excited state is completely damped i.e.,
$p_{1}=1$.

####  IV.0.2 Single decay channel

The single-shot classical capacity $\mathcal{C}_{1}$ for single decay channel
i.e., only one of the three damping parameters $p_{i}$ is non-zero, can be
calculated from Eq. (41) by setting one of the damping parameters $p_{1}$ or
$p_{2}$ equals to zero. Since the Kraus operator for the mappings
$\Phi_{(p_{1},0,0)}$, $\Phi_{(0,p_{2},0)}$ and $\Phi_{(0,0,p_{3})}$ has the
same form, the corresponding Holevo quantity will also be same. So, we can
write the expression of Holevo quantity for the mapping $\Phi_{(p_{1},0,0)}$
as

$\displaystyle\chi(\Phi,\\{\tilde{\xi}_{j},\ket{\tilde{\psi}_{j}}\\})=$ (42)
$\displaystyle\max_{\alpha,\beta,\gamma,\delta}\Big{(}(\alpha+p_{1}\gamma)\log_{2}{(\alpha+p_{1}\gamma)}-6\beta\log_{2}{\beta}$
$\displaystyle-\gamma(1-p_{1})\log_{2}{((1-p_{1})\gamma)}-\delta\log_{2}{\delta}$
$\displaystyle+H_{2}\left\\{\frac{1+\sqrt{1-[2p_{1}\gamma]^{2}}}{2}\right\\}\Big{)}.$

The above equation is the upper bound of the Holevo quantity, which is the
single-shot classical capacity for the mapping $\Phi_{(p_{1},0,0)}$. After
performing the optimization over all possible values of
$\alpha,\beta,\gamma\;\text{and}\;\delta$, one can obtain the variation of
$\mathcal{C}_{1}$ with respect to the damping parameter $p_{1}$, which is
depicted in Fig. (4a). Fig. (4a) displays the values of the parameters
$\alpha,\beta,\gamma\;\text{and}\;\delta$ against $p_{1}$ during the
optimization process.

(a)

(b)

Figure 4: (Color online) (a) The single-shot classical capacity
($\mathcal{C}_{1}$) of channel $\Phi_{(p_{1},0,0)}$ varies according to the
damping parameter $p_{1}$. The value of $\mathcal{C}_{1}$ is obtained through
numerical optimization. (b) The populations $\alpha$, $\beta$, $\gamma$ and
$\delta$ refer to the states that optimize the classical capacity formula for
the $\Phi_{(p_{1},0,0)}$ channel with respect to the damping parameter
$p_{1}$.

##  V Quantum capacity

The quantum capacity, $\mathcal{Q}$ represents the fundamental measure of a
channel’s ability to reliably transmit and convey quantum information. The
closed expression of the quantum capacity is represented by the following
formula [51, 52]

$\mathcal{Q}=\lim_{n\rightarrow\infty}\frac{\mathcal{Q}_{n}}{n},\quad
Q_{n}=\max_{\rho^{(n)}}I_{c}\left(\Phi^{\otimes n},\rho^{(n)}\right),$ (43)

where $\rho^{(n)}$ is the input state for $n$ number of uses of channel and
the coherent information is

$I_{c}\left(\Phi^{\otimes n},\rho^{(n)}\right)=S\left(\Phi^{\otimes
n}\left(\rho^{(n)}\right)\right)-S\left(\tilde{\Phi}^{\otimes
n},\rho^{(n)}\right),$ (44)

with $S(\rho)-\operatorname{tr}[\rho\log_{2}\rho]$ is the von Neuman entropy
of the state $\rho$ and $\tilde{\Phi}$ is the complementary channel
corresponding to the map $\Phi$. For a degradable channel, the quantum
capacity is equal to its single-shot quantum capacity $\mathcal{Q}_{1}$.
However, the quantum capacity of an anti-degradable channel is zero due to no
cloning principle. It is to be noted that the optimization process outlined in
Eq. (43), must be conducted over the set of density matrices $\rho^{(n)}$ of n
channel uses.

Let us consider the generic input state $\\{\xi_{j},\ket{\psi_{j}}\\}$ and the
corresponding density matrix
$\rho=\sum_{j}\xi_{j}\ket{\psi_{j}}\bra{\psi_{j}}$ as defined earlier in Eq.
(29). We aim to identify the category of input states that enables us to find
the quantum capacity, specifically by maximizing the coherent information. To
accomplish this goal, we observe that it is possible to construct a diagonal
density operator for any two-qutrit density operator $\rho$, as follows:

$\rho^{\prime}=\frac{1}{16}\Big{(}\rho+\sum_{i=1}^{15}\mathbf{U}_{i}\rho\mathbf{U}_{i}\Big{)}.$
(45)

Eventually, the density matrix $\rho^{\prime}$ become the diagonal matrix with
elements
{$\alpha,\beta_{1},\beta_{2},\beta_{3},\gamma,\beta_{4},\beta_{5},\beta_{6},\delta$}.
Now , we will show the amount of coherent information of $\rho^{\prime}$ is
equal to or greater than that associated with $\rho$:

$\displaystyle I_{c}(\Phi,\rho^{\prime})$
$\displaystyle=I_{c}\Bigg{(}\Phi,\frac{1}{16}\Big{(}\rho+\sum_{i=1}^{15}\mathbf{U}_{i}\rho\mathbf{U}_{i}\Big{)}\Bigg{)}$
(46)
$\displaystyle\geq\frac{1}{16}I_{c}(\Phi,\rho)+\frac{1}{16}\sum_{i=1}^{15}I_{c}\Big{(}\Phi,\mathbf{U}_{i}\rho\mathbf{U}_{i}\Big{)}$
$\displaystyle=\frac{1}{16}I_{c}(\Phi,\rho)+\frac{1}{16}\sum_{i=1}^{15}S\Big{(}\Phi\Big{(}\mathbf{U}_{i}\rho\mathbf{U}_{i}\Big{)}\Big{)}$
$\displaystyle-\frac{1}{16}\sum_{i=1}^{15}S\Big{(}\tilde{\Phi}\Big{(}\mathbf{U}_{i}\rho\mathbf{U}_{i}\Big{)}\Big{)}=I_{c}(\Phi,\rho).$

We have utilized the property of degradable channels in the inequality above,
which states that the coherent information exhibits concave behaviour. The
reference cited demonstrates that the von Neumann entropy is invariant under
unitary operations. This property was also applied in our analysis, resulting
in the following conclusion:
$S(\Phi(\mathbf{U}_{i}\rho\mathbf{U}_{i}))=S(\Phi(\rho))$.

