# Direct Quantum Communications in the Presence of Realistic Noisy
Entanglement

Daryus Chandra, , Angela Sara Cacciapuoti, , Marcello Caleffi, , Lajos Hanzo
D. Chandra and L. Hanzo are with the School of Electronics and Computer
Science (ECS), University of Southampton, Southampton, SO17 1BJ, UK. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Web: www-mobile.ecs.soton.ac.uk.M.
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.A.S. Cacciapuoti and M. Caleffi 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”.L. Hanzo would like
to acknowledge the financial support of the Engineering and Physical Sciences
Research Council projects EP/P034284/1 and EP/P003990/1 (COALESCE) as well as
of the European Research Council’s Advanced Fellow Grant QuantCom (Grant No.
789028)

###### Abstract

To realize the Quantum Internet, quantum communications require pre-shared
entanglement among quantum nodes. However, both the generation and the
distribution of the maximally-entangled quantum states are inherently
contaminated by quantum decoherence. Conventionally, the quantum decoherence
is mitigated by performing the consecutive steps of quantum entanglement
distillation followed by quantum teleportation. However, this conventional
approach imposes a long delay. To circumvent this impediment, we propose a
novel quantum communication scheme relying on realistic noisy pre-shared
entanglement, which eliminates the sequential steps imposing delay in the
standard approach. More precisely, our proposed scheme can be viewed as a
direct quantum communication scheme capable of improving the quantum bit error
ratio (QBER) of the logical qubits despite relying on realistic noisy pre-
shared entanglement. Our performance analysis shows that the proposed scheme
offers competitive QBER, yield, and goodput compared to the existing state-of-
the-art quantum communication schemes, despite requiring fewer quantum gates.

###### Index Terms:

Quantum communication, quantum entanglement, quantum error-correction, quantum
stabilizer codes, Quantum Internet

##  I Introduction

Enabling quantum communications among quantum devices within the Quantum
Internet [1, 2, 3] will ultimately lead to various groundbreaking
applications. These radically new applications do not necessarily have
classical counterparts [4] and they are not limited to the already well-known
secure classical communications, blind computation, distributed quantum
computing, and quantum secret sharing [5, 2, 6, 7, 8]. Naturally, the reliable
transfer of quantum information is sought across the quantum network relying
on quantum channels [9, 10]. However, the quantum channels inevitably impose
deleterious quantum decoherence, which inflicts quantum errors [11, 12]. In
the classical domain, the errors imposed by the communication channels can be
mitigated using error-control codes [13]. The key idea of error-control codes
is to attach appropriately designed redundancy to the information bits by an
encoding process, which is utilized by the decoder to correct a certain number
of errors. However, observing and/or copying quantum information is not
allowed in the quantum domain due to the no-cloning theorem and the quantum
measurement postulate. This motivates the carefully constructed design of
quantum error-correction codes (QECCs) [14, 15, 16, 17].

QECCs constitute potent error mitigation techniques required for tackling the
deleterious effect of quantum decoherence. Similar to the classical error-
correction codes, QECCs rely on attaching redundant qubits to the logical
qubits to provide additional information that can be exploited for quantum
error-correction during the decoding step [18]. Interestingly, the whole
encoding and decoding process can be completed without actually observing the
physical qubits and thus, preserving the integrity of the quantum information
conveyed by the physical qubits. In the quantum domain, the redundant qubits
can be in form of auxiliary qubits initialized to the $\ket{0}$ or $\ket{+}$
states, or in the form of pre-shared maximally-entangled quantum states, which
are normally assumed to be noise-free. For a two-qubit system, the maximally-
entangled quantum states are represented by the Einstein-Podolsky-Rosen (EPR)
pairs. The state-of-the-art studies typically assume that the EPR pairs are
pre-shared among quantum devices within the quantum networks before any
quantum communication protocol is initiated. Hence, the EPR pairs can be
considered as the primary resource within the Quantum Internet [10].

Having pre-shared entanglement offers several beneficial features for QECCs.
Firstly, it can be used for conveniently transforming some powerful classical
error-correction codes that do not satisfy the symplectic criterion111A pair
of classical error-correction codes having parity-check matrices $H_{x}$ and
$H_{z}$ can be transformed to a quantum error-correction code if they satisfy
$H_{x}H_{z}^{T}+H_{z}H_{x}^{T}=0\mod 2$. into their quantum counterparts [19,
20, 21]. Secondly, they can also be used for increasing the error-correction
capability of quantum stabilizer codes (QSCs) [22]. Indeed, there are several
types of QECCs in the literature that exploit pre-shared entanglement, such as
entanglement-assisted QSCs [23], entanglement-aided canonical codes [24], as
well as teleportation-based QECCs [25]. However, in all the above-mentioned
schemes, the pre-shared entanglement is considered to be noise-free.

In a scenario having realistic noisy pre-shared entanglement, QECCs are
invoked for quantum entanglement distillation (QED) [26, 27, 28, 29, 17],
which is followed by quantum teleportation [30] for transferring the quantum
information. QED can be viewed as a specific application of QECCs, where
several copies of noisy pre-shared EPR pairs are discarded to obtain fewer but
less noisy EPR pairs. In this approach, QED and quantum teleportation have to
be performed subsequently, which typically imposes excessive practical delay.
Additionally, state-of-the-art QED schemes will always have some residual
quantum noise, unless infinitely many noisy pre-shared EPR pairs are discarded
during QED. Unfortunately, this residual quantum noise is carried over to the
logical qubits during the quantum teleportation process and hence it affects
the integrity of the quantum information. In this treatise, we refer to this
specific quantum communication scheme relying on the consecutive steps of QED
and quantum teleportation as QED+QT.

Another QECC-aided technique operating in the presence of noisy pre-shared
entanglement was introduced in [31], which we will refer to as quantum
stabilizer codes using imperfect pre-shared entanglement (QSC-IE). Compared to
the QED-based schemes which apply the QECCs locally on the pre-shared EPR
pairs split between the transmitter and the receiver, the scheme presented in
[31] requires that the pre-shared portion of the EPR pairs at the transmitter
side has to be sent to the receiver in order to apply stabilizer measurements
to both qubits of the EPR pairs. Consequently, a relatively high number of the
two-qubit quantum gates – as exemplified by quantum controlled-NOT (CNOT) gate
– are required for performing these measurements. Furthermore, by reasoning
for a fixed number of pre-shared EPR pairs and logical qubits, the QSC-IE
scheme demands for a higher number of quantum channel uses, to apply
stabilizer measurements to both qubits of the EPR pairs.

Having said that, in this treatise, we propose a novel solution for achieving
a reliable quantum communication, despite using noisy pre-shared entanglement.
Firstly, we eliminate the idealized simplifying assumption of having noise-
free pre-shared EPR pairs. Secondly, we devise a scheme for avoiding the
undesired delay imposed by the consecutive steps of QED and quantum
teleportation in conventional twin-step QED+QT schemes. By contrast, our
proposed scheme can be viewed as a single-step direct quantum communication
scheme, which exploits the quantum noise experienced by the pre-shared EPR
pairs for improving the reliability of quantum communications by encoding the
logical qubits directly with the aid of noisy pre-shared EPR pairs. As it will
become more evident later in this treatise, our proposal may be deemed
philosophically reminiscent of training-based equalization techniques in
classical communications, which rely on pilot sequences for estimating the
channel and then eliminating its impairments. Thirdly, we also eliminate the
necessity of performing stabilizer measurements on both qubits of the pre-
shared EPR pairs for the sake of reducing: (i) the number of quantum gates
required to achieve reliable quantum communications and (ii) the uses of the
considered quantum channel. Indeed, by relying solely on the local
measurements of the pre-shared EPR pairs, our proposal significantly reduces
the number of the required two-qubit quantum gates as well as the number of
quantum channel uses by reasoning with the same number of pre-shared EPR
pairs. Table I boldly and explicitly contrasts our proposed scheme to the
existing schemes of amalgamating pre-shared entanglement and QECCs. Naturally,
our proposal is also suitable for the scenario of noise-free pre-shared
entanglement, similarly to the EA-QECC schemes. In Section VI, we formally
show that our proposed scheme outperforms the state-of-the-art.

TABLE I: Comparison of our proposed scheme with the state-of-the-art schemes. Scheme | Noisy Pre-Shared Entanglement | Direct Communication | Stabilizer Measurement
---|---|---|---
QED+QT | Yes | No | Yes
QSC-IE | Yes | Yes | Yes
EA-QECC | No | Yes | Yes
Proposed | Yes | Yes | No

Our novel contributions can be summarized as follows:

  1. 1.

We propose a new scheme for achieving reliable quantum communications despite
relying on noisy pre-shared entanglement. More specifically,

  2. 2.

We carry out the performance analysis of the proposed scheme for both error-
detection and error-correction based schemes over quantum depolarizing
channels. The results show that the proposed scheme offers competitive
performance in terms of its qubit error ratio, yield, and goodput despite
requiring fewer quantum gates than the existing state-of-the-art schemes.

  3. 3.

In case of noise-free pre-shared entanglement, the proposed scheme outperforms
even the existing entanglement-assisted quantum stabilizer codes.

The rest of the treatise is organized as follows. In Section II, we commence
by presenting the quantum communication model. In Section III, we detail the
explicit formulation of our proposed scheme for direct noiseless quantum
communication over noisy pre-shared entanglement. In Section IV, we exemplify
our scheme proposed for error-detection, while in Section V, we conceive its
counterpart for error-correction. In Section VI, we show the suitability of
our proposal for quantum computing applications. Finally, we conclude in
Section VII by also discussing some future research directions.

