# The superadditivity effects of quantum capacity decrease with the dimension
for qudit depolarizing channels

Josu Etxezarreta Martinez<EMAIL_ADDRESS>Tecnun - University of
Navarra, 20018 San Sebastian, Spain  Antonio deMarti iOlius  Tecnun -
University of Navarra, 20018 San Sebastian, Spain  Pedro M. Crespo  Tecnun -
University of Navarra, 20018 San Sebastian, Spain

###### Abstract

Quantum channel capacity is a fundamental quantity in order to understand how
good can quantum information be transmitted or corrected when subjected to
noise. However, it is generally not known how to compute such quantities,
since the quantum channel coherent information is not additive for all
channels, implying that it must be maximized over an unbounded number of
channel uses. This leads to the phenomenon known as superadditivity, which
refers to the fact that the regularized coherent information of $n$ channel
uses exceeds one-shot coherent information. In this article, we study how the
gain in quantum capacity of qudit depolarizing channels relates to the
dimension of the systems considered. We make use of an argument based on the
no-cloning bound in order to proof that the possible superadditive effects
decrease as a function of the dimension for such family of channels. In
addition, we prove that the capacity of the qudit depolarizing channel
coincides with the coherent information when $d\rightarrow\infty$. We also
discuss the private classical capacity and obain similar results. We conclude
that when high dimensional qudits experiencing depolarizing noise are
considered, the coherent information of the channel is not only an achievable
rate but essentially the maximum possible rate for any quantum block code.

Quantum information theory, quantum channel capacity, superadditivity,
depolarizing channel.

##  I Introduction

Classical communications were revolutionized when Claude Shannon introduced
the noisy-channel coding theorem in his groundbreaking work A Mathematical
Theory of Communication shannon . In such theorem, Shannon introduced the
concept of channel capacity, which refers to the maximum coding rate for which
asymptotically error-free communications are possible over a noisy channel.
The consequences to this result are momentous since it establishes the limit,
in terms of rate, for which error correction makes sense and, thus, the target
that coding theorists should seek when designing their codes. The computation
of such quantity results to be simple due to the fact that the classical
mutual information is additive, implying that the regularization over $n$
channel uses needed to compute the capacity of the channel results in a
single-letter formula, i.e. in the optimization of such quantity over a single
use of the channel shannon .

The development of quantum information theory followed the steps of Shannon,
introducing the concept of quantum channel capacity similarly to its classical
counterpart, i.e. the maximum quantum coding rate for comunication/correction
(note that in the quantum setting the noise can arise from temporal evolution)
with error rates vanishing asymptotically when quantum information is
subjected to noise. In general, the computation of the quantum channel
capacity, $C_{\mathrm{Q}}$, is based on the following regularization lsd1 ;
lsd2 ; lsd3 ; lsd4 ; wildeQIT :

$C_{\mathrm{Q}}(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}Q_{\mathrm{coh}}(\mathcal{N}^{\otimes
n}),$ (1)

where $\mathcal{N}$ denotes the quantum channel and $Q_{\mathrm{coh}}$ refers
to the channel coherent information defined as

$\begin{split}Q_{\mathrm{coh}}(\mathcal{N})&=\max_{\rho}I_{\mathrm{coh}}(\mathcal{N},\rho)\\\
&=\max_{\rho}S(\mathcal{N}(\rho))-S(\mathcal{N}^{c}(\rho)),\end{split}$ (2)

with $I_{\mathrm{coh}}(\mathcal{N},\rho)$ the channel coherent information
when state $\rho$ is the input, $S$ the von Neumann entropy and
$\mathcal{N}^{c}$ is a complementary channel to the environment.

However, in stark contrast to its classical counterpart, the channel coherent
information has been proven not to be additive in general wildeQIT ; super1 ;
super2 ; super3 ; unbounded , implying that the regularization in equation (1)
involves optimizing over an infinite parameter space. Given two arbitrary
quantum channels $\mathcal{N}_{1}$, $\mathcal{N}_{2}$, the most one can say
about the coherent channel information of the parallel channel
$\mathcal{N}_{1}\otimes\mathcal{N}_{2}$ is
$Q_{\mathrm{coh}}(\mathcal{N}_{1}\otimes\mathcal{N}_{2})\geq
Q_{\mathrm{coh}}(\mathcal{N}_{1})+Q_{\mathrm{coh}}(\mathcal{N}_{2})$. When
strict inequality holds, the channels are said to exhibit superaditivity,
otherwise are said to have additive coherent information siddhu . Explicit
examples of superadditivity have been found for several classes of quantum
channels wildeQIT ; unbounded ; siddhu ; super1 ; super2 ; super3 ; super4 ;
super5 ; super6 ; super7 ; super8 ; xzzx ; neuralSuper ; geneticSuper .
Importantly, the non-additivity effects of quantum capacity arise as a result
of entanglement in the input state of the channel since state coherent
information is additive for unentangled input states, i.e.
$I_{\mathrm{coh}}(\mathcal{N}^{\otimes
2},\rho\otimes\sigma)=I_{\mathrm{coh}}(\mathcal{N},\rho)+I_{\mathrm{coh}}(\mathcal{N},\sigma)$
unbounded . This implies that entanglement is a resource that may protect
quantum information from noise in a more efficient way than what it is
classically possible.

