# Resonant Multilevel Amplitude Damping Channels

Stefano Chessa  NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR,
I-56126 Pisa, Italy  Electrical and Computer Engineering, University of
Illinois Urbana-Champaign, Urbana, Illinois, 61801, USA<EMAIL_ADDRESS>Vittorio Giovannetti  NEST, Scuola Normale Superiore and Istituto Nanoscienze-
CNR, I-56126 Pisa, Italy

(16 Jan 2023)

###### Abstract

We introduce a new set of quantum channels: resonant multilevel amplitude
damping (ReMAD) channels. Among other instances, they can describe energy
dissipation effects in multilevel atomic systems induced by the interaction
with a zero-temperature bosonic environment. At variance with the already
known class of multilevel amplitude damping (MAD) channels, this new class of
maps allows the presence of an environment unable to discriminate transitions
with identical energy gaps. After characterizing the algebra of their
composition rules, by analyzing the qutrit case, we show that this new set of
channels can exhibit degradability and antidegradability in vast regions of
the allowed parameter space. There we compute their quantum capacity and
private classical capacity. We show that these capacities can be computed
exactly also in regions of the parameter space where the channels aren’t
degradable nor antidegradable.

##  1 Intro

While two-level quantum systems (qubit) represent the fundamental building
block for any Quantum Information processing, there are indications that
working with qudits (i.e. quantum systems with an Hilbert space of dimension
$d>2$) may bring in some advantages, both in terms of communication and
cryptography (see e.g. [1] and references therein) and of computation (see
e.g. [2] and references therein), with qutrits that recently made their first
appearance on commercial quantum devices [3]. Despite this fact the landscape
of mathematical models describing the physical noises affecting these systems
is still relatively unexplored especially in terms of their associated
information capacities, see [4, 5, 6, 7, 8, 9, 10, 11, 12] for results on
basic noise models in higher dimensions. Capacities are figures of merit
developed in the context of Quantum Shannon Theory [13, 14, 15, 16, 17, 18,
19, 20], which allow one to quantitatively measure the level of deterioration
that a given noise process induces on the quantum system it acts upon. A
formal definition of such quantities is properly constructed in a specific
communication scenario. There one describes the effect of the noise as an
information loss during a signaling process that connects a sender (Alice) who
is controlling the state of the system before the action of the noise, and a
receiver (Bob), who instead has access to the deteriorated version of the
qudit. In this context, depending on the type of messages one is considering
(e.g. classical, private classical, or quantum) and on the type of side
resources one allocates to the task (e.g. shared entanglement, two-way
communication), the capacity of the noise channel is defined. Formally it
corresponds to the optimal rate which gauges the maximum number of bits,
secret bits, or qubtis that Alice can reliably transfer to Bob per use of the
channel in the communication setting. Unfortunately for the vast majority of
models such optimal rates are not computable neither analytically nor
algorithmically [21, 22, 23] due to superadditivity and superactivation
effects [24, 25, 26, 27, 28]. In this sense the literature has evolved to find
capacities bounds and to find channel properties to be leveraged in order to
overcome the hurdles of the direct computation or at least provide meaningful
upper bounds. Among others, concerning the unassisted quantum and private
classical capacities, we find: degradability [6], antidegradability [29], weak
degradability [29], additive extensions [30], conjugate degradability [31],
less noisy or more capable channels [32], partial degradability [33],
approximate degradability [34], teleportation-covariant channels [35], unital
channels [36], low noise approximations [37]. In this sense a corpus of
literature is being built with the aim to produce efficiently computable
bounds and approximations of these capacities, see e.g. [38, 39, 40, 41, 42,
43, 44, 45]. From this perspective, this paper approaches the evaluation of
the quantum capacity $Q$ and the private classical capacity $C_{\text{p}}$ of
a new class of qudit channels, providing their exact expression for a wide
range of noise parameters.

Specifically, we introduce a family of noisy transformations that mimic energy
loss in a multilevel quantum system S (e.g. an atom or a molecule coupled with
a low temperature e.m. field). Our construction builds up from the Multilevel
Amplitude Damping (MAD) proposal of [8] where a $d$-dimensional generalization
of the qubit Amplitude Damping Channel (ADC) [17] is modeled as a collection
of independent two-level processes that induce transitions from higher energy
levels of the system to the lower ones. These transitions are mediated by an
exchange interaction of the type
$\sigma_{\text{S}}^{-}\otimes\sigma_{\text{E}}^{+}+\sigma_{\text{S}}^{+}\otimes\sigma_{\text{E}}^{-}$
between S and its environment E ($\sigma_{\text{X}}^{\pm}$ representing
raising and lowering operators for the system X). The major difference between
the transformations we discuss here and the ones analyzed in [8], is that we
now allow for the possibility that the transitions of S involving identical
energy gaps will couple with the same type of excitations in the environment.
We dub this class of the processes Resonant Multilevel Amplitude Damping
(ReMAD) channels: they behave as usual MAD channels on the populations of each
energy level of S, but exhibit different effects on the coherence terms of the
system. Specifically due to the peculiar nature of the selected
$\text{S}-\text{E}$ coupling, under the action of a ReMAD channel the system S
will be typically slightly less prone to dephasing than under the action of a
MAD channel characterized by the same transition probabilities. At the
mathematical level this corresponds to a net reduction of the minimal number
of Kraus operators [13] required to describe the effect of the associated
noise. This leads to some simplification in the characterization of their
quantum capacities. What makes this noise model interesting is its simplicity
and the fact that it can emerge in a variety of scenarios. To enumerate some
of the ones relevant in quantum information processing and quantum
communications we can mention: atomic systems in quantum memories and quantum
repeaters [46, 47, 48], optical qudits transmitted through lines such as
optical fibers with polarization dependent losses [49], optical qudits
interacting with beamsplitters and qudits encoded in harmonic oscillators [50,
51], see [52, 53] and references therein for applications and implementations
of bosonic codes, quantum computation and simulation with molecular spins [54]
and references therein.

The article is structured as follows: in Sec. 2 we introduce ReMAD channels,
their complementary channels and their composition rules; in Sec. 3 we briefly
review the issues of degradability, antidegradability and the computation of
quantum and private classical capacities $Q$ and $C_{\text{p}}$; in Sec. 4 we
provide an analysis of $Q$ and $C_{\text{p}}$ for qutrit ReMAD channels;
conclusions are drawn in Sec. 5.

##  2 Definitions

Let $\mathcal{H}_{\text{S}}$ be the Hilbert space associated with a
$d$-dimensional quantum system S characterized by an energetically ordered
canonical basis. This basis is represented by a collection of orthonormal
levels
$\\{\ket{0}_{\text{S}},\ket{1}_{\text{S}},\cdots,\ket{d-1}_{\text{S}}\\}$ with
$E_{j}<E_{j+1}$, being $E_{j}$ the energy associated with the $j$-th level. In
this context we can formally describe energy damping processes by specifying a
lower-triangular $d\times d$ transition matrix

$\Gamma:=\begin{pmatrix}\gamma_{0,0}&0&0&0&\cdots&0\\\
\gamma_{1,0}&\gamma_{1,1}&0&0&\cdots&0\\\
\gamma_{2,0}&\gamma_{2,1}&\gamma_{2,2}&0&\cdots&0\\\
\gamma_{3,0}&\gamma_{3,1}&\gamma_{3,2}&\gamma_{3,3}&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\gamma_{d-1,0}&\gamma_{d-1,1}&\gamma_{d,2}&\gamma_{d-1,3}&\cdots&\gamma_{d-1,d-1}\end{pmatrix},$
(1)

whose elements $\gamma_{j,k}$ for $j\leq k\in\\{0,\cdots,d-1\\}$ are positive
semidefinite quantities that define the transition probabilities, i.e. the
probability for the energy level $|j\rangle$ to be mapped into the lower
energy level $|k\rangle$. Consistency requirements are such that

$\displaystyle\sum_{k=0}^{j}\gamma_{j,k}=1\;,\qquad\forall
j\in\\{0,1,\cdots,d-1\\}\;,$ (2)

that for $j\geq 1$ identify $\gamma_{j,j}=1-\sum_{k=0}^{j-1}\gamma_{j,k}$ as
the survival probability of the level $j$ (notice that by construction
$\gamma_{0,0}=1$, being $|0\rangle_{\text{S}}$ a fixed point of the channels
model). The MAD channels introduced in Ref. [8] assign to each matrix $\Gamma$
a special Linear, Completely Positive, Trace Preserving (LCPTP) transformation
$\Phi^{({\text{\tiny MAD}})}_{\Gamma}$ that, at the level of Stinespring
representation [55], can be seen as a coupling with a zero-temperature
external bath B absorbing each individual energy jump $E_{j}\rightarrow E_{k}$
into a distinct (orthogonal) degree of freedom (see left panel of Fig. 1).

Figure 1: Left: depiction of the MAD excitations exchange between the system S
and its environment E. Right: depiction of the ReMAD excitations exchange
between the system S and the environment E, notice the environment size. In
both examples we set the total number of energy level equal to $d=3$.

Specifically, given $\hat{\rho}\in\mathfrak{S}({\cal H}_{\text{S}})$ a generic
density operator of S, we express its evolution under the action of
$\Phi^{({\text{\tiny MAD}})}_{\Gamma}$ as,

$\Phi^{({\text{\tiny
MAD}})}_{\Gamma}(\hat{\rho})=\mbox{Tr}_{\text{B}}[\hat{V}_{\Gamma}(\hat{\rho}\otimes\ket{0}\\!\\!\bra{0}_{\text{B}})\hat{V}^{\dagger}_{\Gamma}]\;,$
(3)

with $\ket{0}_{\text{B}}$ being the ground state of the bath,
$\mbox{Tr}_{\text{B}}[\cdots]$ the partial trace over the environment, and
$\hat{V}_{\Gamma}$ the unitary transformation that induces the mappings

$\displaystyle\hat{V}_{\Gamma}\ket{0}_{\text{S}}\ket{0}_{\text{B}}$
$\displaystyle:=$ $\displaystyle\ket{0}_{\text{S}}\ket{0}_{\text{B}}\;,$
$\displaystyle\hat{V}_{\Gamma}\ket{j}_{\text{S}}\ket{0}_{\text{B}}$
$\displaystyle:=$
$\displaystyle\sqrt{\gamma_{j,j}}\ket{j}_{\text{S}}\ket{0}_{\text{B}}+\sum_{k=1}^{j}\sqrt{\gamma_{j,j-k}}\ket{j-k}_{\text{S}}\ket{j,k}_{\text{B}}\;,\qquad\forall
j\in\\{1,\cdots,d-1\\}\;.$ (4)

In the above expression for $j\in\\{1,\cdots,d-1\\}$ and
$k\in\\{1,\cdots,j\\}$, kets $\ket{j,k}_{\text{B}}$ describe a collection of
vectors that are mutually orthogonal and orthogonal with $\ket{0}_{\text{B}}$
as well. Since they represent the states where the environment stores the
energy $E_{j}-E_{k}$ lost by S when moving from level $\ket{j}_{\text{S}}$ to
level $\ket{k}_{\text{S}}$, the construction in Eq. (4) implicitly assumes
that the environment of the model is capable to discriminate among the
different energy jumps. This condition is physically realized e.g. when the
energy spectrum of S is composed by incommensurable levels. Notice also that
this construction fixes the dimension of the Hilbert space
$\mathcal{H}_{\text{B}}$ of B, as well as the minimum number of Kraus
operators needed to represent $\Phi^{({\text{\tiny MAD}})}_{\Gamma}$ in the
operator-sum representation [56] equal to $d(d-1)/2+1$, i.e.