##  II System Model

Figure 1: The quantum communication model considered for our proposed scheme.

As discussed in [10], both entanglement generation and distribution are the
key for the Quantum Internet. The specific “location” of the device
implementing these functionalities – a.k.a. the entanglement generator and
distributor – varies among the different schemes and solutions [10]. However,
there is a general agreement in the literature that the employment of the so-
called “at both end-points” scheme is vital for the Quantum Internet by
enabling on-demand communication capabilities at the quantum nodes. According
to the “at both end-points” scheme, the entanglement generator and distributor
is embedded within both the transmitter and the receiver [10]. In this light,
we consider the quantum communication model depicted in Fig. 1. The model
includes a transmitter $(A)$, a receiver $(B)$, the entanglement generator and
distributor, a noisy quantum channel and a classical channel. Without loss in
generality, in the figure we only highlight the entanglement generator and
distributor used at the receiver, since it is exploited by the proposed
scheme. The quantum communication session commences with the generation of the
EPR pairs, whose quantum state is

$\displaystyle\ket{\Phi^{\pm}}_{AB}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{00}\pm\ket{11}_{AB}\right),$
$\displaystyle\ket{\Psi^{\pm}}_{AB}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{01}\pm\ket{10}_{AB}\right).$ (1)

In the rest of this treatise, we assume that the pre-shared EPR pairs are
initialized to the quantum state of $\ket{\Phi^{+}}_{AB}$, where the subscript
$AB$ indicates that the first qubit of each EPR pair is held by $A$ and the
second qubit is held by $B$. In Fig. 1, the entanglement generator is located
at $B$. Hence, the first qubit of the EPR pairs $\ket{\Phi^{+}}_{A}$ has to be
sent by $B$ through the quantum channel, while the second qubit of the EPR
pairs $\ket{\Phi^{+}}_{B}$ is available immediately at $B$. After $A$ obtains
the first qubit of the EPR pairs $\ket{\Phi^{+}}_{A}$, it can be exploited for
transmitting the quantum information embedded within the logical quantum qubit
$\ket{\psi}$. In addition to the pre-shared EPR pairs, $A$ and $B$ are also
connected via a classical communication channel, which is considered to be
noise-free222This assumption is not restrictive since we focus our attention
on the quantum noise only. In case of a realistic noisy classical channel, the
well-known classical error-mitigation techniques can be implemented..

The main goal of the quantum communication model of Fig. 1 is to faithfully
transfer the quantum state $\ket{\psi}$ from $A$ to $B$ assisted by the pre-
shared EPR pairs and also by classical communications. To achieve this goal,
$A$ may exploit the noisy pre-shared EPR pairs $\ket{\Phi^{+}}_{A}$ for
appropriately encoding the logical qubits $\ket{\psi}$ into
$\ket{\overline{\psi}}$, which is sent to $B$. In addition to the received
encoded quantum state $\ket{\tilde{\psi}}$, $B$ also obtains the classical
bits gleaned from the measurement of the EPR-pair members $\ket{\Phi^{+}}_{A}$
at $A$. Finally, $B$ performs a decoding procedure to reconstruct the original
quantum state $\ket{\psi}$ of the logical qubits by utilizing the qubits of
the EPR-pair members $\ket{\Phi^{+}}_{B}$ at $B$.

In this treatise, we consider one of the most general quantum channel models,
namely the quantum depolarizing channel $\mathcal{N}(\cdot)$, a type of
quantum Pauli channel. For a single-qubit system, the quantum depolarizing
channel is described by [11]

$\mathcal{N}(\rho)=(1-p)\rho+\frac{p}{3}\left({X}\rho{X}+{Y}\rho{Y}+{Z}\rho{Z}\right),$
(2)

where $\\{I,X,Y,Z\\}$ are the Pauli matrices, $\rho$ denotes the density
matrix of the input quantum state, and $p$ denotes the depolarizing
probability of the quantum channel $\mathcal{N}(\cdot)$. The Kraus operators
of $\mathcal{N}(\cdot)$ are given by $N_{1}=\sqrt{1-p}I$,
$N_{2}=\sqrt{\frac{p}{3}}X$, $N_{3}=\sqrt{\frac{p}{3}}Y$,
$N_{4}=\sqrt{\frac{p}{3}}Z$ [11].

##  III Quantum Communication with Noisy Pre-Shared Entanglement

In this section, we present the general concept of our proposed scheme for
performing both error-detection and error-correction. The schematic of the
proposed scheme is depicted in Fig 2. Its operation commences by preparing the
initialized quantum state as follows:

$\ket{\psi_{p}}=\ket{\psi}^{k}\otimes\ket{\Phi^{+}}_{AB}^{n-k},$ (3)

where $\ket{\psi}^{k}$ represents the quantum state of $k$ logical qubits,
while $\ket{\Phi^{+}}_{AB}^{n-k}$ represents $(n-k)$ pairs of pre-shared EPR
pairs $\ket{\Phi^{+}}$ between $A$ and $B$. The subscripts $A$ and $B$
indicate that half of the EPR pairs are held by $A$ and the other half by $B$.

Figure 2: The scheme proposed for performing noiseless quantum communication
using noisy pre-shared EPR pairs.

As we elucidated in Section II, the generation and the distribution of the EPR
pairs to $A$ are contaminated by the quantum noise imposed by the quantum
channels. Let us denote the $(n-k)$-tuple Pauli operator inflicted by the
quantum channel as $P_{n-k}$. Then, we have

$\ket{\widetilde{\Phi^{+}}}_{A}^{n-k}=P_{n-k}\ket{\Phi^{+}}_{A}^{n-k}.$ (4)

The quantum state $\ket{\psi}^{k}$ of the logical qubits is encoded by a
quantum encoder $\mathcal{V}_{A}$, where we exploit the noisy EPR-pair members
at $A$ $\ket{\widetilde{\Phi^{+}}}_{A}^{n-k}$. The encoded state
$\ket{\overline{\psi}}^{k}$ of the logical qubits is then sent through the
quantum channel $\mathcal{N}(\cdot)$. Let us denote the $k$-tuple Pauli
operator inflicted by the quantum channel as $P_{k}$. Then, we have

$\ket{\widetilde{\psi}}^{k}=P_{k}\ket{\overline{\psi}}^{k}.$ (5)

At the receiver side, the quantum decoder $\mathcal{V}_{B}^{\dagger}$ of Fig.
2 decodes the corrupted quantum state $\ket{\widetilde{\psi}}^{k}$ with the
aid of the $(n-k)$ EPR-pair members $\ket{\Phi^{+}}_{B}^{n-k}$ at $B$. To
design the quantum encoder $\mathcal{V}_{A}$ and the quantum decoder
$\mathcal{V}_{B}^{\dagger}$, we impose the reversible property333We note that
in Fig. 2 there is a little notation-abuse, since we use the symbols
$\mathcal{V}_{A}$ and $\mathcal{V}_{B}^{\dagger}$ to denote the encoding and
decoding performed on the qubits available at $A$ and $B$, respectively.
Instead, in (6), $\mathcal{V}_{A}$ and $\mathcal{V}_{B}^{\dagger}$ denote the
encoder and decoder acting on the global quantum state
$\ket{\psi}^{k}\otimes\ket{\Phi^{+}}_{AB}^{n-k}$. However, this notation abuse
can be tolerated since $\mathcal{V}_{A}$ and $\mathcal{V}_{B}^{\dagger}$ in
(6) leave the qubits unavailable at $A$ and $B$, respectively, unchanged. on
the initialized quantum state in (3), which is formulated as

$\mathcal{V}_{B}^{\dagger}\mathcal{V}_{A}(\ket{\psi}^{k}\otimes\ket{\Phi^{+}}_{AB}^{n-k})=\ket{\psi}^{k}\otimes\ket{\Phi^{+}}_{AB}^{n-k}.$
(6)

######  Remark.

We note that in conventional QECCs, the reversible property of a noise-free
scenario can always be guaranteed, since the quantum encoder $\mathcal{V}$ and
decoder $\mathcal{V}^{\dagger}$ act on the same physical qubits. By contrast,
in our scheme, the quantum encoder $\mathcal{V}_{A}$ only processes the
logical qubits $\ket{\psi}^{k}$ and the EPR-pair members at $A$, whilst the
quantum decoder $\mathcal{V}_{B}^{\dagger}$ only processes the logical qubits
$\ket{\widetilde{\psi}}^{k}$ received via the noisy quantum channel
$\mathcal{N}(\cdot)$ and the EPR-pair members at $B$.

By denoting the density matrix of
$\ket{\psi_{p}}=\ket{\psi}\otimes^{k}\ket{\Phi^{+}}_{AB}^{n-k}$ as
$\overline{\rho}$, it is possible to reformulate the proposed general scheme
of Fig. 2 as the following supermap $\mathcal{S}$:

$\mathcal{S}(\mathcal{V}_{A},\mathcal{V}_{B}^{\dagger},\mathcal{N},\overline{\rho})=\sum_{i,j}(V_{B}N_{j}V_{A}N_{i})\overline{\rho}(V_{B}N_{j}V_{A}N_{i})^{\dagger}.$
(7)

In (7), we take into account the effects of the quantum noise inflicted by the
quantum channels utilized for both the distribution of the EPR-pair members at
$A$ and for the transmission of the encoded state of the logical qubits.
Furthermore, in (7), $N_{i}$, $N_{j}$ represent the Kraus operators of the
quantum channels444To be more precise and with a little notation-abuse,
$N_{i}$, $N_{j}$ denote the extended Kraus operators of the quantum channels,
which account for the specific qubits affected by the quantum channels and for
the increased dimension induced by the supermap of (7), acting on the global
state $\ket{\psi}\otimes^{k}\ket{\Phi^{+}}_{AB}^{n-k}$., while $V_{A}$ and
$V_{B}$ are the matrix representations of the quantum encoder and decoder,
respectively.

