# Fundamental Limitations on Communication over a Quantum Network
Supplemental Material

Junjing Xing  College of Intelligent Systems Science and Engineering, Harbin
Engineering University, Harbin, Heilongjiang 150001, People’s Republic of
China  Tianfeng Feng  State Key Laboratory of Optoelectronic Materials and
Technologies and School of Physics, Sun Yat-sen University, Guangzhou,
Guangdong 510275, People’s Republic of China  Zhaobing Fan
<EMAIL_ADDRESS>College of Intelligent Systems Science and
Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001,
People’s Republic of China  Haitao Ma<EMAIL_ADDRESS>College of
Mathematics Science, Harbin Engineering University, Harbin, Heilongjiang
150001, People’s Republic of China  Kishor Bharti<EMAIL_ADDRESS>Institute of High Performance Computing (IHPC), Agency for Science, Technology
and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632,
Republic of Singapore  Dax Enshan Koh<EMAIL_ADDRESS>Institute of
High Performance Computing (IHPC), Agency for Science, Technology and Research
(A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of
Singapore  Yunlong Xiao<EMAIL_ADDRESS>Institute of High Performance
Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1
Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of Singapore

###### Abstract

Quantum communication is a rapidly advancing and highly promising field that
has the potential to revolutionize information transmission by providing
unprecedented levels of security and efficiency. In this work, we explore the
intrinsic relationship between the entanglement of quantum dynamics and the
performance of quantum communication, from quantum teleportation and quantum
repeaters to general quantum networks. To ensure a comprehensive and rigorous
treatment of the topic, we provide this supplemental material that offers a
detailed examination and rigorous proof of the results presented in the main
text. This supplementary material is designed to be explicit and self-
contained, with a focus on providing a thorough understanding of the
underlying principles and mechanisms that drive quantum communication.
Importantly, the supplementary material provides an extensive array of
supplementary data, encompassing not only the results expounded upon in the
main text but also other pertinent information essential for a comprehensive
understanding of the communication over a quantum network.

###### Contents

  1. I Quantum Dynamics
    1. I.1 Quantum Channels: From Choi-Jamiołkowski Isomorphism to Link Product
    2. I.2 How to Quantify a Quantum Channel: Twist It
    3. I.3 Quantum Superchannels: Pre-Processing, Post-Processing, and Non-Signalling
    4. I.4 Quantum Circuit Fragments: Framework of Non-Markovian Dynamics
  2. II Temporal Entanglement
    1. II.1 Entangled Circuit Fragments: Quantum Correlations across Space and Time
    2. II.2  Fidelity of Circuit Fragment Distillation: Quantifying Temporal Entanglement
  3. III Quantum Communications
    1. III.1 Quantum Teleportation: A Channel-Theoretic Perspective
    2. III.2 Fidelity Benchmarking: The Role of Temporal Entanglement
    3. III.3 Numerical Experiments: Dancing with Data
  4. IV Quantum Network
    1. IV.1 Quantum Repeater: Towards a Global Quantum Communication Network
    2. IV.2 All-In-One Communication Framework: That’s all you need

##  I Quantum Dynamics

In this section, we lay the foundation for the results presented in this work
by introducing our notations and investigating the physical properties of a
quantum channel through its Choi-Jamiołkowski operator (refer to Def. I.1). We
also formulate the sequential composition of channels using the link product
(refer to Def. I.1) in Subsec. I.1. After a bit of these concepts, we delve
into two important quantifiers of a quantum channel: the average fidelity
(refer to Def. I.2) and the entanglement fidelity (refer to Def. I.2). By
employing the link product in Lem. I.2 of Subsec. I.2, we establish their
intrinsic connection. In addition, we discuss the most general way of
manipulating a quantum channel – quantum superchannel via the language of non-
signalling maps (refer to Def. I.3) in Subsec. I.3. Finally, we address the
broadest framework of non-Markovian quantum processes, which we refer to as
quantum circuit fragments in Subsec. I.4. Our aim in this section is to
provide a clear and concise understanding of the concepts and terminologies
that will be used throughout the rest of the work.

###  I.1 Quantum Channels: From Choi-Jamiołkowski Isomorphism to Link Product

Just as Lego bricks can be assembled and connected in countless ways to
construct various objects, including castles, machinery, vehicles, Millennium
Falcon, and even robots, quantum channels are the fundamental building blocks
of the circuit model for quantum computing and information processing. These
channels play a critical role in numerous aspects of quantum theory, including
the preparation of quantum states, implementation of quantum measurements,
handling of noise from system-environment interactions, and execution of
quantum gates in a circuit. Essentially, the concept of quantum channels is
integral to the whole field of quantum theory, acting as a foundational tool
that enables scientists to push the boundaries of what is possible in the
realm of quantum information processing and beyond. In this subsection, we
will briefly review the theory of quantum channels, which forms the backbone
of this work. The main mathematical toolkit employed here is known as Choi-
Jamiołkowski (CJ) isomorphism [1, 2] in literature. While we provide an
overview of the concept and the corresponding mathematical framework, readers
seeking a more in-depth understanding can refer to Refs. [3, 4, 5] and the
cited references therein, which offer comprehensive insights into the topic.

Generally speaking, a linear map $\mathcal{E}_{A\to B}$ from linear operators
to linear operators is called a quantum channel if it satisfies the following
conditions:

  * •

Complete Positivity (CP);

  * •

Trace Preservation (TP).

Here, complete positivity means that by applying $\mathcal{E}_{A\to B}$ to
part of a quantum state $\rho_{A^{{}^{\prime}}A}$, the resultant state is
still positive semidefinite, namely
$\text{id}_{A^{{}^{\prime}}}\otimes\mathcal{E}_{A\to
B}(\rho_{A^{{}^{\prime}}A})\geqslant 0$. Meanwhile, as indicated by the name,
the trace-preserving property guarantees that
$\Tr[\mathcal{E}(\rho)]=\Tr[\rho]$ for all $\rho$. A powerful tool for
investigating the mathematical properties of quantum channels is the so-called
Choi-Jamiołkowski (CJ) isomorphism. Formally, it is defined as

Definition I.1:  Choi-Jamiołkowski Isomorphism [1, 2] Given a quantum channel
$\mathcal{E}_{A\to B}$, there is an isomorphism between map $\mathcal{E}_{A\to
B}$ and matrix $J^{\mathcal{E}}_{AB}$ (known as the Choi-Jamiołkowski operator
of channel $\mathcal{E}_{A\to B}$). Mathematically, $J^{\mathcal{E}}_{AB}$ is
defined as $\displaystyle
J^{\mathcal{E}}_{AB}:=\text{id}_{A}\otimes\mathcal{E}_{A^{{}^{\prime}}\to
B}(\Gamma_{AA^{{}^{\prime}}}),$ (1) where
$\Gamma_{AA^{{}^{\prime}}}:=\outerproduct{\Gamma}{\Gamma}_{AA^{{}^{\prime}}}$,
and $\ket{\Gamma}_{AA^{{}^{\prime}}}:=\sum_{i}\ket{ii}_{AA^{{}^{\prime}}}$
stands for the unnormalized maximally entangled state (UMES) with
$A^{{}^{\prime}}$ being a replica of system $A$, and $\\{\ket{i}\\}$ being an
orthonormal basis on $A$. For maximally entangled state (MES)
$\phi^{+}_{AA^{{}^{\prime}}}$, we define the Choi-Jamiołkowski state of
channel $\mathcal{E}_{A\to B}$ as
$\displaystyle\rho^{\mathcal{E}}_{AB}:=\text{id}_{A}\otimes\mathcal{E}_{A^{{}^{\prime}}\to
B}(\phi^{+}_{AA^{{}^{\prime}}}),$ (2) where
$\phi^{+}_{AA^{{}^{\prime}}}:=\outerproduct{\phi^{+}}{\phi^{+}}_{AA^{{}^{\prime}}}$,
and
$\ket{\phi^{+}}_{AA^{{}^{\prime}}}:=(1/\sqrt{d_{A}})\sum_{i}\ket{ii}_{AA^{{}^{\prime}}}$
with $d_{A}$ being the dimension of system $A$. It is now straightforward to
check that $\rho^{\mathcal{E}}_{AB}=J^{\mathcal{E}}_{AB}/d_{A}$.

Figure 1: (color online) Generations of the Choi-Jamiołkowski operator
$J^{\mathcal{E}}_{AB}$ (see Eq. 1) and state $\rho^{\mathcal{E}}_{AB}$ (see
Eq. 2). In (a), $\Gamma_{AA^{{}^{\prime}}}$ stands for
$\outerproduct{\Gamma}{\Gamma}_{AA^{{}^{\prime}}}$ with
$\ket{\Gamma}_{AA^{{}^{\prime}}}:=\sum_{i}\ket{ii}_{AA^{{}^{\prime}}}$. When
the unnormalized operator $\Gamma$ has been replaced by the maximally
entangled state $\phi^{+}$, we obtain the Choi-Jamiołkowski state
$\rho^{\mathcal{E}}_{AB}$, shown in (b).

See Fig. 1 for an illustration of Eq. 1 and Eq. 2.

######  Remark I.1.

In the fields of quantum information, computing, and condensed matter physics,
conventional diagrammatic techniques, such as tensor networks [6, 7, 8, 9] and
ZX-calculus [10, 11, 12, 13, 14], are widely used. Although our graphical
representation bears similarity to these existing techniques, there exists a
key difference in our approach. Specifically, in the conventional techniques,
a cup symbol (i.e., $\subset$) represents a vector $\ket{\Gamma}$, whereas in
our work, it represents the unnormalized maximally entangled state $\Gamma$.
Similarly, in the conventional approach, a cap symbol (i.e., $\supset$)
represents the effect $\bra{\Gamma}$. However, in our approach, the cup acting
on an operator $M$ represents the trace of $\Gamma\cdot M$, denoted as
$\Tr[\Gamma\cdot M]$. Finally, we note that the _yanking equations_ , which
relate the graphical notations in ZX-calculus, also hold in our graphical
language.

######  Remark I.2.

The Choi-Jamiołkowski operator for quantum effects can be defined by
substituting the quantum channel in Eq. 1 with a completely positive and trace
non-increasing map (CPTNI). This mathematical tool is crucial in understanding
the impact of quantum measurement and probabilistic protocols in quantum
information processing.

Equipped with Choi-Jamiołkowski isomorphism, we can now translate the
properties of completely positive and trace-preserving into the language of
Choi-Jamiołkowski operators. More precisely, a linear map $\mathcal{E}_{A\to
B}$ is completely positive if and only if its Choi-Jamiołkowski operator
$J^{\mathcal{E}}_{AB}$ is positive semidefinite, i.e.,
$J^{\mathcal{E}}_{AB}\geqslant 0$. Furthermore, such a map is trace-preserving
if and only if $J^{\mathcal{E}}_{AB}$ satisfies
$\Tr_{B}[J^{\mathcal{E}}_{AB}]=\mathbb{1}_{A}$. To conclude, given a linear
operator $J^{\mathcal{E}}_{AB}$, it represents a quantum channel from system
$A$ to system $B$ if both $J^{\mathcal{E}}_{AB}\geqslant 0$ and
$\Tr_{B}[J^{\mathcal{E}}_{AB}]=\mathbb{1}_{A}$ are satisfied.