$\Phi^{({\text{\tiny
MAD}})}_{\Gamma}(\hat{\rho})=\hat{M}_{\Gamma}^{(0)}\;\hat{\rho}\;\hat{M}_{\Gamma}^{(0)\dagger}+\sum_{j=1}^{d-1}\sum_{k=1}^{j}\hat{M}_{\Gamma}^{(j,k)}\;\hat{\rho}\;\hat{M}_{\Gamma}^{(j,k)\dagger}\;,$
(5)

with

$\displaystyle\hat{M}_{\Gamma}^{(0)}$ $\displaystyle:=$
$\displaystyle{{}_{\text{B}}\\!\bra{0}}\hat{V}_{\Gamma}\ket{0}_{\text{B}}=\sum_{l=0}^{d-1}\sqrt{\gamma_{l,l}}\ket{l}\\!\\!\bra{l}_{\text{S}}\;,$
$\displaystyle\hat{M}_{\Gamma}^{(j,k)}$ $\displaystyle:=$
$\displaystyle{{}_{\text{B}}\\!\bra{j,k}}\hat{V}_{\Gamma}\ket{0}_{\text{B}}=\sqrt{\gamma_{j,j-k}}\;\ket{j-k}\\!\\!\bra{j}_{\text{S}}\;,\qquad\forall
j\in\\{1,\cdots,d-1\\},\forall k\in\\{1,\cdots,j\\}\;.$ (6)

By introducing ReMAD channels $\Phi_{\Gamma}$ we now allow for the possibility
that some of the transition events of S will excite the same internal degrees
of freedom of the bath, a condition which can be achieved e.g. when the energy
levels of the system are equally spaced, i.e. $E_{j}=j\Delta E$ for all
$j\in\\{0,\cdots,d-1\\}$. In this case we can replace the unitary coupling in
Eq. (4) with the new interaction

$\displaystyle\hat{U}_{\Gamma}\ket{j}_{\text{S}}\ket{0}_{\text{E}}$
$\displaystyle:=$
$\displaystyle\sum_{k=0}^{j}\sqrt{\gamma_{j,j-k}}\ket{j-k}_{\text{S}}\ket{k}_{\text{E}}\;,\qquad\forall
j\in\\{0,\cdots,d-1\\}$ (7)

where
$\\{\ket{0}_{\text{E}},\ket{1}_{\text{E}},\cdots,\ket{d-1}_{\text{E}}\\}$ are
a (possibly energetically ordered) orthonormal basis of the environment E. The
resulting LCPTP transformation is hence obtained by replacing Eqs. (3) and (5)
with

$\Phi_{\Gamma}(\hat{\rho})=\mbox{Tr}_{\text{E}}[\hat{U}_{\Gamma}(\hat{\rho}\otimes\ket{0}\\!\\!\bra{0}_{\text{E}}\hat{U}^{\dagger}_{\Gamma}]=\sum_{i=0}^{d-1}\hat{K}_{\Gamma}^{(i)}\hat{\rho}\hat{K}_{\Gamma}^{(i)^{\dagger}}\;,$
(8)

where

$\displaystyle\hat{K}_{\Gamma}^{(i)}$ $\displaystyle:=$
$\displaystyle{{}_{\text{E}}\\!\bra{i}}\hat{U}_{\Gamma}\ket{0}_{\text{E}}=\sum_{l=0}^{d-i-1}\sqrt{\gamma_{i+l,l}}\;\ket{l}\\!\\!\bra{i+l}_{\text{S}}\;,\qquad\forall
i\in\\{0,\cdots,d-1\\}\;,$ (9)

is a Kraus set characterized by at most $d$ non-zero elements. It is easy to
check that wile for $d=2$, the ReMAD channel $\Phi_{\Gamma}$ coincides with
its corresponding MAD counterpart $\Phi^{({\text{\tiny MAD}})}_{\Gamma}$
(indeed both reduce to the conventional qubit ADC). Also one can easily verify
that for arbitrary $d$, both $\Phi_{\Gamma}$ and $\Phi^{({\text{\tiny
MAD}})}_{\Gamma}$ produce the same diagonal output states per diagonal input
states
$\hat{\rho}^{(\text{diag})}:=\sum_{j=0}^{d-1}\rho_{j,j}\ket{j}\\!\\!\bra{j}_{\text{S}}$,
indeed

$\Phi_{\Gamma}\left(\hat{\rho}^{(\text{diag})}\right)=\Phi^{({\text{\tiny
MAD}})}_{\Gamma}\left(\hat{\rho}^{(\text{diag})}\right)=\sum_{j=0}^{d-1}\sum_{k=0}^{j}\rho_{j,j}\gamma_{j,k}\ket{k}\\!\\!\bra{k}_{\text{S}}\;.$
(10)

However for $d\geq 3$ the two sets of transformations have rather different
impact on the off-diagonal term of the input state. Consider for instance the
qutrit ($d=3$) scenario where, thanks to the constraint (2) the transition
matrix $\Gamma$ can be parametrized by 3 non-negative terms, e.g.
$\gamma_{10}$, $\gamma_{21}$, and $\gamma_{20}$, forming a 3-dimensional
vector $(\gamma_{10},\gamma_{21},\gamma_{20})$ spanning the domain

$\displaystyle{\mathbb{D}}_{3}:=\\{(\gamma_{10},\gamma_{21},\gamma_{20})\in\mathbb{R}^{3}:\gamma_{10},\gamma_{20},\gamma_{21}\in[0,1]\;\mbox{and}\;\gamma_{20}+\gamma_{21}\leq
1\\}\;,$ (11)

represented in Fig. 2 by the green rectangular right wedge delimited by the
vertexes ${\bf A}$, ${\bf B}$, ${\bf C}$, ${\bf D}$, ${\bf E}$, and ${\bf F}$.
In this case Eq. (7) reduces to

$\displaystyle\hat{U}_{\Gamma}\ket{0}_{\text{S}}\ket{0}_{\text{E}}$
$\displaystyle=\ket{0}_{\text{S}}\ket{0}_{\text{E}}\;,$
$\displaystyle\hat{U}_{\Gamma}\ket{1}_{\text{S}}\ket{0}_{\text{E}}$
$\displaystyle=\sqrt{1-\gamma_{10}}\ket{1}_{\text{S}}\ket{0}_{\text{E}}+\sqrt{\gamma_{10}}\ket{0}_{\text{S}}\ket{1}_{\text{E}}\;,$
$\displaystyle\hat{U}_{\Gamma}\ket{2}_{\text{S}}\ket{0}_{\text{E}}$
$\displaystyle=\sqrt{1-\gamma_{21}-\gamma_{20}}\ket{2}_{\text{S}}\ket{0}_{\text{E}}+\sqrt{\gamma_{20}}\ket{0}_{\text{S}}\ket{2}_{\text{E}}+\sqrt{\gamma_{21}}\ket{1}_{\text{S}}\ket{1}_{\text{E}}\;,$
(12)

while, identifying the canonical basis states $|0\rangle_{\text{S}}$,
$|1\rangle_{\text{S}}$, and $|2\rangle_{\text{S}}$ with the column vectors
$(1,0,0)^{T}$, $(0,1,0)^{T}$, and $(0,0,1)^{T}$, we can express the associated
Kraus operators in Eq. (9) as

$\begin{split}&\hat{K}_{\Gamma}^{(0)}=\begin{pmatrix}1&0&0\\\
0&\sqrt{1-\gamma_{10}}&0\\\
0&0&\sqrt{1-\gamma_{21}-\gamma_{20}}\end{pmatrix},\quad\hat{K}_{\Gamma}^{(1)}=\begin{pmatrix}0&\sqrt{\gamma_{10}}&0\\\
0&0&\sqrt{\gamma_{21}}\\\
0&0&0\end{pmatrix},\quad\hat{K}_{\Gamma}^{(2)}=\begin{pmatrix}0&0&\sqrt{\gamma_{20}}\\\
0&0&0\\\ 0&0&0\end{pmatrix}.\end{split}$ (13)

Accordingly the action of $\Phi_{{\Gamma}}$ on a generic density matrix
$\hat{\rho}$ of S produces the output state of the form

$\Phi_{{\Gamma}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+\gamma_{10}\rho_{11}+\gamma_{20}\rho_{22}&\sqrt{1-\gamma_{10}}\rho_{01}+\sqrt{\gamma_{10}\gamma_{21}}\rho_{12}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\\\
\sqrt{1-\gamma_{10}}\rho_{01}^{*}+\sqrt{\gamma_{10}\gamma_{21}}\rho_{12}^{*}&(1-\gamma_{10})\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{12}\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}^{*}&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{12}^{*}&(1-\gamma_{21}-\gamma_{20})\rho_{22}\\\
\end{array}\right)\;,$ (14)

to be compared with the associated transformation induced by the MAD
counterpart

$\Phi^{({\text{\tiny
MAD}})}_{\Gamma}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+\gamma_{10}\rho_{11}+\gamma_{20}\rho_{22}&\sqrt{1-\gamma_{10}}\rho_{01}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\\\
\sqrt{1-\gamma_{10}}\rho_{01}^{*}&(1-\gamma_{10})\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{12}\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}^{*}&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{12}^{*}&(1-\gamma_{21}-\gamma_{20})\rho_{22}\\\
\end{array}\right)\;,$ (15)

with $\rho_{ij}:={{}_{\text{S}}\langle}i|\hat{\rho}|j\rangle_{\text{S}}$. As
one can observe, while on the diagonal elements in both cases we have the
usual decay of populations predicted by Eq. (10), the ReMAD output state in
Eq. (14) exhibits a transfer of coherence that mixes the terms $\rho_{12}$ and
$\rho_{10}$ which is not contemplated in the MAD output of Eq. (15). It is
also worth noticing that in case either $\gamma_{21}=0$ or $\gamma_{10}=0$ the
qutrit map $\Phi^{({\text{\tiny MAD}})}_{\Gamma}$ and the qutrit map
$\Phi_{\Gamma}$ describe the same physical process.

As a final remark we notice that a particular example of ReMAD channel is
provided by the so called beamsplitter type ADC ${\Psi}_{\eta}$ introduced in
Ref. [57]. This subclass of ReMAD channels describes the evolution of a qudit
encoded in the first $d$ states of the Fock basis of an harmonic oscillator
passing through a beamsplitter of transmittance $\eta$. It’s straightforward
to verify that such mappings are a special instance of the ReMAD class
characterized by a transition matrix $\Gamma[\eta]$ whose elements can be
parametrized by the formula

$\displaystyle\gamma_{j,k}[\eta]:=\binom{j}{k}\eta^{j-k}(1-\eta)^{k}\;,\qquad\forall
j\in\\{0,\cdots,d-1\\}\;,\forall k\in\\{0,\cdots,j\\}\;,$ (16)

so that ${\Psi}_{\eta}=\Phi_{\Gamma[\eta]}$. For $d=3$ this corresponds to
having $\gamma_{10}[\eta]=\eta$, $\gamma_{21}[\eta]=2\eta(1-\eta)$, and
$\gamma_{20}[\eta]=\eta^{2}$: a plot of the parameter region spanned by
beamsplitter type ADC for $d=3$ is reported in Fig. 2 .

Figure 2: Plot of the domain ${\mathbb{D}}_{3}$ of Eq. (11) which identifies
the set of ReMAD channels for $d=3$ (green region, the grey volume corresponds
to the inaccessible parameter region); the beamsplitter type ADC of Ref. [57]
corresponds to the blue line that connects the vertexes ${\bf D}$ and ${\bf
B}$.

###  2.1 Complementary maps

At variance with what happens with MAD channels, the complementary map [13]
$\tilde{\Phi}_{\Gamma}$ of a generic ReMAD transformation ${\Phi}_{\Gamma}$ is
also a ReMAD channel (up to an isometry). This can be seen by recalling that
$\tilde{\Phi}_{\Gamma}$ can be obtained from Eq. (8) by replacing the partial
trace over E with a partial trace over S, i.e.

$\tilde{\Phi}_{\Gamma}(\hat{\rho})=\mbox{Tr}_{\text{S}}[\hat{U}_{\Gamma}(\hat{\rho}\otimes\ket{0}\\!\\!\bra{0}_{\text{E}}\hat{U}^{\dagger}_{\Gamma}]=\sum_{i=0}^{d-1}\hat{Q}_{\Gamma}^{(i)}\hat{\rho}\hat{Q}_{\Gamma}^{(i)\dagger}\;,$
(17)

where now $\hat{Q}_{\Gamma}^{(i)}:{\cal H}_{\text{S}}\rightarrow{\cal
H}_{\text{E}}$ are the operators

$\displaystyle\hat{Q}_{\Gamma}^{(i)}$ $\displaystyle:=$
$\displaystyle{{}_{\text{S}}\\!\bra{i}}\hat{U}_{\Gamma}\ket{0}_{\text{E}}=\sum_{l=0}^{d-i-1}\sqrt{\gamma_{i+l,i}}\;\ket{l}_{\text{E\;S}}\\!\bra{i+l}\;,\qquad\forall
i\in\\{0,\cdots,d-1\\}\;.$ (18)

Notice next that up to an isometry $\hat{V}_{SE}$ mapping the energy levels of
E into the corresponding levels of S (i.e.
$\hat{V}_{SE}|l\rangle_{E}\rightarrow|l\rangle_{S}$, $\forall l$),
$\hat{Q}_{\Gamma}^{(i)}$ has exactly the same form of the operators in Eq. (9)
computed with a new lower-triangular transition matrix $\tilde{\Gamma}$. The
non-zero elements of $\tilde{\Gamma}$, row by row, are obtained by a simple
reordering of those of $\Gamma$, i.e.

$\displaystyle\tilde{\gamma}_{j,k}:=\gamma_{j,j-k}\;,\qquad\forall
j\in\\{1,\cdots,d-1\\},\forall k\in\\{1,\cdots,j\\}\;.$ (19)

Specifically we can write

$\displaystyle\hat{V}_{SE}\hat{Q}_{\Gamma}^{(i)}=\hat{K}_{\tilde{\Gamma}}^{(i)}\;,\qquad\forall
i\in\\{0,\cdots,d-1\\}\;.$ (20)

and hence

$\displaystyle\hat{V}_{SE}\tilde{\Phi}_{\Gamma}(\cdots)\hat{V}^{\dagger}_{SE}={\Phi}_{\tilde{\Gamma}}(\cdots)\;.$
(21)

Again, as an example we provide the explicit form of the complementary channel
$\tilde{\Phi}_{{\Gamma}}$ of the qutrit ReMAD map $\Phi_{\Gamma}$ of Eq. (14),
i.e.