The scheme proposed in Fig. 2 is completed by local measurements $\mathcal{M}$
on the EPR pairs whose outcomes control the operator $\mathcal{R}$ depending
the particular error-control strategy implemented. Specifically, to perform
the associated error-control procedure, local measurements of the EPR pairs
are performed for obtaining the classical bits555When $n-k$ EPR pairs are
considered, the local measurements of the EPR pairs produce $2(n-k)$ outcomes.
To denote the associated vectors, we utilize the notation $\underline{s}$.
$\underline{s}_{A,n-k}$ and $\underline{s}_{B,n-k}$. Since no joint
measurements are applied to the EPR pairs for the sake of reducing the number
of quantum channels utilization, a syndrome-like quantity may be constructed
from the modulo-2 addition of the classical measurement results as follows:

$\underline{s}_{n-k}=\underline{s}_{A,n-k}\oplus\underline{s}_{B,n-k}.$ (8)

It is important to note that both $A$ and $B$ have chosen the appropriate pre-
determined measurement basis $\mathcal{M}$ for each of the EPR pairs.

In the case of the proposed error-detection schemes, the operator
$\mathcal{R}$ of Fig. 2 acts as a discard-and-retain unit based on the
syndrome of (8). More specifically, if the syndrome values of (8) indicate the
presence of errors, i.e. the syndrome values are not zeros
$(\underline{s}_{n-k}\neq\underline{0})$, the operator $\mathcal{R}$ will
decide to discard the logical qubits $\ket{\psi}^{k}$, otherwise it will
retain the logical qubits. By contrast, in the case of the proposed error-
correction schemes, the operator $\mathcal{R}$ represents an error-recovery
procedure based on maximum-likelihood decoding relying on the syndrome values
of (8). Specifically, the error-recovery procedure can be formally expressed
as

$\widehat{L}_{k}(\underline{s}_{n-k})=\underset{L_{k}}{\mathrm{arg\,max\,}}P(L_{k}|\underline{s}_{n-k}),$
(9)

where $P(L_{k}|\underline{s}_{n-k})$ denotes the probability of experiencing
the logical error $L_{k}$ imposed on the logical qubits $\ket{\psi}^{k}$,
given that we obtain the syndrome values $\underline{s}_{n-k}$.

##  IV Error-Detection Scheme

In this section, we consider the error-detection of either a single logical
qubit or of two logical qubits and carry out its performance analysis. We rely
on Definition 10 and 11 for characterizing the performance of the proposed
error-detection schemes.

######  Definition 1.

The success probability $p_{s}$ of the proposed error-detection schemes is
defined as the conditional probability of obtaining the legitimate quantum
state $\rho$ of the logical qubits, given that we obtain the all-zero syndrome
values $\underline{s}_{n-k}=\underline{0}$:

$p_{s}=p(\rho|\underline{s}_{n-k}=\underline{0})=\frac{p(\rho\cap\underline{s}_{n-k}=\underline{0})}{p(\underline{s}_{n-k}=\underline{0})}.$
(10)

The relationship between the qubit error ratio (QBER) and the success
probability $p_{s}$ can simply be defined as $\text{QBER}=1-p_{s}$.

######  Definition 2.

The yield $Y$ of the proposed error-detection schemes is defined as the ratio
of $k$ logical qubits retained after the detection to the $n$ uses of the
quantum channel $\mathcal{N}(\cdot)$:

$Y=\left(\frac{k}{n}\right)p(\underline{s}_{n}=\underline{0}).$ (11)

Readers from the classical communication field may notice the relationship
between the yield and goodput metrics. While yield has been widely used in the
QED literature, goodput is a common metric utilized for normalizing the
performance of classical coded communication systems with respect to the
associated coding rate. The notion of goodput in the quantum domain is
clarified in [32], where it is used for comparing the performance of various
QECCs exhibiting different quantum coding rates and for determining their
performance discrepancies with respect to the quantum capacity also known as
the quantum hashing bound. We underline that yield and goodput are not the
same metric, although they are intimately linked. More specifically, the
goodput $G$ is defined as the product of the success probability of a given
QECC by its quantum coding rate [32]. Therefore, the goodput of our proposed
scheme may be reformulated as in Definition 12.

######  Definition 3.

The goodput $G$ of the proposed error-detection schemes is defined as the
product of the success probability $p_{s}$ by the ratio of $k$ logical qubits
to the $n$ uses of the quantum channel $\mathcal{N}(\cdot)$:

$G=\left(\frac{k}{n}\right)p_{s}=\left(\frac{k}{n}\right)(1-\text{QBER}).$
(12)

By comparing Definition 11 and Definition 12 we can observe the intrinsic
relationship between the yield and the goodput.

###  IV-A Error-Detection for A Single Logical Qubit

Figure 3: The quantum circuit conceived for performing a single-qubit error-
detection using a single noisy EPR pair.

Let us consider the proposed single qubit error-detection scheme depicted in
Fig. 3, which utilizes only a single noisy EPR pair. More specifically, the
encoding and decoding circuit of Fig. 2 is detailed in Fig. 3. We design the
quantum encoder and decoder for ensuring that the reversible condition of (6)
is satisfied. The quantum encoder $\mathcal{V}_{A}$ and quantum decoder
$\mathcal{V}_{B}^{\dagger}$ of Fig. 3 can be represented using unitary
matrices as follows:

$\displaystyle V_{A}=\ket{0}\bra{0}\otimes I\otimes I+\ket{1}\bra{1}\otimes
X\otimes I,$ $\displaystyle V_{B}=\ket{0}\bra{0}\otimes I\otimes
I+\ket{1}\bra{1}\otimes I\otimes X.$ (13)

By scrutinizing (IV-A), it is readily seen that the reversible property is
indeed satisfied, i.e.
$\mathcal{V}_{B}^{\dagger}\mathcal{V}_{A}\left(\ket{\psi}\otimes\ket{\Phi^{+}}_{AB}\right)=\ket{\psi}\otimes\ket{\Phi^{+}}_{AB}$.
Finally, the EPR pair is measured in the $Z$ basis
$(\mathcal{M}_{Z}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\})$. The performance of
the scheme proposed in Fig. 3 is characterized by Proposition 1.

######  Proposition 1.

The success probability of the error-detection scheme depicted in Fig. 3 over
quantum depolarizing channels relying on a single noisy EPR pair is given by:

$\displaystyle
p_{s}=1-\frac{2p}{3}-\frac{2p^{2}}{3}-\frac{8p^{3}}{27}+\frac{16p^{4}}{81}+\mathcal{O}(p^{5}),$
(14)

while the yield is given by:

$Y=\frac{1}{2}\left(1-\frac{4p}{3}+\frac{8p^{2}}{9}\right).$ (15)

###### Proof:

Please refer to Appendix A. ∎

First, we compare our proposed scheme to the state-of-the-art QED+QT schemes.
Specifically, we compare the scheme proposed in Fig. 3 to the single-round
recurrence QED of [27] and to the quantum stabilizer code (QSC)-based QED of
[28, 29, 17] having the stabilizer operator of $S=ZZ$. We assume that the
quantum teleportation step is noise-free and therefore the QBER of the
benchmark schemes is directly determined by the QBER of the associated QED
scheme. Note that both the benchmark schemes require two noisy pre-shared EPR
pairs, while our proposed scheme only needs a single noisy pre-shared EPR
pair.

Figure 4: The QBER of our proposed error-detection schemes compared to the
existing schemes for mitigating the effect of quantum depolarizing channels.
The uncoded QBER curve is given by $\text{QBER}=p$. The inset is the QBER
relying on log-log scale.

(a) Yield

(b) Goodput

Figure 5: The (a) yield and the (b) goodput of the proposed error-detection
schemes compared to the existing schemes for mitigating the effect of quantum
depolarizing channels. The insets are the yield and the goodput in the
logarithmic $x$ axis.

The QBER is portrayed in Fig. 4, where we label the performance of the scheme
presented in Fig. 3 as ‘Proposed 1’, the recurrence-based scheme as ‘QED+QT
1’, and the QSC-based scheme as ‘QED+QT 2’. We observe that the QBER of the
scheme presented in Fig. 3 matches that of QED+QT schemes, without requiring
the additional quantum teleportation step, which also relies on the idealized
assumption of being noise-free for both benchmarks. Furthermore, we observe
that all the schemes considered are only capable of detecting a single $X$
error. Additionally, we mark the probability threshold $p_{th}$ using the
vertical black dotted line in Fig. 4, highlighting the particular depolarizing
probability value, below which the proposed error-detection scheme improves
the QBER of the logical qubit. Specifically, in Fig. 4, we obtain the
probability threshold of $p_{th}=0.5$.

In Fig. 5(a), we report the performance of our proposed scheme in terms of its
yield. We observe that our proposed scheme provides an identical yield to the
benchmark schemes. However, two noisy pre-shared EPR pairs are used for
obtaining a single less noisy pre-shared EPR pair for both the recurrence-
based and the QSC-based QED+QT schemes. This means that during the process one
of the noisy pre-shared EPR pairs is discarded. By contrast, our protocol only
needs a single noisy pre-shared EPR pair for achieving the same QBER
performance. Finally, the goodput of our proposed error-detection scheme is
presented in Fig. 5(b), which confirms again the intrinsic relationship
between the yield and the goodput. Specifically, our proposal that provides an
identical yield, gives us also an identical goodput.

