# The Entanglement-Assisted Communication Capacity over Quantum Trajectories

Daryus Chandra, , Marcello Caleffi, , Angela Sara Cacciapuoti D. Chandra, is
with the School of Electronics and Computer Science, University of
Southampton, Southampton, SO17 1BJ, United Kingdom. E-mail:
<EMAIL_ADDRESS>Caleffi and A.S. Cacciapuoti are with the
Department of Electrical Engineering and Information Technology, University of
Naples Federico II, Naples, 80125, Italy. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Web: www.quantuminternet.it.M. Caleffi and
A.S. Cacciapuoti are also with the Laboratorio Nazionale di Comunicazioni
Multimediali, National Inter-University Consortium for Telecommunications
(CNIT), Naples, 80126, Italy.A.S Cacciapuoti, M. Caleffi and D. Chandra would
like to acknowledge the financial support of the project “Towards the Quantum
Internet: A Multidisciplinary Effort”, University of Naples Federico II. A.S
Cacciapuoti and M. Caleffi would like to to acknowledge as well the financial
support of PON project “S4E - Sistemi di Sicurezza e Protezione per l’Ambiente
Mare”.

###### Abstract

The unique and often-weird properties of quantum mechanics allow an
information carrier to propagate through multiple trajectories of quantum
channels simultaneously. This ultimately leads us to quantum trajectories with
an indefinite causal order of quantum channels. It has been shown that
indefinite causal order enables the violation of bottleneck capacity, which
bounds the amount of the transferable classical and quantum information
through a classical trajectory with a well-defined causal order of quantum
channels. In this treatise, we investigate this beneficial property in the
realm of both entanglement-assisted classical and quantum communications. To
this aim, we derive closed-form capacity expressions of entanglement-assisted
classical and quantum communication for arbitrary quantum Pauli channels over
classical and quantum trajectories. We show that by exploiting the indefinite
causal order of quantum channels, we obtain capacity gains over classical
trajectory as well as the violation of bottleneck capacity for various
practical scenarios. Furthermore, we determine the operating region where
entanglement-assisted communication over quantum trajectory obtains capacity
gain against classical trajectory and where the entanglement-assisted
communication over quantum trajectory violates the bottleneck capacity.

###### Index Terms:

quantum communications, quantum trajectory, quantum superposition, quantum
decoherence

##  I Introduction

Classical information theory has been revitalized into the quantum realm,
where it is formally known as quantum Shannon theory or quantum information
theory [1]. In both classical and quantum information theory framework, the
trajectory traversed by the information carrier is causally well-defined.
Interestingly, due to the property of quantum information, the information
carrier can traverse quantum channels in a superposition of multiple
trajectories, characterized by different causal orders. This implies that the
path traversed by the information carrier exhibits an indefinite causal order
of quantum channels [2]. Consequently, there is a novel paradigm of quantum
information theory involving the existence of indefinite causal order of
quantum channels, which leads to a new frontier research field [3].

For instance, let us observe Fig. 1. Assume that in order to transfer the
information from the source to the destination, we have to utilize two quantum
channels denoted by $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$. In the
conventional framework of quantum information theory, the information carrier
can traverse through the quantum channel $\mathcal{D}(\cdot)$ first and then
followed by quantum channel $\mathcal{E}(\cdot)$ or through the quantum
channel $\mathcal{E}(\cdot)$ first and then followed by quantum channel
$\mathcal{D}(\cdot)$. In both cases, the causal order of the quantum channels
are well-defined, it is either
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$ or
$\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$. In other words, it is said
that they have a definite causal order. Furthermore, we may refer to the path
traversed by information carrier with a definite causal order as classical
trajectory.

Figure 1: An illustration of an information carrier traverses two different
classical trajectories with a definite causal order. (a) The information
carrier traverses quantum channels
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$. (b) The information carrier
traverses quantum channels $\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$.

Interestingly, the properties of quantum information allow the information
carrier to traverse the quantum channels in the superposition of both causal
orders as illustrated in Fig. 2. Precisely, in order to transfer the
information from the source to the destination, the information carrier can
traverse both possible combinations of the classical trajectories of the
quantum channels simultaneously, i.e.
$\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$ and
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$. Consequently, the
superposition of both classical trajectories traversed by the information
carrier exhibits an indefinite causal order. We may refer to the path
traversed by an information carrier with an indefinite causal order as quantum
trajectory.

######  Remark.

The term trajectory is used in the relevant literature to denote the path –
generally assumed being constituted by a sequence of quantum channels –
traversed by the information carrier. If the quantum channels are traversed in
a well-defined causal order, the information carrier is said to propagate
through a classical trajectory. Conversely, whenever the order of the quantum
channels cannot be expressed as a well-defined causal order, the information
carrier propagates through a quantum trajectory. In other words, quantum
mechanics allows the information carrier to traverse quantum channels placed
in a quantum configuration.

As counter-intuitive as it seems, the ability of information carrier
traversing a superposition of classical trajectories has been experimentally
verified using a quantum device called quantum switch [4, 5, 6, 7, 8, 9, 10,
11, 12]. Until recently, the alluring benefits of the indefinite casual order
of quantum process have been reported for various aspects of quantum
information processing, including quantum computation [13, 14], noiseless
quantum teleportation [15], communication complexity [16], quantum resource
theory [17, 18], quantum metrology [19], discrimination of quantum process
[20], and ultimately for boosting the channel capacities of quantum and
classical communications over quantum channels [21, 22, 23, 24].

Figure 2: An illustration of a quantum trajectory traversed by information
carrier. The superposition of classical trajectories may be determined using
the control qubit $\omega$. The result is a supermap
$\mathcal{S}_{\omega}(\mathcal{E},\mathcal{D})$ denoted in pink, which
constitutes the superposition of two causal orders,
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$ denoted by red dotted lines
and $\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$ denoted by blue dashed
lines. Within the figure, $\mathcal{V}$ and $\mathcal{R}$ denote a possible
encoding-decoding pair. The whole operation will be elaborated further in
Section IV.

From a communication engineering point of view, the pivotal question is always
how to quantify the advantage that can be obtained from exploiting the
indefinite causal order of quantum channels for enhancing the quality of
classical and quantum communication. In this treatise, we aim for answering
the remaining open questions on how much advantage we can glean from the
indefinite causal order of quantum channels to improve the capacities of both
entanglement-assisted classical and quantum communication. More specifically,
within the Quantum Internet framework, multiple quantum devices are
interconnected via pre-shared entanglement for facilitating various
applications that require the exchange of classical and quantum information
amongst the quantum devices, including quantum communications [25, 26],
quantum cryptography [27], quantum sensing [28, 29], distributed quantum
computation [30, 31], blind quantum computation [32, 33], quantum-secure
direct-communication (QSDC) [34, 35, 36, 37], and quantum-secure secret-
sharing [38]. Therefore, the pre-shared entanglement can be viewed as the
primary consumable resources for enabling entanglement-assisted classical and
quantum communications within the Quantum Internet framework [39, 40, 41, 42,
31]. Ultimately, the advantage gleaned from the indefinite causal order of
quantum channels for entanglement-assisted classical and quantum communication
can be immediately extended to the aforementioned applications. Thus, the
analysis of entanglement-assisted communication over quantum trajectory will
provide a critical milestone for the development of Quantum Internet.

In this treatise, we consider quantum superdense coding protocol [43] as our
model for entanglement-assisted classical communication since a single-letter
capacity formulation can be derived for quantum Pauli channel [44]. More
specifically, quantum Pauli channel constitutes a set of quantum channel
models with various practical applications. Additionally, an entanglement-
assisted classical communication constituted by quantum superdense coding is
known to be the optimal scheme of utilizing a single use of quantum channel
and a pair of pre-shared maximally-entangled quantum state in exchange for two
classical bits [43]. Furthermore, the quantum superdense coding versus quantum
teleportation trade-off suggests that the capacity formulation of
entanglement-assisted quantum communication can be obtained directly from
their classical counterparts [45, 46].

Against this background, our contributions can be summarized as follows:

  * •

We derive the general formulation of entanglement-assisted classical
communication capacity over quantum trajectory for various scenarios involving
quantum Pauli channels.

  * •

We determine the operating region where entanglement-assisted communication
over quantum trajectory obtains capacity gain against classical trajectory.
Additionally, we also portray the operating region where entanglement-assisted
communication over quantum trajectory violates the bottleneck capacity, which
represents stringent upper-bound of communication capacity over definite
causal order.

  * •

We present the achievable capacity of entanglement-assisted quantum
communication over quantum trajectory, which is obtained via quantum
superdense versus quantum teleportation trade-off.

The rest of this treatise is organized as follows. We provide a comprehensive
comparison between our work and the state-of-the-art in Section II. We present
the quantum channel models considered as well as a brief description of
quantum superdense coding in Section III. Furthermore, we also provide the
tools required for evaluating the capacity of entanglement-assisted classical
communication based on the description of quantum channel models and quantum
superdense coding. It is followed by Section IV where we detail the formal
desription of the classical and quantum trajectory. The main results of the
entanglement-assisted classical and quantum communication capacity are
presented in Section V. Finally, we conclude our work in Section VI by also
providing several potential directions for future research.