Now, we can make a new state $\bar{\rho}$:

$\bar{\rho}=\frac{1}{6}\Big{(}\rho+\sum_{i=1}^{5}\mathbf{V}_{i}\rho^{\prime}\mathbf{V}_{i}\Big{)},$

where $\mathbf{V}_{i}$ is defined in Eq. (21). Here the density matrix,
$\bar{\rho}$ becomes a diagonal matrix with elements
{$\alpha,\beta,\beta,\beta,\gamma,\beta,\beta,\beta,\delta$}, where $\beta$ is
defined in Eq. (38).

In the same way as above, using the concavity property of degradable channel
we can show that the coherent information $I_{c}(\Phi,\bar{\rho})\geq
I_{c}(\Phi,\rho^{\prime})$.

We may conclude that by maximizing the coherent information relative to the
diagonal state $\bar{\rho}$, we can derive the quantum capacity. Therefore,
within the degradable region, the expression for the single-shot quantum
capacity can be formulated as:

$\displaystyle\mathcal{Q}\left(\Phi_{m}\right)$
$\displaystyle=\mathcal{Q}_{1}\left(\Phi_{m}\right)=\max_{\bar{\rho}}I_{c}\left(\Phi_{m},\bar{\rho}\right)$
(47)
$\displaystyle=\max_{\bar{\rho}}\left\\{S\left(\Phi_{m}\left(\bar{\rho}\right)\right)-S_{e}\left(\Phi_{m},\bar{\rho}\right)\right\\}.$

Our next task involves calculating the quantum capacity of the fully
correlated MAD channel. Nevertheless, there is an obstacle that we need to
overcome. It has been demonstrated in reference [53] that for any CPTP map
corresponding to a qutrit system to be degradable, a necessary condition is
that its associated Choi rank and hence, the minimum number of Kraus operators
required to represent such a transformation must not exceed three. However,
from Eq. (1), we obtain the transformation from four Kraus operators. So, we
simplify the problem by setting one or two decay parameter in such a way that
the Kraus operators required to represent the map is three. In the following
subsection, we will systematically examine the quantum capacity of a fully
correlated MAD channel under various conditions, one by one.

####  V.0.1 Single decay channel

The instances of the fully correlated qutrit MAD channel that we are analyzing
in this context involve situations where only one of the three damping
parameters, $p_{i}$ has a non-zero value. The associated maps for this single
decay are $\Phi_{(p_{1},0,0)}$, $\Phi_{(0,p_{2},0)}$ and $\Phi_{(0,0,p_{3})}$
respectively.

We observed that two non-zero Kraus operators corresponding to the mapping
$\Phi_{(p_{1},0,0)}$ are

$\displaystyle E_{00}=$
$\displaystyle\ket{00}\bra{00}+\sqrt{1-p_{1}}\ket{11}\bra{11}+\ket{22}\bra{22}$
(48) $\displaystyle+\sum_{\begin{subarray}{c}i,j=0\\\ i\neq
j\end{subarray}}^{2}\ket{ij}\bra{ij}$ $\displaystyle E_{11}=$
$\displaystyle\sqrt{p_{1}}\ket{00}\bra{11}.$

The expression for the transformation $\Phi_{(p_{1},0,0)}(\rho)$ and the
corresponding complementary map $\tilde{\Phi}_{(p_{1},0,0)}(\rho)$ according
to the equation (15) are given in the supplementary note. A comprehensive
analysis of degradability and technique can be found in the supplementary note
and Appendix. From the channel degradability analysis, we have seen that the
channel is degradable for $p_{1}\leq\frac{1}{2}$. Even though the channel is
not degradable or anti-degradable for $p_{1}\geq\frac{1}{2}$, we can still
compute its quantum capacity in this range using the monotonicity constraint
of the quantum capacity function discussed in the Appendix.

Consequently, the quantum capacity in the degradable region is given by

$\displaystyle\mathcal{Q}\left(\Phi_{m}\right)$
$\displaystyle=\max_{\bar{\rho}}I_{c}\left(\Phi_{m},\bar{\rho}\right)$ (49)
$\displaystyle=\max_{\bar{\rho}}\left\\{S\left(\Phi_{m}\left(\bar{\rho}\right)\right)-S_{e}\left(\Phi_{m},\bar{\rho}\right)\right\\}$
$\displaystyle=\max_{\alpha,\beta,\gamma,\delta}\\{-(\alpha+p_{1}\gamma)\log_{2}{(\alpha+p_{1}\gamma)}-6\beta\log_{2}{\beta}$
$\displaystyle-((1-p_{1})\gamma)\log_{2}{((1-p_{1})\gamma)}-\delta\log_{2}{\delta}$
$\displaystyle+(1-p_{1}\gamma)\log_{2}{(1-p_{1}\gamma)}+p_{1}\gamma\log_{2}{(p_{1}\gamma})\\}.$

The results is depicted in Fig. (5). In case of $p_{1}=0$ the value of the the
quantum capacity become $\log_{2}9$.

In the above section, we have calculated the quantum capacity for the mapping
$\Phi_{(p_{1},0,0)}$. It can be readily observed that the other group of
transformation $\Phi_{(0,p_{2},0)}$ and $\Phi_{(0,0,p_{3})}$ can be
transformed into each other by simply swapping energy levels. Therefore, the
quantum capacity of these three groups should be the same as each channel can
be derived from the other, i.e.,
$\mathcal{Q}(\Phi_{(p,0,0)})=\mathcal{Q}(\Phi_{(0,p,0)})=\mathcal{Q}(\Phi_{(0,0,p)})$
for all the value of p between 0 and 1.