Apart from its benefit of utilizing fewer pre-shared EPR pairs, our proposed
scheme also offers a pair of additional advantages:

  * •

It does not suffer from long communication delay, since it does not require
the consecutive steps of performing QED followed by quantum teleportation.

  * •

It requires fewer controlled-NOT (CNOT) quantum gates. Quantitatively, the
proposed scheme of Fig. 3 requires a total of only two CNOT gates. By
contrast, the recurrence-based QED+QT scheme requires a total of three CNOT
gates: two for a single-round recurrence QED and one for quantum
teleportation. As for the QSC-based QED+QT scheme, we need a total of seven
CNOT gates: four for the measurement of stabilizer operators, two for the
quantum inverse encoder, and one for quantum teleportation.

Let us elaborate a little further concerning the delay imposed by each quantum
communication scheme specified in the first bullet point. The quantum
entanglement distillation has to be completed before the quantum teleportation
can be conducted within the QED+QT scheme. Specifically, within the quantum
entanglement distillation step itself, the transmission delay is imposed by
the associated classical communications. Let us assume that each classical
communication takes a duration of $t_{c}$. Therefore, for a recurrence QED
scheme having $m$ rounds of distillation, the total transmission delay is
equal to $mt_{c}$, since for each round of distillation requires a backward-
and a forward-oriented classical communication phase – both of which can be
carried out simultaneously for example using wavelength division multiplexing.
By contrast, the total transmission delay imposed by a QSC-based QED is simply
equal to $t_{c}$, since it only needs a forward-oriented classical
communication. Once the QED step has been completed, quantum teleportation has
to be performed for transferring the quantum information from the transmitter
to the receiver. Since quantum teleportation also requires another forward-
oriented classical information phase, an additional delay of $t_{c}$ is
introduced by the QED+QT scheme. Therefore, we have a total of $2t_{c}$
transmission delay for the QSC-based QED+QT scheme. In case of recurrence-
based QED+QT scheme, we have a transmission delay of $(m+1)t_{c}$. By
contrast, the total transmission delay is only equal to $t_{c}$ for both the
QSC-IE and the proposed schemes. However, it is important to note that we
underestimate the QSC-IE quantum information processing delay, since QSC-IE
scheme utilizes the stabilizer measurements differently from the proposed
scheme.

Arguably, the delay imposed by the transmission of classical information
required by the QED+QT scheme can be avoided by performing QED in an
asynchronous way, implying that the QED is activated before the transmitter
and the receiver have agreed to initiate their quantum communication.
Consequently, an asynchronous QED+QT scheme requires a long-expiry quantum
memory to store the distilled EPR pairs. However, the assumption of having a
long-expiry quantum memory at both the transmitter and the receiver is indeed
a strong one at the current state-of-the-art [33]. In the absence of long-
expiry quantum memory, an asynchronous QED+QT scheme can be employed by
performing continuous QED until both the transmitter and the receiver finally
decide to initiate their quantum communication. However, it is clear that
continuously performing QED consumes a high number of noisy pre-shared EPR
pairs during the waiting period. Therefore, in this treatise, we consider an
on-demand quantum communication model, as we have described in Section II.
This model eliminates the stringent requirement of long-expiry quantum memory
as well as the continuous operation of QED, which is achieved by only
initializing quantum communication once both the transmitter and the receiver
are ready to engage.

Regarding the number of CNOT gates mentioned in the second bullet point, it
has been shown in [34, 35], that the number of CNOT gates provides a
reasonable estimate of the severity of quantum error proliferation effects,
when the realistic quantum encoder $\mathcal{V}_{A}$ and decoder
$\mathcal{V}_{B}^{\dagger}$ are potentially error-infested. Specifically, in
this case, the overall proliferation of quantum errors is heavily dependent on
the number of two-qubit quantum gates – exemplified by the CNOT gates.
However, to fully characterize the performance of quantum communication
schemes under the realistic scenario of having both noisy pre-shared
entanglement and imperfect quantum gates, computer simulations are required.
Thus, we will carry out this full-scale analysis in our future work.

######  Remark.

By invoking the simple scheme presented in Fig 3, we can attain both an
identical yield and a reduced delay, despite relying on a reduced number of
CNOT gates compared to the benchmarks, which is achieved without degrading the
QBER.

Figure 6: The quantum circuit for performing a single-qubit error-detection
using two noisy pre-shared EPR pairs.

In order to further generalize our analysis, let us compare the aforementioned
schemes by using the same number of noisy pre-shared EPR-pairs. More
specifically, we assume having two noisy pre-shared EPR-pairs for all the
QED+QT schemes considered. Specifically, we modify the scheme proposed in Fig.
3 as seen in Fig. 6, where the first EPR pair is measured in the $Z$ basis
$(\mathcal{M}_{Z}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\})$, while the second pair
in the $X$ basis $(\mathcal{M}_{X}=\\{\ket{+}\bra{+},\ket{-}\bra{-}\\})$. Let
us distinguish the components of the syndrome vector in (8) according to the
observation basis used for the measurement. Specifically, let us denote the
syndrome component obtained when the first EPR pair is measured in the $Z$
basis by $s_{Z}=s_{A}\oplus s_{B}$ and that obtained when the second EPR pair
is measured in the $X$ basis by $s_{X}=s_{A}\oplus s_{B}$. The operator
$\mathcal{R}$ acts as follows: if $s_{Z}=0$, the measurement of the second EPR
pair is performed to obtain $s_{X}$. Otherwise, the logical qubit is discarded
immediately, since there is no need to measure the syndrome value $s_{X}$, if
the syndrome value $s_{Z}$ already indicates that the logical qubit is
corrupted. The aforementioned decision strategy is summarized as a look-up
table (LUT) in Table II(a). The performance of the error-detection scheme
depicted in Fig. 6 is quantified in terms of its QBER and yield presented in
Proposition 2.

TABLE II: Syndrome values and associated decision $\mathcal{R}$ for the error-
detection schemes.

(a) Scheme in Fig. 6. $s_{Z}$ | $s_{X}$ | Decision $\mathcal{R}$
---|---|---
0 | 0 | Retain
0 | 1 | Discard
1 | n.a. | Discard

(b) Scheme in Fig. 7. $s_{X}$ | $s_{Z}$ | Decision $\mathcal{R}$
---|---|---
0 | 0 | Retain
0 | 1 | Discard
1 | 0 | Discard
1 | 1 | Discard

######  Proposition 2.

The success probability of the proposed error-detection scheme of Fig. 6
operating over quantum depolarizing channels by utilizing two noisy EPR pairs
is:

$p_{s}=1-\frac{p}{3}-\frac{8p^{2}}{9}-\frac{32p^{3}}{27}-\frac{64p^{4}}{81}+\mathcal{O}(p^{5}),$
(16)

while the yield is expressed as

$Y=\frac{1}{3}\left(1-\frac{8p}{3}+\frac{28p^{2}}{9}-\frac{32p^{3}}{27}\right).$
(17)

###### Proof:

Please refer to Appendix B. ∎

We also compare our proposed scheme to the state-of-the-art QSC-IE scheme.
Specifically, we compare the scheme proposed in Fig. 6 to the QSC-IE scheme of
[31] having the stabilizer operators of $S_{1}=IXX$ and $S_{2}=ZZZ$. The QBER,
the yield, as well as the goodput of the proposed scheme in Fig. 6 are
portrayed in Fig. 4, 5(a), and 5(b), respectively, where it is labeled as
‘Proposed 2’, while the QSC-IE benchmark scheme is labeled as ‘QSC-IE’.
Observe in Fig. 4 that the QBER of the error-detection scheme in Fig. 6
outperforms all the QED+QT benchmark schemes, while providing an identical
QBER to the QSC-IE benchmark scheme. We also obtain the probability threshold
of $p_{th}=0.5$ for the proposed error-detection scheme in Fig. 6. However,
the QSC-IE scheme only requires one pre-shared EPR pair, while our proposed
scheme requires two pre-shared EPR pairs as shown in Fig. 6. Nonetheless, our
proposed scheme requires fewer CNOT gates and the same number of quantum
channel uses by avoiding the need of stabilizer measurements. More
specifically, the scheme presented in Fig. 6 requires a total of only four
CNOT gates, while the QSC-IE scheme requires a total of eight CNOT gates: one
for quantum encoder, five for the measurement of stabilizer operators, and two
for the quantum inverse encoder.

######  Remark.

By maintaining the same maximal yield and goodput as the state-of-the-art
schemes, our proposed scheme provides identical error-detection performance
despite utilizing fewer CNOT gates.

###  IV-B Error-Detection for Two Logical Qubits

Figure 7: The quantum circuit designed for the proposed error-detection for
two logical qubits, which utilizes two noisy pre-shared EPR pairs.

Let us now shift our focus to the scheme presented in Fig. 7, where we use two
noisy EPR pairs for constructing an error-detection scheme for two logical
qubits. Again, the quantum encoder $\mathcal{V}_{A}$ and decoder
$\mathcal{V}^{\dagger}_{B}$ are designed for satisfying the reversible
property. The resultants quantum encoder $\mathcal{V}_{A}$ and decoder
$\mathcal{V}^{\dagger}_{B}$ are seen in Fig. 7. The first EPR pair is measured
in the $X$ basis $(\mathcal{M}_{X}=\\{\ket{+}\bra{+},\ket{-}\bra{-}\\})$,
while the second pair in the $Z$ basis
$(\mathcal{M}_{Z}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\})$. Additionally, the
decision block $\mathcal{R}$ of Fig. 7 is represented by the LUT of Table
II(b). We summarize the performance results in Proposition 3.