##  II Related Works

The first demonstration of the beneficial capacity gains obtained from the
indefinite causal order of quantum channels was presented in [21], which marks
the lower bound of quantum communication capacity over quantum trajectory.
More specifically, this lower bound is represented by unassisted quantum
communication capacity obtained based on the entropy measure of the quantum
channels [47, 48]. By contrast, quantum communication capacity is upper-
bounded by two-way entanglement-assisted quantum communication capacity, whose
formulation is calculated via relative entropy entanglement of the Choi matrix
[49]. Relying on this formulation, the upper-bound of quantum communication
capacity over quantum trajectory has been recently presented in [23].
Additionally, the lower-bound of quantum communication capacity given in [23]
is tighter than that in [21]. Hence, the operating capacity of quantum
communication over quantum trajectory has been established. However, a
noticeable gap can be observed between these lower- and upper- bounds, which
may cause an inconvenience for determining the exact capacity for specific
applications, including those of entanglement-assisted-based described in
Section I. Thus, to navigate the concept of quantum trajectory closer to
practical purposes, in this treatise, we consider the one-way entanglement-
assisted – both classical and quantum – communication in our investigation,
whose capacity conceivably lies between these lower- and upper- bounds111The
term unassisted refers to a scenario where the source and the destination does
not share a pre-shared EPR pair, while the term entanglement-assisted assumes
that the source and the destination have pre-shared maximally-entangled
quantum state such as EPR pairs. The term one-way refers to a scenario where
classical communications can be conducted in one direction only, i.e. from the
source to the destination (forward direction), while the term two-way refers
to a scenario where classical communications can be performed in both
directions, i.e. from the source to the destination and vice versa (forward
and backward directions). Entanglement-assisted communication exhibits higher
capacity compared to the unassisted one and the two-way entanglement-assisted
communication may attain higher capacity than the one-way one..

Since a quantum channel can be used for transferring both classical and
quantum information, the investigation related to indefinite causal order of
quantum channel is extended to the world of classical communication. However,
currently, there is no closed-form formula for determining the exact classical
communication capacity over a quantum channel, except for the widely known
Holevo bound [50, 51], which marks the upper-bound of accessible classical
information of the unassisted classical communication over a quantum channel.
Consequently, it is also generally hard to find a single-letter formula for
establishing the classical communication capacity over an indefinite causal
order of quantum channels for a wide range channel parameters. However, for
very specific cases, such as fully-depolarizing quantum channel and quantum
entanglement-breaking channel, the unassisted classical communication
capacities over quantum trajectory have been determined [22, 52]. On the other
hand, an entanglement-assisted classical communication constituted by quantum
superdense coding is known to be the optimal scheme of utilizing a single use
of quantum channel and a pair of pre-shared maximally-entangled quantum state
in exchange for two classical bits [43]. Therefore, a single-letter formula of
entanglement-assisted classical communication capacity can be derived for a
wide range quantum channel parameters [44]. Relying on these facts,
ultimately, we derive closed-form expressions for entanglement-assisted
classical communication capacity over quantum trajectories.

##  III Preliminaries

In the classical domain, the information is conveyed by binary digit (bit),
which can carry a value of “0” or “1” at a given time. By contrast, the
information in quantum domain is carried by the quantum bit (qubit), where it
can be used to represent “0” or “1” or even the superposition of both values.
Thus, a qubit can also be used to carry classical information if the qubit is
wisely encoded. Similar to the classical domain, the transfer of quantum
information between the source and the destination is affected by the noise
characterized by the quantum channel $\mathcal{N}(\cdot)$. However,
differently from the classical domain, for a given quantum channel
$\mathcal{N}(\cdot)$, we may have several different notions of communication
capacity – the maximal amount of information may be transferred reliably under
a certain encoding and decoding procedure – which include quantum
communication capacity $Q$ as well as classical communication capacity $C$.
More precisely, the quantum communication capacity $Q$ quantifies the maximum
amount of quantum information that can be communicated from the source to the
destination over many independent uses of a quantum channel
$\mathcal{N}(\cdot)$. Similarly, the classical communication capacity $C$
quantifies the amount of classical information that can be reliably
transferred over many independent uses of a quantum channel
$\mathcal{N}(\cdot)$. However, in this case, the information is encapsulated
using a carefully selected classical-to-quantum mapping. We refer the readers
to [53] and [54] for an in-depth overview about classical and quantum
communication capacities. In this treatise, we consider the so-called
entanglement-assisted communication capacities, which differ from unassisted
communication capacities, since maximally-entangled quantum states are pre-
shared between the source and the destination before the communication is
commenced. In this section, we provide the fundamental background required for
establishing the main results of our paper.

###  III-A Quantum Channel

An arbitrary pure quantum state, denoted by $\ket{\psi}$, can be formally
expressed as a superposition of the orthogonal basis states $\\{\ket{i}\\}$ as
follows:

$\ket{\psi}=\sum_{i}\alpha_{i}\ket{i}.$ (1)

The measurement of the quantum state $\ket{\psi}$ in the orthogonal basis
collapses the superposition of the quantum state into one of the basis states,
say $\ket{i}$, with probability $|\alpha_{i}|^{2}$. Hence, the coefficients
$\\{\alpha_{i}\\}$ are subjected to the normalization condition
$\sum_{i}|\alpha_{i}|^{2}=1$. A mixed state, i.e., the statistical mixture of
multiple pure quantum states $\\{\ket{\psi_{i}}\\}$, can be described using
the so-called density matrix $\rho$, which is defined as

$\rho\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i}p_{i}\ket{\psi_{i}}\bra{\psi_{i}},$
(2)

with $\ket{\cdot}\bra{\cdot}$ denoting the outer product between two pure
quantum states.

######  Definition 1.

Quantum channel [55]. A quantum channel $\mathcal{N}(\cdot)$ is a completely-
positive trace-preserving (CPTP) map acting on arbitrary quantum states, which
can be written in the operator-sum representation as follows:

$\mathcal{N}(\rho)=\sum_{i}N_{i}\rho N_{i}^{\dagger},$ (3)

where $\\{N_{i}\\}$ denotes a set of operators – referred to as Kraus
operators – satisfying the following completeness criterion:

$\sum_{i}N_{i}^{\dagger}N_{i}=I.$ (4)

A special class of the quantum channel representation of (3) is constituted by
a quantum channel where the Kraus operators can be expressed in terms of
unitary operators $\\{U_{i}\\}$ as follows:

$\mathcal{N}(\rho)=\sum_{i}p_{i}U_{i}\rho U_{i}^{\dagger},$ (5)

where $U_{i}$ is a unitary operator satisfying $U_{i}^{\dagger}U_{i}=I$ and
$\\{p_{i}\\}_{i}$ is a probability distribution satisfying $\sum_{i}p_{i}=1$.
Hence, the relationship between the Kraus operators and the unitary operators
is given by $N_{i}=\sqrt{p_{i}}U_{i}$.

######  Definition 2.

Quantum Pauli channel [55]. A quantum channel is referred to as quantum Pauli
channel based on the following map:

$\mathcal{N}(\rho)=\sum_{i=0}^{3}p_{i}\sigma_{i}\rho\sigma_{i}^{\dagger},$ (6)

where $\sigma_{0}=I$, $\sigma_{1}=X$, $\sigma_{2}=Y$, and $\sigma_{3}=Z$ are
the Pauli matrices222In the rest of this treatise, we remove the ${\dagger}$
of (6) for quantum Pauli channels since Pauli matrices are Hermitian..

Let us now consider a joint quantum state $\rho_{AB}$ shared between two
parties $A$ and $B$, where quantum channel $\mathcal{N}_{A}(\cdot)$ affects
only the quantum state at $A$. Since we consider that the quantum state at $B$
undergoes an “error-free” identity channel, the extended mapping of
$\mathcal{N}_{A}(\cdot)$ on the joint quantum state $\rho_{AB}$ is given by

$(\mathcal{N}_{A}\otimes I_{B})(\rho_{AB})=\sum_{i}(N_{A,i}\otimes
I_{B})\rho_{AB}(N_{A,i}\otimes I_{B})^{\dagger},$ (7)

where $\\{N_{A,i}\\}$ denotes the set of Kraus operators of the quantum
channel $\mathcal{N}_{A}(\cdot)$ and $\\{N_{A,i}\otimes I_{B}\\}$ denotes the
set of extended Kraus operators.

###  III-B Quantum Superdense Coding

The main objective of this study is to evaluate the entanglement-assisted
classical and quantum communication capacity over quantum trajectories. To
this aim, we can determine the entanglement-assisted classical communication
capacity via quantum superdense coding [43], where a single use of a quantum
channel and a pair of pre-shared maximally-entangled quantum state can be used
for transmitting two classical bits. The general schematic of quantum
superdense coding over quantum channel $\mathcal{N}(\cdot)$ is shown in Fig.
3.

Figure 3: The schematic of quantum superdense coding over noisy quantum
channel $\mathcal{N}(\cdot)$. A single use of quantum channel $\mathcal{N}$
and an EPR pair $\ket{\Phi^{+}}_{AB}$ can be used for transferring two
classical bits $\overline{b}=b_{0}b_{1}$.

The quantum superdense coding protocol is commenced by pre-sharing a
maximally-entangled quantum state between Alice $(A)$ and Bob $(B)$, which is
constituted by the following EPR pair:

$\ket{\Phi^{+}}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right)_{AB},$
(8)

where the subscripts $A$ and $B$ indicate that the first qubit is held by $A$,
while the second is by $B$. Let us assume that the pre-sharing step is error-
free since multiple copies of EPR pairs can be prepared and hence a quantum
entanglement distillation protocol can be invoked to eliminate the quantum
errors [56].