(a)

(b)

Figure 5: (Color online) (a) The quantum capacity ($\mathcal{Q}$) of channel
$\Phi_{(p_{1},0,0)}$ varies according to the damping parameter $p_{1}$. If
$p_{1}$ is less than or equal to 1/2, the channel shows the degradable
behaviour and the value of $\mathcal{Q}$ is obtained through numerical
optimization. However, if $p_{1}$ is greater than 1/2, the channel is neither
degradable nor anti-degradable, and the corresponding value of quantum
capacity is fixed at $\log_{2}8$. It is worth noting that the reported values
adhere to the monotonicity principle as defined in Eq. (65). (b) The
populations $\alpha$, $\beta$, $\gamma$ and $\delta$ refer to the states which
optimize the quantum capacity formula corresponding to the
$\Phi_{(p_{1},0,0)}$ channel with respect to the damping parameter $p_{1}$.

####  V.0.2 Double decay channel

This section focuses primarily on a model that allows for only two possible
transitions. Specifically, we consider a scenario where one of the three
damping parameters, $p_{1}$, $p_{2}$, or $p_{3}$, is equal to zero. Further,
we classify the double decay channel as a V-type or $\Lambda$-type channel
depending upon the conditions $p_{3}=0$ and $p_{1}=0$ [28]. In the following
subsection, we will calculate the quantum capacity of these channels since
they are degradable. However, when $p_{2}=0$, the channel is referred to as
$\Xi$-type channel and can be described by four Kraus operators. Therefore,
since the channel is not degradable, we are unable to calculate the
corresponding quantum capacity of this channel.

###  V.1 V-type channel

If the decay parameter $p_{3}$ is equal to zero, the atoms undergo a V-type
transition, which is commonly referred to as a V-type channel. For this
channel, the accessible values of $p_{1}$ and $p_{2}$ lie on the surface EFH
as shown in Fig. (2). The Kraus operators representing the V-type channel are
given in sec. (IV.0.1), where we computed the classical capacity for this type
of channel. The supplementary materials provide the expression for both the
transformation $\Phi_{(p_{1},p_{2},0)}(\rho)$ and its corresponding
complementary map $\tilde{\Phi}_{(p_{1},p_{2},0)}(\rho)$ and the degradibility
analysis of the channel. From the degradability analysis, we have seen that
the channel is degradable for $p_{1}\leq 1/2$ and $p_{2}\leq 1/2$, and the
quantum capacity becomes the single-shot capacity in this region. In other
range of $p_{1}$ and $p_{2}$, the channel is non-degradable. Still, we can
calculate the quantum capacity in this region by checking the monotonicity
constraint.

The quantum capacity in the degradable region is given by the following
expression:

$\displaystyle\mathcal{Q}\left(\Phi\right)$
$\displaystyle=\max_{\alpha,\beta,\gamma,\delta}\\{-(\alpha+p_{1}\gamma+p_{2}\delta)\log_{2}{(\alpha+p_{1}\gamma+p_{2}\delta)}$
(50)
$\displaystyle-6\beta\log_{2}{\beta}-\gamma(1-p_{1})\log_{2}{((1-p_{1})\gamma)}-\delta(1-p_{2})$
$\displaystyle\times\log_{2}{((1-p_{2})\delta)}+p_{1}\gamma\log_{2}{(p_{1}\gamma)}+(1-p_{1}\gamma-
p_{2}\delta)$ $\displaystyle\times\log_{2}{(1-p_{1}\gamma
p_{2}\delta)}+p_{2}\delta\log_{2}{(p_{2}\delta)}\\}.$

From the above expression we can determine the value of $\mathcal{Q}(\Phi)$ in
the border of the degradable region i.e., $\Phi_{(1/2,p_{2},0)}$ and
$\Phi_{(p_{1},1/2,0)}\forall p_{1},p_{2}\leq 1/2$ is known. Now, we calculate
the quantum capacity at the edge of the parameter space, i.e.,
$\mathcal{Q}(\Phi_{(1,p_{2},0)})$ and $\mathcal{Q}(\Phi_{(p_{1},1,0)})$. The
detailed analysis of the quantum capacity of the map $\Phi_{(1,p_{2},0)}$ is
shown in the supplementary material. We have shown that for a specific value
of $p_{1}$ or $p_{2}$, the quantum capacity has the same value at the border
of the degradable region and on the edges. Thus, based on the monotonicity
constraint described in Eq. (69), it can be concluded that the quantum
capacity remains unchanged in the intermediate region. The behavior of
$\mathcal{Q}(\Phi_{(p_{1},p_{2},0)})$ with respect to $p_{1}$ and $p_{2}$ is
displayed in Fig. (6).

According to the findings presented in Ref. [41], it has been demonstrated
that the uncorrelated V-type channel exhibits anti-degradability property
within the region $p_{1}\geq 1/2$ and $p_{2}\geq 1/2$, which implies that the
quantum capacity in this particular region is effectively zero. Here, we have
shown that fully correlated V-type channel is not anti-degradable and has the
lowest quantum capacity value $\log_{2}7$.

Figure 6: Contour plot of the quantum capacity ($\mathcal{Q}$) for the V-type
channel, $\Phi_{(p_{1},p_{2},0)}$ with the damping parameter $p_{1}$ and
$p_{2}$. In the range $p_{1}$ and $p_{2}$ are less than or equal to 1/2 (area
enclosed by the red line), the channel is degradable. However, if $p_{1}$ and
$p_{2}$ are greater than 1/2, the channel is neither degradable nor anti-
degradable, and the minimum value of the capacity value is $\log_{2}7$.