######  Proposition 3.

The success probability of the proposed error-detection scheme of Fig. 7
operating over quantum depolarizing channels is given by:

$p_{s}=1-2p^{2}-\frac{44p^{3}}{9}-\frac{44p^{4}}{9}+\mathcal{O}(p^{5}),$ (18)

while the yield is expressed as

$Y=\frac{1}{2}\left(1-4p+8p^{2}-\frac{64p^{3}}{9}+\frac{64p^{4}}{27}\right).$
(19)

###### Proof:

Please refer to Appendix C. ∎

To benchmark the performance of the proposed scheme, we have chosen the
following QED+QT schemes. Firstly, for the recurrence-based QED+QT scheme
(QED+QT 1), we carry out two single-round distillations to obtain two less
noisy EPR pairs from four noisy EPR pairs. Secondly, for the QSC-based QED+QT
scheme (QED+QT 2), we choose the stabilizer operators of $S_{1}=XXXX$ and
$S_{2}=ZZZZ$ to apply error-detection to a set of four noisy EPR pairs.
Finally, we also include the QSC-IE scheme having the stabilizer operators of
$S_{1}=XXXX$ and $S_{2}=ZZZZ$ as our benchmark. The uncoded QBER is given by
$\text{QBER}=1-(1-p)^{2}=2p-p^{2}$, which means that any error experienced by
any logical qubit within the two qubits is considered as an error. The
resultant QBER is portrayed in Fig. 8(a), while the yield is quantified in
Fig. 8(b).

(a) QBER

(b) Yield

Figure 8: The (a) QBER and the (b) yield of the proposed error-detection
scheme in Fig. 7 compared to the existing schemes for mitigating the effect of
quantum depolarizing channels. The uncoded QBER curve is given by
$\text{QBER}=1-(1-p)^{2}=2p-p^{2}$. The insets are the QBER and the yield in
the logarithmic $x$ axis.

Let us evaluate the QBER and the yield of the recurrence-based QED+QT scheme
by considering a pair of identical error-detection schemes based on Fig. 3. We
can determine the success probability of this arrangement by taking the square
of (14) of Proposition 1, since the legitimate quantum state of the logical
qubit is retained only when both error-detection schemes make the correct
decision. Consequently, the success probability $p_{s}$ is given by

$\displaystyle p_{s}$
$\displaystyle=\left[\frac{1-2p+\frac{10p^{2}}{9}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}\right]^{2}$
$\displaystyle=1-\frac{4p}{3}-\frac{8p^{2}}{9}+\frac{8p^{3}}{27}+\frac{100p^{4}}{81}+\mathcal{O}(p^{5}).$
(20)

Similarly, the yield $Y$ can be obtained by taking the square of $p(s_{z}=0)$
of (15) in Proposition 1 and then by normalizing it by $k/n$, where we obtain

$\displaystyle Y$
$\displaystyle=\frac{1}{2}\left(1-\frac{4p}{3}+\frac{8p^{2}}{9}\right)^{2}$
$\displaystyle=\frac{1}{2}\left(1-\frac{8p}{3}+\frac{32p^{2}}{9}-\frac{64p^{3}}{27}+\frac{64p^{4}}{81}\right).$
(21)

The QBER and yield results in (20) and (21) are depicted in Fig. 8(a) and
8(b), respectively, as ‘QED+QT 1’.

In Fig. 8(a), our proposed scheme outperforms the recurrence-based QED+QT
scheme (QED+QT 1) for $p<0.5$, while exhibiting an identical QBER to the QSC-
based QED+QT scheme (QED+QT 2) and QSC-IE. Furthermore, we also observe the
probability threshold of $p_{th}=0.5$, which is portrayed using vertical black
dotted line, for the proposed error-detection scheme in Fig. 7. However,
observe in Fig. 8(b) that the recurrence-based QED+QT scheme attains a better
yield. The reason is that the recurrence-based QED+QT scheme exhibits a weaker
error-detection capability than the other schemes. More specifically, each
round of recurrence QED is only capable of detecting a single $X$ error. By
contrast, for the QSC-based QED+QT, QSC-IE, and also our proposed schemes,
they are all capable of detecting a single $X$ error and also a single $Z$
error. Consequently, the recurrence-based QED+QT scheme often makes the wrong
decision of retaining the erroneous logical qubits, instead of discarding
them, which is reflected in higher QBER result. However, these QBER and yield
results can be achieved by utilizing fewer CNOT gates.

Additionally, as shown in Fig. 7, the total number of CNOT gates required by
the entire proposed error-detection scheme is eight. As a comparison, the QSC-
based QED+QT scheme requires a total of 28 CNOT gates, namely 16 for the
stabilizer measurements, 10 for the quantum inverse encoder, and two for
quantum teleportation. The quantum circuit of the QSC-based QED+QT scheme is
portrayed in Fig. 9. Meanwhile, the QSC-IE scheme, whose quantum circuit is
portrayed in Fig. 10, requires a total of 16 CNOT gates, namely four for the
quantum encoder, eight for the stabilizer measurements, and four for the
quantum inverse encoder. Therefore, our proposed scheme requires significantly
fewer CNOT gates while offering an identical QBER and yield.

Figure 9: The quantum circuit designed for performing QSC-based QED+QT scheme
using the stabilizer operators of $\mathcal{C}[4,2,2]$ code followed by a two-
qubit quantum teleportation. Figure 10: The quantum circuit designed for
performing QSC-IE scheme using the stabilizer operators of
$\mathcal{C}[4,2,2]$ code.

We summarize all the resources required by the various quantum communication
schemes of Fig. 8 in Table III. All the schemes considered in Table III –
except for the QED+QT 1 scheme – attain identical QBER and yield. To achieve
this, our proposed scheme requires fewer pre-shared EPR pairs, fewer classical
channel uses, and fewer CNOT gates compared to the QSC-based QED+QT scheme. By
contrast, with the same number of quantum channel uses, our proposed scheme
requires more pre-shared EPR pairs and more classical channel uses than the
QSC-IE scheme, while utilizing fewer CNOT gates.

TABLE III: Comparison of our proposed error-detection scheme with the existing approaches. Scheme | Quantum channel use | Logical qubits | EPR pairs | Classical channel use | CNOT
---|---|---|---|---|---
$n$ | $k$ | $e$ | $c$
QED+QT 1 | 4 | 2 | 4 | 6 | 4
QED+QT 2 | 4 | 2 | 4 | 6 | 28
QSC-IE | 4 | 2 | 1 | 0 | 16
Proposed | 4 | 2 | 2 | 2 | 8

######  Remark.

While providing an identical QBER and yield to the QSC-based QED+QT and QSC-IE
schemes, our error-detection scheme always requires fewer CNOT gates.

##  V Error-Correction Scheme

Error-detection schemes provide dynamic yields, since they rely on a discard-
and-retain action of the operator $\mathcal{R}$, while error-correction
schemes provide a constant yield, since they attempt to recover the legitimate
quantum state of the logical qubits from the received encoded state.
Therefore, a modification of Definition 10 and 11 is required in order to
accurately evaluate the performance of the proposed error-correction scheme.

######  Definition 4.

The success probability $p_{s}$ of the proposed error-correction scheme is
defined as the sum of the conditional probabilities
$p(\widehat{L}_{k}=L_{k}|\underline{s}_{n-k})$, i.e. the sum of the
probabilities that the error-recovery operator $\mathcal{R}$ successfully
applies $\widehat{L}_{k}=L_{k}$ based on the syndrome value
$\underline{s}_{n-k}$:

$p_{s}=\sum_{L_{k}}p(\widehat{L}_{k}=L_{k}|\underline{s}_{n-k}),$ (22)

where the relationship between $p_{s}$ and the QBER can be expressed as
$\text{QBER}=1-p_{s}$.

######  Definition 5.

The yield $Y$ of the proposed error-correction scheme is defined as the ratio
of $k$ logical qubits to the $n$ uses of the quantum channel
$\mathcal{N}(\cdot)$:

$Y=\frac{k}{n},$ (23)

while its goodput is similarly defined to Definition 12.

Figure 11: The quantum encoder $\mathcal{V}_{A}$ and the quantum decoder
$\mathcal{V}_{B}^{\dagger}$ for performing the proposed error-correction
scheme.

Let us now consider the quantum encoder $\mathcal{V}_{A}$ and decoder
$\mathcal{V}_{B}^{\dagger}$ of Fig. 11. To investigate its error-correction
performance, we have to check first that the scheme of Fig. 11 is capable of
discriminating all the single-qubit error patterns based on the measured
syndrome values. In Fig. 11, we can observe that the overall scheme requires
six noisy pre-shared EPR pairs, which means that we have a six-bit syndrome
string denoted by $\underline{s}=s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}$, where the
indices $i\in\\{1,2,3,4,5,6\\}$ represent the EPR pair starting from the top.
Therefore, for each of the single-qubit error patterns, we can evaluate the
syndrome string and the associated error recovery operator, as shown in Table
IV. Observe that the first three elements of the syndrome string
$\underline{s}_{Z}=s_{1}s_{2}s_{3}$ are exclusively used for identifying $X$
errors, which are obtained from $Z$ basis measurements
$(\mathcal{M}_{Z}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\})$. By contrast, the last
three elements $\underline{s}_{X}=s_{4}s_{5}s_{6}$ are used for identifying
$Z$ errors, which are obtained from $X$ basis measurements
$(\mathcal{M}_{X}=\\{\ket{+}\bra{+},\ket{-}\bra{-}\\})$. Finally, the $Y$
errors can be identified based on the combination of $\underline{s}_{Z}$ and
$\underline{s}_{X}$.