A natural question that arises is how to determine the Choi-Jamiołkowski
operator of the sequential composition of two quantum channels, such as
$\mathcal{E}_{A\to B}$ and $\mathcal{F}_{B\to C}$, denoted by
$(\mathcal{F}\circ\mathcal{E})_{A\to C}$, in terms of $J^{\mathcal{E}}_{AB}$
and $J^{\mathcal{F}}_{BC}$. To tackle this question, we can utilize a useful
tool called the _link product_ $\star$. This product allows us to efficiently
compute the Choi-Jamiołkowski operator of the composite channel and is a
powerful technique for analyzing the properties of quantum channels.

Definition I.2:  Link Product [15, 16, 17] Given two Hermitian operators $M$
and $N$ with $X$ being the intersection of the systems on which M and N act,
their link product, denoted by $M\star N$, is defined as $M\star
N:=\Tr_{X}[M^{\mathbf{T}_{X}}\cdot N],$ (3) indicating that the link between
$M$ and $N$, namely system $X$, has been “swallowed” by the product $\star$.
Here $\Tr_{X}$ and $\mathbf{T}_{X}$ represent the partial trace and partial
transpose over the common system $X$ respectively, and the identity operator
$\mathbb{1}$ has been ignored.

With the help of the link product $\star$, we can now address the question
posed earlier regarding the Choi-Jamiołkowski operator of the composite
channel $(\mathcal{F}\circ\mathcal{E})_{A\to C}$ in terms of
$J^{\mathcal{E}}_{AB}$ and $J^{\mathcal{F}}_{BC}$. In this context, system $B$
serves as the link between $J^{\mathcal{E}}_{AB}$ and $J^{\mathcal{F}}_{BC}$,
and we can obtain the resultant Choi-Jamiołkowski operator
$J^{\mathcal{E}\circ\mathcal{F}}_{AC}$ as follows

$\displaystyle J^{\mathcal{F}\circ\mathcal{E}}_{AC}=J^{\mathcal{E}}_{AB}\star
J^{\mathcal{F}}_{BC}=\Tr_{B}[\left((J^{\mathcal{E}}_{AB})^{\mathbf{T}_{B}}\otimes\mathbb{1}_{C}\right)\cdot\left(\mathbb{1}_{A}\otimes
J^{\mathcal{F}}_{BC}\right)].$ (4)

A graphical visualization of link product appeared in Eq. 4 is demonstrated in
Fig. 2, where the equivalence between

Figure 2: (Color online) Physical realization (up to a scalar) of the Choi-
Jamiołkowski operator $J^{\mathcal{F}\circ\mathcal{E}}_{AC}$ (see Eq. 4 for
detail). Here the cup (i.e. $\subset$) stands for the preparation of
$\Gamma_{AA^{{}^{\prime}}}$, and the cap (i.e. $\supset$) represents the
effect of measuring the operator with respect to $\bra{\Gamma}$. In
particular, (a) illustrates the Choi-Jamiołkowski operator of channel
$\mathcal{F}\circ\mathcal{E}$. In (b), we re-derive the Choi-Jamiołkowski
operator $J^{\mathcal{F}\circ\mathcal{E}}$ by using the yanking equation,
which states that the snake line shown in blue dashed box of (b) is equal to
the line drawn in blue dashed box of (c).

Fig. 2(a) and Fig. 2(b) is obtained by using the yanking equation [10, 11, 12,
13]; that is

$\displaystyle\Tr_{B}[\left((J^{\mathcal{E}}_{AB})^{\mathbf{T}_{B}}\otimes\mathbb{1}_{C}\right)\cdot\left(\mathbb{1}_{A}\otimes
J^{\mathcal{F}}_{BC}\right)]=\Tr_{BB^{{}^{\prime\prime}}}[\left(J^{\mathcal{E}}_{AB^{{}^{\prime\prime}}}\otimes
J^{\mathcal{F}}_{BC}\right)\cdot(\mathbb{1}_{A}\otimes\Gamma_{B^{{}^{\prime\prime}}B}\otimes\mathbb{1}_{C})].$
(5)

The definition of the link product (e.g., Eq. 4) involves the partial
transpose $\mathbf{T}_{B}$, which is not completely positive and may seem
unphysical. However, our Eq. 5 and Fig. 2 demonstrate that the partial
transpose can be obtained by implementing Bell state measurements (up to a
scalar factor). This highlights the fact that seemingly unphysical operations
like $\mathbf{T}_{B}$ can have practical interpretation in quantum information
processing.

Let us discuss two specific scenarios to illustrate the usefulness of the link
product $\star$ (see Def. I.1): First, the link product helps us to
demonstrate the action of quantum channel in terms of its Choi-Jamiołkowski
operator. More precisely, given a quantum state $\rho$ acting on system $A$,
the quantum channel $\mathcal{E}_{A\to B}$ will transform it into

$\displaystyle\mathcal{E}(\rho)=\rho\star J^{\mathcal{E}}_{AB},$ (6)

where $J^{\mathcal{E}}_{AB}$ is the Choi-Jamiołkowski operator of channel
$\mathcal{E}$; Second, the _Born rule_ can be rewritten as a link product. In
particular, denote by $p_{x}$ the probability distribution obtained by
measuring $\rho$ with respect to $0\leqslant M_{x}\leqslant\mathbb{1}$, then
we have

$\displaystyle p_{x}=\rho\star M_{x}^{\mathbf{T}}.$ (7)

Here $M_{x}^{\mathbf{T}}$ stands for the Choi-Jamiołkowski operator of effect
$M_{x}$ (see Remark I.2).

######  Remark I.3.

The link product $\star$ is commutative under SWAP [18]. Take operators
$M_{AB}$ and $N_{BC}$ for instance, we have

$\displaystyle M_{AB}\star N_{BC}=\text{SWAP}_{A\leftrightarrow C}(N_{BC}\star
M_{AB}).$ (8)

Once the order of operations is determined, we can disregard the need for the
SWAP operation and treat the resulting link product as a commutative
operation. However, it is important to note that without the inclusion of the
SWAP operation, the property of commutativity no longer holds. For convenience
of presentation, in this work, we will fix the order of systems to make the
link product commutative. Let’s take Eq. 4 for example, when the
lexicographical order is considered, we have

$\displaystyle J^{\mathcal{F}\circ\mathcal{E}}_{AC}=J^{\mathcal{E}}_{AB}\star
J^{\mathcal{F}}_{BC}=J^{\mathcal{F}}_{BC}\star J^{\mathcal{E}}_{AB}.$ (9)

Similarly, Eqs. 6 and 7 can be rewritten as
$\mathcal{E}(\rho)=J^{\mathcal{E}}_{AB}\star\rho$ and
$p_{x}=M_{x}^{\mathbf{T}}\star\rho$ respectively. Furthermore, it is worth
noting that in addition to commutativity, the link product $\star$ is also
associative. This property allows for the grouping of multiple Choi-
Jamiołkowski operators without changing the outcome of their product and is a
useful feature in the analysis of complex network structures.

###  I.2 How to Quantify a Quantum Channel: Twist It

While there is a variety of ways to quantify quantum channels, good channel
quantifiers must have operational interpretations in some quantum information
processing or quantum computation tasks. In this subsection, we review two
such channel quantifiers, namely the average fidelity $f$ and the entanglement
fidelity $F$. Specifically, the average fidelity $f(\mathcal{E})$
characterizes the performance of a quantum channel $\mathcal{E}$ in
transmitting quantum information, and meanwhile the entanglement fidelity
$F(\mathcal{E})$ measures how $\mathcal{E}$ preserves the quantum correlations
between distant parties. Despite the obvious differences in mathematical
expression and physical interpretation, the average fidelity and the
entanglement fidelity enjoy a one-to-one correspondence, which was originally
proved by M. Horodecki, P. Horodecki, and R. Horodecki in Ref. [19]. Here we
provide an alternative proof of the one-to-one correspondence between average
fidelity and entanglement fidelity, showing a different perspective on these
channel quantifiers. Our mathematical toolkit includes the link product (see
Def. I.1) and twirling.

Let us start with the first channel quantifier: the average fidelity. For an
input state $\psi_{A}$ acting on system $A$, the quantum channel
$\mathcal{E}_{A\to B}$ will transform it into the state $\mathcal{E}(\psi)$,
and we can use the _Uhlmann fidelity_ $F_{\text{U}}$ [20, 21] to measure the
similarity between the input and output states of channel $\mathcal{E}_{A\to
B}$, i.e.,

$\displaystyle
F_{\text{U}}(\psi,\mathcal{E}(\psi))=\bra{\psi}\mathcal{E}(\psi)\ket{\psi}=\Tr[\psi\cdot\mathcal{E}(\psi)].$
(10)

In this work, we have adopted the notational convention that
$\psi:=\outerproduct{\psi}{\psi}$. Now the consideration of similarity between
the input and output states over all possible $\psi$ leads to the average
fidelity $f(\mathcal{E})$.

Definition I.3:  Average Fidelity The average fidelity $f(\mathcal{E}_{A\to
B})$ of a quantum channel $\mathcal{E}_{A\to B}$ is defined by $\displaystyle
f(\mathcal{E}_{A\to B}):=\int d\psi F_{\text{U}}(\psi_{B},\mathcal{E}_{A\to
B}(\psi_{A}))=\int d\psi\Tr[\psi_{B}\cdot\mathcal{E}_{A\to B}(\psi_{A})],$
(11) where the integral is over the _Haar measure_ , satisfying $\int
d\psi=1$. Notation $F_{\text{U}}$ stands for the Uhlmann fidelity.

The second channel quantifier considered in this work is the so-called
entanglement fidelity. Applying the quantum channel
$\mathcal{E}_{A^{{}^{\prime}}\to B}$ to part of the maximally entangled state
$\phi^{+}_{AA^{{}^{\prime}}}$ leads to the output state
$\text{id}_{A}\otimes\mathcal{E}_{A^{{}^{\prime}}\to
B}(\phi^{+}_{AA^{{}^{\prime}}})$, whose difference with the maximally
entangled state implies the entanglement fidelity $F(\mathcal{E})$.

Definition I.4:  Entanglement Fidelity The entanglement fidelity
$F(\mathcal{E}_{A\to B})$ of a quantum channel $\mathcal{E}_{A\to B}$ is
defined by $\displaystyle F(\mathcal{E}_{A\to
B}):=F_{\text{U}}(\phi^{+}_{AB},\mathcal{E}_{A^{{}^{\prime}}\to
B}(\phi^{+}_{AA^{{}^{\prime}}}))=\Tr[\phi^{+}_{AB}\cdot\left(\text{id}_{A}\otimes\mathcal{E}_{A^{{}^{\prime}}\to
B}(\phi^{+}_{AA^{{}^{\prime}}})\right)],$ (12) where $F_{\text{U}}$ stands for
the Uhlmann fidelity, with $\phi^{+}$ being the maximally entangled state.

######  Remark I.4.

Since the systems of quantum channels are usually clear from the context, the
notations of average fidelity $f(\mathcal{E}_{A\to B})$ (see Eq. 11) and
entanglement fidelity $F(\mathcal{E}_{A\to B})$ (see Eq. 12) will be
abbreviated to $f(\mathcal{E})$ and $F(\mathcal{E})$ respectively, and the
maximally entangled state will be shortened to $\phi^{+}$. Sometimes we will
also ignore the identity channel id. Adopting these conventions, the
cumbersome expressions can be simplified. For example, Eqs. 11 and 12 can be
rewritten as $f(\mathcal{E})=\int d\psi\Tr[\psi\cdot\mathcal{E}(\psi)]$ and
$F(\mathcal{E})=\Tr[\phi^{+}\cdot\mathcal{E}(\phi^{+})]$ respectively.