###  V.2 $\Lambda$-type channel

In damping channels of this type, energy level $\ket{2}$ interacts with the
lower lying levels $\ket{1}$ and $\ket{0}$. In this case, we examine the
capacity value for $\Phi$ that belongs to the square surface CDEH depicted in
Figure (2), which is defined by the condition $p_{3}=0$. According to Eq.
(II), one can write the Kraus operators that describe this channel as

$\displaystyle E_{00}=$
$\displaystyle\ket{00}\bra{00}+\ket{11}\bra{11}+\sqrt{1-p_{23}}\ket{22}\bra{22}$
$\displaystyle+\sum_{\begin{subarray}{c}i,j=0\\\ i\neq
j\end{subarray}}^{2}\ket{ij}\bra{ij}$ $\displaystyle E_{22}=$
$\displaystyle\sqrt{(1-\Theta)p_{23}}\ket{00}\bra{22}$ $\displaystyle E_{33}=$
$\displaystyle\sqrt{\Theta p_{23}}\ket{11}\bra{22},$

where for convenience we have taken $p_{23}=1-(1-p_{2})(1-p_{3})$ and
$\Theta=\ln{(1-p_{3})}/(\ln{(1-p_{3})}+\ln{(1-p_{2})})$.
The symmetry of the model under the exchange of $p_{2}$ and $p_{3}$ is
apparent from the structure of the Kraus operators. So we can conclude that

$\mathcal{Q}(\Phi_{(0,p_{2},p_{3})})=\mathcal{Q}(\Phi_{(0,p_{3},p_{2})}).$
(51)

In supplementary material we have shown the transformation
$\Phi_{(0,p_{2},p_{3})}$ and complementary map
$\tilde{\Phi}_{(0,p_{2},p_{3})}$. We have seen that the channel is degradable
for $(1-p_{2})(1-p_{3})\geq\frac{1}{2}$, and it is not anti-degradable in any
region. The quantum capacity of the channel in the degradable region is
represented by the given equation:

$\displaystyle\mathcal{Q}\left(\Phi\right)$
$\displaystyle=\max_{\alpha,\beta,\gamma,\delta}\\{-(\alpha+(1-\Theta)p_{23}\delta)\log_{2}{(\alpha+(1-\Theta)p_{23}\delta)}$
(52) $\displaystyle-6\beta\log_{2}{\beta}-(\gamma+\Theta
p_{23}\delta)\log_{2}{(\gamma+\Theta p_{23}\delta)}$
$\displaystyle-((1-p_{23})\delta)\log_{2}{(1-p_{23})\delta)}+(\Theta
p_{23}\delta)\log_{2}(\Theta p_{23}\delta)$
$\displaystyle+((1-\Theta)p_{23}\delta)\log_{2}{((1-\Theta)p_{23}\delta)}+(1-p_{23}\delta)$
$\displaystyle\times\log_{2}{(1-p_{23}\delta)}\\}.$

At the border of the degradable region $(1-p_{2})(1-p_{3})=1/2$, (shown in the
red curve in Fig. (7)) the value of the quantum capacity can be determined,
which is $\log_{2}8$. We also observe that the output density matrix has
8-dimensional noiseless subspace. Hence, the lower bound of the quantum
capacity is $\log_{2}8$. From the composition rule and monotonicity constraint
of capacity function, We can conclude that $\mathcal{Q}(\Phi)=\log_{2}8$ in
the non-degradable region. The behavior of the quantum capacity
$\mathcal{Q}(\Phi_{(0,p_{2},p_{3})})$ with respect to $p_{1}$ and $p_{2}$ is
illustrated in Figure (7). In case of uncorrelated $\Lambda$-type channel, as
shown in Ref. [41], the channel is anti-degradable between $p_{2}+p_{3}\geq
1/2$ and $p_{2}+p_{3}leq1$ indicates zero value of quantum capacity and beyond
the range $p_{2}+p_{3}=1$ the channel is not CPTP. However, fully correlated
$\Lambda$-type channel is CPTP for all values of $p_{2}$ and $p_{3}$, and
lowest value of quantum capacity is $\log_{2}8$.

Figure 7: Contour plot of the quantum capacity ($\mathcal{Q}$) for the
$\Lambda$-type channel, $\Phi_{(0,p_{2},p_{3})}$ with the damping parameter
$p_{2}$ and $p_{3}$. In the range $(1-p_{2})(1-p_{3})\geq\frac{1}{2}$, the
channel is degradable. However, in other regions, the channel is neither
degradable nor anti-degradable, and the corresponding value of quantum
capacity is fixed at $\log_{2}8$, obtained from the monotonicity principle.

####  V.2.1 Three decay rate channel

Let us consider the region $1-p_{1}=(1-p_{2})(1-p_{3})$, which is indicated by
the surface BEFH in Fig. (2). This special three rate decay map satisfying
above mentioned constraint, admits the following Kraus operators:

$\displaystyle E_{00}=$
$\displaystyle\ket{00}\bra{00}+\sqrt{(1-p)}\ket{11}\bra{11}+\sqrt{(1-p)}\ket{22}\bra{22}$
$\displaystyle+\sum_{\begin{subarray}{c}i,j=0\\\ i\neq
j\end{subarray}}^{2}\ket{ij}\bra{ij},$ $\displaystyle E_{11}=$
$\displaystyle\sqrt{p_{23}}\ket{00}\bra{11},\,E_{22}=\sqrt{p_{23}}\ket{00}\bra{22},$

where we have taken $p_{23}=1-(1-p_{2})(1-p_{3})$ for convenience.
From the structural symmetry of the Kraus operators, it is clear that under
the exchange of $p_{2}$ and $p_{3}$, quantum capacity does not change. Hence,
we can write

$\mathcal{Q}(\Phi_{(p_{23},p_{2},p_{3})})=\mathcal{Q}(\Phi_{(p_{23},p_{3},p_{2})}).$
(54)

Similar to the $\Lambda$-type channel this map, $\Phi_{(p_{23},p_{2},p_{3})}$
is also degradable in the region $(1-p_{2})(1-p_{3})\geq\frac{1}{2}$. In the
degradable region the quantum capacity for this channel is,