A two-bit vector $\overline{b}=b_{0}b_{1}$ is used for applying a
controlled-$Z$ and controlled-$X$ operations defined by
$(Z)^{b_{0}}(X)^{b_{1}}$. To elaborate a little further, if the classical bit
$b_{0}=1$, a $Z$ gate is applied to the qubit of the EPR pair on $A$ side,
otherwise, an identity gate is applied. Similarly, if the classical bit
$b_{1}=1$, an $X$ gate is applied, otherwise, an identity gate is applied.
Consequently, the classical-to-quantum mapping of $\overline{b}=b_{0}b_{1}$ to
EPR pairs is given by

$\displaystyle b_{0}b_{1}$
$\displaystyle=00\rightarrow\ket{\Phi^{+}}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right)_{AB},$
$\displaystyle b_{0}b_{1}$ $\displaystyle=01\rightarrow
X_{A}\ket{\Phi^{+}}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{01}+\ket{10}\right)_{AB}=\ket{\Psi^{+}}_{AB},$
$\displaystyle b_{0}b_{1}$ $\displaystyle=10\rightarrow
Z_{A}\ket{\Phi^{+}}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{00}-\ket{11}\right)_{AB}=\ket{\Phi^{-}}_{AB},$
$\displaystyle b_{0}b_{1}$ $\displaystyle=11\rightarrow
X_{A}Z_{A}\ket{\Phi^{+}}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{01}-\ket{10}\right)_{AB}=\ket{\Psi^{-}}_{AB}.$
(9)

Next, the qubit of the EPR pair at $A$ side is sent through a quantum channel
$\mathcal{N}(\cdot)$ as shown in Fig. 3. Assume that we have a quantum Pauli
channel $\mathcal{N}(\cdot)$, which means that the quantum channel inflicts
the bit-flip $(X)$, phase-flip $(Z)$, as well as the simultaneous bit-flip and
phase-flip $(Y)$ errors with the probability of $p_{x}$, $p_{z}$, and $p_{y}$,
respectively. Therefore, the transition probability of every possible
combinations of $\overline{b}=b_{0}b_{1}$ due to $X$, $Z$, and $Y$ errors is
depicted in Fig. 4(a). Finally, as both of the qubits of the EPR pair are now
at $B$ side, a Bell-state measurement is conducted to recover the two-bit
vector $\widehat{b}=\widehat{b}_{0}\widehat{b}_{1}$, which is the corrupted
version of $\overline{b}=b_{0}b_{1}$. With this, the quantum superdense coding
protocol over quantum channel $\mathcal{N}(\cdot)$ is completed.

###  III-C The Capacity of Entanglement-Assisted Classical Communication

Evaluating the capacity of quantum superdense coding protocol can be
reformulated as evaluating the capacity of quaternary discrete classical
communication. To obtain a single-letter formula of its capacity, first, we
provide the general definition of classical communication capacity.

######  Definition 3.

Classical communication capacity [57]. The classical communication capacity is
defined by

$C=\max_{P(x)}I(X;Y)=\max_{P(x)}\\{H(Y)-H(Y|X)\\},$ (10)

where $I(X;Y)$ is the mutual information between random variables $X$ and $Y$,
$P(x)$ is the probability distribution of the source emitting symbol $x\in X$,
$H(Y)$ is the entropy of random variable $Y$, $H(Y|X)$ is the conditional
entropy of random variable $Y$ conditioned by $X$.

Furthermore, the capacity of (10) is also equivalent to

$\displaystyle C$ $\displaystyle=\log_{2}M-\sum_{i,j}\displaystyle
P(x_{j})H(Y|X=x_{j})$ $\displaystyle=\log_{2}M-\sum_{j}\displaystyle
P(x_{j})\sum_{i}P(y_{i}|x_{j})\log_{2}\frac{1}{\displaystyle P(y_{i}|x_{j})}.$
(11)

Since we assume that we have an equiprobable source for the symbols $x\in X$
and $P(y_{i}|x_{j})$ is time invariant, the expression of (11) can be further
simplified to

$C=\log_{2}M+\sum_{i}P(y_{i}|x)\log_{2}P(y_{i}|x),$ (12)

where $P(y_{i}|x)$ is the transition probability of $y_{i}$ for any $x\in X$
characterized by the channel333The transition probability of $P(y_{i}|x)$ for
quantum Pauli channel is known to be symmetric for any $x\in X$..

In order to glean a clearer idea, let us proceed with an example. Consider the
quantum depolarizing channel $\mathcal{N}(\cdot)$ affecting a single-qubit
having the density matrix $\rho$ as follows:

$\mathcal{N}(\rho)=(1-p)\rho+\frac{p}{3}\left({X}\rho{X}+{Y}\rho{Y}+{Z}\rho{Z}\right),$
(13)

where $p$ is the depolarizing probability. Now, we have to make the connection
between the single-qubit depolarizing channel of (13) and the transition
probability of Fig. 4. Utilizing the extended mapping of the quantum channel
of (7) and by considering $\rho_{AB}=\ket{\Phi^{+}}_{AB}\bra{\Phi^{+}}_{AB}$,
the effect of a single-qubit depolarizing channel on an EPR pair $\rho_{AB}$
is given by

$\displaystyle(\mathcal{N}_{A}\otimes I_{B})(\rho_{AB})$
$\displaystyle=\left(1-p\right)\ket{\Phi^{+}}\bra{\Phi^{+}}+\frac{p}{3}\ket{\Psi^{+}}\bra{\Psi^{+}}$
$\displaystyle+\frac{p}{3}\ket{\Psi^{-}}\bra{\Psi^{-}}+\frac{p}{3}\ket{\Phi^{-}}\bra{\Phi^{-}}.$
(14)

Based on (14), we obtain the transition probability $p(y_{i}|x)$ as follows:

$p(y_{i}|x)=\left\\{\begin{array}[]{ll}1-p,&\text{for}\ i=0\\\
p_{x}=p_{y}=p_{z}=\dfrac{p}{3},&\text{for}\ i=1,2,3\end{array},\right.$ (15)

where $i$ represents the decimal representation of binary vector
$\overline{b}=b_{0}b_{1}$. Finally, substituting the value $M=4$ and
$p(y_{i}|x)$ of (15) into (12), we obtain the entanglement-assisted classical
communication capacity in terms of $p$ as follows [44]:

$C_{E,s}=2+(1-p)\log_{2}(1-p)+p\log_{2}\frac{p}{3}.$ (16)

For a single-qubit depolarizing channel, the capacity can be derived for both
symbol-based and bit-based classical communication. The symbol-based capacity
is already given in (16) based on the transition probability of $b_{0}b_{1}$
in Fig. 4(a). By contrast, the bit-based capacity can be determined by summing
the individual capacity of $b_{0}$ and $b_{1}$ based on the transition
probability of Fig. 4(b), where we obtain

$C_{E,b}=2+\left(2-\dfrac{4p}{3}\right)\log_{2}\left(1-\frac{2p}{3}\right)+\dfrac{4p}{3}\log_{2}\frac{2p}{3}.$
(17)

As a benchmark, we provide the unassisted classical communication capacity of
a single-qubit depolarizing channel. Luckily, for a single-qubit depolarizing
channel, the capacity always correspond to the Von Neumann measurements [58],
where in terms of $p$ is given by

$C=1+\left(1-\dfrac{2p}{3}\right)\log_{2}\left(1-\frac{2p}{3}\right)+\dfrac{2p}{3}\log_{2}\frac{2p}{3}.$
(18)

Notice that the unassisted classical capacity is exactly half of the bit-based
entanglement-assisted classical capacity444With reference to the entanglement-
assisted capacity, this statement is true since the cost of distributing
entanglement is not taken into account, as generally assumed in the
theoretical quantum information theory framework..

Figure 4: The transition probability of $\overline{b}=b_{0}b_{1}$ over quantum
Pauli channels $\mathcal{N}(\cdot)$, which will be utilized for deriving the
entanglement-assisted (a) symbol-based and (b) bit-based classical
communication capacity. Compared to (a), where the transition probability is
derived jointly for $\overline{b}=b_{0}b_{1}$, each of the transition
probabilities in (b) is derived for $b_{0}$ and $b_{1}$.

######  Remark.

The inherent inter-bit correlation within the symbols makes the symbol-based
capacity slightly higher than the sum of individual bit-based capacity.
Consequently, the entanglement-assisted classical communication capacity is
also slightly higher than the sum of individual unassisted classical
communication capacity.

Figure 5: The entanglement-assisted classical communication capacity over a
single use of quantum depolarizing channel. The plots are based on (16), (17),
and (18). EA stands for entanglement-assisted.

We portray the symbol-based capacity of (16), the bit-based capacity of (17),
and the unassisted capacity of (18) in Fig. 5. Observe that all the capacities
are equal to $0$ when $p=0.75$. This specific point is associated with the
fully-depolarizing quantum channel where we have $\mathcal{N}(\rho)=I/2$ for
any arbitrary $\rho$. Consequently, there is no classical information can be
transmitted, either by employing entanglement-assisted or unassisted, through
a fully-depolarizing quantum channel.

Finally, following the reasoning we utilized for deriving the entanglement-
assisted classical communication capacity for single-qubit depolarizing
channel, we may also directly derive the entanglement-assisted classical
communication capacity for the general quantum Pauli channel of (6), which is
given by

$C_{E}=2+p_{0}\log_{2}p_{0}+p_{1}\log_{2}p_{1}+p_{2}\log_{2}p_{2}+p_{3}\log_{2}p_{3}.$
(19)

For the rest of treatise, we are going to use the entanglement-assisted
capacity of (19) for evaluating the bottleneck capacity of the quantum
channels arranged in a well-defined causal order, which we will elaborate in
the next section.

##  IV The Classical and Quantum Trajectory

As we have briefly alluded in Section I, the unique properties of quantum
information allow the information carrier to traverse multiple classical
trajectories simultaneously. In this section, we provide the formal
mathematical description of the classical and quantum trajectory of two
quantum channels. More specifically, a classical trajectory of two quantum
channels is characterized by a well-defined causal order of either
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$ or
$\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$. By contrast, a quantum
trajectory of two quantum channels is characterized by the superposition of
two classical trajectories $\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$
and $\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$ implying that the
trajectory exhibits an indefinite causal order of the quantum channels
$\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$.

###  IV-A Classical Trajectory

Consider two quantum channels $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$ as
follows:

$\mathcal{D}(\rho)=\sum_{i}D_{i}\rho
D_{i}^{\dagger},\quad\mathcal{E}(\rho)=\sum_{j}E_{j}\rho E_{j}^{\dagger}.$
(20)

The resultant channel $\mathcal{S}(\cdot)$ of two quantum channels over a
classical trajectory with a definite causal order of
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$ is formulated as

$\mathcal{S}(\mathcal{D},\mathcal{E})(\rho)=\sum_{i,j}W_{i,j}\rho
W_{i,j}^{\dagger},$ (21)

where the Kraus operators are given by

$W_{i,j}=E_{j}D_{i}.$ (22)

The amount of transferable information over a classical trajectory of two
quantum channels $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$ is upper-
bounded by the so-called bottleneck capacity, which applies for both classical
and quantum communications as well as for both unassisted and entanglement-
assisted communications.