Therefore, an important question to be answered is what types of channels have
additive channel coherent information so that their capacity reduces to
single-letter expressions, i.e.,
$C_{\mathrm{Q}}(\mathcal{N})=\max_{\rho}I_{\mathrm{coh}}(\mathcal{N},\rho)$.
At the time of writing, quantum channels with additive channel coherent
information belong to the classes of degradable wildeQIT ; degOrig ; degDep ;
watanabe ; ftvqc , conjugate degradable watanabe ; cdeg  and less noisy than
the environment watanabe  channels. The quantum capacities of antidegradable,
conjugate antidegradable and entanglement-binding channels are also single-
letter characterized, but they are equal to zero wildeQIT ; unbounded ;
watanabe ; entBind ; ftvqc . Recently, examples of quantum channels (the
platypus, multi-level amplitude damping and resonant multi-level amplitude
damping channels), showing additivity while not being degradable have been
found platypus ; mad ; remad .

The depolarizing channel is a widely used quantum channel model in order to
describe the noise that quantum information experiences josuchannels . This
channel is characterized by the depolarizing probability, $p$, and its quantum
channel capacity is still unknown even if it is the simplest and most
symmetric nonunitary quantum channel. In general, $d$-dimensional depolarizing
channels (those acting on $d$-dimensional quantum states referred as qudits)
are antidegradable for $p\geq\frac{d}{2(d+1)}$, while they do not belong to
any of the classes of channels previously mentioned for $p<\frac{d}{2(d+1)}$
depAnti . Several upper bounds on the quantum capacity of $d$-dimensional
depolarizing channels for the non-trivial parameter region have been derived
UB1 ; UB2 ; UB3 ; UB4 ; UB5 ; UB6 . However, the quantum capacity of the
family of $d$-dimensional depolarizing channels remains a mistery for such
region.

In this article, we study how the potential superadditivity effects of the
quantum channel capacity, in qudits per channel use units, relate to the
dimension of the depolarizing channel. Specifically, we want to observe which
is the extra coding rate achievable due to superadditivity when logical qudits
are encoded by physical qudits. We provide an argument based on the no-cloning
bound in order to study how the quantum capacity gain (defined as the
difference between the quantum capacity and the channel coherent information)
caused by potential coherent information superadditivity relates to the
dimension of the depolarizing channel. We conclude that such possible capacity
gain is a monotonically decreasing function with the dimension and, thus, that
the superadditive effects are less and less important when the dimension of
the depolarizing channels increases. In addition, we determine that for the
extremal case in which the dimension of the system is let to grow indefinitely
(in the limit where the qudit becomes a quantum oscillator, i.e., a bosonic
mode qudits ), the depolarizing channel capacity coincides with the channel
coherent information. We also relate the obtained results with the private
capacity of qudit depolarizing channels concluding that such information
theoretic quantity behaves in a similar way as the quatum channel capacity.

##  II Qudit depolarizing channels

The $d$-dimensional or qudit depolarzing channel,
$\Lambda_{p}^{d}:\mathcal{H}_{d}\rightarrow\mathcal{H}_{d}$, is the
completely-positive, trace preserving (CPTP) map defined as UB1 ; UB6 ; dep1 ;
dep2 ; dep3

$\Lambda_{p}^{d}(\rho)=(1-p)\rho+p\mathrm{Tr}(\rho)\frac{I_{d}}{d},$ (3)

where the density matrices $\rho$ are the so-called qudits or quantum states
operating over a $d$-dimensional Hilbert space $\mathcal{H}_{d}$, $I_{d}/d$
refers to the maximally mixed state of dimension $d$ and $p\in[0,1]$ refers to
the depolarizing probability. Consequently, the operation of the qudit
depolarizing channel leaves the state uncorrupted with probability $1-p$ while
transforming it to the maximally mixed state with probability $p$.

The depolarizing channel has a central role in modeling quantum noise in the
theory of quantum information josuchannels . Importantly, depolarizing
channels can be efficiently simulated as an stochastic noise map by classical
means since they fulfill the Gottesman-Knill theorem josuchannels ; GKthm .
This implies that, for example, the performance of quantum error correction
codes, key for fault-tolerant quantum computing and communications, can be
effectively assesed by traditional methods. Furthermore, Clifford twirling an
arbitrary $d$-dimensional CPTP noise map results in a qudit depolarizing
channel josuchannels ; twirldep . Twirling is extensively used in quantum
information theory for studying the average effects of a general noise map by
mapping them to more symmetric versions of themselves josuchannels ; depAnti ;
twirl1 ; twirl2 ; twirl3 . The twirled channel is obtained by averaging the
action of the map over a set of unitaries. Moreover, the following lemma
twirl1  implies that error correction codes for arbitrary noise maps can be
designed by constructing them to correct a twirled map.

######  Lemma 1.

Any correctable code for the twirled channel $\bar{\mathcal{N}}$ is a
correctable code for the original channel $\mathcal{N}$ up to an additional
unitary correction.

Hence, the depolarizing channel is not only interesting because of its nice
properties, but also as error correction codes can be designed by using it.
The depolarizing parameter and the parameters of the original channel are
related in a specific way as a result of the twirl (see josuchannels ; TVQC
for specific details on the qubit case). Notably, twirling channels into Pauli
channels, whose symmetric version is the depolarizing channel, has recently
been used for the quantum error mitigation technique named Probabilisitc Error
Cancellation (PEC) PEC .

Consequently, studying the achievable rates for the different quantum
information theoretical tasks over depolarizing channels is of the outmost
importance. Studying the different capacities of such family of channels is
also interesting from the point of view of quantum information theory too
since the capacities of twirled channels lower bound the capacities of the
channels from which they originated depAnti  and, thus, interesting lower
bounds on the achievable rates of general channels might be obtained.