$\displaystyle\mathcal{Q}\left(\Phi\right)$
$\displaystyle=\max_{\alpha,\beta,\gamma,\delta}\\{-(\alpha+p_{23}\gamma+p_{23}\delta)\log_{2}{(\alpha+p_{23}\gamma+p_{23}\delta)}$
(55)
$\displaystyle-6\beta\log_{2}{\beta}-\gamma(1-p_{23})\log_{2}{((1-p_{23})\gamma)}-(1-p_{23}\delta)$
$\displaystyle\times\log_{2}{(1-p_{23}\delta)}+p_{23}\gamma\log_{2}{(p_{23}\gamma)}+p_{23}\delta\log_{2}{(p_{23}\delta)}$
$\displaystyle+(1-p_{23}\gamma-p_{23}\delta)\log_{2}{(1-p_{23}\gamma-
p_{23}\delta)}\\}.$

In the other region, we can also calculate the quantum capacity using
composition rule and monotonicity constraints like $\Lambda$-type channel. The
output density matrix corresponding to this channel can be verified to possess
a seven dimensional noiseless subspace. This observation indicates that the
lower bound of the quantum capacity for this channel is given by the
$\log_{2}7$. The quantum capacity for this map $\Phi_{(p_{23},p_{2},p_{3})}$
is displayed in Figure (8).

Figure 8: Contour plot of the quantum capacity ($\mathcal{Q}$) of the specific
three decay rate channel, $\Phi_{(p,p_{2},p_{3})}$ with the damping parameter
$p_{1}$ and $p_{2}$. In the range $(1-p_{2})(1-p_{3})\geq\frac{1}{2}$ the
channel is degradable like $\Lambda$-type channel . However, in the other
region, the channel is neither degradable nor anti-degradable, and the
corresponding capacity value is fixed at $\log_{2}7$ obtained from
monotonicity principle.

##  VI Entanglement assisted capacity

For the sake of completeness in this section, we have examined both classical
and quantum capacity of the fully correlated MAD channel in entanglement
assisted scenario. The concept of entanglement-assisted quantum capacity,
$\mathcal{Q}_{E}$, refers to the maximum quantity of quantum information which
can be transferred reliably through a given channel per each use of that
channel with the assumption that the sender and receiver have access to an
unlimited supply of entanglement resources beforehand. It can be expressed as
[7, 54]

$\mathcal{Q}_{E}=\frac{1}{2}\max_{\rho}I\left(\Phi,\rho\right),$ (56)

where the optimization process is carried out with respect to the input state
$\rho$ for a single application of the channel $\Phi$ and

$I\left(\Phi,\rho\right)=S(\rho)+I_{c}\left(\Phi,\rho\right).$ (57)

The mutual information functional $I$ is equal to the coherent information
functional $I_{c}$ with the addition of input state von Neuman entropy
$S(\rho)$. The mutual information functional satisfies the sub-additive
property [55],and because of that, no regularization is needed in the
calculation of $\mathcal{Q}_{E}$. So, we can write,

$I\left(\Phi,\rho\right)=S\left(\rho\right)+S\left(\Phi(\rho)\right)-S(\tilde{\Phi}(\rho)).$
(58)

Covariance property of von Neuman entropy and concavity of the coherent
information also apply here for mutual information.

The entanglement-assisted classical capacity $\mathcal{C}_{E}$ refers to the
maximum quantity of classical information that can be transmitted through each
channel usage with unrestricted entanglement assistance. It is simply equal to
twice the entanglement assisted quantum capacity value, i.e.,

$\mathcal{Q}_{E}=\frac{1}{2}(\mathcal{C}_{E}).$ (59)

The explicit expressions of entanglement assisted quantum capacity for single
decay channel and double decay channel are given in supplementary material. In
the following Fig. (9) and (10), we have shown the entanglement assisted
quantum capacity of the single decay channel and the channel for which all the
decay rates are same, with decay parameter. In addition, the corresponding
populations of $\alpha$, $\beta$, $\gamma$ and $\delta$ during the
optimization process have been plotted. In Fig. (11), the entanglement
assisted quantum capacity of the V-type channel, $\Lambda$-type channel and
special three decay rate channel have been displayed.

(a)

(b)

Figure 9: (Color online) (a) The entanglement assisted quantum capacity
($\mathcal{Q}_{E}$) of channel $\Phi_{(p_{1},0,0)}$ varies according to the
damping parameter $p_{1}$. $\mathcal{C}_{E}$ is obtained by solving the
optimization problem given in supplementary Eq. (23). (b) The populations
$\alpha$, $\beta$, $\gamma$ and $\delta$ refer to the states that optimize the
$\mathcal{C}_{E}$ formula for the $\Phi_{(p_{1},0,0)}$ channel with respect to
the damping parameter $p_{1}$.

(a)

(b)

Figure 10: (Color online) (a) The entanglement assisted quantum capacity
($\mathcal{Q}_{E}$) of channel $\Phi_{(p,p,p)}$ varies according to the
damping parameter $p$. $\mathcal{Q}_{E}$ is obtained by solving the
optimization problem given in supplementary Eq. (26). (b) The populations
$\alpha$, $\beta$, $\gamma$ and $\delta$ refer to the states that optimize the
$\mathcal{C}_{E}$ formula for the $\Phi_{(p,p,p)}$ channel with respect to the
damping parameter $p$.

(a)

(b)

(c)

Figure 11: (a) Contour plot of the entanglement assisted capacity
($\mathcal{Q}_{E}$) of V-type channel, $\Phi_{(p_{1},p_{2},0)}$ varies
according to the damping parameter $p_{1}$ and $p_{2}$. (b) Contour plot of
the entanglement assisted capacity of $\Lambda$-type channel,
$\Phi_{(0,p_{2},p_{3})}$ varies according to the damping parameter $p_{2}$ and
$p_{3}$. (c) Contour plot of the entanglement assisted capacity of three decay
rate channel, $\Phi_{(p,p_{2},p_{3})}$ varies according to the damping
parameter $p_{2}$ and $p_{3}$.