Symmetry is a fundamental concept in nature and is of great importance in
quantum information processing and computing. Twirling is a powerful technique
in quantum information theory that enables the analysis of quantum systems by
exploiting their symmetries. Specifically, twirling transforms a quantum state
in a symmetric manner, preserving its properties under certain operations.
This technique has numerous applications in quantum information theory, such
as the analysis of entanglement [22], quantum error correction [23, 24], and
communication protocols [19]. In this context, we can explore the symmetry of
quantum dynamics by applying different types of twirling. One such type is
channel twirling, denoted by $\mathcal{T}_{\text{ch}}$.

Definition I.5:  Channel Twirling Given a quantum channel $\mathcal{E}$, the
channel twirling of $\mathcal{E}$ is defined by
$\displaystyle\mathcal{T}_{\text{ch}}(\mathcal{E}):=\int
dU\,\mathcal{U}^{{\dagger}}\circ\mathcal{E}\circ\mathcal{U},$ (13) where the
integral is over the _Haar measure_ , with
$\mathcal{U}(\rho):=U\,\rho\,U^{{\dagger}}$ and
$\mathcal{U}^{{\dagger}}(\rho):=U^{{\dagger}}\,\rho\,U$. Notation $\circ$
stands for the composition of channels, and † represents the _Hermitian
adjoint_ of operators.

It is worth mentioning that both the average fidelity $f(\mathcal{E})$ (see
Def. I.2) and entanglement fidelity $F(\mathcal{E})$ (see Def. I.2) are
invariant under channel twirling $\mathcal{T}_{\text{ch}}$. To make our work
self-contained, we will provide the formal statements about these properties,
including their proofs.

Lemma I.6:  Invariance of Average Fidelity under Channel Twirling The average
fidelity $f(\mathcal{E})$ (see Eq. 11) of a quantum channel $\mathcal{E}$ is
invariant under channel twirling $\mathcal{T}_{\text{ch}}$ (see Eq. 13); that
is $\displaystyle f(\mathcal{T}_{\text{ch}}(\mathcal{E}))=f(\mathcal{E}).$
(14)

###### Proof.

According to Def. I.2 and Def. I.2, we have

$\displaystyle f(\mathcal{T}_{\text{ch}}(\mathcal{E}))$ $\displaystyle=\int
d\psi\Tr[\psi\cdot\mathcal{T}_{\text{ch}}(\mathcal{E})(\psi)]$ (15)
$\displaystyle=\int d\psi\int
dU\,\Tr[\psi\cdot\mathcal{U}^{{\dagger}}\circ\mathcal{E}\circ\mathcal{U}(\psi)]$
(16) $\displaystyle=\int d\psi\int
dU\,\Tr[\mathcal{U}(\psi)\cdot\mathcal{E}(\mathcal{U}(\psi))]$ (17)
$\displaystyle=\int dUf(\mathcal{E})$ (18) $\displaystyle=f(\mathcal{E}),$
(19)

which completes the proof. ∎

Lemma I.7:  Invariance of Entanglement Fidelity under Channel Twirling The
entanglement fidelity $F(\mathcal{E})$ (see Eq. 12) of a quantum channel
$\mathcal{E}$ is invariant under channel twirling $\mathcal{T}_{\text{ch}}$
(see Eq. 13); that is $\displaystyle
F(\mathcal{T}_{\text{ch}}(\mathcal{E}))=F(\mathcal{E}).$ (20)

###### Proof.

Note that $\overline{U}\otimes U\ket{\phi^{+}}=\ket{\phi^{+}}$, where
$\overline{U}$ stands for the complex conjugate of $U$. Hence, we obtain

$\displaystyle F(\mathcal{T}_{\text{ch}}(\mathcal{E}))$
$\displaystyle=\Tr[\phi^{+}\cdot\mathcal{T}_{\text{ch}}(\mathcal{E})(\phi^{+})]$
(21)
$\displaystyle=\Tr[\phi^{+}\cdot\left(\mathcal{U}^{{\dagger}}\circ\mathcal{E}\circ\mathcal{U}(\phi^{+})\right)]$
(22) $\displaystyle=\Tr[\left(\mathbb{1}\otimes U\,\phi^{+}\,\mathbb{1}\otimes
U^{{\dagger}}\right)\cdot\left(U^{\mathbf{T}}\otimes\mathbb{1}\,\mathcal{E}(\phi^{+})\,\overline{U}\otimes\mathbb{1}\right)]$
(23) $\displaystyle=\Tr[\left(\overline{U}\otimes
U\,\phi^{+}\,U^{\mathbf{T}}\otimes
U^{{\dagger}}\right)\cdot\mathcal{E}(\phi^{+})]$ (24)
$\displaystyle=\Tr[\phi^{+}\cdot\mathcal{E}(\phi^{+})]$ (25)
$\displaystyle=F(\mathcal{E}),$ (26)

which completes the proof of the lemma. ∎

Another type of twirling is the isotropic twirling $\mathcal{T}_{\text{iso}}$
[25], which was originally introduced by M. Horodecki and P. Horodecki in
formulating the reduction criterion of separability.

Definition I.8:  Isotropic Twirling Given a bipartite state $\rho$, the
isotropic twirling of $\rho$ is defined by
$\displaystyle\mathcal{T}_{\text{iso}}(\rho):=\int
dU\,U\otimes\overline{U}\,\rho\,U^{{\dagger}}\otimes U^{\mathbf{T}},$ (27)
where the integral is over the _Haar measure_.

######  Remark I.5.

Despite their differences in expressions, channel twirling
$\mathcal{T}_{\text{ch}}$ (see Eq. 13) and isotropic twirling
$\mathcal{T}_{\text{iso}}$ (see Eq. 27) are closely related with each other.
In particular, the Choi-Jamiołkowski operator
$J^{\mathcal{T}_{\text{ch}}(\mathcal{E})}$ of channel
$\mathcal{T}_{\text{ch}}(\mathcal{E})$ is equal to
$\mathcal{T}_{\mathrm{iso}}(J^{\mathcal{E}})$, obtained by implementing the
isotropic twirling $\mathcal{T}_{\text{iso}}$ to the Choi-Jamiołkowski
operator of channel $\mathcal{E}$. A graphic proof of above statement, i.e.,
$J^{\mathcal{T}_{\text{ch}}(\mathcal{E})}=\mathcal{T}_{\text{iso}}(J^{\mathcal{E}})$,
is given in Fig. 3.

Figure 3: (Color online) Equivalence between the Choi-Jamiołkowski operators
of $J^{\mathcal{U}^{{\dagger}}\circ\mathcal{E}\circ\mathcal{U}}$ in (a) and
$U^{\mathbf{T}}\otimes U^{{\dagger}}\,J^{\mathcal{E}}\,\overline{U}\otimes U$
in (b), which leads to the equation
$J^{\mathcal{T}_{\text{ch}}(\mathcal{E})}=\mathcal{T}_{\text{iso}}(J^{\mathcal{E}})$
as required.

The isotropic twirling $\mathcal{T}_{\text{iso}}$ is indeed a quantum channel,
and thus its Choi-Jamiołkowski operator $J^{\mathcal{T}_{\text{iso}}}$
completely characterizes the mathematical properties of
$\mathcal{T}_{\text{iso}}$. For more related results, we refer the interested
readers to the work of E. M. Rains [22], which also contains the prototype of
link product (see Def. I.1).

Lemma I.9:  Choi-Jamiołkowski Operator of Isotropic Twirling [22] Assume the
isotropic twirling $\mathcal{T}_{\text{iso}}$ is a map from systems
$A_{1}\otimes B_{1}$ to $A_{2}\otimes B_{2}$, with $d:=\dim A_{1}=\dim
A_{2}=\dim B_{1}=\dim B_{2}$, then its Choi-Jamiołkowski operator is given by
$\displaystyle
J^{\mathcal{T}_{\text{iso}}}_{A_{1}B_{1}A_{2}B_{2}}=\phi^{+}_{A_{1}B_{1}}\otimes\phi^{+}_{A_{2}B_{2}}+\frac{1}{d^{2}-1}(\mathbb{1}_{A_{1}B_{1}}-\phi^{+}_{A_{1}B_{1}})\otimes(\mathbb{1}_{A_{2}B_{2}}-\phi^{+}_{A_{2}B_{2}}),$
(28) where $\phi^{+}$ is the maximally entangled state. By using the
conventional abbreviations $A:=A_{1}A_{2}$, $B:=B_{1}B_{2}$,
$\mathfrak{1}:=A_{1}B_{1}$, and $\mathfrak{2}:=A_{2}B_{2}$, Eq. 28 can be
simplified to
$J^{\mathcal{T}_{\text{iso}}}_{AB}=\phi^{+}_{\mathfrak{1}}\otimes\phi^{+}_{\mathfrak{2}}+(\mathbb{1}_{\mathfrak{1}}-\phi^{+}_{\mathfrak{1}})\otimes(\mathbb{1}_{\mathfrak{2}}-\phi^{+}_{\mathfrak{2}})/(d^{2}-1)$.

We now wish to show that the output state of isotropic twirling
$\mathcal{T}_{\text{iso}}:\mathfrak{1}\to\mathfrak{2}$ is always a noisy
singlet, i.e. in the form of
$p\phi^{+}_{\mathfrak{2}}+(1-p)\mathbb{1}_{\mathfrak{2}}/d^{2}$. To do so, let
us consider the link product between a given state $\rho_{\mathfrak{1}}$
acting on system $\mathfrak{1}$ and the Choi-Jamiołkowski operator of
$\mathcal{T}_{\text{iso}}$. Written out explicitly, we have

$\displaystyle\mathcal{T}_{\text{iso}}(\rho_{\mathfrak{1}})$
$\displaystyle=J^{\mathcal{T}_{\text{iso}}}_{AB}\star\rho_{\mathfrak{1}}$ (29)
$\displaystyle=\left(\phi^{+}_{\mathfrak{1}}\otimes\phi^{+}_{\mathfrak{2}}+\frac{1}{d^{2}-1}(\mathbb{1}_{\mathfrak{1}}-\phi^{+}_{\mathfrak{1}})\otimes(\mathbb{1}_{\mathfrak{2}}-\phi^{+}_{\mathfrak{2}})\right)\star\rho_{\mathfrak{1}}$
(30)
$\displaystyle=\Tr[\phi^{+}_{\mathfrak{1}}\cdot\rho_{\mathfrak{1}}]\phi^{+}_{\mathfrak{2}}+\frac{(1-\Tr[\phi^{+}_{\mathfrak{1}}\cdot\rho_{\mathfrak{1}}])}{d^{2}-1}(\mathbb{1}_{\mathfrak{2}}-\phi^{+}_{\mathfrak{2}})$
(31)
$\displaystyle=\frac{\Tr[\phi^{+}_{\mathfrak{1}}\cdot\rho_{\mathfrak{1}}]d^{2}-1}{d^{2}-1}\phi^{+}_{\mathfrak{2}}+\frac{(1-\Tr[\phi^{+}_{\mathfrak{1}}\cdot\rho_{\mathfrak{1}}])d^{2}}{d^{2}-1}\frac{\mathbb{1}_{\mathfrak{2}}}{d^{2}},$
(32)

implying that

Theorem I.10:  Noisy Singlet Assume the isotropic twirling
$\mathcal{T}_{\text{iso}}$ is a map from systems $A_{1}\otimes B_{1}$ to
$A_{2}\otimes B_{2}$ (i.e., $A_{1}B_{1}\to A_{2}B_{2}$), with $d:=\dim
A_{1}=\dim A_{2}=\dim B_{1}=\dim B_{2}$, and $\rho_{\mathfrak{1}}$ being an
input state acting on system $A_{1}\otimes B_{1}$, then the output state under
$\mathcal{T}_{\text{iso}}$ is a noisy singlet, i.e., $\displaystyle
p\phi^{+}_{\mathfrak{2}}+(1-p)\frac{\mathbb{1}_{\mathfrak{2}}}{d^{2}},$ (33)
with the parameter $p$ being characterized by $\displaystyle
p=\frac{\Tr[\phi^{+}_{\mathfrak{1}}\cdot\rho_{\mathfrak{1}}]d^{2}-1}{d^{2}-1}.$
(34) Here we have adopted the notational conventions that
$\mathfrak{1}:=A_{1}B_{1}$, and $\mathfrak{2}:=A_{2}B_{2}$.