The channel coherent information, $Q_{\mathrm{coh}}$, defined in equation (2),
for qudit depolarizing channels is UB1

$\begin{split}&Q_{\mathrm{coh}}(\Lambda_{p}^{d})=\max\left\\{0,\log_{2}{d}\right.\\\
&+\left(1-p\frac{d^{2}-1}{d^{2}}\right)\log_{2}{\left(1-p\frac{d^{2}-1}{d^{2}}\right)}\\\
&\left.+p\frac{d^{2}-1}{d^{2}}\log_{2}{\left(\frac{p}{d^{2}}\right)}\right\\},\end{split}$
(4)

with units of qubits per channel use. It provides a lower bound for the
quantum channel capacity, $C_{\mathrm{Q}}(\mathcal{N})\geq
Q_{\mathrm{coh}}(\mathcal{N})$. Note that by changing the $\log_{2}$ in the
above expression by $\log_{d}$, the units of
$Q_{\mathrm{coh}}(\Lambda_{p}^{d})$ are qudits per channel use. The reason to
consider this units is that we are interested in studying the logical qudits
per physical qudits, i.e. coding rate, that can be achieved for a qudit error
correction scheme, and not the amount of logical qubits that can be encoded by
means of qudits. For the sake of notation we will denote the channel coherent
information in such units by $Q^{d}_{\mathrm{coh}}(\Lambda_{p}^{d})$.

Recall that for $p<\frac{d}{2(d+1)}$ the channel does not belong to any of the
classes with proven additive channel coherent information degDep , implying
that the quantum channel capacity is not known and may exhibit superadditivity
gains. In fact, these gains have been obtained in previous works wildeQIT ;
super1 ; super2 ; neuralSuper . Several techniques have been developed in
order to obtain upper bounds for the quantum channel capacity of
$d$-dimensional depolarizing channels UB1 ; UB2 ; UB3 ; UB4 ; UB5 . Each of
those upper bounds are tighter depending on the region of depolarizing
probability considered in $p\in\left[0,\frac{d}{2(d+1)}\right]$. The tightest
upper bound is usually obtained by using the fact that the convex hull of the
upper bounds is itself an upper bound UB2 . However, for the purposes of this
work, we will consider the so called no-cloning bound, $Q_{\mathrm{nc}}$. The
no-cloning bound on quantum capacity is based on combining Cerf’s no-cloning
bounds nc2  and the degradable extension technique of UB6 . Cerf’s results lay
on the no-cloning theorem111A unitary operator that perfectly copies arbitrary
quantum states cannot be constructed. of quantum mechanics for determining
that Pauli channels (depolarizing channels are an specific instance of this)
cannot have a positive capacity under certain conditions. By using this
result, the bound can be obtained by the techniques in UB6 . A proof for this
can be found in depAnti . The no-cloning bound upper bounds the quantum
channel capacity of qudit depolarizing channels as depAnti ; UB2 ; nc1 ; nc2 ;
nc3

$C_{\mathrm{Q}}(\Lambda_{p}^{d})\leq
Q_{\mathrm{nc}}(\Lambda_{p}^{d})=\left(1-2p\frac{d+1}{d}\right)\log_{2}{d},$
(5)

with units of qubits per channel use. Note that the expresion of
$Q_{\mathrm{nc}}(\Lambda_{p}^{d})$ in qudits per channel use reduces to

$Q_{\mathrm{nc}}^{d}(\Lambda_{p}^{d})=\left(1-2p\frac{d+1}{d}\right).$ (6)

##  III Superadditivity gain

As explained in the previous section, the potential superadditive nature of
the coherent information may lead to quantum channel capacities that are
higher than the one-shot channel coherent information. In other words, there
exists a gain in quantum channel capacity if several quantum channel uses are
considered. Remarkably, it has been proven that even an unbounded number of
channels uses may be required for this effect to happen unbounded . In order
to quantify this gain we define the superadditivity gain, $\xi$, as

$\xi(\mathcal{N})=C_{\mathrm{Q}}(\mathcal{N})-Q_{\mathrm{coh}}(\mathcal{N}),$
(7)

which gives the additional qubits per channel that the channel capacity has
when compared the achievable rate of the channel coherent information.
Clearly, if the coherent information of the channel is additive, then
$\xi(\mathcal{N})=0$. Knowledge about the quantum channel capacity is needed
in order to compute the superadditivity gain in equation (7) and, as stated
before, the quantum capacity of qudit depolarizing channels is still unknown.
However, upper bounds on such quantity can be obtained using the upper bounds
derived in UB1 ; UB2 ; UB3 ; UB4 ; UB5 ; UB6 . For the purposes of this work
we will upper bound the superadditivity gain by using the no-cloning bound as

$\xi_{\mathrm{nc}}(\Lambda_{p}^{d})=Q_{\mathrm{nc}}(\Lambda_{p}^{d})-Q_{\mathrm{coh}}(\Lambda_{p}^{d})\geq\xi(\Lambda_{p}^{d}).$
(8)

The units in the above expression are qubits per channel use. However, we will
study the capacity gain with qudits per channel use units in order to have a
fair comparison of the extra capacity that is obtained via superadditive
effects. In this way, we will be able to see how many more qudits per channel
use can be potentially obtained due to superadditive effects, which is more
fair to compare those effects for different dimensions, since operating in
more dimensions trivially implies that more information (in terms of qubits)
can be encoded in a single quantum state. For example, consider $d_{1}<d_{2}$
and assume that their superadditivity gains in qudits per channel use (coding
rate) for both cases is the same. That is,
$\xi(\Lambda_{p}^{d_{1}})=\xi(\Lambda_{p}^{d_{2}})=g$. However, these gains
become $g\log_{2}(d_{1})<g\log_{2}(d_{2})$ when expressed in qubits per
channel use, making the impression that the capacity of for $d_{2}$ increases
more. Note that whenever qudit error correction codes are constructed, their
coding rate will have logical qudits per physical qudits units, implying that
the extra rate obtained via superadditivity should be quantified in such
terms.