######  Definition 4.

The bottleneck capacity [59, 23]. The communication capacity over classical
trajectory of two quantum channels is upper-bounded by

$C_{\text{B}}=\min\\{C(\mathcal{D}),C(\mathcal{E})\\},$ (23)

where $C(\mathcal{D})$ and $C(\mathcal{E})$ are the capacities of the
individual quantum channels $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$,
respectively. When $C(\mathcal{D})$ and $C(\mathcal{E})$ represent the
entanglement-assisted communication capacities of $\mathcal{D}(\cdot)$ and
$\mathcal{E}(\cdot)$, respectively, the notation $C_{\text{E,B}}$ is used for
portraying the associated bottleneck capacity.

In this treatise, we consider $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$ to
be general quantum Pauli channels of (6). Therefore, we may evaluate the
individual $C(\mathcal{D})$ and $C(\mathcal{E})$ using the entanglement-
assisted classical communication capacity for quantum Pauli channels of (19)
to obtain the bottleneck capacity $C_{\text{E,B}}$.

###  IV-B Quantum Trajectory

Given two quantum channels $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$, the
resultant quantum channel $\mathcal{S}(\cdot)$ over quantum trajectory is
formulated as [21]

$\mathcal{S}_{\omega}(\mathcal{D},\mathcal{E})(\rho)=\sum_{i,j}W_{i,j}(\rho\otimes\omega)W_{i,j}^{\dagger},$
(24)

where $\omega$ is the control qubit and the Kraus operators are given by

$W_{i,j}=E_{j}D_{i}\otimes\ket{0}\bra{0}+D_{i}E_{j}\otimes\ket{1}\bra{1}.$
(25)

The formulation of the Kraus operators over quantum trajectory implies that we
may create a superposition of classical trajectories using the control qubit
$\omega$. More specifically, when $\omega=\ket{0}\bra{0}$, the information
carrier of the quantum state $\rho$ traverses the classical trajectory of
$\mathcal{D}(\cdot)\rightarrow\mathcal{E}(\cdot)$. By contrast, when
$\omega=\ket{1}\bra{1}$, the information carrier of the quantum state $\rho$
traverses the classical trajectory of
$\mathcal{E}(\cdot)\rightarrow\mathcal{D}(\cdot)$. For instance, we may create
an equal superposition of both classical trajectories by initializing
$\omega=\ket{+}\bra{+}$, which means that the quantum state $\rho$ is
traversed through both classical trajectories simultaneously, which results in
an indefinite causal order of quantum channels. Thus, throughout this
treatise, we assume that the control qubit is always initialized in the
quantum state of $\omega=\ket{+}\bra{+}$ since it gives us the capability of
detecting the superposition of causal orders from anti-commuting Kraus
operators and ultimately provides us with the highest possible capacity gain
[21].

More specifically, let us assume that $D_{1}$ and $E_{1}$ are two anti-
commuting Kraus operators, i.e. $D_{1}E_{1}=-E_{1}D_{1}$. Thus, the Kraus
operators of Eq (25) can be expressed as follows:

$\displaystyle W_{1,1}$
$\displaystyle=E_{1}D_{1}\otimes\ket{0}\bra{0}-E_{1}D_{1}\otimes\ket{1}\bra{1}$
$\displaystyle=E_{1}D_{1}\otimes(\ket{0}\bra{0}-\ket{1}\bra{1})=E_{1}D_{1}\otimes
Z.$ (26)

Therefore, the action of Kraus operators $W_{1,1}$ of (26) on the initial
quantum state of $\rho\otimes\ket{+}\bra{+}$ is given by

$\displaystyle W_{1,1}\left(\rho\otimes\ket{+}\bra{+}\right)W_{1,1}^{\dagger}$
$\displaystyle=(E_{1}D_{1})\rho(E_{1}D_{1})^{\dagger}\otimes\left(Z\ket{+}\bra{+}Z\right)$
$\displaystyle=(E_{1}D_{1})\rho(E_{1}D_{1})^{\dagger}\otimes\ket{-}\bra{-}.$
(27)

According to this result, when we measure the quantum state $\ket{-}\bra{-}$
on the control qubit, we can infer that a superposition of causal orders from
two anti-commuting Kraus operators has taken place. Thus, we can utilize the
measurement result to our advantage for improving the performance of classical
and quantum communication.

######  Remark.

If the control qubit is initialized in the quantum state of
$\omega=\ket{+}\bra{+}$, the superposition of causal orders from two anti-
commuting Kraus operators transforms the control qubit into $\ket{-}\bra{-}$.

Based on the formal description of the quantum trajectory in (24), indeed we
require an additional auxiliary qubit $\omega$ to control the superposition of
the causal orders of the quantum channels $\mathcal{D}(\cdot)$ and
$\mathcal{E}(\cdot)$. Intuitively, we have the inclination to make a
comparison between the advocated scheme presented in this treatise to another
auxiliary-qubits assisted scheme, such as quantum error-correction codes [60,
61]. Nevertheless, viewing the control qubit $\omega$ of (24) in the same
light as the auxiliary qubits in quantum error-correction codes can be very
problematic. More specifically, the auxiliary qubits in quantum error-
correction codes are encoded together with the logical qubits and sent through
the quantum channels carrying a certain amount of information. Consequently,
the incorporation of auxiliary qubits in quantum error-correction codes
increases the total number of quantum channel uses. By contrast, the
utilization of the auxiliary qubit $\omega$ of (24) does not increase the
number of quantum channel uses. In fact, as shown in [18], the encoding
operation of (24) must be considered as a non-side-channel generating
operation, since the control qubit $\omega$ does not carry – or embed in any
way – the information from the source to the destination. The initial state of
the control qubit $\omega$ is fixed as part of the placement and thus, it is
deemed inaccessible to the sender for encoding information. Having said that,
we refer enthusiastic readers to [18] for the full discourse on quantum
resource theories.

##  V The Entanglement-Assisted Capacity over Classical and Quantum Trajectory

We employ the formulation of classical and quantum trajectories of (22) and
(25) and incorporate them into (7) for devising the transition probability
required for calculating the classical communication capacity of (12).
Consider the following quantum Pauli channels $\mathcal{D}(\cdot)$ and
$\mathcal{E}(\cdot)$:

$\mathcal{D}(\rho)=p_{0}\rho+p_{1}X\rho X+p_{2}Y\rho Y+p_{3}Z\rho Z,$ (28)
$\mathcal{E}(\rho)=q_{0}\rho+q_{1}X\rho X+q_{2}Y\rho Y+q_{3}Z\rho Z.$ (29)

Our results of the entanglement-assisted classical communication capacity over
classical and quantum trajectories are summarized in the following
propositions.

######  Proposition 1.

The entanglement-assisted classical communication capacity $C_{\text{E,C}}$ of
the two arbitrary quantum Pauli channels in (28) and (29) over a classical
trajectory is given by

$C_{\text{E,C}}=2+{A}_{0}\log_{2}{A}_{0}+{A}_{1}\log_{2}{A}_{1}+{A}_{2}\log_{2}{A}_{2}+{A}_{3}\log_{2}{A}_{3},$
(30)

where ${A}_{0}=p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3}$,
${A}_{1}=p_{0}q_{1}+p_{1}q_{0}+p_{2}q_{3}+p_{3}q_{2}$,
${A}_{2}=p_{0}q_{2}+p_{2}q_{0}+p_{3}q_{1}+p_{1}q_{3}$, and
${A}_{3}=p_{0}q_{3}+p_{3}q_{0}+p_{1}q_{2}+p_{2}q_{1}$.

###### Proof:

Please refer to Appendix A. ∎

######  Proposition 2.

The entanglement-assisted classical communication capacity $C_{\text{E,Q}}$ of
the two arbitrary quantum Pauli channels of (28) and (29) over a quantum
trajectory is given by

$\displaystyle C_{\text{E,Q}}$
$\displaystyle=2+H(\alpha)+{A}_{0}\log_{2}{A}_{0}$
$\displaystyle+{A}_{1}^{+}\log_{2}{A}_{1}^{+}+{A}_{2}^{+}\log_{2}{A}_{2}^{+}+{A}_{3}^{+}\log_{2}{A}_{3}^{+}$
$\displaystyle+{A}_{1}^{-}\log_{2}{A}_{1}^{-}+{A}_{2}^{-}\log_{2}{A}_{2}^{-}+{A}_{3}^{-}\log_{2}{A}_{3}^{-},$
(31)

where $\alpha=p_{\ket{+}}={A}_{0}+{A}_{1}^{+}+{A}_{2}^{+}+{A}_{3}^{+}$,
${A}_{0}=p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3}$,
${A}_{1}^{+}=p_{0}q_{1}+p_{1}q_{0}$, ${A}_{2}^{+}=p_{0}q_{2}+p_{2}q_{0}$,
${A}_{3}^{+}=p_{0}q_{3}+p_{3}q_{0}$, ${A}_{1}^{-}=p_{2}q_{3}+p_{3}q_{2}$,
${A}_{2}^{-}=p_{3}q_{1}+p_{1}q_{3}$, ${A}_{3}^{-}=p_{1}q_{2}+p_{2}q_{1}$, and
$H(\alpha)$ is the binary entropy of $\alpha$ defined by
$-\alpha\log_{2}\alpha-(1-\alpha)\log_{2}(1-\alpha)$.