Let us turn our attention to the output channel of channel twirling
$\mathcal{T}_{\text{ch}}:A_{2}\to A_{1},B_{1}\to B_{2}$. For any input channel
$\mathcal{E}:A_{1}\to B_{1}$, the Choi-Jamiołkowski operator of output channel
$\mathcal{T}_{\text{ch}}(\mathcal{E})$ is then given by

$\displaystyle J^{\mathcal{T}_{\text{ch}}(\mathcal{E})}_{\mathfrak{2}}$
$\displaystyle=\mathcal{T}_{\text{iso}}(J^{\mathcal{E}}_{\mathfrak{1}})$ (35)
$\displaystyle=\left(\phi^{+}_{\mathfrak{1}}\otimes\phi^{+}_{\mathfrak{2}}+\frac{1}{d^{2}-1}(\mathbb{1}_{\mathfrak{1}}-\phi^{+}_{\mathfrak{1}})\otimes(\mathbb{1}_{\mathfrak{2}}-\phi^{+}_{\mathfrak{2}})\right)\star
J^{\mathcal{E}}_{\mathfrak{1}}$ (36)
$\displaystyle=\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}]\phi^{+}_{\mathfrak{2}}+\frac{(d-\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}])}{d^{2}-1}(\mathbb{1}_{\mathfrak{2}}-\phi^{+}_{\mathfrak{2}})$
(37) $\displaystyle=\frac{\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}]d^{2}-d}{d(d^{2}-1)}\Gamma_{\mathfrak{2}}+\frac{(d-\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}])d}{d^{2}-1}\frac{\mathbb{1}_{\mathfrak{2}}}{d},$
(38)

where the first equality (i.e., Eq. 35) has been investigated in Remark. I.5,
the second equality (i.e., Eq. 36) is a result of Lem. I.2, and the third
equality (i.e., Eq. 37) follows from the property of trace-preservation;
namely,

$\displaystyle\mathbb{1}_{\mathfrak{1}}\star
J^{\mathcal{E}}_{\mathfrak{1}}=\Tr_{A_{1}}[\mathbb{1}_{A_{1}}]=d.$ (39)

Thanks to the Choi-Jamiołkowski isomorphism, we know that
$\Gamma_{\mathfrak{2}}$ represents the Choi-Jamiołkowski operator of the
noiseless quantum channel $\text{id}_{A_{2}\to B_{2}}$. Meanwhile,
$\mathbb{1}_{\mathfrak{2}}/d$ is the Choi-Jamiołkowski operator of the
replacement channel $\mathbb{1}_{B_{2}}/d\otimes\Tr_{A_{2}}$. Written in full,
that is

$\displaystyle\mathcal{T}_{\text{ch}}(\mathcal{E})=\left(\frac{\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}]d^{2}-d}{d(d^{2}-1)}\right)\text{id}_{A_{2}\to
B_{2}}+\left(\frac{(d-\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}])d}{d^{2}-1}\right)\frac{\mathbb{1}_{B_{2}}}{d}\otimes\Tr_{A_{2}}.$
(40)

To conclude, we have the following theorem for channel twirling
$J^{\mathcal{T}_{\text{ch}}}$.

Theorem I.11:  Depolarizing Channel Given a channel $\mathcal{E}:A_{1}\to
B_{1}$, the channel twirling $\mathcal{T}_{\text{ch}}:A_{2}\to A_{1},B_{1}\to
B_{2}$ will transform it into a depolarizing channel, i.e.,
$\displaystyle\mathcal{T}_{\text{ch}}(\mathcal{E})=p\,\text{id}_{A_{2}\to
B_{2}}+(1-p)\,\frac{\mathbb{1}_{B_{2}}}{d}\otimes\Tr_{A_{2}},$ (41) where the
parameter $p$ is given by $\displaystyle
p=\frac{\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}]d^{2}-d}{d(d^{2}-1)},$ (42) with $d:=\dim
A_{1}=\dim A_{2}=\dim B_{1}=\dim B_{2}$.

Before the end of this subsection, let us return to the average fidelity
$f(\mathcal{E})$ (see Def. I.2) and the entanglement fidelity $F(\mathcal{E})$
(see Def. I.2). In Ref. [19], M. Horodecki, P. Horodecki, and R. Horodecki
find the essential connection between $f(\mathcal{E})$ and $F(\mathcal{E})$,
in particular that

Lemma I.12:  Connection between Average Fidelity and Entanglement Fidelity
[19] Given a quantum channel $\mathcal{E}:A\to B$, its average fidelity
$f(\mathcal{E})$ (see Def. I.2) and entanglement fidelity $F(\mathcal{E})$
(see Def. I.2) satisfy the following equation, $\displaystyle
f(\mathcal{E})=\frac{dF(\mathcal{E})+1}{d+1},$ (43) with $d:=\dim A=\dim B$.

######  Remark I.6.

There are several ways to prove Lem. I.2. The original proof was given by M.
Horodecki, P. Horodecki, and R. Horodecki in Ref. [19], which had later been
simplified by M. A. Nielsen in Ref. [26]. Here we will provide an alternative
proof based on the Choi-Jamiołkowski operator of isotropic twirling
$\mathcal{T}_{\text{iso}}$ (see Lem. I.2) and the link product $\star$ (see
Def. I.1). In particular, our method relies heavily on Thm. I.2 presented in
this subsection, whose proof is completed by using the Choi-Jamiołkowski
operator of isotropic twirling and the link product. Hence, we conclude that
our proof of Lem. I.2 is based on $J^{\mathcal{T}_{\text{iso}}}$ and $\star$.
Besides its fundamental importance in understanding channel quantifiers, Lem.
I.2 is also a key to characterizing general quantum teleportation [19].

###### Proof.

To prove this claim, we start with the input state $\psi$ and consider the
corresponding output under quantum channel
$\mathcal{T}_{\text{ch}}(\mathcal{E})$. Thanks to Eq. 41 of Thm. I.2, we
immediately obtain

$\displaystyle\mathcal{T}_{\text{ch}}(\mathcal{E})(\psi)=p\,\psi+(1-p)\,\frac{\mathbb{1}}{d},$
(44)

with $p$ given by Eq. 42. Thus, by using Lem. I.2, we see that

$\displaystyle
f(\mathcal{E})=f(\mathcal{T}_{\text{ch}}(\mathcal{E}))=p+\frac{1-p}{d}=\frac{p(d-1)+1}{d}.$
(45)

On the other hand, the entanglement fidelity $F(\mathcal{E})$ of channel
$\mathcal{E}$ is equal to $\Tr[\phi^{+}_{\mathfrak{1}}\cdot
J^{\mathcal{E}}_{\mathfrak{1}}]/d$, implying that

$\displaystyle p=\frac{F(\mathcal{E})d^{2}-1}{d^{2}-1}.$ (46)

Combining Eq. 45 with Eq. 46 leads to Eq. 43, as desired. ∎

###  I.3 Quantum Superchannels: Pre-Processing, Post-Processing, and Non-
Signalling

Exploring quantum channels primarily involves manipulating them in specific
contexts. For instance, when dealing with a noisy quantum channel, determining
its capacity to transmit information through the use of encoder and decoder
leads to the development of quantum information theory – a rapidly evolving
and dynamic field that seeks to unleash the extraordinary capabilities of
quantum mechanics in revolutionizing information processing. Mathematically,
the sender’s encoder and the receiver’s decoder can be represented as a
bipartite quantum operation known as a _superchannel_ [27]. This concept
encapsulates the most general way of manipulating a quantum channel, including
the channel twirling $\mathcal{T}_{\text{ch}}$ (see Def. I.2) discussed in
Subsec. I.2. Quantum superchannels also constitute a fundamental tool for
understanding dynamical resources and the associated resource measures [28,
29, 30]. In this subsection, we provide a rigorous introduction to the basic
concepts of quantum superchannels and investigate their properties using the
powerful Choi-Jamiołkowski isomorphism (see Def. I.1).

A quantum superchannel is formed by combining two quantum channels, which can
be implemented in sequence and connected through a quantum memory in
accordance with the principle of causality in the theory of relativity [31,
32, 33]. This causal structure is formally known as _non-signalling_ (NS)
between the parties involved in quantum coding protocols. A comprehensive
review of non-signalling codes is beyond the scope of this discussion.
Interested readers are referred to Ref. [34] and the references therein for a
more detailed introduction. In the context of causal boxes, the causality
property of quantum superchannels is referred to as _semicausality_ [35, 36,
37]. The use of the Choi-Jamiołkowski isomorphism has proven to be a powerful
tool in analyzing the properties of semicausal maps and their relation to
_semilocalizability_ , as discussed in previous works [38]. Now let us start
with the definition of non-signalling.

Definition I.13:  Non-Signalling A bipartite channel $\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}$ is non-signalling from $B:=B_{1}B_{2}$ to $A:=A_{1}A_{2}$,
denoted as $B\not\to A$, if
$\displaystyle\Tr_{B_{2}}\circ\,\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}=\Tr_{B_{1}}\otimes\,\mathcal{E}_{A_{1}\to A_{2}},$ (47) holds for
some quantum channel $\mathcal{E}_{A_{1}\to A_{2}}$. Similarly, a bipartite
channel $\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ is non-signalling (NS) from $A\to
B$, denoted as $A\not\to B$, if
$\displaystyle\Tr_{A_{2}}\circ\,\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}=\Tr_{A_{1}}\otimes\,\mathcal{F}_{B_{1}\to B_{2}},$ (48) holds for
some quantum channel $\mathcal{F}_{B_{1}\to B_{2}}$. A map $\Theta$ is non-
signalling if it is both $A\not\to B$ and $B\not\to A$.

Physically, non-signalling from $B$ to $A$ means that no action on system $B$
can cause any detectable effect on system $A$. A visualization of non-
signalling bipartite channels is provided in Fig. 4.

Figure 4: (Color online) Demonstration of non-signalling bipartite channel
$\Theta$. A general bipartite quantum channel $\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}$ is depicted in (a), where systems $A_{1}$ and $A_{2}$ belongs to
Alice, systems $B_{1}$ and $B_{2}$ belongs to Bob. If the bipartite channel
$\Theta$ is non-signalling from $B$ to $A$, i.e., Bob’s action cannot affect
Alice, we arrive at the top of (b). In other words, tracing out system $B_{2}$
will lead to the form of $\mathcal{E}_{A_{1}\to A_{2}}\otimes\Tr_{B_{1}}$ (see
Eq. 47) for some quantum channel $\mathcal{E}$ on Alice’s side, shown in the
top box of (c). On the other hand, if the bipartite channel $\Theta$ is non-
signalling from $A$ to $B$, i.e. Alice’s behavior won’t influence Bob, the
bipartite channel $\Theta$ can be transformed into the form illustrated in the
bottom of (b). In this case, discarding system $A_{2}$ will imply the
decomposition of bipartite operation between Alice and Bob; that is
$\Tr_{A_{1}}\otimes\mathcal{F}_{B_{1}\to B_{2}}$ (see Eq. 48) with
$\mathcal{F}$ being a quantum acting on Bob’s side, painted in the bottom box
of (c).