Therefore, in what follows, the units of the superadditive gains will be given
in qudits per channel use, that is

$\xi_{\mathrm{nc}}(\Lambda_{p}^{d})=Q^{d}_{\mathrm{nc}}(\Lambda_{p}^{d})-Q^{d}_{\mathrm{coh}}(\Lambda_{p}^{d})\geq\xi(\Lambda_{p}^{d}).$
(9)

##  IV Superadditivity effects of quantum capacity decrease as a function of
the dimension

We now provide the main result of this article.

######  Theorem 1.

Let $d_{l}$ be an arbitrary positive integer higher than $2$ and
$p_{0}^{d_{l}}\in\mathbb{R}$ defined as

$p_{0}^{d_{l}}=\min_{p}\left(\left\\{p\in\left(0,\frac{d_{l}}{2(d_{l}+1)}\right):Q_{\mathrm{coh}}^{d_{l}}(\Lambda_{p}^{d_{l}})=0\right\\}\right).$
(10)

That is, $p_{0}^{d_{l}}$ is the smallest depolarizing probability that makes
the coherent information of the $d_{l}$-dimensional depolarizing channel equal
to zero. Then, for any depolarizing probability $p$ in the range
$p\in(0,p_{0}^{d_{l}})$, the superadditivity gain,
$\xi_{\mathrm{nc}}(\Lambda_{p}^{d})$, in qudits per channel use units is a
monotonically decreasing function of the channel dimension, $d$, for $d\geq
d_{l}$.

###### Proof.

To prove the theorem, we must prove that

$\frac{\partial\xi_{\mathrm{nc}}(\Lambda_{p}^{d})}{\partial d}<0,\quad\forall
p\in\left(0,p_{0}^{d_{l}}\right).$ (11)

Thus, the derivative of $\xi_{\mathrm{nc}}(\Lambda_{p}^{d})$ over the
dimension in the range $p\in\left(0,p_{0}^{d_{l}}\right)$

$\begin{split}\frac{\partial\xi_{\mathrm{nc}}(\Lambda_{p}^{d})}{\partial
d}&=-1-4\frac{p}{d}+4p\frac{(d^{2}-1)}{d^{3}}-p\frac{(d^{2}-1)\log_{2}{\left(\frac{p}{d^{2}}\right)}}{d^{2}\log_{2}{d}}\\\
&-\frac{(1-p\frac{d^{2}-1}{d^{2}})\log_{2}{\left(1-p\frac{d^{2}-1}{d^{2}}\right)}}{\log_{2}(d)}\\\
&=-4\frac{p}{d}+4p\frac{(d^{2}-1)}{d^{3}}-Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})<0.\end{split}$
(12)

The last inequality follows from the fact that
$4\frac{p}{d}>4p\frac{(d^{2}-1)}{d^{3}},\forall d$ (this inequality reduces to
$\frac{1}{d}>\frac{1}{d}-1$ which is true for all $d>0$), and the fact that
$\forall d\geq d_{l}$, $Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})\geq 0$, since
$p_{0}^{d}$ increases with $d$ and we are considering the range
$p\in\left(0,p_{0}^{d_{l}}\right)$. ∎

Figure 1 graphically shows the results of this theorem. It plots the no-
cloning superadditivity gain versus depolarizing probability, $p$, for four
different $d_{l}$ dimensions. For a given $d_{l}$, the vertical dashed lines
give the value of the corresponding $p_{0}^{d_{l}}$.

Figure 1: No-cloning superadditivity gain as a function of depolarizing
probability $p$. Channel dimensions $d\in\\{2,2^{2},2^{5},2^{20}\\}$ are
plotted.

Note the result of Theorem 1 states that for an initial dimension $d_{l}$, the
no-cloning superadditive gain, $\xi_{\mathrm{nc}}(\Lambda_{p}^{d})$ is a
decreasing function with respect to the dimension $d\geq d_{l}$ in the
depolarizing probability range $p\in(0,p_{0}^{d_{l}})$. It is noteworthy that
the result of the theorem can be extended to a non-tirival region where the
coherent information vanishes. However, since the point of maximum potential
superadditivity lays in the region considered, expanding the analysis to such
parameter space would result in similar conclusions. Additionally, the upper
limit of such range, $p_{0}^{d_{l}}$, increases with respect to the initial
dimension in consideration. This value saturates to $1/2$ when the dimension
of the system is left to grow indefinitely since

$\begin{split}\lim_{d\rightarrow\infty}Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})=&\lim_{d\rightarrow\infty}\left(1+\frac{\left(1-p\frac{d^{2}-1}{d^{2}}\right)\log_{2}{\left(1-p\frac{d^{2}-1}{d^{2}}\right)}}{\log_{2}{d}}\right.\\\
&\left.+\frac{p\frac{d^{2}-1}{d^{2}}\log_{2}{\left(\frac{p}{d^{2}}\right)}}{\log_{2}{d}}\right)=1-2p,\end{split}$
(13)

which vanishes at the value of $1/2$.