###### Proof:

Please refer to Appendix B. ∎

Observe from (30) and (31), the following inequality always holds:
$C_{\text{E,Q}}\geq C_{\text{E,C}}$. The equality holds when
${A}_{1}^{-}={A}_{2}^{-}={A}_{3}^{-}=0$ implying ${A}_{0}^{+}={A}_{0}$,
${A}_{1}^{+}={A}_{1}$, ${A}_{2}^{+}={A}_{2}$, ${A}_{3}^{+}={A}_{3}$, and
$H(\alpha)=0$. It means that the advantage of quantum trajectory can only be
observed when the coefficient pair of $\\{p_{i},q_{j}|i,j=1,2,3\\}$ for non-
commuting Pauli matrices is non zero. More specifically, the two classical
trajectories of two non-commuting Pauli matrices, for example $XZ=-ZX$,
transforms the control qubit $\omega=\ket{+}\bra{+}$ into
$\omega=\ket{-}\bra{-}$, which consequently increases the entanglement-
assisted classical communication capacity compared to that of classical
trajectories.

######  Remark.

In order to glean the advantage of quantum trajectory for increasing the
entanglement-assisted classical communication capacity of two arbitrary
quantum Pauli channels $\mathcal{D}(\cdot)$ and $\mathcal{E}(\cdot)$, the
coefficient pair of $\\{p_{i},q_{j}|i,j=1,2,3\\}$ for non-commuting Pauli
matrices has to be non-zero.

###  V-A Classical Communication Capacity

In the following subsections, we provide several derivative results of
Proposition 1 and 2 for various types of quantum Pauli channels, which are
widely considered in the practical applications of quantum communications.
More specifically, we consider the combination of bit-flip and phase flip
quantum channels, quantum entanglement-breaking channels, as well as quantum
depolarizing channels. Moreover, the experimental implementation of the
indefinite causal orders of the aforementioned quantum channels can be found
in [8, 10, 11].

####  V-A1 Quantum Bit-Flip and Phase-Flip Channels

We consider two quantum Pauli channels constituted by bit-flip and phase-flip
quantum channels. A quantum bit-flip channel is defined as follows:

$\mathcal{D}(\rho)=(1-p)\rho+pX\rho X,$ (32)

where the Kraus operators are given by $D_{1}=\sqrt{1-p}I$ and
$D_{2}=\sqrt{p}X$. A quantum phase-flip channel is defined as follows:

$\mathcal{E}(\rho)=(1-q)\rho+qZ\rho Z,$ (33)

where the Kraus operators are given by $E_{1}=\sqrt{1-q}I$ and
$E_{2}=\sqrt{q}Z$. Based on the Kraus operators of the bit-flip and phase-flip
quantum channels of (32) and (33), respectively, and based on the formulation
of Kraus operators for classical trajectory of (22), we obtain the following
corollary.

######  Corollary 1.

The entanglement-assisted classical communication capacity $C_{\text{E,C}}$ of
bit-flip and phase-flip quantum channels over classical trajectory is given by

$\displaystyle C_{\text{E,C}}$
$\displaystyle=2+\left[(1-p)(1-q)\right]\log_{2}\left[(1-p)(1-q)\right]$
$\displaystyle+\left[p(1-q)\right]\log_{2}\left[p(1-q)\right]+\left[(1-p)q\right]\log_{2}\left[(1-p)q\right]$
$\displaystyle+pq\log_{2}pq.$ (34)

###### Proof:

By substituting $p_{0}=1-p$, $q_{0}=1-q$, $p_{1}=p$, $q_{3}=q$, and the rest
of the coefficients equal to $0$ into (30) of Proposition 1, we obtain the
result in (34). ∎

Similarly, based on the formulation of Kraus operators for quantum trajectory
of (25), we also obtain the following corollary.

######  Corollary 2.

The entanglement-assisted classical communication capacity $C_{\text{E,Q}}$ of
bit-flip and phase-flip quantum channels over quantum trajectory is given by

$\displaystyle C_{\text{E,Q}}$
$\displaystyle=2+H(\alpha)+\left[(1-p)(1-q)\right]\log_{2}\left[(1-p)(1-q)\right]$
$\displaystyle+\left[p(1-q)\right]\log_{2}\left[p(1-q)\right]+\left[(1-p)q\right]\log_{2}\left[(1-p)q\right]$
$\displaystyle+pq\log_{2}pq,$ (35)

where $\alpha=p_{\ket{+}}=1-pq$.

###### Proof:

By substituting $p_{0}=1-p$, $q_{0}=1-q$, $p_{1}=p$, $q_{3}=q$, and the rest
of the coefficients equal to $0$ into (31) of Proposition 2, we obtain the
result in (35). ∎

Figure 6: The entanglement-assisted classical communication capacity for the
combination of bit-flip and phase-flip quantum channels. These plots are based
on (34) and (35).

To demonstrate the advantage of the quantum trajectory compared to the
classical trajectory, let us consider a scenario where we have $p=q$. We plot
the entanglement-assisted capacity of (34) and (35) in a scenario of $p=q$ in
Fig. 6. In addition to these results, we include the bottleneck capacity of
the two quantum channels. Observe that the capacity of the resultant channel
over classical trajectory indeed cannot violate the stringent bottleneck
capacity. However, the capacity over quantum trajectory violates the
bottleneck capacity in the region of $0.618<p<1$. This violation of bottleneck
capacity is induced by the combination of two causal orders of two non-
commuting Pauli matrices from the Kraus operators of the quantum channels,
$XZ=-ZX$. This combination transforms the control qubit $\omega$, which is
initialized in the quantum state of $\ket{+}\bra{+}$, into quantum state of
$\ket{-}\bra{-}$. Consequently, this gives us a perfect entanglement-assisted
classical communication with a probability of $p^{2}$ since every time we
measure $\ket{-}\bra{-}$ on the control qubit, we can apply $XZ$ operation to
perfectly recover the quantum state of information qubit $\rho$. This is very
important to highlight that the measurement of the control qubit $\omega$ does
not give us any information about which causal order the information carrier
has traversed, or in other words it means the measurement does not collapse
the superposition of the classical trajectories. Instead, what the measurement
tells us is if the quantum channels have inflicted two non-commuting Pauli
matrices in two different causal orders, which can only occur when the
information carrier has actually traversed both classical trajectories
simultaneously.

######  Remark.

The result in Corollary 2 implies that the indefinite causal order of the
quantum channels allows us to violate the bottleneck capacity, which
constrains the capacity of classical and quantum communication traversed over
a classical trajectory with a definite causal order of quantum channels.

For the sake of demonstration, the results presented in in Fig. 6 is valid for
a scenario of $p=q$. However, the results presented in Corollary 1 and 2 are
actually general for a wider range of $p$ and $q$. To enrich our discussion
and to explore the advantages gleaned by the indefinite causal order of
quantum channels, we introduce two additional metrics, namely the capacity
gain and the bottleneck violation based on the following definitions.

######  Definition 5.

The capacity gain $G$ is defined as the difference between the entanglement-
assisted classical communication capacity over quantum trajectory and that of
classical trajectory, which depicts the amount of additional classical
information that can be sent by exploiting the indefinite causal order of
quantum channels. Formally, the capacity gain $G$ can be expressed as

$G=C_{\text{E,Q}}-C_{\text{E,C}},$ (36)

where $C_{\text{E,Q}}$ is the entanglement-assisted classical communication
capacity over quantum trajectory presented in (31) of Proposition 2, while
$C_{\text{E,C}}$ is that of over classical trajectory obtained from (30) of
Proposition 1.

######  Definition 6.

The bottleneck violation $V$ is defined as the non-negative gain obtained by
entanglement-assisted classical communication capacity over quantum trajectory
against the bottleneck capacity. The bottleneck violation $V$ depicts the
amount of capacity gain which cannot be attained through any definite causal
order of quantum channels, which is formally expressed as

$V=\max\\{0,C_{\text{E,Q}}-C_{\text{E,B}}\\},$ (37)

where $C_{\text{E,Q}}$ is the entanglement-assisted classical communication
capacity over quantum trajectory presented in (31) of Proposition 2, while
$C_{\text{E,B}}$ is the bottleneck capacity of entanglement-assisted classical
communication based on (23) of Definition 23.

(a) Capacity gain ($G$)

(b) Bottleneck violation ($V$)

Figure 7: (a) The entanglement-assisted classical capacity gain of the bit-
flip and phase-flip quantum channels over quantum trajectory against classical
trajectory. (b) The violation of bottleneck capacity due to the indefinite
causal order of quantum channels.

We portray the capacity gain for the bit-flip and phase-flip quantum channels
over quantum trajectory in Fig. 7(a), while the bottleneck violation in Fig.
7(b). Observe that based on the color map in Fig. 7(a), we obtain the benefit
of capacity gain at all range values of $0<p,q<1$. However, the significant
capacity gain is more prominent in the region of $0.2\leq p,q\leq 1$ portrayed
by the dark red color area. The maximum gain observed is $G=1$, which is
attained for the values of $(p,q)$ satisfying $pq=0.5$ for $p,q\leq 1$. We are
also interested in the region where we can observe the violation of the
bottleneck capacity as well as the associated magnitude. In Fig. 7(b), we
observe that the violation of bottleneck capacity occurs in the region of
$p+q\geq 1$ for $p,q\leq 1$. The maximum violation observed is $V=1$, which is
attained for the values of $(p,q)=(0.5,1)$ and $(p,q)=(1,0.5)$. This can be
verified directly because the bottleneck capacities for $(p,q)=(0.5,1)$ and
$(p,q)=(1,0.5)$ are $C_{\text{E,B}}=1$. By contrast, for these given values
$p$ and $q$, we have $C_{\text{E,Q}}=2$, which means a perfect entanglement-
assisted classical communication over quantum trajectory.