TABLE IV: Syndrome values and the associated error-recovery operator $\mathcal{R}$ of the error-correction scheme presented in Fig. 11. Error pattern | Syndrome | Error recovery | Error pattern | Syndrome | Error recovery
---|---|---|---|---|---
$P_{k},P_{n-k}$ | $\underline{s}$ | $\mathcal{R}$ | $P_{k},P_{n-k}$ | $\underline{s}$ | $\mathcal{R}$
$XIIIIII$ | $(1\ 1\ 0\ 0\ 0\ 0)$ | $X$ | $ZIIIIII$ | $(0\ 0\ 0\ 1\ 1\ 0)$ | $Z$
$IXIIIII$ | $(1\ 0\ 0\ 0\ 0\ 0)$ | $I$ | $IZIIIII$ | $(0\ 0\ 0\ 1\ 0\ 1)$ | $Z$
$IIXIIII$ | $(0\ 1\ 0\ 0\ 0\ 0)$ | $I$ | $IIZIIII$ | $(0\ 0\ 0\ 0\ 1\ 1)$ | $Z$
$IIIXIII$ | $(0\ 0\ 1\ 0\ 0\ 0)$ | $I$ | $IIIZIII$ | $(0\ 0\ 0\ 1\ 1\ 1)$ | $I$
$IIIIXII$ | $(0\ 1\ 1\ 0\ 0\ 0)$ | $X$ | $IIIIZII$ | $(0\ 0\ 0\ 1\ 0\ 0)$ | $I$
$IIIIIXI$ | $(1\ 0\ 1\ 0\ 0\ 0)$ | $X$ | $IIIIIZI$ | $(0\ 0\ 0\ 0\ 1\ 0)$ | $I$
$IIIIIIX$ | $(1\ 1\ 1\ 0\ 0\ 0)$ | $I$ | $IIIIIIZ$ | $(0\ 0\ 0\ 0\ 0\ 1)$ | $I$

For the quantum depolarizing channel, we have a total of $4^{7}=16,384$ error
patterns represented by the total number of combinations in terms of bit-flip
$(X)$, phase-flip $(Z)$, as well as simultaneous bit-flip and phase-flip $(Y)$
errors, where we observe a total of $4,096$ correctable error patterns. After
scrutinizing all $4,096$ error patterns, we obtain the Pauli weight
distribution of the error patterns in quantum depolarizing channels as
follows: one error pattern is the all-identity operator (weight = $0$); 21
error patterns having weight = 1; 42 error patterns having weight = 2; 252
error patterns having weight = 3; 609 error patterns having weight = 4; 1281
having weight = 5; 1428 error patterns having weight = 6; 462 error patterns
having weight = 7. This distribution is identical to that of a QSC-based
QED+QT scheme utilizing the stabilizer operators of the Steane code. Given
that we have $p_{x}=p_{z}=p_{y}=\frac{p}{3}$, the success probability of the
proposed error-correction scheme of Fig. 11 in quantum depolarizing channels
is given by

$\displaystyle p_{s}$
$\displaystyle=\sum_{i=0}^{n}W_{i}\left(\frac{p}{3}\right)^{i}(1-p)^{(n-i)}$
$\displaystyle=1-\frac{49p^{2}}{3}+56p^{3}-\frac{2380p^{4}}{27}$
$\displaystyle+\frac{6160p^{5}}{81}-\frac{8512p^{6}}{243}+\frac{4824p^{7}}{729},$
(24)

where
$W=\\{W_{0},W_{1},W_{2},W_{3},W_{4},W_{5},W_{6},W_{7}\\}=\\{1,21,42,252,609,1281,1428,462\\}$
is the Pauli weight of the correctable error patterns. Notice that our
proposed scheme is capable of correcting not only the error patterns
exhibiting a Pauli weight = 1, but also several error patterns having higher
Pauli weights. This is due to the degeneracy property of quantum information
inherited by QECCs. Naturally, by exploiting the degeneracy property, the QBER
of quantum error-control schemes, including our proposed schemes, can be
improved.

Let us now compare the QBER of our proposed scheme to those of the QSC-based
QED+QT and QSC-IE scheme. Indeed for a fair comparison, we do not consider the
recurrence-based QED+QT scheme, since it is an error-detection scheme, not an
error-correction one. For the QSC-based QSC+QT scheme, we utilized the
stabilizer operators of Steane code [36, 17] over seven noisy pre-shared EPR
pairs. By contrast, for the QSC-IE scheme, we also utilized the stabilizer
operators of Steane code, but only by using three noisy pre-shared EPR pairs
[31]. The resultant QBER of the proposed scheme is depicted in Fig. 12(a),
which is identical to the QBER of the QSC-based QED+QT and QSC-IE schemes
employing the stabilizer operators of the Steane code. Here, we obtain the
probability threshold of $p_{th}=0.081$ for the proposed error-correction
scheme in Fig. 11, which is indicated by the vertical black dotted line in
Fig. 12(a). The proposed, the QSC-based QED+QT, and the QSC-IE schemes all
provide a yield of $Y=\frac{1}{7}$, since they perform error-correction,
instead of error-detection. Consequently, as reported in Fig. 12(b), the
proposed error-correction scheme also provides an identical goodput to the
QSC-based QED+QT and QSC-IE schemes.

(a) QBER

(b) Goodput

Figure 12: The (a) QBER and the (b) goodput of our error-correction scheme
proposed in Fig. 11 compared to the existing QSC-based scheme for mitigating
the effect of quantum depolarizing channels. The insets are the QBER and the
goodput in the logarithmic $x$ axis.

As for their quantum circuit implementations, our proposed scheme requires a
total of 22 CNOT gates as seen in Fig 11. By contrast, the QSC-based QED+QT
scheme requires a total of 71 CNOT gates, namely 48 for stabilizer
measurements, 22 for the quantum inverse encoder, and one for quantum
teleportation. To elaborate a little further on the quantum circuit
implementation required for performing QED+QT scheme using the stabilizer
operators of Steane code, please refer to Fig. 13. Meanwhile, the QSC-IE
scheme requires a total of 43 CNOT gates, namely eight for quantum encoder, 24
for the stabilizer measurements, and 11 for quantum inverse encoder. To
provide a clear picture about the quantum circuit implementation of the QSC-IE
scheme using the stabilizer operators of Steane code, please refer to Fig. 14.
We summarize all the physical resources required for performing the error
correction schemes to achieve reliable quantum communication in the presence
of noisy pre-shared EPR pairs in Table V.

Figure 13: The quantum circuit designed for performing QSC-based QED+QT scheme using the stabilizer operators of Steane code followed by a single qubit quantum teleportation. Figure 14: The quantum circuit designed for performing QSC-IE scheme using the stabilizer operators of Steane code. TABLE V: Comparison of our proposed error-correction scheme with the existing approaches. Scheme | Quantum channel use | Logical qubits | EPR pairs | Classical channel use | CNOT
---|---|---|---|---|---
$n$ | $k$ | $e$ | $c$
QED+QT | 7 | 1 | 7 | 8 | 71
QSC-IE | 7 | 1 | 3 | 0 | 43
Proposed | 7 | 1 | 6 | 6 | 22

######  Remark.

While attaining an identical QBER and yield to that of QSC-based QED+QT
scheme, our proposed arrangement requires fewer pre-shared EPR pairs, fewer
classical channel uses, and fewer CNOT gates. However, the proposed
arrangement requires more pre-shared EPR pairs and more classical channel uses
than the QSC-IE scheme in exchange for fewer CNOT gates and the same number of
quantum channel uses.

##  VI Discussion: A Quantum Computing Perspective

In the previous sections, we have shown the advantages of our proposal in
quantum communication applications. In this section, we demonstrate that the
proposed scheme can also be adopted for quantum computing applications. In
quantum computing applications, the quantum information is usually protected
with the aid of noise-free auxiliary qubits, which may also take form of pre-
shared entanglement [23, 24, 19, 20, 21, 25]. A prime example is constituted
by the family of entanglement-assisted quantum stabilizer codes (EA-QSCs).
Compared to the conventional QSCs, which are unassisted by noise-free pre-
shared entanglement, EA-QSCs offer an error-correction capability improvement.
This is reminiscent of having an additional error-free side channel between
the transmitter and the receiver in the classical domain. The argument that we
can always have noise-free pre-shared entanglement relies on the assumption
that EPR pairs can be created abundantly and quantum entanglement distillation
can be applied to them. The concept of EA-QSCs is favourable in the realms of
quantum computation, since the EA-QSCs can be readily amalgamated both with
transversal implementation of quantum gates [37, 38] as well as with magic
state distillation [39] for creating a universal set of fault-tolerant quantum
gates. In the following, we propose an error-correction scheme that
outperforms the state-of-the-art EA-QSC.