In this way, by starting with the minimum dimension of a quantum system, i.e.
a qubit $d_{l}=2$, we can always find another initial higher dimension for
which the no-cloning superadditive gain decreases in all the range of
depolarizing probabilities $p\in(0,1/2)$. For example, see that in Figure 1 we
can change from $d_{l}=2$ to $d_{l}=4$ once we reach $p_{0}^{d_{l}=2}$, and
the gain will still be decreasing for $d>d_{l}=4$. This can be done each time
we reach a particular $d_{l}$. Thus, we effectively prove that whenever the
dimension of the system increases, the room left for superadditive effects in
qudits per channel use units decreases. Note also that the region
$p\in(0,1/2)$ is actually the only region where superadditivity may happen for
every $d$-dimensional depolarizing channels since for $p=0$ there is no noise,
implying that $C_{\mathrm{Q}}^{d}(\Lambda_{p}^{d})=1$, while for $p>1/2$ every
qudit depolarizing channel is antidegradable since
$\lim_{d\rightarrow\infty}d/(2(d+1))=1/2$.

Figure 2 showcases the decrease of the no-cloning superadditive gain for
different depolarizing probabilities $p\in\\{0.01,0.05,0.1,0.2,0.25\\}$ as a
function of the dimension of the system considered.

Figure 2: No-cloning superadditivity gain as a function of dimension and
depolarizing probability. We plot the superadditivity gain in terms of qudits
per channel use as a function of the dimension of the depolarizing channel for
$p\in\\{0.01,0.05,0.1,0.2,0.25\\}$.

Two important conclusions are derived from Theorem 1, which are clearly
appreciated in the above two figures. The first conclusion is that whenever
quantum systems of high dimensions are corrupted by the operation of a qudit
depolarizing channel, the non-additive behaviour of the coherent information
is less relevant. That is, the potential superadditivity gain in terms of
qudits per channel use decreases. This is an important result for the
depolarizing channel since it implies that for very high dimensional systems,
the channel coherent information and the quantum channel capacity will be
close together. Note that tighter bounds than the no-cloning bound can be used
to bound the superadditivity gain, implying that the actual gain will be much
smaller. This yields to the second conclusion which states that for high
dimensional systems, the capacity of the depolarizing channel is close to the
single-letter coherent information of the channel, that is, one can state that
$C_{\mathrm{Q}}^{d}(\Lambda_{p}^{d})\approx
Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})$. Therefore, we can conclude that for
such high dimensional systems, random block codes on the typical subspace of
the optimal input (for the one-shot coherent information) will essentially
achieve quantum channel capacity wildeQIT ; leditzkyRandom . This means that
the best strategy to achieve the capacity of a depolarizing channels with
sufficiently large dimension is by randomly selecting a stabilizer code
wildeQIT .

We have observed that the superadditive behaviour of coherent information
loses importance when the dimensions of the qudit depolarizing channel
increase. In particular, in the limit when $d$ is let to be infinite, the
qudit becomes a quantum oscillator or bosonic mode qudits , and the quantum
channel capacity of the $\infty$-dimensional or bosonic depolarizing channel
is given by $1-2p$, as it is shown in the following Corollary.

######  Corollary 1.

The quantum channel capacity of the $\infty$-dimensional or bosonic
depolarizing channel is

$C_{\mathrm{Q}}^{d}(\Lambda_{p}^{\infty})=Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{\infty})=1-2p,$
(14)

with bosonic modes per channel use units for $p\in[0,1/2]$ and $0$ for
$p\in[1/2,1]$.

###### Proof.

We use a sandwich argument to prove the corollary. We know from equation (13)
that the coherent information of the depolarizing channel has the following
asymptotic behaviour in the region $p\in[0,1/2]$

$C_{\mathrm{Q}}^{d}(\Lambda_{p}^{\infty})\geq
Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{\infty})=\lim_{d\rightarrow\infty}Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})=1-2p.$
(15)

In addition, if we study the aymptotic behaviour of the no-cloning bound in
equation (6), then

$C_{\mathrm{Q}}^{d}(\Lambda_{p}^{\infty})\leq\lim_{d\rightarrow\infty}\left(1-2p\frac{d+1}{d}\right)=1-2p,$
(16)

which completes the sandwich and, thus,

$C_{\mathrm{Q}}^{d}(\Lambda_{p}^{\infty})=Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{\infty})=1-2p.$
(17)

For the complementary region, $p\in[1/2,1]$, we know that this channel is
antidegradable. Therefore, the quantum channel capacity vanishes. ∎

Consequently, it can be seen that the superadditive nature of the coherent
information of the qudit depolarizing channel is lost when the dimension of
the system is let to grow indefinitely, i.e.
$\xi(\Lambda_{p}^{\infty})=0,\forall p$. This result is specially interesting
since it is an example of a channel not belonging to the degradable or
conjugate degradable classes (the depolarizing channel does not belong to
these families of channels), but showing channel coherent information with an
additive behaviour. Knowledge about quantum channels presenting additive
coherent information but not belonging to the classes of channels known to
exhibit additivity is important to obtain a better understanding about the
behaviour of quantum channel capacity. In this sense, understanding the
structure of particular channels exhibiting additive coherent information may
provide hints to understand general classes of channels with such property.