####  V-A2 Quantum Entanglement-Breaking Channels

Any quantum channel $\mathcal{N}(\cdot)$ is said to be entanglement breaking
if it transforms a maximally-entangled quantum state $\rho_{AB}$, for example
$\ket{\Phi^{+}}_{AB}\bra{\Phi^{+}}_{AB}$, into a mixture of two separable
quantum states $\rho_{A}$ and $\rho_{B}$. Formally, a quantum entanglement-
breaking channel is defined as follows [62]:

$(\mathcal{N}_{A}\otimes
I_{B})(\rho_{AB})=\sum_{i}p_{i}\left(\rho_{A,i}\otimes\rho_{B,i}\right).$ (38)

An example of quantum entanglement-breaking channels that can be written as a
linear combination of Pauli matrices is given by

$\mathcal{N}(\rho)=\frac{1}{2}\left(X\rho X+Y\rho Y\right).$ (39)

Now, let us consider two partially entanglement-breaking channels, where the
first quantum channel is formulated as

$\mathcal{D}(\rho)=(1-p)X\rho X+pY\rho Y,$ (40)

where the Kraus operators are given by $D_{1}=\sqrt{1-p}X$ and
$D_{2}=\sqrt{p}Y$, where $p=1/2$ gives us the quantum entanglement-breaking
channel of (39). Similarly, the second quantum channel is formulated as

$\mathcal{E}(\rho)=(1-q)X\rho X+qY\rho Y,$ (41)

where the Kraus operators are given by $E_{1}=\sqrt{1-q}X$ and
$E_{2}=\sqrt{q}Y$. Based on the Kraus operators of the partially entanglement-
breaking channels of (40) and (41) and based on the formulation of Kraus
operators for classical trajectory of (22), we have the following corollary.

######  Corollary 3.

The entanglement-assisted classical communication capacity $C_{\text{E,C}}$ of
two partially entanglement-breaking channels over classical trajectory is
given by

$\displaystyle C_{\text{E,C}}$
$\displaystyle=2+\left(1-p-q+2pq\right)\log_{2}\left(1-p-q+2pq\right)$
$\displaystyle+\left(p+q-2pq\right)\log_{2}\left(p+q-2pq\right).$ (42)

###### Proof:

By substituting $p_{1}=1-p$, $q_{1}=1-q$, $p_{2}=p$, $q_{2}=q$, and the rest
of the coefficients equal to $0$ into (30) of Proposition 1, we obtain the
result in (42). ∎

Similarly, based on the formulation of Kraus operators for quantum trajectory
of (25), we also have the following corollary.

######  Corollary 4.

The entanglement-assisted classical communication capacity $C_{\text{E,Q}}$ of
two partially entanglement-breaking channels over quantum trajectory is given
by

$C_{\text{E,Q}}=2,$ (43)

for every value of $p\in[0,1]$ and $q\in[0,1]$.

###### Proof:

By substituting $p_{1}=1-p$, $q_{1}=1-q$, $p_{2}=p$, $q_{2}=q$ and the rest of
the coefficients equal to $0$ into (31) of Proposition 2, we obtain the result
in (43). ∎

Figure 8: The entanglement-assisted classical capacity for two identical
partially entanglement-breaking channels. The plots are based on (42) and
(43).

To provide a clearer picture, we portray the entanglement-assisted capacity of
(42) and (43) in a scenario where we have $p=q$ in Fig. 8. Once again, we also
include the bottleneck capacity of the two partially entanglement-breaking
channels. Here, we observe an interesting phenomenon where we can always
achieve a perfect entanglement-assisted classical communication over quantum
trajectory, which demonstrates a full violation of bottleneck capacity for
every value of $p\in(0,1)$. Compared to the result presented in Subsection
V-A1, the Kraus operators of the partially entanglement-breaking channels
contain two non-commuting Pauli matrices given by $XY=-YX$. Therefore, the two
different causal orders of two non-commuting Pauli matrices can always be
detected by the control qubit $\omega$, since it transforms the initialized
$\omega=\ket{+}\bra{+}$ into $\omega=\ket{-}\bra{-}$. By contrast, the
remaining combinations of the Pauli matrices given by $XX=YY=I$, leave the
control qubit $\omega$ unchanged.

(a) Capacity gain ($G$)

(b) Bottleneck violation ($V$)

Figure 9: (a) The entanglement-assisted classical capacity gain of partially
entanglement-breaking channels over quantum trajectory against classical
trajectory. (b) The violation of bottleneck capacity due to the indefinite
causal order of quantum channels.

We portray the capacity gain for two partially entanglement-breaking channels
over quantum trajectory in Fig. 9(a), while the bottleneck violation in Fig.
9(b). Since we always attain a perfect entanglement-assisted classical
communication over quantum trajectory for two partially entanglement-breaking
channels, the magnitude of capacity gain, which may be observed from the
dominantly dark red color area portrayed in Fig. 9(a), is significantly higher
than that of bit-flip and phase-flip quantum channels. Consequently, the
benefit from capacity gain from Fig. 9(a) is directly translated to the
capacity violation displayed by the color map in Fig. 9(b). In this case, we
have shown that the quantum trajectory of two entanglement-breaking channels
is capable of achieving the full violation of bottleneck capacity for every
value $p\in(0,1)$ and $q\in(0,1)$, as described in (43) of Corollary 4.

######  Remark.

The result in Corollary 4 implies that the indefinite causal order of the
quantum channels allows us to always enable a perfect entanglement-assisted
classical communication over two partially entanglement-breaking channels.

####  V-A3 Quantum Depolarizing Channels

Let us now consider two quantum depolarizing channels. For the first quantum
channel, we have the following description:

$\mathcal{D}(\rho)=(1-p)\rho+\frac{p}{3}\left({X}\rho{X}+{Y}\rho{Y}+{Z}\rho{Z}\right),$
(44)

where the Kraus operators are given by $D_{1}=\sqrt{1-p}I$,
$D_{2}=\sqrt{\frac{p}{3}}X$, $D_{3}=\sqrt{\frac{p}{3}}Y$, and
$D_{4}=\sqrt{\frac{p}{3}}Z$. Similarly, for the second quantum channel, we
have

$\mathcal{E}(\rho)=(1-q)\rho+\frac{q}{3}\left({X}\rho{X}+{Y}\rho{Y}+{Z}\rho{Z}\right),$
(45)

where the Kraus operators are given by $E_{1}=\sqrt{1-q}I$,
$E_{2}=\sqrt{\frac{q}{3}}X$, $E_{3}=\sqrt{\frac{q}{3}}Y$, and
$E_{4}=\sqrt{\frac{q}{3}}Z$. Based on the Kraus operators of the quantum
depolarizing channels of (44) and (45) and based on the formulation of Kraus
operators for classical trajectory of (22), we arrive at the following
corollary.

######  Corollary 5.

The entanglement-assisted classical communication capacity $C_{\text{E,C}}$ of
two quantum depolarizing channels over classical trajectory is given by

$\displaystyle C_{\text{E,C}}$
$\displaystyle=2+\left(1-p-q+\dfrac{4pq}{3}\right)\log_{2}\left(1-p-q+\dfrac{4pq}{3}\right)$
$\displaystyle+\left(p+q-\dfrac{4pq}{3}\right)\log_{2}\left(\dfrac{3p+3q-4pq}{9}\right).$
(46)

###### Proof:

By substituting $p_{0}=q_{0}=1-p$, $p_{1}=p_{2}=p_{3}=q_{1}=q_{2}=q_{3}=p/3$,
into (30) of Proposition 1, we obtain the result in (46). ∎

Similarly, based on the the formulation of Kraus operators for quantum
trajectory of (25), we also arrive at the following corollary.

######  Corollary 6.

The entanglement-assisted classical communication capacity $C_{\text{E,Q}}$ of
two quantum depolarizing channels over quantum trajectory is given by

$\displaystyle C_{\text{E,Q}}$ $\displaystyle=2+H(\alpha)$
$\displaystyle+\left(1-p-q+\frac{4pq}{3}\right)\log_{2}\left(1-p-q+\frac{4pq}{3}\right)$
$\displaystyle+(p+q-2pq)\log_{2}\left(\dfrac{p+q-2pq}{3}\right)+\frac{2pq}{3}\log_{2}\frac{2pq}{9},$
(47)

where $\alpha=p_{\ket{+}}=1-{2pq}/{3}$.

###### Proof:

By substituting $p_{0}=1-p$, $q_{0}=1-q$, $p_{1}=p_{2}=p_{3}=p/3$,
$q_{1}=q_{2}=q_{3}=q/3$, into (31) of Proposition 2, we obtain the result in
(47). ∎

Figure 10: The entanglement-assisted classical communication capacity for two
identical quantum depolarizing channels. The plots are based on (46) and (47).

As a special case, we portray the entanglement-assisted capacity of (46) and
(47) in a scenario where we have $p=q$ in Fig. 10. In Subsection V-A1 and
V-A2, the bottleneck capacities are always strictly positive. Instead, here we
have a condition where we have a zero-capacity at $p=0.75$. At this point, we
have the so-called fully-depolarizing quantum channel, which we have mentioned
briefly in Section III-C. It means that no classical information can be sent
by the means of entanglement-assisted classical communication through the
individual quantum channel. Clearly, this is also the case for the classical
trajectory since it cannot violate the bottleneck capacity. Interestingly,
observe in Fig. 10 that we have a non-zero capacity for entanglement-assisted
communication utilizing two fully-depolarizing quantum channels over quantum
trajectory. More specifically, we have $C_{\text{E,Q}}=0.204$ for $p=0.75$.
The ability to enable a non-zero capacity communication using two zero-
capacity quantum channels over quantum trajectory is referred to as the causal
activation [22, 21, 24, 18].

(a) Capacity gain ($G$)

(b) Bottleneck violation ($V$)

Figure 11: (a) The entanglement-assisted classical capacity gain of two
quantum depolarizing channels over quantum trajectory against classical
trajectory. (b) The violation of bottleneck capacity due to the indefinite
causal order of quantum channels.

######  Remark.

The entanglement-assisted classical communication capacity over quantum
trajectory is always strictly positive, which means that we can always send
classical information through quantum channels even when we cannot send any
classical information through the individual quantum channel.