Any EA-QSC can be defined as $\mathcal{C}[n,k,d,e]$, where $n$ is the number
of physical qubits, $k$ is the number of logical qubits, $d$ is the minimum
distance of the code, and $e$ is the number of noise-free pre-shared
maximally-entangled qubits. The error-detection and error-correction
capability of any EA-QSC can be determined by its minimum distance $d$. An EA-
QSC exhibiting a minimum distance $d$ is capable of detecting $(d-1)$ quantum
errors or correcting $t=\lfloor(d-1)/2\rfloor$ quantum errors. Based on the
quantum Singleton bound of EA-QSCs [23], there exists a EA-QSC capable of
correcting a single-qubit error $(d=3)$, which encodes one logical qubit
$(k=1)$ into three physical qubits $(n=3)$ with the aid of two noise-free pre-
shared maximally-entangled qubits $(e=2)$. This specific code is denoted by
$\mathcal{C}[n,k,d,e]=\mathcal{C}=[3,1,3,2]$. In the following, we will show
that by utilizing two noise-free pre-shared EPR pairs, instead of error-
correction, we can achieve error elimination, implying that in this specific
context, we can always obtain a noise-free logical qubit.

(a)

(b)

Figure 15: (a) The quantum circuit of the proposed scheme utilizing two noise-
free pre-shared EPR pairs. (b) The rearranged quantum circuit of (a) for
analysis.

Let us now discuss our proposed scheme portrayed in Fig. 15(a), which is
rearranged into Fig. 15(b) for facilitating our analysis. The quantum channel
$\mathcal{N}(\cdot)$ in Fig. 15(a) and 15(b) represents a quantum channel
contaminating the logical qubit. According to Fig. 15(b), the quantum encoder
$\mathcal{V}_{A}$ is represented by the following unitary matrix:

$\displaystyle V_{A}$ $\displaystyle=\left(\ket{0}\bra{0}\otimes I\otimes
I\otimes I\otimes I+\ket{1}\bra{1}\otimes X\otimes I\otimes I\otimes I\right)$
$\displaystyle\left(I\otimes I\otimes I\otimes\ket{0}\bra{0}\otimes I+X\otimes
I\otimes I\otimes\ket{1}\bra{1}\otimes I\right),$ (25)

while the quantum decoder $\mathcal{V}_{B}^{\dagger}$ is described by the
following unitary matrix:

$\displaystyle V_{B}$ $\displaystyle=\left(I\otimes I\otimes I\otimes
I\otimes\ket{0}\bra{0}+X\otimes I\otimes I\otimes
I\otimes\ket{1}\bra{1}\right)$ $\displaystyle\left(I\otimes I\otimes
I\otimes\ket{0}\bra{0}\otimes I+X\otimes I\otimes
I\otimes\ket{1}\bra{1}\otimes I\right).$ (26)

It can be readily verified that the reversible property is satisfied, i.e. we
have
$\mathcal{V}_{B}^{\dagger}\mathcal{V}_{A}\left(\ket{\psi}\otimes\ket{\Phi^{+}}_{AB}^{2}\right)=\ket{\psi}\otimes\ket{\Phi^{+}}_{AB}^{2}$.

Upon denoting the density matrix of the initial global quantum state of
$\ket{\psi}\otimes\ket{\Phi^{+}}_{AB}^{2}$ by $\overline{\rho}$, the proposed
scheme can be formulated with the aid of the following supermap:

$\mathcal{S}(\mathcal{V}_{A},\mathcal{N},\mathcal{V}_{B}^{\dagger},\overline{\rho})=\sum_{i}(V_{B}N_{i}V_{A})\overline{\rho}(V_{B}N_{i}V_{A})^{\dagger},$
(27)

where $N_{i}$ is the Kraus operator describing the quantum channel, while
$V_{A}$ and $V_{B}$ represent the unitary matrices of (25) and (26).
Therefore, (27) can be rewritten as:

$\displaystyle\mathcal{S}(\overline{\rho})$
$\displaystyle=(1-p)\rho\otimes\ket{\Phi^{+}}\bra{\Phi^{+}}\otimes\ket{\Phi^{+}}\bra{\Phi^{+}}$
$\displaystyle+\frac{p}{3}\left(X\rho
X\right)\otimes\ket{\Psi^{+}}\bra{\Psi^{+}}\otimes\ket{\Phi^{+}}\bra{\Phi^{+}}$
$\displaystyle+\frac{p}{3}\left(Y\rho
Y\right)\otimes\ket{\Psi^{+}}\bra{\Psi^{+}}\otimes\ket{\Phi^{-}}\bra{\Phi^{-}}$
$\displaystyle+\frac{p}{3}\left(Z\rho
Z\right)\otimes\ket{\Phi^{+}}\bra{\Phi^{+}}\otimes\ket{\Phi^{-}}\bra{\Phi^{-}}.$
(28)

After the decoding operation, we perform the measurement of the EPR pairs.
Observe that we can apply $Z$ basis measurement to the first EPR pair and $X$
basis measurement to the second EPR pair for determining the type of Pauli
error experienced by the logical qubit $\ket{\psi}$. To elaborate a little
further, we design a scheme so that requiring a joint measurement of the EPR
pairs can be avoided to reduce the complexity of the quantum encoder and
decoder. We combine the classical bits of $A$ and $B$ of Fig. 15(b) to
determine the error recovery operator $\mathcal{R}$.

TABLE VI: Syndrome values and associated error recovery $\mathcal{R}$ for the scheme in Fig. 15(a). $s_{Z}$ | $s_{X}$ | Error recovery $\mathcal{R}$
---|---|---
0 | 0 | $I$
1 | 0 | $X$
1 | 1 | $Y$
0 | 1 | $Z$

To expound a little further, let us denote the syndrome string as
$\underline{s}=s_{Z}s_{X}$, where $s_{Z}$ is obtained from the measurement of
the first EPR pair in $Z$ basis and $s_{X}$ is gleaned from the measurement of
the second EPR pair in $X$ basis. The error recovery operator associated with
the syndrome value $\underline{s}=s_{Z}s_{X}$ is portrayed in Table VI.
Finally, it may be inferred from (28), that after the error recovery operator
$\mathcal{R}$ of Fig. 15(a), we always obtain the legitimate quantum state
$\rho$ of the logical qubit. Hence, we have demonstrated that with the aid of
two noise-free EPR pairs, instead of correcting a single-qubit achievable by
an EA-QSC, we can always recover a noise-free logical qubit. Observe that when
we replace the quantum channel $\mathcal{N}(\cdot)$ by realistic noisy quantum
Pauli gates, we can modify the LUT of Table VI to benefit from the noise-free
operation of the quantum Pauli gates.

##  VII Conclusions and Future Research

In this treatise, we have conceived a novel direct quantum communication
scheme using noisy pre-shared EPR pairs. Conventionally, achieving a reliable
quantum communication tends to rely on the consecutive steps of QED followed
by quantum teleportation (QED+QT). One of the salient benefits that we can
offer is the elimination of the long communication delay imposed by the
aforementioned consecutive steps, despite relying on noisy pre-shared EPR
pairs. Additionally, our proposed schemes offer better QBER than the
recurrence-based QED+QT schemes and provide identical QBER and yield to the
QSC-based QED+QT schemes. Moreover, compared to the QSC-based QED+QT schemes,
our proposal requires fewer pre-shared entanglement, fewer classical channel
uses, and fewer CNOT gates. We have also included the quantum stabilizer code
using imperfect pre-shared entanglement (QSC-IE) scheme as our benchmark. Our
results show that despite attaining the same level of error-detection and
error-correction capability, our proposed scheme requires more pre-shared EPR
pairs and more classical channel uses, however, in exchange for fewer CNOT
gates requirement and the same number of quantum channel uses. Finally, we
have also compared our proposed scheme to EA-QSC, which requires noise-free
pre-shared EPR pairs. Again, EA-QSCs require joint eigenvalue measurements
relying on all the qubits gleaned from the EPR pairs for performing error-
correction. Despite relying only on the local measurements of the EPR pairs
and classical communications, we can always obtain a noise-free logical qubit
using our proposed scheme.

In our future research, we are interested in finding a systematic way of
constructing the quantum encoder and decoder pair. In fact, we found that an
arbitrary quantum encoder and decoder pair cannot always satisfy the
reversible property of (6). Therefore, the sufficient and necessary conditions
of generating the quantum encoder and decoder pair should be found. Since our
proposed scheme performs identically to the QSC-based QED+QT schemes, it
remains to be shown whether a wider range of QSCs can be directly embedded
into our scheme. Furthermore, since our proposed scheme requires fewer CNOT
gates compared to all state-of-the-art schemes, we are also interested in
investigating the performance of the quantum communication schemes under
additional realistic assumption of having imperfect quantum gates as well as
imperfect measurements and looking at the possibility of creating a fault-
tolerant quantum communication protocol.

##  VIII Appendix A: Proof of Proposition 1

By exploiting the quantum depolarizing channel model of Section II and by
utilizing the expressions of (IV-A), (7) can be reformulated as shown in (29),
where $\rho$ is the density matrix of the logical qubit and we assume that the
quantum depolarizing channels experienced by $\ket{\psi}$ and
$\ket{\Phi^{+}}_{A}$ exhibit an identical depolarizing probability $p$.