Other implication of Theorem 1 and Corollary 1 is that whenever the dimension
of the qudit depolarizing channels is big enough, the advantage that utilizing
entangled inputs may provide for protecting quantum information losses
importance. This comes from the fact that the non-additive effects of coherent
information are a result of considering input states that are entangled222Note
that this does not refer to entanglement-assistance, but to the fact that the
inputs used over sequential uses of the same channel are entangled among them.
unbounded . Since the channel coherent information of is approximately
additive for suffficiently high system dimensions, then the use of entangled
input states will provide almost no net gain. As entanglement is an expensive
resource, this significantly relaxes the required resources for optimal
quantum communication/correction over such channels.

To finish with this section, it is important to discuss what happens with the
superadditive gain whenever it is considered in qubits per channel use units.
As we discussed, we have considered qudits per channel use units since we
wanted to study the extra rate achievable due to superadditivity whenever
qudit error correction codes are considered, i.e. protecting logical qudits
using physical qudits. However, sometimes the information rate in terms of
qubits per channel use is also an important thing to study as, for example,
when logical qubits want to be encoded by means of qudits quditQubit ;
quditQubit2  or when the noise in a system composed by $n$ qubits experiences
a depolarizing channel of dimension equal to the Hilbert space of the whole
system, i.e. $d=2^{n}$. The last example would refer to noise that is
correlated, that is, noise that cannot be seen as independent noises acting
over each of the qubits of the system (See section VI for further
discussions). In this case, it is more convenient (in terms of calculations)
to redefine the superaddivity gain as
$\zeta(\mathcal{N})=C_{\mathrm{Q}}(\mathcal{N})/Q_{\mathrm{coh}}(\mathcal{N})$
for obtaining the same results as before. Note for example that it is
straightforward to see that
$\lim_{d\rightarrow\infty}\zeta(\Lambda_{p}^{d})=(C_{\mathrm{Q}}^{d}(\Lambda_{p}^{d})\log_{2}{d})/(Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})\log_{2}{d})=1$,
implying that the coherent information is additive. Unluckily, this
redefinition of the gain poses some problems since it diverges for the region
where the coherent information vanishes but the capacity is still strictly
positive. However, since Theorem 1 considers only the region of positive
coherent information and in Corollary 1 such thing is true for the whole
region, the same results are obtained. Anyway, we still consider
$\xi(\mathcal{N})$ to be the appropriate way to define the superadditivity
gain since it is able to capture the non-additivity effects of coherent
information for the whole parameter region and, thus, discussed such quantity
in terms of qudits per channel use.

##  V Relationship with other capacities

Generally speaking, quantum channels have many other quantum capacities
associated with the optimal rate at which some information theoretic task can
be performed. Therefore, in this section we discuss the superadditive gain for
the classical and private capacities of qudit depolarizing channels.

The classical capacity of a quantum channel, $C_{\chi}(\mathcal{N})$, is
defined as the asymptotically achievable rate of reliable transmission of
classical information through the noisy channel classCap ; classCap2 ;
partialorders . The classical capacity of a quantum channel is geven by the
following regularized formula

$C_{\chi}(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}\chi(\mathcal{N}^{\otimes
n}),$ (18)

where $\chi(\mathrm{N})$ is named the Holevo quantity partialorders  and it is
calculated as

$\chi(\mathcal{N})=\sup_{\rho_{XA}}I(X;B)_{\rho},$ (19)

where $\rho_{XA}$ refer to pure classical-quantum states partialorders  and
$I(A;B)_{\rho_{AB}}=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$ is the quantum
mutual information wildeQIT . The mutual information is evaluated with the
state $(\mathbb{I}_{X}\otimes\mathcal{N})(\rho_{XA})$. For arbitrary channels,
the Holevo information is superadditive, implying that the regularization in
equation (18) is necessary wildeQIT ; partialorders . However, it is well
known that the Holevo information of qudit depolarizing channels is additive,
implying that the classical capacity of such families of channels is equal to
the Holevo quantity dep1 . Therefore, the superadditivity gain of the
classical capacity vanishes for all depolarizing probabilities.

The private capacity, $P(\mathcal{N})$, referres to the maximum achievable
rate for private transmission of information over a quantum channel with an
asymptotically vanishing error rate lsd3 ; partialorders ; privateCap . Such
quantity can be evaluated as

$P(\mathcal{N})=\lim_{n\rightarrow\infty}\frac{1}{n}P_{1}(\mathcal{N}^{\otimes
n}),$ (20)

where the one-shot private information is calculated as

$P_{1}(\mathcal{N})=\sup_{\rho_{UA}}I(U;B)_{\rho}-I(U;E),$ (21)

where $\rho_{UA}$ refer to mixed classical-quantum states partialorders . The
mutual informations are evaluated for states
$(\mathbb{I}_{U}\otimes\mathcal{N})(\rho_{UA})$ and
$(\mathbb{I}_{U}\otimes\mathcal{N}^{c})(\rho_{UA})$, respectively. Private
capacity has also shown to be a superadditive quantity partialorders ;
superPriv . Importantly, the private capacity upper bounds the unassisted
quantum capacity of a quantum channel partialorders ; lamiWilde ; privUBcap ,
i.e.

$P(\mathcal{N})\geq C_{\mathrm{Q}}(\mathcal{N}),$ (22)

which also holds for the one-one shot capacities, i.e. $P_{1}(\mathcal{N})\geq
Q_{\mathrm{coh}}(\mathcal{N})$. The upper bound saturates for the class of
more capable channels (which includes less noisy and degradable channels)
watanabe .