We portray the capacity gain for two quantum depolarizing channels over
quantum trajectory in Fig. 11(a), while the bottleneck violation in Fig.
11(b). Observe that based on the color map in Fig. 11(a), the capacity gain
gleaned at range values of $0<p,q<1$ are relatively modest compared to that of
bit-flip and phase-flip quantum channels. The capacity gain are more prominent
in the region of $0.5\leq p,q\leq 1$ denoted by the light blue to dark red
color area. The maximum gain observed is $G=0.918$, which is attained for the
value of $(p,q)=(1,1)$. In Fig. 11(b), we also observe modest violation of
bottleneck capacity. Nonetheless, the bottleneck violation can be observed for
a wide range of $p$ and $q$ values. The maximum violation observed is
$V=0.528$, which is also attained for the value of $(p,q)=(1,1)$.

######  Remark.

The quantum trajectory, which leads to indefinite causal order of quantum
channels, enables the violation of bottleneck capacity, implying that the
phenomenon inducing this violation cannot be obtained by any process
exhibiting a well-defined causal order.

###  V-B Quantum Communication Capacity

In the previous subsections, we have provided a thorough analysis on the
entanglement-assisted classical communication capacity via classical and
quantum trajectories. In this section, we extend the analysis to the
entanglement-assisted quantum communication. In [44] and [45], the
relationship between the entanglement-assisted capacity of classical
communication $(C_{E})$ and that of quantum capacity $(Q_{E})$ is readily
given by

$Q_{E}=\frac{C_{E}}{2},$ (48)

based on quantum superdense coding versus quantum teleportation trade-off,
which dictates that a pair of pre-shared maximally-entangled quantum state and
a single use of quantum channel $\mathcal{N}(\cdot)$ can be exchanged for a
single qubit or two classical bits. Consequently, by exploiting (48), the
results in Proposition 1 and 2 for entanglement-assisted classical
communication over classical and quantum trajectories can be extended directly
to the entanglement-assisted quantum communications as shown in the following
corollaries.

######  Corollary 7.

The entanglement-assisted quantum communication capacity $Q_{\text{E,Q}}$ of
the two arbitrary quantum Pauli channels of (28) and (29) over classical
trajectory is given by

$\displaystyle Q_{\text{E,C}}$
$\displaystyle=\dfrac{C_{\text{E,C}}}{2}=1+\frac{1}{2}\left[{A}_{0}\log_{2}{A}_{0}+{A}_{1}\log_{2}{A}_{1}\right.$
$\displaystyle+\left.{A}_{2}\log_{2}{A}_{2}+{A}_{3}\log_{2}{A}_{3}\right].$
(49)

###### Proof:

The proof follows directly by accounting for Proposition 1 and (48). ∎

######  Corollary 8.

The entanglement-assisted quantum communication capacity $Q_{\text{E,Q}}$ of
the two arbitrary quantum Pauli channels of (28) and (29) over quantum
trajectory is given by

$\displaystyle Q_{\text{E,Q}}=\dfrac{C_{\text{E,Q}}}{2}$
$\displaystyle=1+\frac{1}{2}\left[H(\alpha)+{A}_{0}\log_{2}{A}_{0}+{A}_{1}^{+}\log_{2}{A}_{1}^{+}\right.$
$\displaystyle+\left.{A}_{2}^{+}\log_{2}{A}_{2}^{+}+{A}_{3}^{+}\log_{2}{A}_{3}^{+}+{A}_{1}^{-}\log_{2}{A}_{1}^{-}\right.$
$\displaystyle+\left.{A}_{2}^{-}\log_{2}{A}_{2}^{-}+{A}_{3}^{-}\log_{2}{A}_{3}^{-}\right].$
(50)

###### Proof:

The proof follows directly by accounting for Proposition 2 and (48). ∎

It may be interesting to see the relationship between the capacity bounds of
quantum communication via quantum trajectory presented in [23] to the results
presented here. Let us take the example of bit-flip and phase flip quantum
channels of (32) and (33). Based on (50) in Corollary 8, we obtain the
entanglement-assisted quantum communication capacity $Q_{\text{E,Q}}$ of the
bit-flip and phase-flip quantum channels over quantum trajectory as follows:

$\displaystyle Q_{\text{E,Q}}$
$\displaystyle=1+\dfrac{1}{2}\left[H(\alpha)+(1-p)^{2}\log_{2}(1-p)^{2}\right.$
$\displaystyle+\left.(2p-2p^{2})\log_{2}(p-p^{2})+p^{2}\log_{2}p^{2}\right],$
(51)

for $p=q$, where $\alpha=1-p^{2}$. Meanwhile, the lower bound of the quantum
communication capacity over quantum trajectory is given by [23]

$Q_{LB}=p^{2}+\max\\{0,1-p^{2}-2H(p)+H(p^{2})\\}.$ (52)

This lower bound is obtained from the entropy measure of the quantum Pauli
channels, which represents the capacity of unassisted quantum communication
[47, 48]. By contrast, quantum communication capacity over quantum trajectory
is upper-bounded by the capacity of two-way entanglement-assisted quantum
communication [49]. In case of bit-flip and phase-flip quantum channels, this
upper bound is given by [23]

$Q_{UB}=1-\left[(1-p)H(p)\right],$ (53)

which is obtained from the relative entropy of entanglement of the Choi matrix
[49].

Figure 12: The capacity of quantum communication for the combination of bit-
flip and phase-flip quantum channels.

We plot the entanglement-assisted capacity of (51) and the corresponding
quantum communication capacity bounds of (52) and (53) in Fig. 12. Again, we
include the bottleneck entanglement-assisted quantum communication capacity of
the bit-flip and phase-flip quantum channels, which is given by

$Q_{\text{E,B}}=1-\dfrac{H(p)}{2},$ (54)

for $p=q$. The result presented in Fig. 12 suggests the confirmation of the
following relationship between various capacities of quantum communication
over quantum trajectory: $Q_{LB}\leq Q_{\text{E,Q}}\leq Q_{UB}$. Additionally,
the violation of bottleneck capacity for entanglement-assisted quantum
communication can be observed within the range of $0.618<p<1$.

(a) Capacity gain ($G$)

(b) Bottleneck violation ($V$)

Figure 13: (a) The entanglement-assisted quantum capacity gain of the bit-flip
and phase-flip quantum channels over quantum trajectory against classical
trajectory. (b) The violation of bottleneck capacity due to the indefinite
causal order of quantum channels.

Following the same line of investigation presented for entanglement-assisted
classical communication, here we also provide the results of capacity gain and
bottleneck violation of entanglement-assisted quantum communication over
quantum trajectory. More specifically, let us observe Fig. 13. We plot the
capacity gain for the bit-flip and phase-flip quantum channels over quantum
trajectory in Fig. 13(a) and the bottleneck violation in Fig. 13(b). We can
observe in Fig. 13(a) that we obtain the capacity gain at all range values of
$0<p,q<1$. Similar to the results for entanglement-assisted classical
communication, the capacity gain is more profound in the region of $0.2\leq
p,q\leq 1$ portrayed by the dark red color area. The maximum gain observed is
$G=0.5$, which is attained for the values of $(p,q)$ satisfying $pq=0.5$ for
$p,q\leq 1$. Notice that the maximum gain attained in entanglement-assisted
quantum communication is half of its classical counterpart due to the trade-
off of (48). In Fig. 13(b), we observe that the violation of bottleneck
capacity occurs in the region of $p+q\geq 1$ for $p,q\leq 1$ with the maximum
violation observed is $V=0.5$, which is attained for the values of
$(p,q)=(0.5,1)$ and $(p,q)=(1,0.5)$. This can be verified directly since the
bottleneck capacities for $(p,q)=(0.5,1)$ and $(p,q)=(1,0.5)$ are given by
$Q_{\text{E,B}}=0.5$. Meanwhile, for the values $(p,q)=(0.5,1)$ and
$(p,q)=(1,0.5)$, we have $Q_{\text{E,Q}}=1$, which means we can achieve a
perfect entanglement-assisted quantum communication over quantum trajectory.

To conclude this section, we would like to highlight some potential practical
advantages that may be gained based on the resultant $G$ and $V$ gleaned from
the indefinite causal order of the quantum channels. As an instance, in the
area of quantum error-correction codes, the capacity gain may be utilized for
improving the quantum coding rate or the quantum bit error ratio of the
designed codes for mitigating the quantum errors imposed by the quantum Pauli
channels. Another prospective application may be constituted by quantum-secure
direct-communication, where the indefinite causal order of the quantum
channels can be utilized for increasing the transmission rate. Finally, the
resultant capacity gain may also be used for increasing the fidelity of a
remote two-qubit quantum gate in a distributed quantum computing framework.
However, it is important to note that the application aspect of the quantum
configuration resulting in indefinite causal order within the quantum
communication framework is still in its infancy and thus, further
investigation is mandatory. It is also crucial to underline that the
motivation for studying the advantage of the indefinite causal order is not
directly a practical one, since such motivation is related to explore how the
theory of quantum communication – as we know it – would be affected by the
possibility of combining quantum channels in a superposition of causal orders,
as clarified in [18].

##  VI Conclusions and Future Works

In this contribution, we have presented the general formulation of both
entanglement-assisted classical and quantum communication capacities over
quantum trajectory. Our analysis included several examples from the family of
quantum Pauli channels. Furthermore, we have explicitly portrayed the region
in which the indefinite causal structure of quantum channels violate the
bottleneck capacity constraining the capacity of quantum channels with a
definite causal structure. Additionally, for two fully-depolarizing quantum
channels, we have witnessed the causal activation of the zero capacity quantum
channels induced by quantum trajectory.

For our next works, we may consider extending our analysis for more than two
quantum channels [63, 64, 65, 66]. In this treatise, the control qubit
$\omega$ exhibits only one degree of freedom and therefore only one
superposition of two classical trajectories can be included. When the number
of quantum channels increases $(N>2)$, we can no longer use a qubit as our
control qubit $\omega$, but a qudit with $d$ quantum level. The critical point
arising from this scenario is the ultimate or asymptotic limit of capacity
gain can be attained by exploiting indefinite causal order of infinitely many
quantum channels. Finally, we are also interested in investigating the effect
of the indefinite causal order of quantum channels in a more realistic
scenario of quantum superdense coding, where the quantum channels affect both
the forward and backward communication between $A$ and $B$ [67].