$\tilde{\Phi}_{{\Gamma}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+(1-\gamma_{10})\rho_{11}+(1-\gamma_{21}-\gamma_{20})\rho_{22}&\sqrt{\gamma_{10}}\rho_{01}+\sqrt{(1-\gamma_{10})\gamma_{21}}\rho_{12}&\sqrt{\gamma_{20}}\rho_{02}\\\
\sqrt{\gamma_{10}}\rho_{01}^{*}+\sqrt{(1-\gamma_{10})\gamma_{21}}\rho_{12}^{*}&\gamma_{10}\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{\gamma_{10}\gamma_{20}}\rho_{12}\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&\sqrt{\gamma_{10}\gamma_{20}}\rho_{12}^{*}&\gamma_{20}\rho_{22}\\\
\end{array}\right)\;.$ (22)

Notice finally that, as a consequence of Eq. (19), it follows that in the case
of a beamsplitter type ADC ${\Psi}_{\eta}$ defined by Eq. (16), one gets
$\tilde{\gamma}_{j,k}[\eta]={\gamma}_{j,k}[1-\eta]$. This, thanks to Eq. (21),
allows us to recover the well known fact that the complementary map
$\tilde{\Psi}_{\eta}$ of ${\Psi}_{\eta}$ is isometrically equivalent to
${\Psi}_{1-\eta}$.

###  2.2 Composition rules

In Ref. [8] it was shown that MAD channels are closed under composition rules,
a property that proves useful in studying their information capacities. In
this section we investigate whether ReMAD channels behave similarly. Maybe
surprisingly, it turns out that it’s not always the case. To begin with, let’s
observe that from the composition rules of beamsplitters one can easily verify
that the following identity holds true

$\displaystyle\Phi_{\Gamma[\eta^{\prime}]}\circ\Phi_{\Gamma[\eta]}=\Phi_{\Gamma[\eta^{\prime}\eta]}\;,\qquad\forall\eta^{\prime},\eta\in[0,1]\;,$
(23)

(the symbol “$\circ$” represents channel composition). Apart from this special
case, the analysis is slightly convoluted. We therefore focus on the simplest,
nontrivial case of $d=3$ qutrit systems defined by the input-output relations
of Eq. (14). By explicit computation it follows that being $\Gamma$ and
$\Gamma^{\prime}$ two $3\times 3$ transition matrices as in Eq. (1) we have

$\displaystyle[\Phi_{\Gamma^{\prime}}\circ\Phi_{\Gamma}(\hat{\rho})]_{00}=\rho_{00}+(\gamma^{\prime}_{10}\gamma_{10}+1-\gamma_{10})\rho_{11}+(\gamma^{\prime}_{10}\gamma_{21}+\gamma^{\prime}_{20}\gamma_{20}+1-\gamma_{21}-\gamma_{20})\rho_{22}\;,$
$\displaystyle[\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}(\hat{\rho})]_{01}=\sqrt{1-\gamma^{\prime}_{10}}\sqrt{1-\gamma_{10}}\rho_{01}+(\sqrt{\gamma^{\prime}_{10}\gamma^{\prime}_{21}(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}+\sqrt{\gamma_{10}\gamma_{21}(1-\gamma^{\prime}_{10})})\rho_{12}\;,$
$\displaystyle[\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}(\hat{\rho})]_{02}=\sqrt{1-\gamma^{\prime}_{21}-\gamma^{\prime}_{20}}\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\;,$
$\displaystyle[\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}(\hat{\rho})]_{11}=(1-\gamma^{\prime}_{10})(1-\gamma_{10})\rho_{11}+[\gamma^{\prime}_{21}(1-\gamma_{21}-\gamma_{20})+(1-\gamma^{\prime}_{10})\gamma_{21}]\rho_{22}\;,$
$\displaystyle[\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}(\hat{\rho})]_{12}=\sqrt{(1-\gamma^{\prime}_{10})(1-\gamma^{\prime}_{21}-\gamma^{\prime}_{20})}\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{12}\;,$
$\displaystyle[\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}(\hat{\rho})]_{22}=(1-\gamma^{\prime}_{21}-\gamma^{\prime}_{20})(1-\gamma_{21}-\gamma_{20})\rho_{22}\;,$
(24)

plus the Hermitian conjugate of off-diagonal elements. We are interested in
understanding under which conditions a new transition matrix
${\Gamma^{\prime\prime}}$ of elements $\gamma^{\prime\prime}_{j,k}$ exists
such that

$\displaystyle\Phi_{\Gamma^{\prime\prime}}=\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}\;.$
(25)

Among the equations above, those referring to elements 00, 02, 11, 12, 22 are
all consistent with setting

$\displaystyle\gamma^{\prime\prime}_{10}=\gamma_{10}+\gamma^{\prime}_{10}(1-\gamma_{10})\;,$
$\displaystyle\gamma^{\prime\prime}_{20}=\gamma_{20}+\gamma^{\prime}_{10}\gamma_{21}+\gamma^{\prime}_{20}(1-\gamma_{21}-\gamma_{20})\;,$
$\displaystyle\gamma^{\prime\prime}_{21}=(1-\gamma^{\prime}_{10})\gamma_{21}+\gamma^{\prime}_{21}(1-\gamma_{21}-\gamma_{20})\;.$
(26)

(observe that such definitions are compatible with the request that
$\gamma^{\prime\prime}_{j,k}\in[0,1]$, as well as with the normalization
constraint $\sum_{k=0}^{j}\gamma^{\prime\prime}_{j,k}=1$ for all $j$). Element
01 instead forces an additional constraint, i.e.

$\sqrt{\gamma^{\prime\prime}_{21}\gamma^{\prime\prime}_{10}}=\sqrt{\gamma^{\prime}_{10}\gamma^{\prime}_{21}(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}+\sqrt{\gamma_{10}\gamma_{21}(1-\gamma^{\prime}_{10})}\;,$
(27)

which is not necessarily granted. As a matter of fact by substituting in Eq.
(27) the values for $\gamma^{\prime\prime}_{21}$ and
$\gamma^{\prime\prime}_{10}$ obtained in Eq. (2.2) we get that in order to
satisfy it we need

$\gamma_{10}\gamma^{\prime}_{21}(1-\gamma_{21}-\gamma_{20})=\gamma_{21}\gamma^{\prime}_{10}(1-\gamma_{10})(1-\gamma^{\prime}_{10})\;.$
(28)

This identifies a specific region for the parameters $\Gamma$ and
${\Gamma^{\prime}}$ where the composition
$\Phi_{\Gamma^{\prime}}\circ\Phi_{\Gamma}$ corresponds to a ReMAD channel.
Notice in particular that, in agreement with Eq. (23), the identity in Eq.
(28) is always fulfilled for beamsplitter type ADC, i.e. for
$\Gamma=\Gamma[\eta]$ and $\Gamma^{\prime}=\Gamma[\eta^{\prime}]$. Furthermore
we observe that if $\Gamma^{\prime}$ is such that either
$\gamma_{21}^{\prime}=\gamma^{\prime}_{10}=0$ or
$\gamma_{21}^{\prime}=1-\gamma^{\prime}_{10}=0$, then Eq. (28) is always
satisfied for all the choices of $\gamma^{\prime}_{20}$. Of particular
interest for our analysis is the first of these two configurations: here
$\Phi_{{\Gamma^{\prime}}}$ describes a ReMAD channel where the first two
levels of S are untouched by the noise, while the third level gets mapped into
the ground state with probability $\gamma^{\prime}_{20}$. Looking at Eq. (2.2)
we notice that the transition probabilities $\gamma^{\prime\prime}_{10}$ and
$\gamma^{\prime\prime}_{21}$ of the ReMAD channel
$\Phi_{\Gamma^{\prime\prime}}$, which emerge by the concatenation, coincide
with those of $\Phi_{\Gamma}$ (i.e. $\gamma^{\prime\prime}_{10}=\gamma_{10}$
and $\gamma^{\prime\prime}_{21}=\gamma_{21}$), while the transition
probability $\gamma^{\prime\prime}_{20}$ increases with respect to
$\gamma_{20}$. Specifically
$\gamma^{\prime\prime}_{20}=\gamma_{20}+\gamma^{\prime}_{20}(1-\gamma_{21}-\gamma_{20})$.
Observing this, by varying $\gamma_{20}^{\prime\prime}\in[0,1]$, the last
quantity can span over the entire interval $[1-\gamma_{21},1]$. We can use
this observation to claim that being $\Phi_{\Gamma}$ and
$\Phi_{\Gamma^{\prime\prime}}$ two qutrits ReMAD channels having the same
transition probabilities connecting level 2 to level 1 and level 1 to level 0,
but with $\gamma^{\prime\prime}_{20}$ larger than or equal to $\gamma_{20}$,
then they can be connected via a third ReMAD channel as in Eq. (25).

###  2.3 Covariance

The ReMAD channels exhibit a covariance property under suitable unitary
transformations. Specifically consider the set of unitary gates

$\displaystyle\hat{U}_{S}(\theta)=\sum_{j=0}^{d-1}e^{-ij\theta}\ket{j}\\!\\!\bra{j}_{\text{S}}\;,$
(29)
$\displaystyle\hat{U}_{E}(\theta)=\sum_{j=0}^{d-1}e^{-ij\theta}\ket{j}\\!\\!\bra{j}_{\text{E}}\;,$
(30)

with $\theta$ real. It then follows that

${\Phi}_{\Gamma}(\hat{U}_{\text{S}}(\theta)\hat{\rho}\hat{U}_{\text{S}}^{\dagger}(\theta))=\sum_{k=0}^{d-1}\sum_{j,j^{\prime}=k}^{d-1}\;\rho_{j,j^{\prime}}\;e^{-i(j-j^{\prime})\theta}\sqrt{\gamma_{j,k}\gamma_{j^{\prime},k}}\;\ket{j-k}\\!\\!\bra{j^{\prime}-k}_{\text{S}}=\hat{U}_{\text{S}}(\theta){\Phi}_{\Gamma}(\hat{\rho})\hat{U}_{\text{S}}^{\dagger}(\theta)\;,$
(31)

and similarly for the complementary channel $\tilde{\Phi}$ we get

$\tilde{\Phi}_{\Gamma}(\hat{U}_{\text{S}}(\theta)\hat{\rho}\hat{U}_{\text{S}}^{\dagger}(\theta))=\sum_{k=0}^{d-1}\sum_{j,j^{\prime}=k}^{d-1}\;\rho_{j,j^{\prime}}\;e^{-i(j-j^{\prime})\theta}\sqrt{\tilde{\gamma}_{j,k}\tilde{\gamma}_{j^{\prime},k}}\;\ket{j-k}\\!\\!\bra{j^{\prime}-k}_{\text{E}}=\hat{U}_{\text{E}}(\theta)\tilde{\Phi}_{\Gamma}(\hat{\rho})\hat{U}_{\text{E}}^{\dagger}(\theta)\;.$
(32)

As we’ll see in the following the identities above turn out to be extremely
useful in the computation of the capacities of our channels.

##  3 Review of quantum and private classical capacities

This section is dedicated to reviewing some basic notions about channel
capacities for those readers who may be not familiar with these quantities.
While the plethora of capacities for quantum channels includes a large
collection of different functionals, our analysis will be focused on the
quantum capacity $Q$ and the private classical capacity $C_{\text{p}}$.