Moreover, the no-cloning bound in equation (5) also upper bounds the private
capacity of qudit depolarizing channels. We are unaware of a manuscript
including this results and, thus, we provide a proof for it.

######  Corollary 2.

The no-cloning bound, $Q_{\mathrm{nc}}(\Lambda_{p}^{d})$ is an upper bound for
the private quantum capacity of qudit depolarizing channels, i.e.

$P(\mathcal{N})\leq
Q_{\mathrm{nc}}(\Lambda_{p}^{d})=\left(1-2p\frac{d+1}{d}\right)\log_{2}{d}.$
(23)

###### Proof.

Note that the $d$-dimensional depolarizing channel is both degradable and
antidegradable when $p=\frac{d}{2(d+1)}$. Following the rationale in depAnti ,
we can invoke Smith and Smolin’s technique of degradable extensions UB6  to
obtain the upper bound given in the corollary by noting that if the additive
extension is degradable, then its coherent information does also upper bound
the private capacity of the channel (Theorem 3 in UB6 ). ∎

Similarly as done with the quantum channel capacity, we will define the
normalized private capacity as
$P^{d}(\mathcal{N})=P(\mathcal{N})/\log_{2}{d}$. Here, the operational meaning
of this quantity will be the number of private bits that can be reliably sent
per qubit channel use. Note that, since having a higher dimensional system
implies that more classical information can be packed, by normalizing this
quantity we can more fairly compare how many extra private bits can be
achieved due to superadditivity effects when comparing different dimensional
depolarizing channels.

Using the no-cloning bound upper bound on the private capacity, we extend the
results of Theorem 1 and Corollary 1 for the private capacity of qudit
depolarizing channels.

######  Corollary 3.

The normalized private capacity superadditivity gain of qudit depolarizing
channels, $\xi^{P}(\Lambda_{p}^{d})$, in units of private bits per two-
dimensional channel use is upper bounded by
$\xi_{\mathrm{nc}}(\Lambda_{p}^{d})$, which is a monotonically decreasing
function with $d$ for any depolarizing probability, $p$, in the range
$p\in(0,p_{0}^{d_{l}})$, where $p_{0}^{d_{l}}$ is defined as in Theorem 1 with
$d_{l}$ an arbitrary positive integer higher than $2$. Therefore, the
potential gain that can be obtained from superadditive effects for the private
capacity decrease with the dimension of the system.

Moreover, the normalized private channel capacity of the $\infty$-dimensional
or bosonic depolarizing channel coincides with its quantum capacity and is
given by

$P^{d}(\Lambda_{p}^{\infty})=C_{\mathrm{Q}}^{d}(\Lambda_{p}^{\infty})=1-2p,$
(24)

with private bits per two-dimensional channel use units for $p\in[0,1/2]$ and
$0$ for $p\in[1/2,1]$.

###### Proof.

Since we are restricting the depolarizing probabilities to the range
$p\in(0,p_{0}^{d_{l}})$, we know that the superaddivity gain of the quantum
capacity, $\xi_{\mathrm{nc}}(\Lambda_{p}^{d})$, is a monotonically decreasing
function with $d$ from Theorem 1. Therefore, by taking into account the
following chain of inequalities:

$\begin{split}\xi_{\mathrm{nc}}(\Lambda_{p}^{d})&=Q_{\mathrm{nc}}^{d}(\Lambda_{p}^{d})-Q_{\mathrm{coh}}^{d}(\Lambda_{p}^{d})\geq
P^{d}(\Lambda_{p}^{d})-Q_{\mathrm{coh}}(\Lambda_{p}^{d})\\\ &\geq
P^{d}(\Lambda_{p}^{d})-P_{1}(\Lambda_{p}^{d})=\xi^{P}(\Lambda_{p}^{d}),\end{split}$
(25)

and, therefore, the upper bound for the superadditivity gain of the quantum
channel capacity upper bounds the superadditivity gain of the private capacity
of qudit depolarizing channels too. Consequently, the possible room for
increasing the achievable rate in a task of private classical communication
over a qudit depolarizing channel decrases as the dimension of the system
increases.

The second part of the corollary is straightforward from Corollary 1 due to
the fact that

$\lim_{d\rightarrow\infty}\xi_{\mathrm{nc}}(\Lambda_{p}^{d})=0,$ (26)

thus, $\xi^{P}(\Lambda_{p}^{\infty})=0$. This implies that
$P^{d}(\Lambda_{p}^{\infty})=\lim_{d\rightarrow\infty}Q_{\mathrm{nc}}(\Lambda_{p}^{\infty})=1-2p$,
for $p\in[0,1/2]$. The complementary depolarizing parameter region is trivial
from the fact that the channel is antidegradable and, thus, the private
capacity vanishes. ∎

Note that the result imlying that the quantum channel capacity and the private
capacity of the qudit depolarizing channels coincide when the dimension of the
system is let to grow indefinitely is an interesting result since, at the time
of writing, only the class of more capable (which includes the class of
degradable channels) quantum channels present such equality watanabe ;
partialorders ; singh , while depolarizing channels are not more capable.