Given a quantum channel $\mathcal{E}_{A_{2}\to B_{1}}$, the general quantum
supermap $\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ that maps $\mathcal{E}_{A_{2}\to
B_{1}}$ to a channel $\Theta(\mathcal{E})_{A_{1}\to B_{2}}$, called
superchannel, satisfies the following conditions [27]:

  * •

Completely Positivity (CP);

  * •

Trace Preservation (TP);

  * •

Non-Signalling (NS) from $B$ to $A$.

Similar to the construction of Choi-Jamiołkowski operator for a quantum
channel (see Def. I.1 and Fig. 1), we can introduce the Choi-Jamiołkowski
operator for superchannels. In particular, the Choi-Jamiołkowski operator for
superchannel $\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ is defined as

$\displaystyle
J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}:=\text{id}_{A_{1}}\otimes\Theta_{A^{{}^{\prime}}_{1}B^{{}^{\prime}}_{1}\to
A_{2}B_{2}}\otimes\text{id}_{B_{1}}(\Gamma_{A_{1}A^{{}^{\prime}}_{1}}\otimes\Gamma_{B_{1}B^{{}^{\prime}}_{1}}),$
(49)

where $\Gamma_{A_{1}A^{{}^{\prime}}_{1}}$ and
$\Gamma_{B_{1}B^{{}^{\prime}}_{1}}$ are unnormalized maximally entangled
states acting on systems $A_{1}A^{{}^{\prime}}_{1}$ and
$B_{1}B^{{}^{\prime}}_{1}$ respectively. An illustration of Eq. 49 is given in
Fig. 5(a), including the corresponding quantum circuit realization (see Fig.
5(b)).

Figure 5:  (Color online) Pictorial representation of the Choi-Jamiołkowski
operator $J^{\Theta}_{A_{1}B_{1}\to A_{2}B_{2}}$ (see Eq. 49) for superchannel
$\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ that is non-signalling from $B$ to $A$
(a), and circuit realization of normalized Choi-Jamiołkowski state
$\rho^{\Theta}_{A_{1}B_{1}\to A_{2}B_{2}}$ (b). Here
$\rho^{\Theta}_{A_{1}B_{1}\to A_{2}B_{2}}=J^{\Theta}_{A_{1}B_{1}\to
A_{2}B_{2}}/d_{A_{1}}d_{B_{1}}$.

Using the language of Choi-Jamiołkowski isomorphism, we say that a quantum
supermap $\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ is completely positive if and
only if its Choi-Jamiołkowski operator $J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}$ is
positive semidefinite, namely $J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}\geqslant 0$.
It is said to be trace-preserving if and only if
$\Tr_{B_{1}B_{2}}J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}=\mathbb{1}_{A_{1}A_{2}}$.
Finally, such a supermap is non-signalling from $B$ to $A$ if and only if its
Choi-Jamiołkowski operator $J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}$ satisfies

$\displaystyle\Tr_{B_{2}}[J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}]=J^{\mathcal{E}}_{A_{1}A_{2}}\otimes\mathbb{1}_{B_{1}},$
(50)

where $J^{\mathcal{E}}_{A_{1}A_{2}}$ stands for the Choi-Jamiołkowski operator
of some quantum channel $\mathcal{E}$ from $A_{1}$ to $A_{2}$.

Equivalently, a superchannel $\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ can also be
viewed as the sequential composition of two quantum channels; that is

$\displaystyle\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}=\Theta^{\text{post}}_{B_{1}E\to
B_{2}}\circ\Theta^{\text{pre}}_{A_{1}\to A_{2}E},$ (51)

where $E$ is a memory system between $\Theta^{\text{pre}}$ and
$\Theta^{\text{post}}$ (see Fig. 6).

Figure 6:  (Color online) Decomposition of superchannel $\Theta_{A_{1}B_{1}\to
A_{2}B_{2}}$ (see Eq. 51). In other words, superchannel $\Theta$ (a) can be
realized by preparing a pre-processing $\Theta^{\text{pre}}_{A_{1}\to A_{2}E}$
and a post-processing $\Theta^{\text{post}}_{B_{1}E\to B_{2}}$, which are
connected by a hidden memory system $E$ (b).

Here, quantum channels $\Theta^{\text{pre}}$ and $\Theta^{\text{post}}$ are
known as the pre-processing and post-processing of $\Theta$ respectively.
Denote their Choi-Jamiołkowski operators as $J^{\text{pre}}$ and
$J^{\text{post}}$, then it is straightforward to write down the Choi-
Jamiołkowski operator of superchannel $\Theta$ by using link product $\star$
(see Def. I.1), i.e.,

$\displaystyle
J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}=J^{\text{pre}}_{A_{1}A_{2}}\star
J^{\text{post}}_{B_{1}B_{2}}.$ (52)

For any quantum channel $\mathcal{E}_{A_{2}\to B_{1}}$, the Choi-Jamiołkowski
operator of resultant channel $\Theta(\mathcal{E})_{A_{1}\to B_{2}}$ under
$\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$ is characterized by

$\displaystyle
J^{\Theta(\mathcal{E})}_{A_{1}B_{2}}=J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}\star
J^{\mathcal{E}}_{A_{2}B_{1}}=\Tr_{A_{2}B_{1}}[J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}\cdot\left((J^{\mathcal{E}}_{A_{2}B_{1}})^{\mathbf{T}}\otimes\mathbb{1}_{A_{1}B_{2}}\right)].$
(53)

Then, for an input state $\rho_{A_{1}}$, the output state of channel
$\Theta(\mathcal{E})_{A_{1}\to B_{2}}$ reads

$\displaystyle\Theta(\mathcal{E})_{A_{1}\to
B_{2}}(\rho_{A_{1}})=J^{\Theta(\mathcal{E})}_{A_{1}B_{2}}\star\rho_{A_{1}}=J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}\star
J^{\mathcal{E}}_{A_{2}B_{1}}\star\rho_{A_{1}}=\Tr_{A_{2}B_{1}}[J^{\Theta}_{A_{1}B_{1}A_{2}B_{2}}\cdot\left(\rho_{A_{1}}^{\mathbf{T}}\otimes(J^{\mathcal{E}}_{A_{2}B_{1}})^{\mathbf{T}}\otimes\mathbb{1}_{B_{2}}\right)].$
(54)

The formula presented above is well-defined due to the commutativity and
associativity properties of the link product $\star$, as noted in Remark I.3.

###  I.4 Quantum Circuit Fragments: Framework of Non-Markovian Dynamics

Quantum channels are a central concept in quantum mechanics, characterizing
the time-evolution of quantum systems between two different time points (see
Fig. 1). The most general way of manipulating quantum channels is through the
use of quantum superchannels, which describe the non-Markovian dynamics of
four time points (see Fig. 5). These superchannels capture the complex
interplay between system and memory, allowing for a more comprehensive
understanding of quantum dynamics. A notable example is the superchannel
$\Theta_{A_{1}B_{1}\to A_{2}B_{2}}$, shown in Fig. 6(b). This superchannel
captures the system-memory dynamics from $A_{1}$ to $A_{2}$ and from $B_{1}$
to $B_{2}$, via the composition of two channels, i.e.,
$\Theta^{\text{pre}}_{A_{1}\to A_{2}E}$ and $\Theta^{\text{post}}_{B_{1}E\to
B_{2}}$. The study of non-Markovian quantum processes that involve multiple
time points is critical for various fields, such as distributed quantum
computing [39, 40, 41], cloud-based quantum computing [42, 43, 44, 45],
quantum network communication [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58], quantum-enhanced agents [59, 60, 61, 62], quantum metrology [63, 64, 65],
open quantum systems [66, 67, 68], quantum clocks and sensors [69, 70, 71, 72,
73]. Furthermore, it has significant implications for emerging areas such as
quantum biology [74], making it a topic of utmost importance. Generally, non-
Markovian quantum processes offer a window into the interplay between system
and environment, allowing us to explore the rich landscape of quantum
phenomena that arise from it. In this subsection, we introduce the concept of
_quantum circuit fragment_ as a general framework for understanding these
processes. The corresponding Choi-Jamiołkowski operator, which characterizes
the quantum circuit fragment, is known as the _quantum comb_ , and has been
extensively investigated in literature [15, 16, 17].

Consider the collection of all quantum circuit fragments consisting of $k$
sequential quantum channels, denoted as $\mathfrak{F}_{k}$. It then follows
that for any given point-to-point quantum channel $\mathcal{E}$, we have

$\displaystyle\mathcal{E}\in\mathfrak{F}_{1}.$ (55)

Here, the set $\mathfrak{F}_{1}$ represents the exact set of all quantum
channels, encompassing the full range of physical operations that can be used
to manipulate quantum states. Notably, both the set of quantum states, denoted
by $\mathfrak{S}_{\text{State}}$, and the set of positive operator-valued
measures (POVMs), denoted by $\mathfrak{S}_{\text{POVM}}$, are subsets of
$\mathfrak{F}_{1}$ [4]. Writing everything out explicitly, we have

$\displaystyle\mathfrak{S}_{\text{State}}\cup\mathfrak{S}_{\text{POVM}}\subset\mathfrak{F}_{1}=\mathfrak{S}_{\text{Channel}},$
(56)

where $\mathfrak{S}_{\text{Channel}}$ stands for the collection of all quantum
channels. The set $\mathfrak{F}_{2}$, on the other hand, refers to the
collection of all quantum superchannels. In the context of quantum causal
inference [75, 76, 77], the set of causal maps under investigation is denoted
as $\mathfrak{S}_{\text{CMap}}$. By examining the relationship between these
two sets, it becomes evident that $\mathfrak{S}_{\text{CMap}}$ is naturally
embedded within $\mathfrak{F}_{2}$, as all causal maps can be realized as
special cases of superchannels.

$\displaystyle\mathfrak{S}_{\text{CMap}}\subset\mathfrak{F}_{2}=\mathfrak{S}_{\text{SChannel}}.$
(57)

Here $\mathfrak{S}_{\text{SChannel}}$ is the set of all quantum superchannels.
Let us now turn our attention to the case of a general set $\mathfrak{F}_{k}$,
which comprises elements that consist of $k$ sequential quantum channels.
Consider a sequence of quantum channels
${\mathcal{E}_{1},\mathcal{E}_{2},\ldots,\mathcal{E}_{k}}$ that belongs to
$\mathfrak{F}_{1}$ and possesses the following defining characteristics:

$\displaystyle\mathcal{E}_{1}:\mathcal{H}_{0}\to\mathcal{H}_{1}E_{1},$ (58)
$\displaystyle\mathcal{E}_{i}:\mathcal{H}_{2i-2}E_{i-1}\to\mathcal{H}_{2i-1}E_{i},\,\,\,\,\,\,\,\,2\leqslant
i\leqslant k-1$ (59)
$\displaystyle\mathcal{E}_{k}:\mathcal{H}_{2k-2}E_{k-1}\to\mathcal{H}_{2k-1}.$
(60)

In this context, it is important to note that intermediate channels typically
have both an environment input system and an environment output system. In
contrast, the first channel in the sequence generally lacks an environment
input system, while the final channel lacks an environment output system. In
our analysis of quantum circuits, we adopt the term “quantum circuit fragment”
[77] to refer to the overall dynamics of the circuit, which is composed of a
sequence of $k$ quantum channels. In full, a quantum circuit fragment can be
expressed as the composite of $k$ quantum channels, denoted as

$\displaystyle\mathcal{E}:=\mathcal{E}_{k}\circ\mathcal{E}_{k-1}\circ\cdots\circ\mathcal{E}_{2}\circ\mathcal{E}_{1}.$
(61)

Quantum circuit fragment (see Eq. 61) offers us the most general way of
demonstrating quantum dynamics of non-Markovian processes [78, 79, 80, 81,
82], including the time evolution of quantum systems under environment
coupling and designing and operating quantum devices in quantum computing. The
corresponding Choi-Jamiołkowski operator is called quantum comb [15, 16, 17].
See Fig. 7 for an illustration of the dynamics of quantum circuit fragment
$\mathcal{E}$ introduced in Eq. 61 and its Choi-Jamiołkowski state
$\rho^{\mathcal{E}}$.