###  3.1 Quantum Capacity and Private Classical Capacity

The quantum capacity $Q(\Phi)$ of a quantum channel $\Phi$ defines the maximum
rate of transmitted quantum information achievable per channel use, assuming
$\Phi$ to act in the regime of i.i.d. noise [13, 14, 15, 16, 19, 17, 18, 20].
Intuitively it tells you how faithfully a quantum state, possibly correlated
with an external system, can be sent and received by two communication parties
if the communication line is noisy. Recently $Q$ has also been showed to
provide a lower bound to the space overhead necessary for fault tolerant
quantum computation in presence of noise [58]. The formal definition of $Q$
relies on the coherent information $I_{\text{coh}}$ [59]. Assuming $n$ uses of
the channel $\Phi$ and a generic state $\hat{\rho}^{(n)}\in\mathfrak{S}({\cal
H}_{\text{S}}^{\otimes n})$ we have

$I_{\text{coh}}\left(\Phi^{\otimes
n},\hat{\rho}^{(n)}\right):=S\left(\Phi^{\otimes
n}(\hat{\rho}^{(n)})\right)-S\left(\tilde{\Phi}^{\otimes
n}(\hat{\rho}^{(n)})\right).$ (33)

with $S(\hat{\rho}):=-\mbox{Tr}[\hat{\rho}\log_{2}\hat{\rho}]$ the von Neumann
entropy of the state $\hat{\rho}$, and $\tilde{\Phi}$ the complementary
channel of $\Phi$. Maximizing over $\mathfrak{S}({\cal H}_{\text{S}}^{\otimes
n})$ we get the maximized coherent information $Q^{(n)}$ for $n$ instances of
the channel

$Q^{(n)}(\Phi):=\max_{\hat{\rho}^{(n)}\in{\mathfrak{S}({\cal
H}_{\text{S}}^{\otimes n})}}I_{\text{coh}}\left(\Phi^{\otimes
n},\hat{\rho}^{(n)}\right)\;,$ (34)

and by regularization of this expression we get the maximized coherent
information per channel use, that is the quantum capacity $Q(\Phi)$ [60, 61,
62]

$Q(\Phi)=\lim_{n\rightarrow\infty}\frac{Q^{(n)}(\Phi)}{n}\;.$ (35)

The private classical capacity $C_{\text{p}}(\Phi)$ quantifies the maximum
rate of classical information achievable per channel use assuming also the
privacy of communication (where privacy is intended as limiting to arbitrarily
small the amount of information that an eavesdropper can extract from the
environment during the communication). The fundamental tool needed to compute
$C_{\text{p}}(\Phi)$ is the Holevo information functional $\chi$ [63]. Being
${\cal E}_{n}:=\\{p_{i},\hat{\rho}^{(n)}_{i}\\}$ an ensemble of quantum states
$\hat{\rho}^{(n)}_{i}\in\mathfrak{S}({\cal H}_{\text{S}}^{\otimes n})$, we
have

$\chi(\Phi^{\otimes n},{\cal E}_{n}):=S\left(\Phi^{\otimes
n}\left(\sum_{i}p_{i}\hat{\rho}_{i}^{(n)}\right)\right)-\sum_{i}p_{i}S\left(\Phi^{\otimes
n}\left(\hat{\rho}_{i}^{(n)}\right)\right)\;.$ (36)

Through the Holevo information we define next the private information for $n$
uses $C_{\text{p}}^{(n)}(\Phi)$, that involves a maximization over all
ensembles ${\cal E}_{n}$

$C_{\text{p}}^{(n)}(\Phi):=\max_{{\cal E}_{n}}\left(\chi\left(\Phi^{\otimes
n},{\cal E}_{n}\right)-\chi\left(\tilde{\Phi}^{\otimes n},{\cal
E}_{n}\right)\right)\;,$ (37)

from which finally $C_{\text{p}}(\Phi)$ can be computed via regularization
over $n$, i.e. [62, 64]:

$C_{\text{p}}(\Phi)=\lim_{n\to\infty}\frac{C_{\text{p}}^{(n)}(\Phi)}{n}\;.$
(38)

We conclude our brief review by recalling that $Q$ and $C_{\text{p}}$ obey
data processing inequalities [14]. This means that, for (any) channel $\Phi$
that can be expressed as a composition of other two LCPTP maps $\Phi_{1}$ and
$\Phi_{2}$ (i.e. $\Phi=\Phi_{1}\circ\Phi_{2}$) it follows

$\displaystyle Q(\Phi)\leq\min\\{Q(\Phi_{1}),Q(\Phi_{2})\\}\;,$ (39)
$\displaystyle
C_{\text{p}}(\Phi)\leq\min\\{C_{\text{p}}(\Phi_{1}),C_{\text{p}}(\Phi_{2})\\}\;.$
(40)

###  3.2 Degradability and antidegradability

The need of regularization in the evaluation of $Q(\Phi)$ and
$C_{\text{p}}(\Phi)$ poses a well known problem which ultimately is the
underlying reason of our efforts here. An exception to this predicament is
given by degradable [6] and antidegradable [29] channels, whose definitions
and properties are reviewed here. A quantum channel
$\Phi:\mathcal{L}(\mathcal{H}_{\text{S}})\rightarrow\mathcal{L}(\mathcal{H}_{\text{S}^{\prime}})$
is said degradable if a LCPTP map
$\mathcal{D}:\mathcal{L}(\mathcal{H}_{\text{S}^{\prime}})\rightarrow\mathcal{L}(\mathcal{H}_{\text{E}})$
exists s.t.

$\tilde{\Phi}=\mathcal{D}\circ\Phi\;,$ (41)

while it’s said antidegradable if it exists a LCPTP map
$\mathcal{A}:\mathcal{L}(\mathcal{H}_{\text{E}})\rightarrow\mathcal{L}(\mathcal{H}_{\text{S}^{\prime}})$
s.t.

$\Phi=\mathcal{A}\circ\tilde{\Phi}\;.$ (42)

In case $\Phi$ is mathematically invertible, a simple direct way to determine
whether it is degradable or not is to formally invert the composition in Eq.
(41). This is done (if possible) by constructing the super-operator
$\mathcal{D}=\tilde{\Phi}\circ\Phi^{-1}$ and by checking whether such object
is LCPTP [65, 66]. The check can be performed by studying the positivity of
its associated Choi matrix $\text{C}_{\mathcal{D}}$, i.e.

$\mbox{$\Phi$ invertible}\Longrightarrow\mbox{$\Phi$ degradable iff
$\text{C}_{\mathcal{D}}\geq 0$}\;,$ (43)

where given
$\ket{\Gamma}_{\text{RS}}=\sum_{i=0}^{d-1}\ket{i}_{\text{R}}\ket{i}_{\text{S}}$
we set
$\text{C}_{\mathcal{D}}:=(I_{\text{R}}\otimes\mathcal{D}_{\text{S}})\ket{\Gamma}\\!\\!\bra{\Gamma}_{\text{RS}}$
(see App. A for details). A completely similar argument can be made for
antidegradability: in this case we can claim

$\mbox{$\tilde{\Phi}$ invertible}\Longrightarrow\mbox{$\Phi$ antidegradable
iff $\text{C}_{\mathcal{A}}\geq 0$}\;,$ (44)

where now $\text{C}_{\mathcal{A}}$ is the Choi operator of the map
$\mathcal{A}={\Phi}\circ\tilde{\Phi}^{-1}$.

As already mentioned the evaluation of the quantum a private classical
capacity simplifies for degradable channels. To begin with, in this case $Q$
and $C_{\text{p}}$ result to be additive, so the regularization over $n$ in
Eq. (35) isn’t needed, leading to the single-letter formula [67]

$C_{\text{p}}(\Phi)=Q(\Phi)=Q^{(1)}(\Phi):=\max_{\hat{\rho}\in{\mathfrak{S}({\cal
H}_{\text{S}})}}I_{\text{coh}}\left(\Phi,\hat{\rho}\right)\;.$ (45)

Another important simplification arises from the fact that under degradability
conditions the coherent information functional is concave w.r.t. the input
state [68]. Accordingly if $\Phi$ and $\tilde{\Phi}$ turn out to be covariant
under the action of some unitary group, the maximization in Eq. (45) can be
further simplified. In the case of ReMAD channels where the properties (31)
and (32) hold true, under degradability conditions we can write

$\displaystyle
I_{\text{coh}}\left(\Phi_{\Gamma},\int\frac{d\theta}{2\pi}\hat{U}_{S}(\theta)\hat{\rho}\hat{U}^{\dagger}_{S}(\theta)\right)$
$\displaystyle\geq$
$\displaystyle\int\frac{d\theta}{2\pi}I_{\text{coh}}\left(\Phi_{\Gamma},\hat{U}_{S}(\theta)\hat{\rho}\hat{U}^{\dagger}_{S}(\theta)\right)$
(46) $\displaystyle=$
$\displaystyle\int\frac{d\theta}{2\pi}S(\Phi_{\Gamma}(\hat{U}_{S}(\theta)\hat{\rho}\hat{U}^{\dagger}_{S}(\theta))-S(\tilde{\Phi}_{\Gamma}(\hat{U}_{S}(\theta)\hat{\rho}\hat{U}^{\dagger}_{S}(\theta))$
$\displaystyle=$ $\displaystyle
I_{\text{coh}}\left(\Phi_{\Gamma},\hat{\rho}\right)\;,$

where in the last equality we made use of the invariance of von Neumann
entropy under unitary transformations. Observing then that

$\displaystyle\int\frac{d\theta}{2\pi}\hat{U}_{S}(\theta)\hat{\rho}\hat{U}^{\dagger}_{S}(\theta)=\hat{\rho}^{(\text{diag})}:=\sum_{j=0}^{d-1}\rho_{j,j}\;\ket{j}_{\text{S}}\\!\bra{j}\;,$
(47)

we can conclude that for degradable ReMAD channels the maximization over of
(45) can be restricted on the set of states $\hat{\rho}^{(\text{diag})}$ which
are diagonal in the canonical basis, i.e.

$C_{\text{p}}(\Phi_{\Gamma})=Q(\Phi_{\Gamma})=Q^{(1)}(\Phi_{\Gamma})=\max_{\hat{\rho}^{(\text{diag})}\in{\mathfrak{S}({\cal
H}_{\text{S}})}}I_{\text{coh}}\left(\Phi_{\Gamma},\hat{\rho}^{(\text{diag})}\right)\;.$
(48)

For antidegradable channels instead, due to a no-cloning argument [69],
$Q(\Phi)=0$. Similarly, $C_{\text{p}}(\Phi)=0$: the environment can
reconstruct the channel output simply by applying the antidegrading channel,
so no private information can be transmitted. Therefore for channels
exhibiting antidegradability no maximizations are needed.

##  4 Degradability and antidegradability regions for $d=3$ ReMAD channels

In this section we study the degradability/antidegradability properties for
qutrits ReMAD channels. Mimicking the approach used for other amplitude
damping channels, we tackle the problem working under the heuristic assumption
that if a degrading (or antidegrading) channel of a ReMAD channel exists, it
is itself a ReMAD channel. While this choice is potentially suboptimal,
numerical tests based on the more rigorous (but analytically impractical)
matrix inversion method discussed in Sec. 3.2, reveals that this is not the
case.

Reminding the isometric connection given in Eq. (21), we hence translate the
degradability condition on $\Phi_{\Gamma}$ into the problem of identifying a
transition matrix ${\Gamma^{\prime}}$ s.t.

$\displaystyle{\Phi}_{\tilde{\Gamma}}=\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\Gamma}\;.$
(49)

With a procedure similar to the one employed in Sec. 2.2 such equation can be
mapped into the following constraints

$\displaystyle\tilde{\gamma}_{10}=\gamma_{10}+\gamma^{\prime}_{10}(1-\gamma_{10})\;,$
$\displaystyle\tilde{\gamma}_{20}=\gamma_{20}+\gamma^{\prime}_{10}\gamma_{21}+\gamma^{\prime}_{20}(1-\gamma_{21}-\gamma_{20})\;,$
$\displaystyle\tilde{\gamma}_{21}=(1-\gamma^{\prime}_{10})\gamma_{21}+\gamma^{\prime}_{21}(1-\gamma_{21}-\gamma_{20})\;,$
(50)

which using the connection as in Eq. (19) to replace
$\tilde{\gamma}_{10}=\gamma_{11}=1-\gamma_{10}$,
$\tilde{\gamma}_{20}=\gamma_{22}=1-\gamma_{20}-\gamma_{21}$, and
$\tilde{\gamma}_{21}=\gamma_{21}$, leads to

$\displaystyle\gamma^{\prime}_{10}=\frac{1-2\gamma_{10}}{1-\gamma_{10}}\;,$
$\displaystyle\gamma^{\prime}_{21}=\frac{1-2\gamma_{10}}{1-\gamma_{10}}\frac{\gamma_{21}}{1-\gamma_{21}-\gamma_{20}}\;,$
$\displaystyle\gamma^{\prime}_{20}=\frac{1-\gamma_{21}-2\gamma_{20}}{1-\gamma_{21}-\gamma_{20}}-\frac{\gamma_{21}}{1-\gamma_{21}-\gamma_{20}}\frac{1-2\gamma_{10}}{1-\gamma_{10}}\;.$
(51)

Imposing now the vector
$(\gamma^{\prime}_{10},\gamma^{\prime}_{21},\gamma^{\prime}_{20})$ to belong
to the domain ${\mathbb{D}}_{3}$ of Eq. (11) we get the degradability region
which we depict as the yellow region of Fig. 3. In such region we can compute
exactly the value of $Q(\Phi_{\Gamma})$ and $C_{\text{p}}(\Phi_{\Gamma})$
using the single-letter formula in Eq. (48) only involving diagonal input
matrices. Results of such optimization are reported in Fig. 4 for different
values of the parameters $\gamma_{10},$ $\gamma_{21}$ and $\gamma_{20}$. In
the cases of $\gamma_{21}=0$ and $\gamma_{10}=0$ the obtained values were
already known in the whole resulting parameter space (including some non-
degradable regions), since there ReMAD channels reduce to double decay MAD
channels, already studied in [8].

Figure 3: Degradability region (yellow) and antidegradability region (blue)
for a qutrit ReMAD channel, the grey volume representing the non-physical
region of the parameter space (the two panels show different perspectives).
Notice that the beamsplitter type ADC $\Psi_{\eta}$ (blue line of Fig. 2) is
fully contained in the above regions (it lies in the degradability region for
$\eta\geq 1/2$, $\Phi_{\eta}$ and lies in the antidegradability region for
$\eta<1/2$).