##  VI Implications on quantum error correction with correlated depolarizing
noise

As stated before, the $d$-dimensional depolarizing channel can be used to
describe a noise map over a set of $n$ qubits for which the noise occurs in a
very correlated manner. In this sense, this channel will have a dimension that
is equal to the whole qubit system, i.e. $d=2^{n}$. In order to better
understand the correlated noise model descirbed by the $d$-dimensional
depolarizing channel for this systems, note that expression (3) can be
rewritten as kathri

$\Lambda_{p}^{d=2^{n}}(\rho)=(1-p+\frac{p}{2^{2n}})\rho+\frac{p}{2^{2n}}\sum_{\\{\bar{j},\bar{k}\\}\setminus\\{\bar{0},\bar{0}\\}}\mathrm{X}^{\bar{j}}\mathrm{Z}^{\bar{k}}\rho\mathrm{Z}^{\bar{k}}\mathrm{X}^{\bar{j}},$
(27)

where
$\mathrm{X}^{\bar{j}}=\mathrm{X}^{j_{1}}\otimes\mathrm{X}^{j_{2}}\otimes\cdots\mathrm{X}^{j_{n}}$
and
$\mathrm{Z}^{\bar{k}}=\mathrm{Z}^{k_{1}}\otimes\mathrm{Z}^{k_{2}}\otimes\cdots\mathrm{Z}^{k_{n}}$,
and $\mathrm{X},\mathrm{Z}$ are the bit and phase flip Pauli matrices,
respectively. By inspecting this expression, it can be observed that the
$d$-dimensional depolarizing channel refers to a channel in which all the non-
trivial Pauli elements of the $n$-fold Pauli group are applied in an
equiprobable manner. Therefore, this channel represents a channel in which
there exists a visible correlation in the Pauli errors that each of the qubits
of the system experiences. A visual example of why this is said to be
correlate can be seen in the fact that for this channel an error or weight $n$
would occur with a probability $p/2^{2n}$ while in an independent depolarizing
channel, such probability would be gien by the product of the probability of
error of the channel, i.e. $(p/4)^{n}$. Thus, such even is much more
infrequent for the uncorrelated depolarizing channel.

Following this logic, consider for example a rotated planar surface code with
distance $d=21$ surface  or a length $1000$ quantum turbo code qtc . Those
lengths refer to quantum error correction codes with a good performance. Note
that for the dimensions of the whole system for such codes will be $d=2^{441}$
and $2^{1000}$, respectively. Thus, those codes have humongous dimensionality.
Quantum error correction codes are presumed to operate over a large quantity
of qubits, similarly as the examples provided, hence, if the noise experienced
by those systems ha a signifcant correlation, that is, similar to the
depolarizing channel presented in Eq. (27); then there will no be possible
superadditive effects and the optimal communication/correction rates will be
achieved by random stabilizer codes.

##  VII Conclusion

In this article we have studied how the potential superadditivity effects of
the quantum capacity of the qudit depolarizing channel relate to the dimension
of the quantum systems in consideration. We proved that whenever the dimension
of the $d$-dimensional depolarizig channel increases, the potential gain in
terms of qudits per channel use decreases. This is an important result since
it implies that for very high dimensional systems the channel coherent
information and the quantum channel capacity will be very similar for the
depolarizing channel, which results in the fact that random block codes on the
typical subspace of the optimal input will be capacity achieving. We also
observed that when $\infty$-dimensional or bosonic depolarizing channels are
considered, the coherent information results to be an additive quantity,
making the superadditivity gain to vanish for all depolarizing probabilities.
We proved that the private capacity of qudit depolarizing channels behaves
similarly in the sense that its potential superadditivity gain decrases with
the dimension of the system. Asymptotically, the ability of sending private
classical information over such family of channels is also an additive
quantity and, interestingly, it coicides with the previously discussed quantum
channel capacity. We also discussed the fact that since high dimensional
depolarizing channels exhibit additive coherent information, the use of
entangled input states is not required for optimal quantum information
protection for such cases, significantly relaxing the resources required.

We have conducted this analysis of the reduction of superadditivity effects
for depolarizing channels, but we consider that this type of arguments can be
used in order to study how superadditivity behaves in high dimensions for
other quantum channels. Similar proofs for other general qudit channels could
be potentially obtained by squeezing upper bounds for their capacities with
their coherent informations. Also, it is noteworthy to state that the Clifford
twirl of a general $d$-dimensional channel results in a qudit depolarizing
channel twirldep , implying that since its capacity lower bounds the capacity
of the original channel, the results obtained here may be somehow extended.
This way it could be concluded if the gain in qudits per channel use decreases
with respect to the dimension for every quantum channel that admits a seamless
extension to $d$-dimensions, implying that seeking such effects should be
restricted for low dimensional quantum channels. Additionally, the behaviour
of other channel capacities such as the Local Operations and Classical
Communications (LOCC)-assisted quantum capacity lamiWilde ; partialorders ,
$Q_{\leftrightarrow}$, or the secret-key agreement capacity (LOCC-assisted
private capacity) lamiWilde ; partialorders , $P_{\leftrightarrow}$, can also
be studied for the family of depolarizing channels and for general maps too.
These thoughts are conjectures and are deemed as future work.

###### Acknowledgements.

This work was supported by the Spanish Ministry of Economy and Competitiveness
through the ADELE (Grant No. PID2019-104958RB-C44) and MADDIE projects (Grant
No. PID2022-137099NB-C44), by the Spanish Ministry of Science and Innovation
through the proyect Few-qubit quantum hardware, algorithms and codes, on
photonic and solid-state systems (PLEC2021-008251), by the Ministry of
Economic Affairs and Digital Transformation of the Spanish Government through
the QUANTUM ENIA project call - QUANTUM SPAIN project, and by the European
Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the Digital Spain 2025 Agenda.

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