##  VII Conclusion

In quantum information theory, the qubit Amplitude damping Channel (ADC) model
is a well-known example of quantum noise. It has been shown that correlated
ADC channel has higher information transmission capacity and it can protect
quantum correlation efficiently. In our work, we have investigated the
information capacity for multi-dimensional version of the correlated ADC model
with a special focus on dimension d=3. We have explicitly calculated the
single-shot classical and quantum capacities of these channels for fully
correlated qutrit MAD channels. This computation has significantly expanded
the set of models whose capacity is known. We compare our results with the
memoryless MAD channel [41] and observed that, unlike the memoryless MAD
channel, the fully correlated channel does not exhibit anti-degradability. The
fully correlated channel always have a positive quantum capacity, whereas the
uncorrelated channel has zero capacity in the anti-degradable region. A
correlated MAD channel holds greater potential for applications in quantum
communication compared to an uncorrelated channel.

The insights provided by this research can be useful in designing and
optimizing quantum communication systems to operate under noisy environments.
The findings of this study provide a basis for further exploration of the
information capacity for MAD channels with arbitrary degrees of memory.

###### Acknowledgements.

R.S acknowledge the financial support provided by the institute. We also
acknowledge Abhinash Kumar Roy for fruitful discussion.

##  Appendix A Appendixes

### Composition rule

The study presented in Ref. [41] established that under composition rules, MAD
channels are closed, a useful property for examining their information
capacities. Here, we examine whether fully correlated MAD channels exhibit
similar behaviour. We notice that if
$\Phi_{(}p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime})$ and
$\Phi_{(}p_{1}^{\prime\prime},p_{2}^{\prime\prime},p_{3}^{\prime\prime})$ are
two maps such that they fulfil the CPTP conditions, then we have

$\Phi_{(}p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime})\circ\Phi_{(}p_{1}^{\prime\prime},p_{2}^{\prime\prime},p_{3}^{\prime\prime})=\Phi_{(}p_{1},p_{2},p_{3})$
(60)

The significance of the above equation in addressing our current problem lies
in the channel data-processing inequalities [56, 24]. If a CPTP map
$\Phi_{(p_{1},p_{2},p_{3})}$ is obtained by combining two CPTP maps
$\Phi_{(p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3})}$ and
$\Phi_{(p^{\prime\prime}_{1},p^{\prime\prime}_{2},p^{\prime\prime}_{3})}$ then
the channel data processing inequality indicates that any information capacity
function $\mathcal{F}$ such as classical capacity $\mathcal{C}$, quantum
capacity $\mathcal{Q}$, entanglement-assisted quantum capacity
$\mathcal{Q}_{E}$, etc., must satisfy the relation described below [26]

$\mathcal{F}(\Phi_{(p_{1},p_{2},p_{3})})\leq\min\left\\{\mathcal{F}\left(\Phi_{(p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3})}\right),\mathcal{F}\left(\Phi_{(p^{\prime\prime}_{1},p^{\prime\prime}_{2},p^{\prime\prime}_{3})}\right)\right\\}.$
(61)

The new rate vector $\left(p_{1},p_{2},p_{3}\right)$ for V-type channel from
Eq. (60) is

$\displaystyle p_{1}$
$\displaystyle=p_{1}^{\prime}+p_{1}^{\prime\prime}-p_{1}^{\prime}p_{1}^{\prime\prime},$
(62) $\displaystyle p_{2}$
$\displaystyle=p_{2}^{\prime}+p_{2}^{\prime\prime}-p_{2}^{\prime}p_{2}^{\prime\prime},$

and the new rate vector of components for $\Lambda$-type channel

$\displaystyle p_{2}$
$\displaystyle=p_{2}^{\prime}+p_{2}^{\prime\prime}-p_{2}^{\prime}p_{2}^{\prime\prime},$
(63) $\displaystyle p_{3}$
$\displaystyle=p_{3}^{\prime}+p_{3}^{\prime\prime}-p_{3}^{\prime}p_{3}^{\prime\prime},$

which also satisfy CPTP conditions. The aforementioned inequality (61) can be
utilized to predict the monotonic behaviour of the capacity
$\mathcal{F}(\Phi_{(p_{1},p_{2},p_{3})})$ with respect to the decay rates
$\\{p_{1},p_{2},p_{3}\\}$ by applying it to Eq. (60). The lower and upper
bounds obtained from this inequality can prove to be valuable in expanding the
capacity formula to domains where other approaches, such as degradability
analysis, fail to be effective. In specific cases, for a single-decay fully
correlated MAD channels characterized by single non-zero rate vector (for
example, $p_{1}$), we obtain

$\Phi_{\left(p_{1}^{\prime},0,0\right)}\circ\Phi_{\left(p_{1}^{\prime\prime},0,0\right)}=\Phi_{\left(p_{1}^{\prime\prime},0,0\right)}\circ\Phi_{\left(p_{1}^{\prime},0,0\right)}=\Phi_{\left(p_{1},0,0\right)}.$
(64)

In this way we can infer that the cpacity functional
$\mathcal{F}(\Phi_{(p_{1},p_{2},p_{3})})$ is a non-increasing function with
respect to decay parameter $p_{1}$:

$\mathcal{F}\left(\Phi_{\left(p_{1},0,0\right)}\right)\geq\mathcal{F}\left(\Phi_{\left(p^{\prime},0,0\right)}\right),\quad\forall
p_{1}\leq p^{\prime},$ (65)

Similary for other single decay maps $\Phi_{(0,p_{2},0)}$ and
$\Phi_{(0,0,p_{3})}$ we can deduce the above inequality and prove their non-
increasing behaviour. Now, we write the mapping $\Phi_{(p_{1},p_{2},p_{3})}$
in terms of the composition form:

$\Phi_{(p_{1},p_{2},p_{3})}=\Phi_{(0,0,\bar{p}_{3})}\circ\Phi_{(0,\bar{p}_{2},0)}\circ\Phi_{(p_{1},0,0)}.$
(66)

where, $\bar{p}_{2}=p_{1}+(1-\Theta)p_{123}$ and $\bar{p}_{3}=\frac{\Theta
p_{123}}{1-p_{1}-(1-\Theta)p_{123}}$. Alternatively, we can write the given
composition:

$\Phi_{(p_{1},p_{2},p_{3})}=\Phi_{(0,\bar{p}_{2},0)}\circ\Phi_{(0,0,\bar{p}_{3})}\circ\Phi_{(p_{1},0,0)}.$
(67)

where, $\bar{p}_{2}=\frac{p_{1}+(1-\Theta)p_{123}}{1-\Theta p_{123}}$ and
$\bar{p}_{3}=\Theta p_{123}$. To make the notation simpler in the above
equation we have written $\Theta(\Gamma)\equiv\Theta$ and
$p_{123}=(1-p_{1})-(1-p_{2})(1-p_{3})$. For V-type channel setting $p_{3}=0$,
we obtain the relation

$\Phi_{(p_{1},p_{2},0)}=\Phi_{(0,p_{2},0)}\circ\Phi_{(p_{1},0,0)}=\Phi_{(p_{1},0,0)}\circ\Phi_{(0,p_{2},0)}.$
(68)

So the Eq. (61) gives the following condition:

$\mathcal{F}(\Phi_{(p_{1},p_{2},0)})\leq\min\left\\{\mathcal{F}\left(\Phi_{(0,p_{2},0)}\right),\mathcal{F}\left(\Phi_{(p_{1},0,0)}\right)\right\\}.$
(69)

For, $\Lambda$-type channel, $\Phi_{(0,p_{2},p_{3})}$ can be written as a
following composition:

$\Phi_{(0,p_{2},p_{3})}=\Phi_{(0,0,\tilde{p}_{3})}\circ\Phi_{(0,\tilde{p}_{2},0)}=\Phi_{(0,\tilde{p_{2}},0)}\circ\Phi_{(0,0,\tilde{p}_{3})}.$
(70)

and we obtain the following inequality

$\mathcal{F}(\Phi_{(0,p_{2},p_{3})})\leq\min\left\\{\mathcal{F}\left(\Phi_{(0,\tilde{p}_{2},0)}\right),\mathcal{F}\left(\Phi_{(0,0,\tilde{p}_{3})}\right)\right\\}.$
(71)

Where, $\tilde{p_{2}}=(1-\Theta)p_{23}$ and $\tilde{p_{3}}=\frac{\Theta
p_{23}}{1-(1-\Theta)p_{23}}$ and $p_{23}=1-(1-p_{2})(1-p_{3})$.