Figure 7: (Color online) The quantum circuit fragment $\mathcal{E}$ (a) and
the implementation of its Choi-Jamiołkowski state $\rho^{\mathcal{E}}$ (b).
Here $\mathcal{E}$ is constituted of $k$ sequential quantum channels
$\mathcal{E}_{1},\mathcal{E}_{2},\ldots,\mathcal{E}_{k}$ (see Eq. 61).

Denote the Choi-Jamiołkowski operator of channel $\mathcal{E}_{i}$
($1\leqslant i\leqslant k$) as $J^{\mathcal{E}_{i}}$, then the quantum comb
$J^{\mathcal{E}}$ can be re-expressed as the link product between channels,
namely

$\displaystyle J^{\mathcal{E}}=J^{\mathcal{E}_{1}}\star
J^{\mathcal{E}_{2}}\star\cdots\star J^{\mathcal{E}_{k-1}}\star
J^{\mathcal{E}_{k}}.$ (62)

The process of measuring a quantum circuit fragment is referred to as an
_interactive measurement_ , which is an essential tool for witnessing non-
Markovianity and inferring causality. While the topic of interactive
measurements is complex and important, a full discussion of this topic is
beyond the scope of this work. We therefore leave the exploration of
interactive measurements to future research endeavors [77]. Nevertheless, it
is worth noting that interactive measurements represent a promising avenue for
advancing our understanding of quantum systems over multiple time points. By
leveraging the principles of interactive measurement, we can gain valuable
insights into the behavior of quantum systems and develop new approaches to
solving complex problems in various fields.

##  II Temporal Entanglement

In this section, we present a comprehensive framework of temporal entanglement
from a quantum resource theoretical perspective, extending the entanglement
theory from states and channels to the most general form of quantum circuit
fragments (refer to Subsec. I.4). In Subsec. II.1, we demonstrate the
framework of quantum resource theory and examine various options for free
morphism in the context of temporal entanglement. Additionally, we introduce
the fidelity of quantum circuit fragments (refer to Def. II.2) concerning
various free morphisms in Subsec. II.2. It is essential to develop this
framework to investigate the concepts of quantum teleportation, quantum
repeaters, and communication in quantum networks thoroughly. The connection
between temporal entanglement and quantum network communication lies at the
heart of our research, and we will unravel this relationship in greater detail
in the following sections.

###  II.1 Entangled Circuit Fragments: Quantum Correlations across Space and
Time

Quantum entanglement is one of the most remarkable and puzzling features of
quantum mechanics, where two or more particles become intrinsically connected
in such a way that the state of one particle instantaneously affects the state
of the other, even when they are separated by significant distances. This
extraordinary phenomenon has emerged as a key resource in various fields [83],
ranging from communication [84, 85, 86, 87, 88, 89], cryptography [90, 91,
92], to computation [93, 94, 95, 96, 97]. The study of entangled states, which
represent a form of quantum entanglement at a given moment, has become a major
focus of research since its inception, owing to their indispensable role in
understanding the fundamentals of quantum mechanics and their potential for
advancing a wide range of quantum technologies. As such, entangled states have
been the subject of numerous research articles in the literature. In the realm
of physical systems, change is the only constant. As famously stated by Chuck
Palahniuk in Fight Club, “nothing is static, everything is evolving”. Recent
progress has extended the theory of entanglement [98, 99, 100, 101, 102] to
encompass quantum channels [103, 104, 105, 106, 107], enabling a comprehensive
investigation into the intricate entanglement of dynamical processes. Both
point-to-point and bipartite channels have been explored, offering a
multifaceted approach to understanding quantum dynamics. Notably, there is no
fundamental difference between quantum states and quantum channels, as both
are elements of $\mathfrak{F}_{1}$, i.e., quantum circuit fragments with a
single process. Despite these advancements, the generic entanglement theory of
quantum dynamics over multiple time points [108, 109, 110, 111, 112, 113,
114], represented by the quantum circuit fragment $\mathfrak{F}_{k}$ (see
Subsec. I.4), is still absent. Deeper insights into the intricate interactions
of quantum systems, particularly as they evolve over time, hold immense
potential to illuminate fundamental theoretical concepts and practical
applications. The continued expansion of our understanding of quantum dynamics
over multiple time points will undoubtedly shape the future of quantum
computing and technology. In order to tackle this issue, our attention is
directed towards examining temporal entanglement - that is, the entanglement
of general quantum circuit fragments - in this subsection. Through our
analysis, we aim to develop a quantum resource theory that aligns with our
findings, obtained through a meticulous analysis.

Quantum resource theory is a vibrant and rapidly developing field at the
forefront of modern quantum theories. At its core, the theory focuses on free
states or channels, which can be prepared or manipulated without the need for
any additional resources [115, 116]. On the contrary, the use of resources can
provide quantum advantages in a wide range of tasks, including channel (or
phase) discrimination [117, 118, 119, 120, 121] and exclusion [122, 123].
Entangled states [83] and coherent states [124] are among the most important
and widely studied quantum resources in quantum resource theory. They
represent distinct types of resources that can be used to perform a variety of
tasks with advantages that cannot be achieved using classical resources alone.
For example, entangled states can be used to perform tasks such as quantum
teleportation, superdense coding, and quantum cryptography, while coherent
states can be used in applications such as quantum metrology and quantum
computing. In order to streamline the nomenclature of quantum resource
theories and clarify the distinction between the subject of study and its
manipulation, we adopt the language of category theory [125]. Specifically, we
refer to the subject of study as _free objects_ and the corresponding physical
manipulation of these objects as _free morphisms_ in this work. For instance,
in the conventional resource theory of entanglement, the SWAP operation serves
as a morphism. On the other hand, in the investigation of dynamical
entanglement, the SWAP becomes an object. In summary, a quantum resource
theory $\mathcal{R}$ is a framework that allows researchers to study the
properties and behaviors of quantum resources in a rigorous and systematic
way. It is constituted of three fundamental components that are essential to
its functioning:

  * •

Set of Free Objects $\text{ob}(\mathcal{R})$: These are quantum circuit
fragments that can be prepared or manipulated without the need for any
additional resources. The free objects are the starting point for
understanding the properties of more complex quantum systems over multiple
time points, such as entangled or coherent non-Markovian quantum processes;

  * •

Set of Free Morphisms $\text{hom}(\mathcal{R})$: These are the physical
transformations that can be applied to the free objects without creating new
resources. Free morphisms play a crucial role in the manipulation and
transformation of quantum resources, and they provide a framework for
understanding the ways in which quantum circuit fragments can be transformed
and manipulated;

  * •

Golden Rule
$\Theta(\text{ob}(\mathcal{R}))\subset\text{ob}(\mathcal{R}),\forall\,\Theta\in\text{hom}(\mathcal{R})$:
This is a fundamental principle that guides the manipulation and
transformation of quantum resources. It states that in any given quantum
resource theory, the set of free objects should be closed under the action of
the free morphisms.

A prominent example is the study of conventional resource theory of
entanglement for quantum states, where $\text{ob}(\mathcal{R})$ is the set of
all separable states. Meanwhile, set $\text{hom}(\mathcal{R})$ is the
collection of all local operations and classical communication (LOCC). The use
of free morphisms has been instrumental in establishing the resource theory of
entanglement of quantum states. By considering the action of free morphisms,
it is possible to generate all separable states using local operations and
classical communication. This means that the entire resource theory of
entanglement can be built upon the framework of free morphisms. In practical
laboratory settings, there are physical limitations on the types of operations
that can be performed, and as a result, the set of LOCC can be divided into
different classes [126]:

  * •

Local Operations (LO): quantum operations that can be performed on individual
subsystems of a composite quantum system. These operations can be performed
independently by each party without any communication between them [127]. A
typical example is the local unitary (LU) operations [128, 129, 130], which
form a special class of local operations, namely $\text{LU}\subset\text{LO}$,
that involve applying a unitary transformation to one or more subsystems of a
composite quantum system. These operations are commonly used in the context of
entangled quantum states to manipulate the entanglement between subsystems;

  * •

$\text{LOCC}_{1}(\text{poly}(d))$: local quantum operations and one-way
classical communication with respect to a fixed party, followed by a coarse-
graining map. Here, the communication complexity is polynomially dependent on
an integral parameter $d$. This parameter $d$ represents a measure of the
dimensionality or complexity of the quantum system under consideration, such
as the dimension of the message system. By incorporating the concept of
$\text{LOCC}_{1}(\text{poly}(d))$, the communication complexity is effectively
reduced, making it more feasible to implement in real-world scenarios.

  * •

$\text{LOCC}_{1}$: local quantum operations and one-way classical
communication with respect to a fixed party, followed by a coarse-graining
map. $\text{LOCC}_{1}$ is a specific subclass of LOCC operations that has been
extensively studied in the field of quantum information theory. It has been
shown to be strictly less powerful than the set of all LOCC operations, which
means that there are entangled states that cannot be transformed into one
another using $\text{LOCC}_{1}$ operations but can be transformed using more
general LOCC operations;

  * •

$\text{LOCC}_{k}$: quantum channels that are LOCC-linked – obtaining by
implementing $\text{LOCC}_{1}$ and coarse-graining maps – to a element of
$\text{LOCC}_{k-1}$, where $k\geq 2$. $\text{LOCC}_{k}$ operations are a
generalization of $\text{LOCC}_{1}$ and provide a way to perform more complex
entanglement manipulation tasks. However, as $k$ increases, the set of
$\text{LOCC}_{k}$ operations becomes more powerful and less well-understood;

  * •

$\text{LOCC}_{\mathds{N}}$: a element of $\text{LOCC}_{k}$ for some
$k\in\mathds{N}$;

  * •

LOCC: quantum channel $\mathcal{E}$ is a LOCC channel if there exists a
sequence of quantum channels $\\{\mathcal{E}_{1},\mathcal{E}_{2},\ldots\\}$
satisfying (i) each of them belongs to the set $\text{LOCC}_{\mathds{N}}$,
(ii) $\mathcal{E}_{k}$ is LOCC-linked to $\mathcal{E}_{k-1}$, and (iii)
$\mathcal{E}_{k}^{{}^{\prime}}$, obtained by implementing coarse-graining maps
on $\mathcal{E}_{k}$, converges to $\mathcal{E}$. LOCC operations are
important in quantum information theory because they are considered to be the
most basic and physically realistic class of operations that can be performed
on entangled quantum systems. They are also used as a benchmark for the study
of several important tasks, such as entanglement distillation, entanglement
cost, and quantum teleportation;

  * •

$\overline{\text{LOCC}_{\mathds{N}}}$: topological closure of
$\text{LOCC}_{\mathds{N}}$.