$\displaystyle\mathcal{S}(\overline{\rho})$
$\displaystyle=\Big{[}(1-p)^{2}\rho+\frac{p^{2}}{9}X\rho
X+\frac{p^{2}}{9}Y\rho Y+\frac{p(1-p)}{3}Z\rho
Z\Big{]}\otimes\ket{\Phi^{+}}\bra{\Phi^{+}}$
$\displaystyle+\Big{[}\frac{p^{2}}{9}\rho+\frac{p^{2}}{9}X\rho
X+\frac{p^{2}}{9}Y\rho Y+\frac{p(1-p)}{3}Z\rho
Z\Big{]}\otimes\ket{\Phi^{-}}\bra{\Phi^{-}}$
$\displaystyle+\Big{[}\frac{p^{2}}{9}\rho+\frac{p^{2}}{9}X\rho
X+\frac{p^{2}}{9}Y\rho Y+\frac{p(1-p)}{3}Z\rho
Z\Big{]}\otimes\ket{\Psi^{-}}\bra{\Psi^{-}}$
$\displaystyle+\Big{[}\frac{p(1-p)}{3}\rho+\frac{p(1-p)}{3}X\rho
X+\frac{p(1-p)}{3}Y\rho Y+\frac{p^{2}}{9}Z\rho
Z\Big{]}\otimes\ket{\Psi^{+}}\bra{\Psi^{+}},$ (29)

After the decoding, a measurement in the $Z$ basis of the EPR pair shared
between $A$ and $B$ is performed. Every time we find a disagreement in the
classical measurement results from the EPR pair $(s=s_{A}\oplus s_{B}=1)$, the
associated logical qubit is discarded, otherwise, it is retained. We note that
the syndrome value of $s=0$ is obtained if the EPR pair is in the quantum
state $\ket{\Phi^{+}}$ or $\ket{\Phi^{-}}$, while the EPR pair in the state
$\ket{\Psi^{+}}$ or $\ket{\Psi^{-}}$ gives us a syndrome value of $s=1$.
Hence, the probability of retaining the logical qubit is equal to the
probability of obtaining the syndrome value $s=0$. Based on these
considerations and by accounting for (29), we can determine the probability of
obtaining the syndrome value $s=0$:

$p(s=0)=1-\frac{4p}{3}+\frac{8p^{2}}{9},$ (30)

which is obtained from the following error operators
$P\in\\{II,IZ,XX,XY,YX,YY,ZI,ZZ\\}$. Then, based on this set of error
operators, we can determine the probability of obtaining the syndrome value of
$s=0$ and obtain the legitimate quantum state of the logical qubit $\rho$ :

$p(\rho\cap(s=0))=1-2p+\frac{10p^{2}}{9},$ (31)

which is obtained from the following error operators $P\in\\{II,ZZ\\}$. More
specifically, the error operator $ZZ$ imposed by the quantum channels is
transformed into an error operator $IZ$ at the output of the quantum decoder
$\mathcal{V}^{\dagger}_{B}$ of Fig. 3. Consequently, we have the final quantum
state of $\rho\otimes\ket{\Phi^{-}}\bra{\Phi^{-}}$. Since the measurement of
the EPR pair is performed in $Z$ basis, we obtain the syndrome value of $s=0$,
while retaining the legitimate quantum state of the logical qubit $\rho$.
Finally, the success probability of the scheme presented in Fig. 3 can be
determined according to Definition 10 as follows:

$\displaystyle p_{s}$
$\displaystyle=p(\rho|(s=0))=\frac{1-2p+\frac{10p^{2}}{9}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}$
$\displaystyle=1-\frac{2p}{3}-\frac{2p^{2}}{3}-\frac{8p^{3}}{27}+\frac{16p^{4}}{81}+\mathcal{O}(p^{5}),$
(32)

which gives us an approximately linear performance improvement over the
uncoded QBER as a function of $p$. By accounting for Definition 11, the yield
is $Y=p(s=0)$ and the proof follows.

##  IX Appendix B: Proof of Proposition 2

After applying the first CNOT of the decoder $\mathcal{V}_{B}^{\dagger}$ of
Fig. 6, we can determine the probability of obtaining the syndrome value
$s_{Z}=0$ from the first EPR pair as follows:

$p(s_{Z}=0)=1-\frac{4p}{3}+\frac{8p^{2}}{9}.$ (33)

If $s_{Z}=1$, we discard the logical qubit. If $s_{Z}=0$, the first-two
quantum depolarizing channels are reduced into a single depolarizing channel
having the following Kraus operators:

$\displaystyle
N_{1}=\sqrt{\frac{1-2p+\frac{10p^{2}}{9}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}}I,$
$\displaystyle
N_{2}=\sqrt{\frac{\frac{2p^{2}}{9}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}}X,$
$\displaystyle
N_{3}=\sqrt{\frac{\frac{2p^{2}}{9}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}}Y,$
$\displaystyle
N_{4}=\sqrt{\frac{\frac{2p}{3}-\frac{2p^{2}}{3}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}}Z.$
(34)

Now, by applying the second CNOT of the decoder $\mathcal{V}_{B}^{\dagger}$ of
Fig. 6, we can determine the probability of obtaining the syndrome value
$s_{X}=0$ from the second EPR pair as follows:

$p(s_{X}=0)=\frac{1-\frac{8p}{3}+\frac{28p^{2}}{9}-\frac{32p^{3}}{27}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}.$
(35)

If $s_{Z}=1$, we discard the logical qubit, while If $s_{Z}=0$, we retain the
logical qubit. Therefore, the probability of arriving at the legitimate
quantum state $\rho$ and measuring the syndrome value $s_{X}=0$ is given by

$p(\rho\cap(s_{X}=0))=\frac{1-3p+\frac{28p^{2}}{9}-\frac{28p^{3}}{27}}{1-\frac{4p}{3}+\frac{8p^{2}}{9}}.$
(36)

Finally, by using Definition 10 and 11, the proof follows.

##  X Appendix C: Proof of Proposition 3

Similar to the proof of Proposition 1, By exploiting the error model of
Section II and relying on the quantum encoder and decoder of Fig. 7, the
supermap of (7) can be readily obtained. After the decoding operation, the
first EPR pair is measured in the $X$ basis while the second one in the $Z$
basis. Let us distinguish the components of the syndrome string in (8)
according to the basis used for the measurement. Specifically, let us denote
the syndrome component obtained when the second EPR pair is measured in the
$Z$ basis by $s_{Z}=s_{A}\oplus s_{B}$, while the syndrome component obtained
when the first EPR pair is measured in the $X$ basis by $s_{X}=s_{A}\oplus
s_{B}$. The overall syndrome string is $\underline{s}=s_{X}s_{Z}$. Since no
error operators exhibiting even numbers of $X$ errors and even numbers of $Z$
errors can be detected, which gives us the syndrome vector of
$\underline{s}=00$, the probability we retain the logical qubits is equal to
the sum of the probabilities of all these possible error patterns resulting in
the syndrome vector of $\underline{s}=00$. After observing $4^{4}=256$ error
patterns, we found 64 error patterns that generate the syndrome vector of
$\underline{s}=00$: one all-identity error pattern (weight = 0); 18 error
patterns having Pauli weight = 2; 24 error patterns having Pauli weight = 3;
and 21 error patterns having Pauli weight = 4. Therefore, by accounting for
(7), we have

$\displaystyle p(\underline{s}=00)$
$\displaystyle=\sum_{i=0}^{n}W_{i}\left(\frac{p}{3}\right)^{i}(1-p)^{(n-i)}$
$\displaystyle=1-4p+8p^{2}-\frac{64p^{3}}{9}+\frac{64p^{4}}{27},$ (37)

where $W=\\{W_{0},W_{1},W_{2},W_{3},W_{4}\\}=\\{1,0,18,24,21\\}$. However,
from all of those error patterns, only four are actually associated with the
legitimate quantum state $\rho$ of the logical qubits, which are
$P\in\\{IIII,XXXX,YYYY,ZZZZ\\}$. Hence, the probability of obtaining the
legitimate quantum state $\rho$ while measuring $\underline{s}=00$ is given by

$\displaystyle p(\rho\cap(\underline{s}=00))$
$\displaystyle=(1-p)^{4}+3\left(\frac{p}{3}\right)^{4}$
$\displaystyle=1-4p+6p^{2}-4p^{3}+\frac{28p^{4}}{27}.$ (38)

Finally, by using Definition 10 and 11, the proof follows.

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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 current research interests include
classical and quantum error-correction codes, quantum information, and quantum
communications.
---|---
|  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.
---|---
|  Marcello Caleffi (M’12, SM’16) is associate professor with DIETI
Department, University of Naples Federico II. Previously, he was with the
Broadband Wireless Networking Laboratory, Georgia Institute of Technology,
Atlanta, GA, USA, and also with the NaNoNetworking Center in Catalunya
(N3Cat), Universitat Politecnica de Catalunya (UPC), Barcelona, as visiting
researcher. Since July 2018, he held the Italian National Habilitation as full
professor in telecommunications engineering. He currently 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. In 2017, he has been appointed as a Distinguished
Lecturer for the IEEE Computer Society. In December 2018, he has been an
appointed member of the IEEE New Initiatives Committee.
---|---
|  Lajos Hanzo (FIEEE’04) received his Master degree and Doctorate in 1976 and
1983, respectively from the Technical University (TU) of Budapest. He was also
awarded the Doctor of Sciences (DSc) degree by the University of Southampton
(2004) and Honorary Doctorates by the TU of Budapest (2009) and by the
University of Edinburgh (2015). He is a Foreign Member of the Hungarian
Academy of Sciences and a former Editor-in-Chief of the IEEE Press. He has
served several terms as Governor of both IEEE ComSoc and of VTS. He has
published 1970 contributions at IEEE Xplore, 19 Wiley-IEEE Press books and has
helped the fast-track career of 123 PhD students. Over 40 of them are
Professors at various stages of their careers in academia and many of them are
leading scientists in the wireless industry. He is also a Fellow of the Royal
Academy of Engineering (FREng), of the IET and of EURASIP. (http://www-
mobile.ecs.soton.ac.uk, https://en.wikipedia.org/wiki/Lajos-Hanzo)
---|---