To check antidegradability of a ReMAD channel we follow the same path, looking
for a ${\Gamma^{\prime}}$ that fulfills the identity

$\displaystyle{\Phi}_{{\Gamma}}=\Phi_{{\Gamma^{\prime}}}\circ\Phi_{\tilde{\Gamma}}\;,$
(52)

leading to a constraint that can be obtained from the one in Eq. (4) by
exchanging $\gamma_{j,k}$ with $\tilde{\gamma}_{j,k}$. This leads finally to

$\displaystyle\gamma^{\prime}_{10}=\frac{2\gamma_{10}-1}{\gamma_{10}}\;,$
$\displaystyle\gamma^{\prime}_{21}=\frac{\gamma_{21}}{\gamma_{20}}\frac{2\gamma_{10}-1}{\gamma_{10}}\;,$
$\displaystyle\gamma^{\prime}_{21}=2+\frac{\gamma_{21}(1-\gamma_{10})-\gamma_{10}}{\gamma_{10}\gamma_{20}}\;.$
(53)

Imposing to have
$(\gamma^{\prime}_{10},\gamma^{\prime}_{21},\gamma^{\prime}_{20})$ in
${\mathbb{D}}_{3}$ brings us to the antidegradability region depicted in blue
in Fig. 3.

Figure 4: $Q$ (and $C_{\text{p}}$) for a qutrit ReMAD $\Phi_{\Gamma}$ at
different $\gamma_{10},$ $\gamma_{21}$ and $\gamma_{20}$ in the degradable
region. The white regions correspond to points associated with not degradable
(nor antidegradable) channels, grey regions are those outside the allowed
parameter space ${\mathbb{D}}_{3}$. As we’ll see in Sec. 4.1, at least for
$\gamma_{10}=0.000$ (left-top panel) $Q$ and $C_{\text{p}}$ can also be
extended in the white region.

###  4.1 Capacities in non-degradable and non-antidegradable regions

In those regions of parameters for which degradability or antidegradability
are not achieved a closed accessible expression for $Q$ and $C_{\text{p}}$ is
lacking. A notable exception is provided by the two planar regions identified
respectively by $\gamma_{10}=0$ (triangular surface of Fig. 2, identified by
the vertexes ${\bf ADE}$) and $\gamma_{21}=0$ (rectangular region ${\bf
ABCD}$). As one can observe from Fig. 3 they overlap only partially with the
degradable and antidegradable regions: yet, thanks to the fact that there
$\Phi_{\Gamma}$ and $\Phi^{({\text{\tiny MAD}})}_{\Gamma}$ coincide (see
comments in Sec. 2), we can use the same techniques of [8], see Appendix C, to
compute the quantum capacity of the associated ReMAD channels. Even for those
points that aren’t explicitly degradable (or antidegradable), specifically:

  * •

Planar region $\gamma_{10}=0$: here the channel $\Phi_{\Gamma}$ is neither
degradable nor antidegradable for all points with $\gamma_{12}+\gamma_{20}\geq
1/2$, see Appendix A.1. With a similar approach as the one in [8], see
Appendix C.1, we can conclude that in this region the capacities are constant
and equal to 1, i.e.

$\displaystyle
Q(\Phi_{\Gamma})=C_{\text{p}}(\Phi_{\Gamma})=1\;,\qquad\forall\;\gamma_{12}+\gamma_{20}\geq
1/2\;,$ (54)

see left panel of Fig. 5.

  * •

Planar region $\gamma_{21}=0$: here the channel $\Phi_{\Gamma}$ is neither
degradable nor antidegradable for all points which verify the conditions
$\gamma_{01}\geq 1/2\geq\gamma_{20}$ (right-lower quadrant in Fig. 5, right
panel) or $\gamma_{20}\geq 1/2\geq\gamma_{01}$ (left-upper quadrant in Fig. 5,
right panel), see Appendix A.1. Applying techniques of [8], see Appendix C.2,
we can conclude that in these regions the capacities assume symmetric values,
i.e. give $0\leq x\leq 1/2\leq y\leq 1$, we have

$\displaystyle Q(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=x\\\
\gamma_{21}=0\\\
\gamma_{20}=y\end{subarray}}=C_{\text{p}}(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=x\\\
\gamma_{21}=0\\\
\gamma_{20}=y\end{subarray}}=Q(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=y\\\
\gamma_{21}=0\\\
\gamma_{20}=x\end{subarray}}=C_{\text{p}}(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=y\\\
\gamma_{21}=0\\\ \gamma_{20}=x\end{subarray}}={\cal Q}(x)\;,$ (55)

with

$\displaystyle{\cal Q}(x)$ $\displaystyle:=$
$\displaystyle\max_{p_{1},p_{2}}\\{-[1-(1-x)p_{1})]\log_{2}[1-(1-x)p_{1})]-[(1-x)p_{1}]\log_{2}[(1-x)p_{1}]$
(56)
$\displaystyle\qquad\quad+(1-xp_{1}-p_{2})\log_{2}(1-xp_{1}-p_{2})+xp_{1}\log_{2}(xp_{1})+p_{2}\log_{2}(p_{2})\\}$
$\displaystyle=$
$\displaystyle\max_{p_{1}\in[0,1]}\\{H_{2}((1-x)p_{1})-H_{2}(xp_{1})\\}\;,$

see right panel of Fig. 5. In the first expression the maximization is
performed over the populations $p_{1},p_{2}\in[0,1]$ of the input state with
respect to levels $|1\rangle_{\text{S}}$ and $|2\rangle_{\text{S}}$ under the
consistency constraint $p_{1}+p_{2}\leq 1$; the second identity instead
follows from the observation that for $p_{0}$ fixed, the maximum of the r.h.s.
term is always achieved by $p_{2}=0$ (no population assigned to the second
excited level). Observe that the final expression in Eq. (61) exactly matches
the quantum capacity of a qubit ADC channel with damping probability equal to
$1-x$ [70].

Figure 5: Left: $Q(\Phi_{\Gamma})$ (and $C_{\text{p}}(\Phi_{\Gamma})$) for
$\gamma_{10}=0$. The capacities are known also in the region
$\gamma_{20}+\gamma_{21}\geq\frac{1}{2}$ that is not degradable nor
antidegradable. Right: $Q(\Phi_{\Gamma})$ (and $C_{\text{p}}(\Phi_{\Gamma})$)
for $\gamma_{21}=0$. The capacities are known also in two regions that are not
degradable nor antidegradable, see left-top panel ($\gamma_{10}=0.000$) of
Fig. 4

  * •

With a little additional effort we can also determine the exact value of
$Q(\Phi_{\Gamma})$ for the elements on the boundary of the ${\mathbb{D}}_{3}$
region identified by the constraints

$\displaystyle\gamma_{10}=1\;,\qquad\gamma_{21}+\gamma_{20}=1\;,$ (57)

(notice that while for $\gamma_{21}\leq 1/2$ the points belong to the
antidegradable region, for $\gamma_{21}>1/2$ they are neither degradable nor
antidegradable, see Appendix A.1). For these values Eq. (14) becomes

$\Phi_{{\Gamma}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+\rho_{11}+(1-\gamma_{21})\rho_{22}&\sqrt{\gamma_{21}}\rho_{12}&0\\\
\sqrt{\gamma_{21}}\rho_{12}^{*}&\gamma_{21}\rho_{22}&0\\\ 0&0&0\\\
\end{array}\right)\;,$ (58)

which bears a close resemblance to the MAD channel $\Phi^{({\text{\tiny
MAD}})}_{\Gamma^{\prime}}$ with values $\gamma_{21}^{\prime}=0$,
$\gamma_{20}^{\prime}=1$, i.e.

$\Phi^{({\text{\tiny
MAD}})}_{\Gamma^{\prime}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+\gamma^{\prime}_{10}\rho_{11}+\rho_{22}&\sqrt{1-\gamma^{\prime}_{10}}\rho_{01}&0\\\
\sqrt{1-\gamma_{10}}\rho_{01}^{*}&(1-\gamma^{\prime}_{10})\rho_{11}&0\\\
0&0&0\\\ \end{array}\right)\;,$ (59)

for which capacities $Q$ and $C_{\text{p}}$ have been computed for all values
of $\gamma_{10}^{\prime}$ [8], see Appendix C.3. The explicit connection
between Eq. (58) and Eq. (59) follows by observing that, setting
$\gamma_{10}^{\prime}=1-\gamma_{21}$ we can map the former into the latter via
a unitary transformation on the input of the channel via the identity

$\displaystyle\Phi_{\Gamma}(\hat{\rho})=\Phi^{({\text{\tiny
MAD}})}_{\Gamma^{\prime}}(\hat{V}_{\text{S}}\hat{\rho}\hat{V}^{\dagger}_{\text{S}})\;,$
(60)

being $\hat{V}_{\text{S}}$ the unitary operator reordering the canonical basis
$\ket{0}_{\text{S}}$, $\ket{1}_{\text{S}}$, $\ket{2}_{\text{S}}$, into
$\ket{1}_{\text{S}}$, $\ket{2}_{\text{S}}$, $\ket{0}_{\text{S}}$. Accordingly
using Eq. (C.2) of Appendix C.2, in the region $\frac{1}{2}\leq\gamma_{21}\leq
1$ we can write

$\displaystyle
Q(\Phi_{\Gamma})=C_{\text{p}}(\Phi_{\Gamma})=\max_{p_{0},p_{1}}\\{$
$\displaystyle-(1-\gamma_{21}p_{0})\log_{2}(1-\gamma_{21}p_{0})-\gamma_{21}p_{0}\log_{2}(\gamma_{21}p_{0})$
$\displaystyle+(1-(1-\gamma_{21})p_{0}-p_{1})\log_{2}(1-(1-\gamma_{21})p_{0}-p_{1})$
$\displaystyle+(1-\gamma_{21})p_{0}\log_{2}((1-\gamma_{21})p_{0})+p_{1}\log_{2}p_{1}\\}$
$\displaystyle={\cal Q}(1-\gamma_{21})\;,$ (61)

in the region $0\leq\gamma_{21}\leq\frac{1}{2}$ instead, $Q=C_{\text{p}}=0$ as
the channel is antidegradable. The overall profile of the capacities in this
case reduces then to that of a qubit ADC with damping parameter
$1-\gamma_{21}$ [70]. We report its evaluation in Fig. 6, left panel. Notice
that in the first expression the maximization is performed over the
populations $p_{0},p_{1}\in[0,1]$ of the input state with respect to levels
$|0\rangle_{\text{S}}$ and $|1\rangle_{\text{S}}$ under the consistency
constraint $p_{0}+p_{1}\leq 1$. The second identity then follows by noticing
that, similarly to what happens in Eq. (56), for $p_{0}$ fixed the maximum of
the r.h.s. term is always achieved by $p_{1}=0$ (no population assigned to the
first excited level). Comparing Eq. (61) with Eq. (55) we observe that on the
edges ${\bf BC}$ and ${\bf BF}$, that bound $\mathbb{D}_{3}$ on the plane
$\gamma_{10}=1$, the capacity of the channel takes the same values as those
for fixed $\gamma_{20}$. Indeed we get

$\displaystyle Q(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=1\\\
\gamma_{21}=0\\\
\gamma_{20}\end{subarray}}=Q(\Phi_{\Gamma})\Big{|}_{\begin{subarray}{l}\gamma_{01}=1\\\
\gamma_{21}=1-\gamma_{20}\\\ \gamma_{20}\end{subarray}}={\cal
Q}(\gamma_{20})\;,$ (62)

which again is the capacity profile of a qubit ADC with damping parameter
$\gamma_{20}$. See 6, right panel.

Figure 6:  Left: $Q(1-\gamma_{21})$ as in Eq. (61), in the region
$\gamma_{21}\geq\frac{1}{2}$ the capacity is found for provably not degradable
nor antidegradable channels. Right: $Q(\gamma_{20})$ as in Eq. (62), in the
region $\gamma_{20}\geq\frac{1}{2}$ the capacity is found for provably not
degradable nor antidegradable channels.

  * •

In the other sectors of the parameter space it is still possible to exploit
some information theoretic properties to obtain computationally efficient
upper bounds. For instance exploiting the considerations underlined in the
final paragraphs of Sec. 2.2 and the data processing inequality (39) we can
claim that, for fixed values of $\gamma_{10}$ and $\gamma_{21}$, both
$Q(\Phi_{\Gamma})$ and $C_{\text{p}}(\Phi_{\Gamma})$ are non-increasing
functions of $\gamma_{20}$, e.g.