The formal definitions of coarse-graining maps, LOCC-link, and convergence of
channels can be found in Ref. [126], which is a comprehensive reference on
LOCC. These concepts are important for studying the behavior of entangled
quantum systems under local operations and classical communication. In
addition to these concepts, there are many other related operations in
entanglement theory that are of interest to researchers. These include:

  * •

Separable Operations (SEP): quantum channels that can be expressed using a
product form of _Kraus operators_. It is noteworthy that when considering two
bipartite pure quantum states, the existence of a separable operation capable
of transforming one into the other implies the existence of a LOCC operation
that can achieve the same result. Specifically, Gheorghiu and Griffiths have
demonstrated that any transformation of a pure state using separable
operations satisfies the same necessary and sufficient conditions as an LOCC
transformation [131]. It should be noted, however, that this equivalence does
not hold for mixed states, as illustrated by Chitambar and Duan in Ref. [132].
In fact, every LOCC operation can be expressed in terms of a Kraus operator-
sum representation, using product Kraus operators only. However, the converse
is not necessarily true;

  * •

Separability Preserving Operations (SEPP): quantum channels that preserve the
separability of states. To further elaborate, SEPP is a set of operations that
are non-entangling, meaning they do not create any entanglement from separable
states. The study of SEPP, especially its approximate versions, is important
in the context of entanglement manipulation, as it provides a framework for
the reversible manipulation of entanglement. A deeper exploration of
approximate SEPP can be found in various literature references, including Ref.
[133, 134, 135, 136], which discuss the generalization of quantum Stein’s
lemma and the second law of entanglement manipulation. These references
provide detailed explanations and mathematical formulations of the properties
and applications of SEPP, and can be a valuable resource for those interested
in studying this topic;

  * •

Positive-Partial-Transpose Preserving Operations (PPT): quantum channels that
completely preserve the positivity of the partial transpose of states. More
precisely, any quantum state that has a positive partial transpose will remain
positive under a PPT operation. This property distinguishes PPT operations
from other types of quantum operations, such as LOCC operations, because they
can create bound entangled states that cannot be created using only LOCC
operations. One of the key advantages of using PPT operations over LOCC
operations is that PPT operations are generally easier to characterize. This
is because the set of PPT operations is a convex set that can be described by
using semidefinite programming (SDP) techniques [137, 138, 139], whereas the
set of LOCC operations is a non-convex set that is much harder to handle.

The models under consideration follow a hierarchical structure, with each
model being more general and encompassing than the previous one. This
hierarchy allows for a systematic study of the different models and their
relationships to one another in entanglement theory. Specifically, the models
considered in this work can be organized in the following hierarchy, starting
from the most restrictive:

$\displaystyle\text{LO}\subsetneq\text{LOCC}_{1}(\text{poly}(d))\subset\text{LOCC}_{1}\subsetneq\text{LOCC}_{k}\subsetneq\text{LOCC}_{k+1}\subsetneq\text{LOCC}_{\mathds{N}}\subsetneq\text{LOCC}\subsetneq\overline{\text{LOCC}_{\mathds{N}}}\subsetneq\text{SEP}\subsetneq\text{SEPP}\subsetneq\text{PPT},$
(63)

for any $k>1$.

######  Remark II.1.

While $\text{LOCC}_{1}(\text{poly}(d))$ is clearly a subset of
$\text{LOCC}_{1}$, encompassing operations with a narrower range of
communication complexities, it is still unclear whether this inclusion is
strict. This ambiguity arises from the possibility that there are scenarios
where $\text{LOCC}_{1}(\text{poly}(d))$ and $\text{LOCC}_{1}$ result in the
same set of achievable transformations. To gain a comprehensive understanding
of the relationship between $\text{LOCC}_{1}(\text{poly}(d))$ and
$\text{LOCC}_{1}$, more thorough examination and in-depth analysis are
required.

We will now expand the scope of these definitions to encompass quantum circuit
fragments – the carrier of temporal entanglement. The first new and important
concept is called the $\mathcal{S}$-simulable quantum circuit fragment.

Definition II.1:  $\mathcal{S}$-Simulable Quantum Circuit Fragment A quantum
circuit fragment $\mathcal{E}\in\mathfrak{F}_{k}$ can be referred to as a
$\mathcal{S}$-simulable quantum circuit fragment if it can be decomposed into
a series of quantum channels
$\\{\mathcal{E}_{1},\mathcal{E}_{2},\ldots,\mathcal{E}_{k}\\}$, where each
$\mathcal{E}_{i}\in\mathcal{S}$. The set $\mathcal{S}$ is chosen from
$\\{\text{LO},\text{LOCC}_{1}(\text{poly}(d)),\text{LOCC}_{k},\text{LOCC}_{\mathds{N}},\text{LOCC},\overline{\text{LOCC}_{\mathds{N}}},\text{SEP},\text{SEPP},\text{PPT}\\}$.
We denote the set of all $\mathcal{S}$-simulable quantum circuit fragments as
$\mathfrak{F}_{k}(\mathcal{S})$.

######  Remark II.2.

When examining $\text{LOCC}_{1}$-simulable quantum circuit fragments, namely
$\mathfrak{F}_{k}(\text{LOCC}_{1})$, we impose the restriction that all of its
components are $\text{LOCC}_{1}$ with respect to the same party.

Having introduced $\mathcal{S}$-simulable quantum circuit fragments in Def.
II.1, we can now present the formal definition of an entangled circuit
fragment, which captures the essence of temporal entanglement over multiple
time points.

Definition II.2:  Entangled Circuit Fragment We refer to a quantum circuit
fragment $\mathcal{E}$ as an entangled circuit fragment if it does not belong
to the set $\mathcal{E}\notin\mathfrak{F}_{k}(\text{SEPP})$.

######  Remark II.3.

The significance of Def. II.1 lies in its ability to extend the concept of
entanglement to its most general form, which includes not just quantum states
and channels but also quantum circuit fragments. In doing so, it provides a
crucial framework for characterizing and understanding entanglement in the
context of non-Markovian quantum processes. In particular, by introducing

$\displaystyle\text{Sta}(\mathcal{S}):=\\{\rho\,|\,\rho\in\mathfrak{F}_{1}(\mathcal{S}),\Tr[\rho]=1\\},$
(64)

with
$\mathcal{S}\in\\{\text{LO},\text{LOCC}_{k},\text{LOCC}_{\mathds{N}},\text{LOCC},\overline{\text{LOCC}_{\mathds{N}}},\text{SEP},\text{SEPP},\text{PPT}\\}$,
the collection of separable states is completely captured by

$\displaystyle\text{Sta}(\text{LOCC})=\text{Sta}(\text{SEP})=\text{Sta}(\text{SEPP}).$
(65)

If a state $\rho\notin\text{Sta}(\text{SEPP})$, it is entangled. Therefore, an
entangled state can be regarded as a specific instance of an entangled quantum
circuit fragment, which consists of a single quantum process.

While we trigger the investigation of entangled quantum circuit fragments , or
directly temporal entanglement, our objective of this work is only to explore
their connection to quantum network communication, including quantum
teleportation and quantum repeater-based protocols. We will not dive into the
complete framework here, but our upcoming works will present a more
comprehensive understanding of temporal entanglement.

###  II.2  Fidelity of Circuit Fragment Distillation: Quantifying Temporal
Entanglement

Entanglement distillation is a critical component of quantum information
processing, involving the conversion of a large number of weakly entangled
quantum states into a smaller number of highly entangled ones. This
transformation is essential for numerous quantum communication and computation
applications, as the strength of entanglement between parties constrains their
operational effectiveness. In this subsection, we will delve into the topic of
entanglement distillation for quantum circuit fragments, and introduce the
notion of $k$-fidelity of $\mathcal{S}$ distillation.

With a focus on practicality, our attention in this work is directed towards
single-shot entanglement distillation of quantum circuit fragments. More
precisely, we are interested in transforming a single instance of quantum
circuit fragment $\mathcal{E}\in\mathfrak{F}_{k}$ into a state that closely
approximates the maximally entangled state $\phi^{+}_{d}$ through the
application of operation $\Theta\in\mathfrak{F}_{k+1}(\mathcal{S})$ with
$\mathcal{S}\in\\{\text{LO},\text{LOCC}_{k},\text{LOCC}_{\mathds{N}},\text{LOCC},\overline{\text{LOCC}_{\mathds{N}}},\text{SEP},\text{SEPP},\text{PPT}\\}$;
that is

Definition II.3:  $d$-Fidelity of $\mathcal{S}$ Distillation The $d$-fidelity
$F_{d,\mathcal{S}}(\mathcal{E})$ of a quantum circuit fragment
$\mathcal{E}\in\mathfrak{F}_{k}$ under operation
$\Theta\in\mathfrak{F}_{k+1}(\mathcal{S})$ is defined by $\displaystyle
F_{d,\mathcal{S}}(\mathcal{E}):=\max_{\Theta\in\mathfrak{F}_{k+1}(\mathcal{S})}F_{\text{U}}(\Theta(\mathcal{E}),\phi^{+}_{d})=\max_{\Theta\in\mathfrak{F}_{k+1}(\mathcal{S})}\Tr[\Theta(\mathcal{E})\cdot\phi^{+}_{d}],$
(66) where $F_{\text{U}}$ stands for the Uhlmann fidelity, with $\phi^{+}_{d}$
being the $d$-dimensional maximally entangled state.

The $d$-Fidelity of $\mathcal{S}$ distillation offers a quantitative
assessment of how closely the output channel – $\Theta(\mathcal{E})$ –
resembles the ideal, maximally entangled state $\phi^{+}_{d}$ under operations
$\Theta\in\mathfrak{F}_{k+1}(\mathcal{S})$, characterizing their similarity
through the overlap between $\Theta(\mathcal{E})$ and $\phi^{+}_{d}$. The
functional $F_{d,\mathcal{S}}(\mathcal{E})$ is a generalization of the concept
of single-shot entanglement distillation, where the goal is to generate states
that are as close as possible to a maximally entangled state under LOCC from
initial object, which can be a quantum state or a quantum channel. When
$\mathcal{E}$ is a quantum state and the set of allowed operations are LOCC,
i.e., $\mathcal{S}=\text{LOCC}$, then $F_{d,\mathcal{S}}(\mathcal{E})$
recovers the concept of single-shot entanglement distillation under LOCC. On
the other hand, when the set of operations that can be applied to state
$\mathcal{E}$ are PPT channels, namely $\mathcal{S}=\text{PPT}$, then
$F_{d,\mathcal{S}}(\mathcal{E})$ gives the fidelity of $d$-state PPT
distillation [22]. Moreover, when $\mathcal{E}$ is a quantum channel and
$\Theta$ represents a superchannel, the function
$F_{d,\mathcal{S}}(\mathcal{E})$ captures the concept of single-shot
entanglement distillation of dynamical entanglement [103, 104, 105, 106, 107].