## References

  * Nielsen and Chuang [2002] M. A. Nielsen and I. Chuang, Quantum computation and quantum information (2002).
  * Holevo [2019] A. S. Holevo, Quantum systems, channels, information, in  _Quantum Systems, Channels, Information_ (de Gruyter, 2019).
  * Shannon [2001] C. E. Shannon, A mathematical theory of communication, ACM SIGMOBILE mobile computing and communications review 5, 3 (2001).
  * Watrous [2018] J. Watrous,  _The theory of quantum information_ (Cambridge university press, 2018).
  * Wilde [2013] M. M. Wilde,  _Quantum information theory_ (Cambridge University Press, 2013).
  * Gyongyosi  _et al._ [2018] L. Gyongyosi, S. Imre, and H. V. Nguyen, A survey on quantum channel capacities, IEEE Communications Surveys & Tutorials 20, 1149 (2018).
  * Bennett  _et al._ [1999] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels, Physical Review Letters 83, 3081 (1999).
  * Poshtvan and Karimipour [2022] A. Poshtvan and V. Karimipour, Capacities of the covariant pauli channel, Physical Review A 106, 062408 (2022).
  * Buscemi and Datta [2010] F. Buscemi and N. Datta, The quantum capacity of channels with arbitrarily correlated noise, IEEE Transactions on Information theory 56, 1447 (2010).
  * Fanizza  _et al._ [2021] M. Fanizza, F. Kianvash, and V. Giovannetti, Estimating quantum and private capacities of gaussian channels via degradable extensions, Physical Review Letters 127, 210501 (2021).
  * Singh and Datta [2022] S. Singh and N. Datta, Detecting positive quantum capacities of quantum channels, npj Quantum Information 8, 50 (2022).
  * Macchiavello and Sacchi [2016a] C. Macchiavello and M. F. Sacchi, Detecting lower bounds to quantum channel capacities, Physical Review Letters 116, 140501 (2016a).
  * Macchiavello and Sacchi [2016b] C. Macchiavello and M. F. Sacchi, Witnessing quantum capacities of correlated channels, Physical Review A 94, 052333 (2016b).
  * Cuevas  _et al._ [2017] Á. Cuevas, M. Proietti, M. A. Ciampini, S. Duranti, P. Mataloni, M. F. Sacchi, and C. Macchiavello, Experimental detection of quantum channel capacities, Physical Review Letters 119, 100502 (2017).
  * Banaszek  _et al._ [2004] K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Experimental demonstration of entanglement-enhanced classical communication over a quantum channel with correlated noise, Physical Review Letters 92, 257901 (2004).
  * Paladino  _et al._ [2002] E. Paladino, L. Faoro, G. Falci, and R. Fazio, Decoherence and 1/f noise in josephson qubits, Physical Review Letters 88, 228304 (2002).
  * Makhlin  _et al._ [2001] Y. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with josephson-junction devices, Reviews of modern physics 73, 357 (2001).
  * Caruso  _et al._ [2014] F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, Quantum channels and memory effects, Reviews of Modern Physics 86, 1203 (2014).
  * D’Arrigo  _et al._ [2007] A. D’Arrigo, G. Benenti, and G. Falci, Quantum capacity of dephasing channels with memory, New Journal of Physics 9, 310 (2007).
  * Plenio and Virmani [2007] M. Plenio and S. Virmani, Spin chains and channels with memory, Physical Review Letters 99, 120504 (2007).
  * Lupo  _et al._ [2010] C. Lupo, V. Giovannetti, and S. Mancini, Capacities of lossy bosonic memory channels, Physical Review Letters 104, 030501 (2010).
  * Giovannetti and Fazio [2005] V. Giovannetti and R. Fazio, Information-capacity description of spin-chain correlations, Physical Review A 71, 032314 (2005).
  * Jahangir  _et al._ [2015] R. Jahangir, N. Arshed, and A. Toor, Quantum capacity of an amplitude-damping channel with memory, Quantum Information Processing 14, 765 (2015).
  * Khatri  _et al._ [2020] S. Khatri, K. Sharma, and M. M. Wilde, Information-theoretic aspects of the generalized amplitude-damping channel, Physical Review A 102, 012401 (2020).
  * Sk and Panigrahi [2022] R. Sk and P. K. Panigrahi, Protecting quantum coherence and entanglement in a correlated environment, Physica A: Statistical Mechanics and its Applications 596, 127129 (2022).
  * Wolf and Perez-Garcia [2007] M. M. Wolf and D. Perez-Garcia, Quantum capacities of channels with small environment, Physical Review A 75, 012303 (2007).
  * D’Arrigo  _et al._ [2013] A. D’Arrigo, G. Benenti, G. Falci, and C. Macchiavello, Classical and quantum capacities of a fully correlated amplitude damping channel, Physical Review A 88, 042337 (2013).
  * Macchiavello  _et al._ [2020] C. Macchiavello, M. F. Sacchi, and T. Sacchi, Bounding the classical capacity of multilevel damping quantum channels, Advanced Quantum Technologies 3, 2000013 (2020).
  * Chessa and Giovannetti [2023] S. Chessa and V. Giovannetti, Resonant multilevel amplitude damping channels, Quantum 7, 902 (2023).
  * Muthukrishnan and Stroud Jr [2000] A. Muthukrishnan and C. R. Stroud Jr, Multivalued logic gates for quantum computation, Physical Review A 62, 052309 (2000).
  * Lanyon  _et al._ [2009] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional hilbert spaces, Nature Physics 5, 134 (2009).
  * Ivanov  _et al._ [2012] S. Ivanov, H. Tonchev, and N. Vitanov, Time-efficient implementation of quantum search with qudits, Physical Review A 85, 062321 (2012).
  * Cozzolino  _et al._ [2019] D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, High-dimensional quantum communication: benefits, progress, and future challenges, Advanced Quantum Technologies 2, 1900038 (2019).
  * Looi  _et al._ [2008] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths, Quantum-error-correcting codes using qudit graph states, Physical Review A 78, 042303 (2008).
  * Muralidharan  _et al._ [2017] S. Muralidharan, C.-L. Zou, L. Li, J. Wen, and L. Jiang, Overcoming erasure errors with multilevel systems, New Journal of Physics 19, 013026 (2017).
  * Klimov  _et al._ [2003] A. Klimov, R. Guzmán, J. Retamal, and C. Saavedra, Qutrit quantum computer with trapped ions, Physical Review A 67, 062313 (2003).
  * Kaszlikowski  _et al._ [2003] D. Kaszlikowski, D. K. Oi, M. Christandl, K. Chang, A. Ekert, L. C. Kwek, and C. Oh, Quantum cryptography based on qutrit bell inequalities, Physical Review A 67, 012310 (2003).
  * Lanyon  _et al._ [2008] B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’Brien, K. J. Resch, A. Gilchrist, and A. White, Manipulating biphotonic qutrits, Physical Review Letters 100, 060504 (2008).
  * Bocharov  _et al._ [2017] A. Bocharov, M. Roetteler, and K. M. Svore, Factoring with qutrits: Shor’s algorithm on ternary and metaplectic quantum architectures, Physical Review A 96, 012306 (2017).
  * Luo  _et al._ [2019] Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, _et al._ , Quantum teleportation in high dimensions, Physical Review Letters 123, 070505 (2019).
  * Chessa and Giovannetti [2021] S. Chessa and V. Giovannetti, Quantum capacity analysis of multi-level amplitude damping channels, Communications Physics 4, 22 (2021).
  * Yeo and Skeen [2003] Y. Yeo and A. Skeen, Time-correlated quantum amplitude-damping channel, Physical Review A 67, 064301 (2003).
  * Daffer  _et al._ [2003] S. Daffer, K. Wódkiewicz, and J. K. McIver, Quantum markov channels for qubits, Physical Review A 67, 062312 (2003).
  * Choi [1975] M.-D. Choi, Completely positive linear maps on complex matrices, Linear algebra and its applications 10, 285 (1975).
  * Stinespring [1955] W. F. Stinespring, Positive functions on c*-algebras, Proceedings of the American Mathematical Society 6, 211 (1955).
  * Devetak and Shor [2005] I. Devetak and P. W. Shor, The capacity of a quantum channel for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics 256, 287 (2005).
  * Smith and Smolin [2007] G. Smith and J. A. Smolin, Degenerate quantum codes for pauli channels, Physical Review Letters 98, 030501 (2007).
  * Schumacher and Westmoreland [1997] B. Schumacher and M. D. Westmoreland, Sending classical information via noisy quantum channels, Physical Review A 56, 131 (1997).
  * Dorlas and Morgan [2008] T. Dorlas and C. Morgan, Calculating a maximizer for quantum mutual information, International Journal of Quantum Information 6, 745 (2008).
  * Xu  _et al._ [2022] R. Xu, R.-G. Zhou, Y. Li, S. Jiang, and H. Ian, Enhancing robustness of noisy qutrit teleportation with markovian memory, EPJ Quantum Technology 9, 1 (2022).
  * Lloyd [1997] S. Lloyd, Capacity of the noisy quantum channel, Physical Review A 55, 1613 (1997).
  * Barnum  _et al._ [1998] H. Barnum, M. A. Nielsen, and B. Schumacher, Information transmission through a noisy quantum channel, Physical Review A 57, 4153 (1998).
  * Cubitt  _et al._ [2008] T. S. Cubitt, M. B. Ruskai, and G. Smith, The structure of degradable quantum channels, Journal of Mathematical Physics 49, 102104 (2008).
  * Bennett  _et al._ [2002] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem, IEEE transactions on Information Theory 48, 2637 (2002).
  * Adami and Cerf [1997] C. Adami and N. J. Cerf, von neumann capacity of noisy quantum channels, Physical Review A 56, 3470 (1997).
  * Keyl [2002] M. Keyl, Fundamentals of quantum information theory, Physics reports 369, 431 (2002).