$\displaystyle Q(\Phi_{\Gamma})\geq
Q(\Phi_{\Gamma^{\prime}})\;,\qquad\forall\gamma_{10}^{\prime}=\gamma_{10},\gamma_{21}^{\prime}=\gamma_{21},\gamma_{20}^{\prime}\geq\gamma_{20}\;.$
(63)

##  5 Conclusions

We introduced and characterized a new class of physically relevant noise
models for high-dimensional quantum systems. ReMAD channels enlarge the
limited class of quantum channels for which an exact analytical or numerical
approach in terms of quantum and private classical capacities is feasible.
This has been shown by focusing on the simplest, yet nontrivial, case in which
the system of interest is a qutrit. In such case we have shown the existence
of a substantial region of the noise parameter space where the channels
$\Phi_{\Gamma}$ are degradable/antidegradable, allowing hence the direct
expression of the associated quantum and private classical capacities. Also,
by resorting to formal mapping to the case of MAD channels discussed in [8],
we computed the value of these capacities in regions where $\Phi_{\Gamma}$ is
provably neither degradable nor antidegradable. The full characterization in
terms of information capacities, such as for instance classical capacity and
two-way capacities, is still missing for ReMAD channels and will require
further investigations. A semi-analytical treatment is instead achievable for
the characterization of entanglement assisted quantum and classical capacities
$Q_{E}$ and $C_{E}$ of ReMAD channels, see Appendix B.

We acknowledge financial support by MIUR (Ministero dell’ Istruzione, dell’
Universitá e della Ricerca) by PRIN 2017 Taming complexity via Quantum
Strategies: a Hybrid Integrated Photonic approach (QUSHIP) Id. 2017SRN- BRK,
and via project PRO3 Quantum Pathfinder.

##  Appendix A Channel inversion

To compute $C_{\cal D}$ we need to identify the inverse of the channel $\Phi$.
Concretely the inversion of ${\Phi}$ can be done by exploiting the fact that
quantum channels are linear maps connecting vector spaces of linear operators.
They can in turn be represented as matrices acting on vector spaces. This
through the following “vectorization” isomorphism, also called Liouville
representation:

$\displaystyle\hat{\rho}_{\text{S}}=\sum_{ij}\rho_{ij}\ket{i}\\!\\!\bra{j}_{\text{S}}$
$\displaystyle\longrightarrow$
$\displaystyle\ket{\rho\rangle}=\sum_{ij}\rho_{ij}\ket{i}_{\text{S}}\otimes\ket{j}_{\text{S}}\in{\cal
H}_{\text{S}}^{\otimes 2}\;,$ $\displaystyle\Phi(\hat{\rho}_{\text{S}})$
$\displaystyle\longrightarrow$
$\displaystyle\hat{\text{M}}_{\Phi}\ket{\rho\rangle}\;,$

where now $\hat{\text{M}}_{\Phi}$ is a $d_{\text{S}^{\prime}}^{2}\times
d_{\text{S}}^{2}$ matrix connecting ${\cal H}_{\text{S}}^{\otimes 2}$ and
${\cal H}_{\text{S}^{\prime}}^{\otimes 2}$ ($d_{\text{S}}$ and
$d_{\text{S}^{\prime}}$ being respectively the dimensions of ${\cal
H}_{\text{S}}$ and ${\cal H}_{\text{S}^{\prime}}$). Given a Kraus set
$\\{\hat{K}_{i}\\}_{i}$ for $\Phi$, $\hat{\text{M}}_{\Phi}$ can be explicitly
expressed as

$\hat{\text{M}}_{\Phi}=\sum_{i}\hat{K}_{i}\otimes\hat{K}_{i}^{*}\;.$ (65)

Following Eq. (41) we have hence that for a degradable channel the following
identity must apply

$\hat{\text{M}}_{\tilde{\Phi}}=\hat{\text{M}}_{\mathcal{D}}\hat{\text{M}}_{\Phi}\;,$
(66)

with $\hat{\text{M}}_{\mathcal{D}}$ the matrix representation of the LCPTP
degrading channel ${\cal D}$, implying that the super-operator
$\tilde{\Phi}\circ\Phi^{-1}$ is now represented by

$\hat{\text{M}}_{\mathcal{D}}=\hat{\text{M}}_{\tilde{\Phi}}\hat{\text{M}}_{\Phi}^{-1}\;.$
(67)

In the case of the qutrits ReMAD channel of (14), the inverse channel can also
be expressed in the form

$\Phi^{-1}_{{\Gamma}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}-\tfrac{\gamma_{10}}{1-\gamma_{10}}\rho_{11}+\tfrac{\gamma_{10}\gamma_{21}-\gamma_{20}(1-\gamma_{10})}{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}\rho_{22}&\tfrac{\rho_{01}}{\sqrt{1-\gamma_{10}}}-\tfrac{\sqrt{\gamma_{10}\gamma_{21}}\rho_{12}}{(1-\gamma_{10})\sqrt{(1-\gamma_{20}-\gamma_{21})}}&\tfrac{\rho_{02}}{\sqrt{1-\gamma_{21}-\gamma_{20}}}\\\
\tfrac{\rho^{*}_{01}}{\sqrt{1-\gamma_{10}}}-\tfrac{\sqrt{\gamma_{10}\gamma_{21}}\rho^{*}_{12}}{(1-\gamma_{10})\sqrt{(1-\gamma_{20}-\gamma_{21})}}&\tfrac{\rho_{11}}{1-\gamma_{10}}-\tfrac{\gamma_{21}\rho_{22}}{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}&\tfrac{\rho_{12}}{\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}}\\\
\tfrac{\rho_{02}^{*}}{\sqrt{1-\gamma_{21}-\gamma_{20}}}&\tfrac{\rho_{12}^{*}}{\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}}&\tfrac{\rho_{22}}{1-\gamma_{21}-\gamma_{20}}\\\
\end{array}\right)\;,$ (68)

as one can check by direct computation.

###  A.1 Uniqueness of the degrading channel

In the most general case, even when the inverse of the channel considered
exists, degrading maps may not be unique [71]. We show here how in the case of
qutrit ReMAD channels, if these are degradable, the degrading channel must be
unique and in the form of Eq. (67). For our analysis this is also equivalent
to say that if the degrading channel $\mathcal{D}$ can be found as in Eq. (67)
but it’s not LCPTP, then the channel is not degradable. A proof of can be
derived from [71, Theorem 3]:

######  Theorem 4 ([71]).

Let $\Phi:\mathfrak{S}({\cal H}_{\text{A}})\rightarrow\mathfrak{S}({\cal
H}_{\text{B}})$ be a quantum channel and $\tilde{\Phi}:\mathfrak{S}({\cal
H}_{\text{A}})\rightarrow\mathfrak{S}({\cal H}_{\text{E}})$ its complementary
channel and let the corresponding super-operator $\hat{\text{M}}_{\Phi}$ of
$\Phi$ be full rank:
$\text{rank}\;\hat{\text{M}}_{\Phi}=\min[d_{A}^{2},d_{B}^{2}]$. Then, if a
degrading map $\mathcal{D}:\mathfrak{S}({\cal
H}_{\text{B}})\rightarrow\mathfrak{S}({\cal H}_{\text{E}})$ exists, it is
unique iff $d_{B}\leq d_{A}$.

In our case $d_{B}=d_{A}$ and the ‘superoperator’ $\hat{\text{M}}_{\Phi}$ is
always full rank except when $\gamma_{10}=1$ or $\gamma_{20}+\gamma_{21}=1$,
since

$\mathsf{M}=\hat{\text{M}}_{\Phi_{\Gamma}}=\scalebox{0.7}{$\begin{pmatrix}1&0&0&0&\gamma_{10}&0&0&0&\gamma_{20}\\\
0&\sqrt{1-\gamma_{10}}&0&0&0&\sqrt{\gamma_{10}\gamma_{21}}&0&0&0\\\
0&0&\sqrt{1-\gamma_{21}-\gamma_{20}}&0&0&0&0&0&0\\\
0&0&0&\sqrt{1-\gamma_{10}}&0&0&0&\sqrt{\gamma_{10}\gamma_{21}}&0\\\
0&0&0&0&1-\gamma_{10}&0&0&0&\gamma_{21}\\\
0&0&0&0&0&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}&0&0&0\\\
0&0&0&0&0&0&\sqrt{1-\gamma_{21}-\gamma_{20}}&0&0\\\
0&0&0&0&0&0&0&\sqrt{(1-\gamma_{10})(1-\gamma_{21}-\gamma_{20})}&0\\\
0&0&0&0&0&0&0&0&1-\gamma_{21}-\gamma_{20}\end{pmatrix}$}$

is upper triangular and full rank, where we ordered the $\rho_{ij}$ ‘basis’ as
$\\{\rho_{00},\rho_{01},\rho_{02},\rho_{10},\rho_{11},\rho_{12},\rho_{20},\rho_{21},\rho_{22}\\}$.
So, whenever $\gamma_{10}\neq 1$ and $\gamma_{21}+\gamma_{20}\neq 1$ we have
that $\hat{\text{M}}_{\Phi_{\Gamma}}$ is full rank (hence invertible) and if
$\hat{\text{M}}_{\mathcal{D}}=\hat{\text{M}}_{\tilde{\Phi}_{\Gamma}}\hat{\text{M}}_{\Phi_{\Gamma}}^{-1}$
doesn’t represent a LCPTP map then $\Phi_{\Gamma}$ is not degradable.

Now the cases $\gamma_{10}=1$ and $\gamma_{20}+\gamma_{21}=1$ remain, but it’s
straightforward to show that these channels can’t degradable. Specifically we
just need to show that
$\text{ker}\;\Phi_{\Gamma}\not\subset\text{ker}\;\tilde{\Phi}_{\Gamma}$, see
[11, Sec. 2.1]. Looking at the expressions of the channels and their
complementaries

$\displaystyle\Phi^{(\gamma_{10}=0)}_{{\Gamma}}(\hat{\rho})$
$\displaystyle=\left(\begin{array}[]{ccc}\rho_{00}+\rho_{11}+\gamma_{20}\rho_{22}&\sqrt{\gamma_{21}}\rho_{12}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\\\
\sqrt{\gamma_{21}}\rho_{12}^{*}&\gamma_{21}\rho_{22}&0\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}^{*}&0&(1-\gamma_{21}-\gamma_{20})\rho_{22}\\\
\end{array}\right)\;$ (72)
$\displaystyle\tilde{\Phi}^{(\gamma_{10}=0)}_{{\Gamma}}(\hat{\rho})$
$\displaystyle=\left(\begin{array}[]{ccc}\rho_{00}+(1-\gamma_{21}-\gamma_{20})\rho_{22}&\rho_{01}&\sqrt{\gamma_{20}}\rho_{02}\\\
\rho_{01}^{*}&\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{\gamma_{20}}\rho_{12}\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&\sqrt{\gamma_{20}}\rho_{12}^{*}&\gamma_{20}\rho_{22}\\\
\end{array}\right)\;,$ (76)

$\displaystyle\Phi^{(\gamma_{21}+\gamma_{20}=1)}_{{\Gamma}}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+\gamma_{10}\rho_{11}+\gamma_{20}\rho_{22}\sqrt{1-\gamma_{10}}\rho_{01}+\sqrt{\gamma_{10}\gamma_{21}}\rho_{12}&0\\\
\sqrt{1-\gamma_{10}}\rho_{01}^{*}+\sqrt{\gamma_{10}\gamma_{21}}\rho_{12}^{*}&(1-\gamma_{10})\rho_{11}+\gamma_{21}\rho_{22}&0\\\
0&0&0\\\ \end{array}\right)\;$ (80)
$\displaystyle\tilde{\Phi}^{(\gamma_{21}+\gamma_{20}=1)}_{{\Gamma}}(\hat{\rho})=\scalebox{1.0}{$\left(\begin{array}[]{ccc}\rho_{00}+(1-\gamma_{10})\rho_{11}&\sqrt{\gamma_{10}}\rho_{01}+\sqrt{(1-\gamma_{10})\gamma_{21}}\rho_{12}&\sqrt{\gamma_{20}}\rho_{02}\\\
\sqrt{\gamma_{10}}\rho_{01}^{*}+\sqrt{(1-\gamma_{10})\gamma_{21}}\rho_{12}^{*}&\gamma_{10}\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{\gamma_{10}\gamma_{20}}\rho_{12}\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&\sqrt{\gamma_{10}\gamma_{20}}\rho_{12}^{*}&\gamma_{20}\rho_{22}\\\
\end{array}\right)$}\;,$ (84)

we see that the element
$\ket{1}\\!\\!\bra{2}\in\text{ker}\;\Phi^{(\gamma_{10}=0)}_{\Gamma}\not\subset\text{ker}\;\tilde{\Phi}^{(\gamma_{10}=0)}_{\Gamma}$
and that the element
$\ket{0}\\!\\!\bra{2}\in\text{ker}\;\Phi^{(\gamma_{21}+\gamma_{20}=1)}_{\Gamma}\not\subset\text{ker}\;\tilde{\Phi}^{(\gamma_{21}+\gamma_{20}=1)}_{\Gamma}$.
Therefore both cannot be degradable.

In synthesis, for ReMAD channels the degrading map can be uniquely found via
the inverse matrix of the channel. If it doesn’t exist or the derived
degrading map is not LCPTP then the channel is not degradable.