##  Appendix A Proof of Proposition 1

Let us consider two quantum Pauli channels $\mathcal{D}(\cdot)$ and
$\mathcal{E}(\cdot)$. By substituting the Kraus operators description of
quantum Pauli channels, into the Kraus operators formulation for classical
trajectory given in (22), we obtain the resultant quantum channel
$\mathcal{S}(\cdot)$ as follows:

$\displaystyle\mathcal{S}(\mathcal{D},\mathcal{E})(\rho)$
$\displaystyle=(p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3})\rho$
$\displaystyle+(p_{0}q_{1}+p_{1}q_{0}+p_{2}q_{3}+p_{3}q_{2})X\rho X$
$\displaystyle+(p_{0}q_{2}+p_{2}q_{0}+p_{3}q_{1}+p_{1}q_{3})Y\rho Y$
$\displaystyle+(p_{0}q_{3}+p_{3}q_{0}+p_{1}q_{2}+p_{2}q_{1})Z\rho Z.$ (55)

Based on the resultant quantum channel $\mathcal{S}(\cdot)$ of (55), we obtain
the following transition probability:

$p(y_{i}|x)=\left\\{\begin{array}[]{ll}p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3},&\text{for}\
i=0\\\ p_{0}q_{1}+p_{1}q_{0}+p_{2}q_{3}+p_{3}q_{2},&\text{for}\ i=1\\\
p_{0}q_{2}+p_{2}q_{0}+p_{3}q_{1}+p_{1}q_{3},&\text{for}\ i=2\\\
p_{0}q_{3}+p_{3}q_{0}+p_{1}q_{2}+p_{2}q_{1},&\text{for}\
i=3\end{array}.\right.$ (56)

Substituting the transition probability of (56) into (12), we obtain the
entanglement-assisted classical capacity over classical trajectory of (30)
presented in Proposition 1. Hence, we complete our proof.

##  Appendix B Proof of Proposition 2

By substituting the Kraus operators description of quantum Pauli channels,
into the Kraus operators formulation for quantum trajectory given in (25), we
obtain the resultant channel $\mathcal{S}_{\omega}(\cdot)$ as follows:

$\displaystyle\mathcal{S}(\mathcal{D},\mathcal{E})_{\omega}(\rho)$
$\displaystyle=\big{[}(p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3})\rho$
$\displaystyle+(p_{0}q_{1}+p_{1}q_{0})X\rho X+(p_{0}q_{2}+p_{2}q_{0})Y\rho Y$
$\displaystyle+(p_{0}q_{3}+p_{3}q_{0})Z\rho Z\big{]}\otimes\ket{+}\bra{+}$
$\displaystyle+\big{[}(p_{2}q_{3}+p_{3}q_{2})X\rho
X+(p_{3}q_{1}+p_{1}q_{3})Y\rho Y$ $\displaystyle+(p_{1}q_{2}+p_{2}q_{1})Z\rho
Z\big{]}\otimes\ket{-}\bra{-}.$ (57)

Observe that we may measure the control qubit $\omega$ in the Hadamard basis
$\\{\ket{+}\bra{+},\ket{-}\bra{-}\\}$. The measurement of the control qubit
$\omega$ collapses the resultant channel $\mathcal{S}_{\omega}(\cdot)$ into
either $\mathcal{S}_{\ket{+}}(\cdot)$ or $\mathcal{S}_{\ket{-}}(\cdot)$
depending on the result. Let us denote $p_{\ket{+}}$ as the probability of
measuring the control qubit in $\ket{+}\bra{+}$ state and $p_{\ket{-}}$ as the
probability of measuring the control qubit in $\ket{-}\bra{-}$ state.
Therefore, the entanglement-assisted classical communication capacity
$C_{\text{E,Q}}$ over quantum trajectory can be generalized as

$C_{\text{E,Q}}=p_{\ket{+}}C_{E}(\mathcal{S}_{\ket{+}})+p_{\ket{-}}C_{E}(\mathcal{S}_{\ket{-}}),$
(58)

where $C_{E}(\mathcal{S}_{\ket{+}})$ and $C_{E}(\mathcal{S}_{\ket{-}})$ are
the capacity of quantum channel $\mathcal{S}_{\ket{+}}(\cdot)$ and quantum
channel $\mathcal{S}_{\ket{-}}(\cdot)$, respectively. Explicitly, based on
(57), we measure $\omega=\ket{+}\bra{+}$ with a probability of
$p_{\ket{+}}=p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3}+p_{0}q_{1}+p_{1}q_{0}+p_{0}q_{2}+p_{2}q_{0}+p_{0}q_{3}+p_{3}q_{0}$,
which consequently collapses the resultant channel
$\mathcal{S}_{\omega}(\cdot)$ to

$\displaystyle\mathcal{S}_{\ket{+}}(\mathcal{D},\mathcal{E})(\rho)$
$\displaystyle=\dfrac{1}{p_{\ket{+}}}\big{[}(p_{0}q_{0}+p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3})\rho$
$\displaystyle+(p_{0}q_{1}+p_{1}q_{0})X\rho X+(p_{0}q_{2}+p_{2}q_{0})Y\rho Y$
$\displaystyle+(p_{0}q_{3}+p_{3}q_{0})Z\rho Z\big{]}.$ (59)

By contrast, we measure $\omega=\ket{-}\bra{-}$ with a probability of
$p_{\ket{-}}=p_{2}q_{3}+p_{3}q_{2}+p_{3}q_{1}+p_{1}q_{3}+p_{1}q_{2}+p_{2}q_{1}$,
which consequently collapses the resultant channel
$\mathcal{S}_{\omega}(\cdot)$ to

$\displaystyle\mathcal{S}_{\ket{-}}(\mathcal{D},\mathcal{E})(\rho)$
$\displaystyle=\dfrac{1}{p_{\ket{-}}}\big{[}(p_{2}q_{3}+p_{3}q_{2})X\rho X$
$\displaystyle+(p_{3}q_{1}+p_{1}q_{3})Y\rho Y+(p_{1}q_{2}+p_{2}q_{1})Z\rho
Z\big{]}.$ (60)

Based on (59) and (60), we can devise the transition probability $p(y_{i}|x)$
and determine the capacity of $C_{E}(\mathcal{S}_{\ket{+}})$ and
$C_{E}(\mathcal{S}_{\ket{-}})$ accordingly. Substituting them into (58), we
obtain the entanglement-assisted classical communication capacity of quantum
two quantum Pauli channels over quantum trajectory of (31) presented in
Proposition 2. Hence, we complete our proof.

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|  Daryus Chandra (S’15, M’20) received the M.Eng. degree in electrical
engineering from Universitas Gadjah Mada, Indonesia, in 2014 and the Ph.D.
degree in electronics and electrical engineering from University of
Southampton, UK, in 2020. He was a research fellow with the Future
Communications Laboratory, University of Naples Federico II, Italy. Currently,
he is a research fellow with the Next-Generation Wireless Research Group,
University of Southampton, UK. His research interests include classical and
quantum error correction codes, quantum information, and quantum
communications.
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|  Marcello Caleffi (M’12, SM’16) received the M.S. degree with the highest
score (summa cum laude) in computer science engineering from the University of
Lecce, Lecce, Italy, in 2005, and the Ph.D. degree in electronic and
telecommunications engineering from the University of Naples Federico II,
Naples, Italy, in 2009. Currently, he is associate professor with the DIETI
Department, University of Naples Federico II. From 2010 to 2011, he was with
the Broadband Wireless Networking Laboratory at Georgia Institute of
Technology, Atlanta, as visiting researcher. In 2011, he was also with the
NaNoNetworking Center in Catalunya (N3Cat) at the Universitat Politecnica de
Catalunya (UPC), Barcelona, as visiting researcher. Since July 2018, he held
the Italian national habilitation as Full Professor in Telecommunications
Engineering. His work appeared in several premier IEEE Transactions and
Journals, and he received multiple awards, including best strategy award, most
downloaded article awards and most cited article awards. Currently, he serves
as associate editor/associate technical editor for IEEE Communications
Magazine, IEEE Trans. on Wireless Communications, IEEE Trans. on Quantum
Engineering and IEEE Communications Letters. He served as Chair, TPC Chair,
Session Chair, and TPC Member for several premier IEEE conferences. In 2016,
he was elevated to IEEE Senior Member and in 2017 he has been appointed as
Distinguished Lecturer from the IEEE Computer Society. In December 2017, he
has been elected Treasurer of the Joint IEEE VT/ComSoc Chapter Italy Section.
In December 2018, he has been appointed member of the IEEE New Initiatives
Committee.
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|  Angela Sara Cacciapuoti (M’10, SM’16) is an Associate Professor at the
University of Naples Federico II, Italy. Since July 2018, she held the
national habilitation as “Full Professor” in Telecommunications Engineering.
Her work has appeared in first tier IEEE journals and she has received
different awards, including the elevation to the grade of IEEE Senior Member
in 2016. Currently, Angela Sara serves as Area Editor for IEEE Communications
Letters, and as Editor/Associate Editor for the journals: IEEE Trans. on
Communications, IEEE Trans. on Wireless Communications, IEEE Trans. on Quantum
Engineering, IEEE Network and IEEE Open Journal of Communications Society. She
was a recipient of the 2017 Exemplary Editor Award of the IEEE Communications
Letters. In 2016 she has been an appointed member of the IEEE ComSoc Young
Professionals Standing Committee. From 2017 to 2018, she has been the Award
Co-Chair of the N2Women Board. From 2017 to 2020, she has been the Treasurer
of the IEEE Women in Engineering (WIE) Affinity Group of the IEEE Italy
Section. In 2018, she has been appointed as Publicity Chair of the IEEE ComSoc
Women in Communications Engineering (WICE) Standing Committee. And since 2020,
she is the vice-chair of WICE. Her current research interests are mainly in
Quantum Communications, Quantum Networks and Quantum Information Processing.
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