A similar analysis can be applied to antidegradability but typically we don’t
need it in our discussion: where we can’t prove degradability, in those
parameter regions we consider, ReMAD channels have positive capacity and hence
cannot be antidegradable.

##  Appendix B Entanglement assisted quantum and classical capacities

The discovery of protocols such as quantum teleportation [72] and superdense
coding [73] showed how entanglement could be leveraged as an additional
resource in order to boost the communication performance between two
communicating parties. The formalization of these entanglement-assisted
protocols in Shannon-theoretic terms was given in [74, 75], where the
entanglement-assisted classical capacity $C_{E}$ and entanglement-assisted
quantum capacity $Q_{E}$ were introduced. The peculiarity and the advantage
with the definition of these capacities is that they are additive quantities
and don’t need a regularization. Specifically, recalling the definition of the
quantum mutual information $I(\Phi,\hat{\rho})$

$I(\Phi,\hat{\rho})=S(\hat{\rho})+I_{\text{coh}}(\Phi,\hat{\rho})\;,$ (85)

we have:

$C_{E}(\Phi)=\max\limits_{\hat{\rho}\in\mathfrak{S}({\cal
H}_{\text{S}})}I(\Phi,\hat{\rho})\;,\qquad
Q_{E}(\Phi)=\frac{1}{2}C_{E}(\Phi)\;,$ (86)

where the definition of $Q_{E}(\Phi)$ is justified by the fact that in
presence of entanglement a qudit quantum state can be teleported ‘spending’
two classical $d$its (quantum teleportation) and viceversa two classical
$d$its can be communicated by sending a single qudit (superdense coding).

To effectively compute these quantities we notice that the Shannon entropy is
always concave w.r.t. the input state but, as we saw before, the coherent
information can be proved concave only when the channel is degradable.
Nevertheless the quantum mutual information is always concave in the input
state [14, chapter 13.4.2]. Therefore also $I(\Phi)$ can be maximized just
over diagonal states if $\Phi$ is a covariant channel by following steps as in
Eq. (46). We report then in Fig. 7 the evaluation of $C_{E}(\Phi_{\Gamma})$
for a qutrit ReMAD channel at varying $\gamma_{20}$ and $\gamma_{21}$ for some
instances of $\gamma_{10}$ values.

Figure 7: Entanglement assisted classical capacity $C_{E}(\Phi_{\Gamma})$ for
different $\gamma_{10},$ $\gamma_{21}$ and $\gamma_{20}$. The grey area
represents values of $\gamma_{20}$ and $\gamma_{21}$ s.t.
$\gamma_{20}+\gamma_{21}>0$ for which the channel is not defined.

##  Appendix C $Q$ and $C_{\text{p}}$ in non-degradable regions

###  C.1 $Q$ and $C_{\text{p}}$ for $\gamma_{10}=0$

In case $\gamma_{10}=0$ the actions of a generic qutrit ReMAD channel
$\Phi_{\Gamma}^{(\gamma_{10}=0)}$ and of its complementary channel
$\tilde{\Phi}_{\Gamma}^{(\gamma_{10}=0)}$ on a generic density matrix
$\hat{\rho}$ reduce to

$\displaystyle\Phi_{\Gamma}^{(\gamma_{10}=0)}(\hat{\rho})=\begin{pmatrix}\rho_{00}+\gamma_{20}\rho_{22}&\rho_{01}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\\\
\rho_{01}^{*}&\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{12}\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}^{*}&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{12}^{*}&(1-\gamma_{21}-\gamma_{20})\rho_{22}\end{pmatrix},$
(87)
$\displaystyle\tilde{\Phi}_{\Gamma}^{(\gamma_{10}=0)}(\hat{\rho})=\left(\begin{array}[]{ccc}1-(\gamma_{21}+\gamma_{20})\rho_{22}&\sqrt{\gamma_{21}}\rho_{12}&\sqrt{\gamma_{20}}\rho_{02}\\\
\sqrt{\gamma_{21}}\rho_{12}^{*}&\gamma_{21}\rho_{22}&0\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&0&\gamma_{20}\rho_{22}\\\ \end{array}\right),$
(91)

It’s immediate to notice that $\Phi_{\Gamma}^{(\gamma_{10}=0)}$ has a
noiseless subspace given by $\\{\ket{0},\ket{1}\\}$ and consequently can
establish the following lower bound:

$\displaystyle C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{10}=0)})\geq
Q(\Phi_{\Gamma}^{(\gamma_{10}=0)})\geq\log_{2}(2)=1\;.$ (92)

This implies that $\Phi_{\Gamma}^{(\gamma_{10}=0)}$ can’t be antidegradable
(we can reach the same conclusion by noticing that
$\tilde{\Phi}_{\Gamma}^{(\gamma_{10}=0)}$ has a kernel that is not included
into the kernel of $\Phi_{\Gamma}^{(\gamma_{10}=0)}$ [11]. For instance the
former contains $\ket{0}\\!\\!\bra{1}$ while the latter doesn’t).

To compute the capacities we follow the channel inversion method described in
Appendix A. We find that $\Phi_{\Gamma}^{(\gamma_{10}=0)}$ can be inverted
when $\gamma_{21}+\gamma_{20}=1$, and that
$\tilde{\Phi}_{\Gamma}^{(\gamma_{10}=0)}\circ\Phi_{\Gamma}^{(\gamma_{10}=0)-1}$
is LCPTP when $\gamma_{21}+\gamma_{20}\leq\frac{1}{2}$, identifying then the
region of degradability for the channel. There, exploiting the channel
covariance described in Sec. 2.3, we can compute
$Q(\Phi_{\Gamma}^{(\gamma_{10}=0)})$ and as  $\displaystyle
Q(\Phi_{\Gamma}^{(\gamma_{10}=0)})$
$\displaystyle=C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{10}=0)})$
$\displaystyle=\max_{p_{1},p_{2}}\Big{\\{}-(p_{1}+\gamma_{21}p_{2})\log_{2}(p_{1}+\gamma_{21}p_{2})-[1-p_{1}-(1-\gamma_{20})p_{2}]\log_{2}[1-p_{1}-(1-\gamma_{20})p_{2}]$
$\displaystyle\qquad\qquad-(1-\gamma_{21}-\gamma_{20})p_{2}\log_{2}((1-\gamma_{21}-\gamma_{20})p_{2})+(1-(\gamma_{21}+\gamma_{20})p_{2})\log_{2}(1-(\gamma_{21}+\gamma_{20})p_{2})$
$\displaystyle\qquad\qquad+\gamma_{21}p_{2}\log_{2}(\gamma_{21}p_{2})+\gamma_{20}p_{2}\log_{2}(\gamma_{20}p_{2})\Big{\\}}\;.$
(93)

We are therefore able to evaluate numerically the value of $Q$ and
$C_{\text{p}}$ on the boundary of the degradability region
$\gamma_{21}+\gamma_{20}=\frac{1}{2}$, where we are able to showing that there
it’s equal to the lower bound in Eq. (92). This, together with the closure
under composition of channels $\Phi_{\Gamma}^{(\gamma_{10}=0)}$ and data
processing inequalities, allows us to conclude that
$Q(\Phi_{\Gamma}^{(\gamma_{10}=0)}),\;C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{10}=0)})=1$
in the entire parameter region $\gamma_{21}+\gamma_{20}\geq 1/2$.

###  C.2 $Q$ and $C_{\text{p}}$ for $\gamma_{21}=0$

In the case of ReMAD channels with $\gamma_{21}=0$ we have that channels and
complementary channels actions reduce to

$\displaystyle\Phi_{\Gamma}^{(\gamma_{21}=0)}(\hat{\rho})=\begin{pmatrix}\rho_{00}+\gamma_{10}\rho_{11}+\gamma_{20}\rho_{22}&\sqrt{1-\gamma_{10}}\rho_{01}&\sqrt{1-\gamma_{20}}\rho_{02}\\\
\sqrt{1-\gamma_{10}}\rho_{01}^{*}&(1-\gamma_{10})\rho_{11}&\sqrt{1-\gamma_{10}}\sqrt{1-\gamma_{20}}\rho_{12}\\\
\sqrt{1-\gamma_{20}}\rho_{02}^{*}&\sqrt{1-\gamma_{10}}\sqrt{1-\gamma_{20}}\rho_{12}^{*}&(1-\gamma_{20})\rho_{22}\end{pmatrix},$
(94)
$\displaystyle\tilde{\Phi}_{\Gamma}^{(\gamma_{10}=0)}(\hat{\rho})=\left(\begin{array}[]{ccc}1-\gamma_{10}\rho_{11}-\gamma_{20}\rho_{22}&\sqrt{\gamma_{10}}\rho_{01}&\sqrt{\gamma_{20}}\rho_{02}\\\
\sqrt{\gamma_{10}}\rho_{01}^{*}&\gamma_{10}\rho_{11}&\sqrt{\gamma_{10}}\sqrt{\gamma_{20}}\rho_{12}\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&\sqrt{\gamma_{10}}\sqrt{\gamma_{20}}\rho_{12}^{*}&\gamma_{20}\rho_{22}\\\
\end{array}\right).$ (98)

To compute the capacities we follow the channel inversion method described in
Appendix A. We find that $\Phi_{\Gamma}^{(\gamma_{21}=0)}$ is invertible when
$\gamma_{10},\gamma_{20}<1$, while
$\tilde{\Phi}_{\Gamma}^{(\gamma_{21}=0)}\circ\Phi_{\Gamma}^{(\gamma_{21}=0)-1}$
is LCPTP when $\gamma_{10},\gamma_{20}\leq\frac{1}{2}$. From this follows that
in this range of parameters the channels are degradable. Comparing Eqs. (94)
with (98) we can also see that

$\displaystyle\tilde{\Phi}_{\Gamma}^{({\gamma_{10},0,\gamma_{20}})}=\Phi_{\Gamma}^{(1-\gamma_{10},0,1-\gamma_{20})}\;.$
(99)

Therefore, by the same argument above, we have that
$\Phi_{\Gamma}^{(\gamma_{21}=0)}$ is antidegradable for
$\gamma_{10},\gamma_{20}\geq\frac{1}{2}$ and that
$Q(\Phi_{\Gamma}^{(\gamma_{21}=0)}),C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{21}=0)})=0$
for that range of values. To compute $Q(\Phi_{\Gamma}^{(\gamma_{21}=0)})$ and
$C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{21}=0)})$ in the degradable region we
exploit the channels covariance described in Sec. 2.3 to maximize only over
diagonal inputs, getting

$\displaystyle
Q(\Phi_{\Gamma}^{(\gamma_{21}=0)})=C_{\text{p}}(\Phi_{\Gamma}^{(\gamma_{21}=0)})$
$\displaystyle=\max_{\begin{subarray}{c}p_{1},p_{2}\in[0,1]\\\ p_{1}+p_{2}\leq
1\end{subarray}}\Big{\\{}-[1-(1-\gamma_{10})p_{1}+(1-\gamma_{20})p_{2}]\log_{2}[1-(1-\gamma_{10})p_{1}+(1-\gamma_{20})p_{2}]$
$\displaystyle\qquad\qquad\qquad-(1-\gamma_{10})p_{1}\log_{2}((1-\gamma_{10})p_{1})-(1-\gamma_{20})p_{2}\log_{2}((1-\gamma_{20})p_{2})$
$\displaystyle\qquad\qquad\qquad+(1-\gamma_{10}p_{1}-\gamma_{20}p_{2})\log_{2}(1-\gamma_{10}p_{1}-\gamma_{20}p_{2})+\gamma_{10}p_{1}\log_{2}(\gamma_{10}p_{1})+\gamma_{20}p_{2}\log_{2}(\gamma_{20}p_{2})\Big{\\}}\;.$
(100)

The capacities are also known on the boundaries of the parameters space, since
when one of the remaining damping parameters is 0
$\Phi_{\Gamma}^{(\gamma_{21}=0)}$ reduces to a single-decay qutrit MAD, for
which $Q$ and $C_{\text{p}}$ are known [8]. When one of the damping parameters
is instead 1 we reduce to the MAD channel discussed in Appendix C.3, for which
$Q$ is already available. More precisely in Appendix. C.3 we compute
$Q(\Phi^{(1,0,\gamma_{20})})$, verifying that it coincides with the capacity
of a qubit ADC.

Since the value of $Q$ is known also on the boundaries of the degradable
region, we can compare $Q({\Phi}_{\Gamma}^{(\gamma_{10},0,\gamma_{20})})$ at
$\gamma_{20}=\frac{1}{2}$ and $\gamma_{20}=1$, for all $\gamma_{21}\geq 1/2$.
We find that the two are the same, i.e.
$Q({\Phi}_{\Gamma}^{(\gamma_{10},0,1)})=Q({\Phi}_{\Gamma}^{(\gamma_{10},0,1/2})$,
Accordingly, invoking the data-processing inequality and closure under
composition, we can finally conclude that

$\displaystyle
Q({\Phi}_{\Gamma}^{(\gamma_{10},0,1)})=Q({\Phi}_{\Gamma}^{(\gamma_{10},0,\gamma_{20}})\;\qquad\forall\gamma_{20}\geq\frac{1}{2}\;,$
(101)

that allows us to evaluate the $Q$ and $C_{\text{p}}$ over the entire
parameters region.

###  C.3 $Q$ and $C_{\text{p}}$ for $\gamma_{10}=1$

We work here under the assumption of qutrit ReMAD channels with
$\gamma_{10}=1$, for which the expression of channels and their complementary
becomes

$\displaystyle\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}(\hat{\rho})=\begin{pmatrix}1-(1-\gamma_{20})\rho_{22}&0&\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}\\\
0&\gamma_{21}\rho_{22}&0\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\rho_{02}^{*}&0&(1-\gamma_{21}-\gamma_{20})\rho_{22}\end{pmatrix},$
(102)
$\displaystyle\tilde{\Phi}_{{\Gamma}}^{(1,\gamma_{21},\gamma_{20})}(\hat{\rho})=\left(\begin{array}[]{ccc}\rho_{00}+(1-\gamma_{21}-\gamma_{20})\rho_{22}&\rho_{01}&\sqrt{\gamma_{20}}\rho_{02}\\\
\rho_{01}^{*}&\rho_{11}+\gamma_{21}\rho_{22}&\sqrt{\gamma_{20}}\rho_{12}\\\
\sqrt{\gamma_{20}}\rho_{02}^{*}&\sqrt{\gamma_{20}}\rho_{12}^{*}&\gamma_{20}\rho_{22}\\\
\end{array}\right)\;,$ (106)

for $\gamma_{21},\gamma_{20}\in[0,1]$ such that $\gamma_{21}+\gamma_{20}\leq
1$. The channel cannot be degradable since
$\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ has a kernel that is not
included into the kernel of
$\tilde{\Phi}_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ [11]. We are still able
though to express exactly the value of its quantum capacity. Specifically we
are able to state that

$\displaystyle Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$
$\displaystyle=$ $\displaystyle
C_{\text{p}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})=Q^{(1)}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})={\cal
Q}(\gamma_{21},\gamma_{20})\;,$ (107)

with ${\cal Q}(\gamma_{21},\gamma_{20})$ defined as

$\displaystyle{\cal Q}(\gamma_{21},\gamma_{20})$ $\displaystyle\equiv$
$\displaystyle\max_{\hat{\tau}_{\text{diag}}}\left\\{S(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))-S(\tilde{\mathcal{D}}_{(1,\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))\right\\}$
(108) $\displaystyle=$
$\displaystyle\max_{p\in[0,1]}\Big{\\{}-(1-(1-\gamma_{20})p)\log_{2}(1-(1-\gamma_{20})p)-(1-\gamma_{21}-\gamma_{20})p\log_{2}(1-\gamma_{21}-\gamma_{20})p)$
$\displaystyle\qquad\quad\;+(1-(\gamma_{21}+\gamma_{20})p)\log_{2}(1-(\gamma_{21}+\gamma_{20})p)+\gamma_{20}p\log_{2}\gamma_{20}p)\Big{\\}}\;,$

where we are able to restrict the maximization to diagonal density matrices of
form $\hat{\tau}_{\text{diag}}=(1-p)|0\rangle\\!\langle
0|+p|2\rangle\\!\langle 2|$ of $\text{A}^{\prime}$ associated with the
subspace ${\cal H}_{\text{A}^{\prime}}\equiv\mbox{Span}\\{\ket{0},\ket{2}\\}$.
We plot ${\cal Q}(\gamma_{21},\gamma_{20})$ in Fig. 8: we notice that when
$\gamma_{20}\geq\frac{1-\gamma_{21}}{2}$ we have ${\cal
Q}(\gamma_{21},\gamma_{20})=0$, coherently with the fact that in that
parameter region the channel $\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ has
zero capacity, i.e.

$\displaystyle Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$
$\displaystyle=$ $\displaystyle
C_{\text{p}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})=0\quad\forall\;1-\gamma_{21}\geq\gamma_{20}\geq\tfrac{1-\gamma_{21}}{2}\;.$
(109)

Figure 8:  Numerical evaluation of $Q(\gamma_{21},\gamma_{20})$ obtained by
performing the maximization in Eq. (108).

To prove Eq. (107) let us start by observing that ${\cal
Q}(\gamma_{21},\gamma_{20})$ provides a natural lower bound for
$Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$ and hence for
$C_{\text{p}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$:

$\displaystyle Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$
$\displaystyle\geq$
$\displaystyle\max_{\hat{\rho}}I_{\text{coh}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})},\hat{\rho})\geq\max_{\hat{\tau}_{\text{diag}}}I_{\text{coh}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})},\hat{\tau}_{\text{diag}})={\cal
Q}(\gamma_{21},\gamma_{20})\;.$

We need now to show that ${\cal Q}(\gamma_{21},\gamma_{20})$ can also upper
bound $Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$. To do this we
construct a new channel $\Phi^{\prime(\gamma_{21},\gamma_{20})}$ with larger
capacity than $\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$, i.e.

$\displaystyle Q(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$
$\displaystyle\leq$ $\displaystyle
Q(\Phi^{\prime(\gamma_{21},\gamma_{20})})\;,$ (110) $\displaystyle
C_{\text{p}}(\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})})$ $\displaystyle\leq$
$\displaystyle C_{\text{p}}(\Phi^{\prime(\gamma_{21},\gamma_{20})})\;,$ (111)

and for which we can show that

$\displaystyle
Q(\Phi^{\prime(\gamma_{21},\gamma_{20})})=C_{\text{p}}(\Phi^{\prime(\gamma_{21},\gamma_{20})})={\cal
Q}(\gamma_{21},\gamma_{20})\;.$ (112)

For this purpose notice that, since the population of level $|1\rangle$
completely depleted, the output produced by
$\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ can be reproduced by the channel
$\Phi^{\prime(\gamma_{21},\gamma_{20})}:{\cal L}({\cal
H}_{\text{A}^{\prime}})\rightarrow{\cal L}({\cal H}_{\text{A}})$ operating on
the two-levels Hilbert space ${\cal
H}_{\text{A}^{\prime}}\equiv\mbox{Span}\\{\ket{0},\ket{2}\\}$, and producing
qutrit states of A as outputs. In particular, calling $\hat{\tau}$ a generic
density matrix on ${\cal H}_{\text{A}^{\prime}}$ we have

$\Phi^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau})=\begin{pmatrix}1-(1-\gamma_{20})\tau_{22}&0&\sqrt{1-\gamma_{21}-\gamma_{20}}\tau_{02}\\\
0&\gamma_{21}\tau_{22}&0\\\
\sqrt{1-\gamma_{21}-\gamma_{20}}\tau_{02}^{*}&0&(1-\gamma_{21}-\gamma_{20})\tau_{22}\end{pmatrix}$
(113)

with associated complementary channel

$\tilde{\Phi}^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau})=\left(\begin{array}[]{ccc}1-(\gamma_{21}+\gamma_{20})\tau_{22}&0&\sqrt{\gamma_{20}}\tau_{02}\\\
0&\gamma_{21}\tau_{22}&0\\\
\sqrt{\gamma_{20}}\tau_{02}^{*}&0&\gamma_{20}\tau_{22}\end{array}\right),$
(114)

where for $i,j=0,2$ we set $\tau_{ij}\equiv\langle i|\hat{\tau}|j\rangle$.

$\Phi^{\prime(\gamma_{21},\gamma_{20})}$ fulfills the inequality (110) because
$\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$, while producing the same output
of $\Phi^{\prime(\gamma_{21},\gamma_{20})}$, is also ‘wasting’ resources in
the useless level $|1\rangle$. Explicitly, notice that we can write

$\displaystyle\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}=\Phi^{\prime(\gamma_{21},\gamma_{20})}\circ{\cal
A}\;,$ (115)

being $\mathcal{A}:{\cal L}({\cal H}_{\text{A}})\rightarrow{\cal L}({\cal
H}_{\text{A}^{\prime}})$ is the LCPTP map bringing the input state of the
qutrit A to the qubit $\text{A}^{\prime}$ by transfer of the population of the
level $\ket{1}$ to $\ket{0}$ followed by the erasure of $\ket{1}$ , i.e.

$\displaystyle\mathcal{A}(\hat{\rho})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\rho_{00}+\rho_{11}&\rho_{02}\\\
\rho_{20}&\rho_{22}\\\ \end{pmatrix},$ (116)

where $\rho_{ij}=\langle i|\hat{\rho}|j\rangle$ with
$\hat{\rho}\in\mathfrak{S}({\cal H}_{\text{A}})$. Equation (110) can then be
derived as a consequence of the data processing inequality applied to $Q$ (and
$C_{\text{p}}$). The second part of the argument, i.e. Eq. (112), can instead
be proved by noticing that, differently of the original channel
$\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ that is not degradable,
$\Phi^{\prime(\gamma_{21},\gamma_{20})}$ is degradable if

$\displaystyle 0\leq\gamma_{20}\leq({1-\gamma_{21}})/{2}\;,$ (117)

and antidegradable otherwise, i.e. for
$(1-\gamma_{21})/2\leq\gamma_{20}\leq{1-\gamma_{21}}$. This can be shown by
observing that in the region identified by the inequality (117) the quantity

$\displaystyle\bar{\gamma}_{3}\equiv\frac{1-\gamma_{21}-2\gamma_{20}}{1-\gamma_{21}-\gamma_{20}}\;,$
(118)

belongs to $[0,1]$ and can be used to build a single-decay qutrit MAD channel
$\Phi_{\Gamma}^{(0,0,\bar{\gamma}_{3})}$ . Furthermore by direct calculation
we get

$\displaystyle\Phi_{\Gamma}^{(0,0,\bar{\gamma}_{3})}\circ\Phi^{\prime(\gamma_{21},\gamma_{20})}=\tilde{\Phi}^{\prime(\gamma_{21},\gamma_{20})}\;,$
(119)

which shows that $\Phi_{\Gamma}^{(0,0,\bar{\gamma}_{3})}$ acts as the
degrading channel of $\Phi^{\prime(\gamma_{21},\gamma_{20})}$. From Eqs. (113)
and (114) it is also evident that $\Phi^{\prime(\gamma_{21},\gamma_{20})}$ can
be recovered from $\tilde{\Phi}^{\prime(\gamma_{21},\gamma_{20})}$ by the
substitution $\gamma_{20}\rightarrow 1-\gamma_{21}-\gamma_{20}$. Consequently,
using the same construction of Eq. (119), we can conclude that
$\Phi^{\prime(\gamma_{21},\gamma_{20})}$ is antidegradable when
$(1-\gamma_{21})/2\leq\gamma_{20}\leq{1-\gamma_{21}}$.

Finally, to derive Eq. (112) we observe that as the original channel
$\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}$ also
$\Phi^{\prime(\gamma_{21},\gamma_{20})}$ is covariant and accordingly we can
express its capacity as

$\displaystyle
Q(\Phi^{\prime(\gamma_{21},\gamma_{20})})=C_{\text{p}}(\Phi^{\prime(\gamma_{21},\gamma_{20})})=\max_{\hat{\tau}_{\text{diag}}}\Big{\\{}S(\Phi^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))-S(\tilde{\Phi}^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))\Big{\\}}={\cal
Q}(\gamma_{21},\gamma_{20}),$ (120)

where the last identity follows from the fact that
$\Phi^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))$ coincides
with $\Phi_{\Gamma}^{(1,\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}})$
and by the fact that the positive component of the spectrum of
$\tilde{\Phi}^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}})$
coincides with the one of
$\tilde{\Phi}^{(1,\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}})$
(strictly speaking the above derivation holds true only in the degradable
region (117) of
$\Phi^{\prime(\gamma_{21},\gamma_{20})}(\hat{\tau}_{\text{diag}}))$: still
since ${\cal Q}(\gamma_{21},\gamma_{20})$ nullifies for
$1-\gamma_{21}\geq\gamma_{20}\geq({1-\gamma_{21}})/{2}$, we can apply (120)
also in the antidegradability region of the channel where
$Q(\Phi^{\prime(\gamma_{21},\gamma_{20})})=0$).

Finally, we notice that for $\gamma_{21}=0$ the effective channel in Eq. (113)
can be replaced by the channel

$\Phi^{\prime\gamma_{20}}(\hat{\tau})=\begin{pmatrix}1-(1-\gamma_{20})\tau_{22}&\sqrt{1-\gamma_{20}}\tau_{02}\\\
\sqrt{1-\gamma_{20}}\tau_{02}^{*}&(1-\gamma_{20})\tau_{22}\end{pmatrix}\;,$
(121)

mapping the two-level system $\text{A}^{\prime}$ into itself via a qubit ADC
with damping parameter $\gamma_{20}$. Accordingly, following the same analysis
done above it follows that $Q(\Phi_{\Gamma}^{(1,0,\gamma_{20})})$ and
$C_{\text{p}}(\Phi_{\Gamma}^{(1,0,\gamma_{20})})$ coincide with the respective
capacities of a qubit ADC, computed in Ref. [70].

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