######  Remark II.4.

Here, $F_{d,\mathcal{S}}(\mathcal{E})$ is not an entanglement monotone as it
is not faithful, i.e., $F_{d,\mathcal{S}}(\mathcal{E})>0$ if and only if
$\mathcal{E}$ is an entangled circuit fragment (see Def. II.1). But, it is
still monotonically non-increasing under free morphism
$\Xi\in\mathfrak{F}_{k+1}(\mathcal{S})$. That is, given a quantum circuit
fragment $\mathcal{E}$, we have $F_{d,\mathcal{S}}(\mathcal{E})\geqslant
F_{d,\mathcal{S}}(\Xi(\mathcal{E}))$ holds for all
$\Xi\in\mathfrak{F}_{k+1}(\mathcal{S})$.

This subsection introduced the basic idea of $d$-Fidelity of $\mathcal{S}$
distillation $F_{d,\mathcal{S}}(\mathcal{E})$ and explored its importance in
single-shot entanglement distillation. However, the comprehensive scope of
temporal entanglement extends far beyond what was discussed here. This raises
many questions, such as: How can we formulate the standard entanglement cost
and distillable entanglement in the asymptotic case of quantum circuit
fragments under $\mathfrak{F}_{k}(\text{LOCC})$? Is it possible to solve the
entanglement cost of a quantum circuit fragment and find a quantity that
unifies both $\kappa$ entanglement [104] for states and max-logarithmic
negativity [106] for bipartite channels in the case of
$\mathfrak{F}_{k}(\text{PPT})$? These questions exceed the boundaries of this
work and will be presented in our upcoming works.

##  III Quantum Communications

In this section, we will provide a fresh perspective on quantum teleportation
by focusing on the role of resourceful quantum channels in the protocol.
Instead of simply considering the entanglement of a static state, we will
examine the crucial role of the entanglement-generating channel that connects
the sender and receiver. To explore the optimal performance of a bipartite
channel in quantum teleportation, we will introduce a new superchannel-
assisted teleportation protocol (shown in Fig. 8(f)) in Subsec. III.1. In the
subsequent Subsec. III.2, we will dive deeper into the general mathematical
framework that underlies these protocols. By doing so, we will gain a more
complete understanding of the principles that govern a variety of quantum
teleportation protocols (illustrated in Fig. 8) and the role of entanglement-
generating channel in superchannel-assisted quantum teleportation. Finally, in
Subsec. III.3, we will demonstrate the superior performance of our newly
proposed teleportation protocol compared to conventional methods in certain
noise models. This result highlights the practical significance of our
approach and its potential for real-world applications.

Figure 8: (Color online) Quantum teleportation protocols: (a) standard
teleportation, a fundamental process in quantum information theory where a
quantum state is transmitted from one location to another. The protocol
involves the sharing of a maximally entangled state between two parties, Alice
and Bob, who are situated at distinct locations. The quantum information that
needs to be transmitted from Alice to Bob is stored in the state $\psi_{R}$
that acts on system $R$. Alice has access to systems $RA$, while Bob has
access to system $B$. The protocol is enabled by implementing a standardized
operation, which is represented by the pink box in the diagram and denoted as
$\Theta^{\text{post}}_{0}\in\mathfrak{F}_{1}(\text{LOCC}(RA:B))$; (b) noisy
teleportation, where the entangling gates are followed by channels
$\mathcal{N}_{1}(p)$ and $\mathcal{N}_{2}(q)$ (see Eq. 68), resulting in a
noisy entangling gate $\mathcal{E}(p,q)$ (see Eq. 70). Despite these noises,
the transmission of quantum information $\psi_{R}$ remains through
$\Theta^{\text{post}}_{0}$; (c) LOCC teleportation [19]. Rather than relying
on a standardized operation $\Theta^{\text{post}}_{0}$ executed by remote
parties, a more versatile LOCC channel
$\Theta_{1}^{\text{post}}\in\mathfrak{F}_{1}(\text{LOCC}(RA:B))$ has been
utilized to facilitate the transmission of quantum information between the
composite systems $RA$ and $B$. After optimizing over all possible LOCC
channels, this protocol demonstrates superior performance compared to the
previous protocol (b); (d) PPT teleportation. The set of allowable operations
has been expanded to include all PPT channels, which means that the LOCC
channel $\Theta^{\text{post}}_{1}$ has been replaced by a PPT channel
$\Theta^{\text{post}}_{2}\in\mathfrak{F}_{1}(\text{PPT}(RA:B))$. As a result,
the performance of quantum state teleportation has been further enhanced; (e)
pre-processing-assisted teleportation. When the entangling gate (represented
by the brown box) is seen as a shared dynamical resource between Alice and
Bob, there is no justification for limiting consideration solely to the
implementation of post-processing. In this case, we consider a special pre-
processing protocol
$\Theta^{\text{pre}}_{0}:=\mathcal{U}_{1}(\outerproduct{0}{0})_{A}\otimes\mathcal{U}_{2}(\outerproduct{0}{0})_{B}$,
followed by the standard teleportation protocol $\Theta^{\text{post}}_{0}$;
(f) superchannel-assisted teleportation, achieved by implementing a
superchannel $\Theta\in\mathfrak{F}_{2}(\text{LOCC})$. Here, gate
$\mathcal{E}(p,q)$ has been transformed into channel
$\Theta(\mathcal{E}(p,q))_{R\to B}$. This consideration leads to superior
performance compared to the case with only post-processing, i.e., protocol
(c); From protocols (b) to (f), the output is represented by $\sigma$. These
diagrams distinguish between systems belonging to Alice, which are located
above the dashed red line, and those belonging to Bob, which are situated
below the dashed line.

###  III.1 Quantum Teleportation: A Channel-Theoretic Perspective

At the heart of quantum communication is the ability to securely transfer
information between parties [140]. This is crucial for a variety of
applications, from quantum cryptography to quantum computing. The most widely
recognized and effective method for achieving secure transfer of quantum
information is through the use of quantum teleportation [141, 142, 143, 144,
145]. This protocol, which was first introduced by Bennett et al. in Ref.
[142], enables the transfer of quantum states from one location to another,
without physically transporting the particles that carry the information. Its
effectiveness has been experimentally demonstrated in Refs. [144, 145].
Recently, efforts to build a global-scale quantum network have resulted in the
successful ground-to-satellite teleportation over distances of up to $1,400$
kilometers as reported in Ref. [146]. To facilitate a thorough exploration of
related topics, we highly recommend consulting Ref. [147, 148, 149, 150, 151].
Theoretically, the transfer of any quantum information from Alice to Bob can
be achieved perfectly through the use of a maximally entangled state shared
between them (see Fig. 8(a)). However, the physical implementation of
teleportation is subject to various sources of quantum noise resulting from
interactions between the system and its environment (see Fig. 8(b)) [147, 148,
149, 150, 151, 152, 153, 154, 155]. The conventional approach to combating the
noise is to replace the standard teleportation protocol (i.e.,
$\Theta_{0}^{\text{post}}$ of Fig. 8(b)) with a general LOCC protocol (i.e.,
$\Theta_{1}^{\text{post}}$ of Fig. 8(c)) [19], as $\Theta_{0}^{\text{post}}$
is only a special case of LOCC. The power of teleportation originates from the
entangling gate (brown boxes in Fig. 8) between different parties. One might
then wonder: does the conventional approach (see Fig. 8(c)) fully leverage the
capabilities of the entangling gate?

Regrettably, the answer to the question posed above is negative. There is
concrete evidence that the use of a superchannel-based protocol is
demonstrably superior to one that relies solely on post-processing techniques.
To provide a more concrete illustration, we demonstrate that there exist
examples where a straightforward application of local unitary operations in
the pre-processing stage (see Fig. 8(e)) can outperform any possible PPT-post-
processing technique (see Fig. 8(d)), which naturally encompasses all LOCC-
post-processing protocols as a subset (see Fig. 8(c)).

To provide a comprehensive explanation, we will evaluate each of the protocols
depicted in Fig. 8 systematically, along with their corresponding performance.

  * •

Protocol a: Standard Teleportation (see Fig. 8(a)). The mechanism of
teleportation can be traced back to classical textbooks, such as Refs. [3, 4,
5]. Here, we visually demonstrate the standard teleportation through a diagram
in Fig. 9.

Figure 9: (Color online) Graphical illustration of standard teleportation
procedure (see Fig. 8(a)). Here,
$(U_{i}\otimes\mathbb{1}\ket{\phi^{+}})^{{\dagger}}$ stands for the standard
Bell measurements with $U_{0}=\mathbb{1}$, $U_{1}=Z$, $U_{2}=X$ and
$U_{3}=XZ$. Thanks to $U\otimes\mathbb{1}\ket{\phi^{+}}=\mathbb{1}\otimes
U^{\mathbf{T}}\ket{\phi^{+}}$, we have (a) = (b). By further using the yanking
equation, we obtain the equivalence between (b) and (c). The technique
employed here is akin to that illustrated in Fig. 2.

  * •

Protocol b: Noisy Teleportation (see Fig. 8(b)). In this case, the entangling
gate is subject to local noise, which affects its performance. Denote by
$\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ the local noise affecting systems $A$
and $B$, respectively. In the subsequent analysis, we will make use of the
following noise models; that are

$\displaystyle\mathcal{N}_{1},\mathcal{N}_{2}\in\\{\text{Bit
Flip}\,\,\,\mathcal{N}_{\text{BF}},\text{Phase
Flip}\,\,\,\mathcal{N}_{\text{PF}},\text{Depolarizing}\,\,\,\mathcal{N}_{\text{D}},\text{Amplitude
Damping}\,\,\,\mathcal{N}_{\text{AD}}\\}.$ (67)

As a fundamental type of error in quantum computing and communication, the bit
flip channel $\mathcal{N}_{\text{BF}}$ can cause a qubit state to flip from
$0$ to $1$ or from $1$ to $0$ with a certain probability, which is typically
described by a single parameter known as the error probability. This type of
error can arise from a variety of noise sources, such as thermal fluctuations
and electromagnetic interference. The phase flip channel
$\mathcal{N}_{\text{PF}}$, which arises from fluctuations in the local
magnetic field, can cause a qubit phase to flip between $|0\rangle$ and
$|1\rangle$ states with a certain probability. Specifically, if the qubit is
initially in the state $\alpha\ket{0}+\beta\ket{1}$, then the phase flip
channel can transform this state to $\alpha\ket{0}-\beta\ket{1}$ with a
certain probability. As a combination of the bit flip, phase flip, and the
bit-phase flip channels, the quantum depolarizing channel, denoted as
$\mathcal{N}_{\text{D}}$, takes its initial state to a completely mixed state
with some probability. The amplitude damping channel $\mathcal{N}_{\text{AD}}$
is a type of noise that is commonly encountered in various physical systems,
such as superconducting qubits and trapped ions. This channel causes a qubit
to lose energy and transition to a lower energy state, resulting in a decay of
the amplitude of its quantum state. Using the Kraus decomposition
$\mathcal{N}(\cdot)=\sum_{i}E_{i}\cdot E^{{\dagger}}_{i}$, we can characterize
these noise models in Tab. 1. For more details on this topic, see Ref. [3].

Table 1: Kraus decomposition of the fundamental noisy channels, including bit
flip channel $\mathcal{N}_{\text{BF}}$, phase flip channel
$\mathcal{N}_{\text{PF}}$, quantum depolarizing channel
$\mathcal{N}_{\text{D}}$, and amplitude damping channel
$\mathcal{N}_{\text{AD}}$. In this context, the symbols $\mathbb{1}$, $X$,
$Y$, and $Z$ represent the _Pauli operators_.

To differentiate the parameters associated with the quantum channels
$\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, we denote their corresponding
parameters as $p$ and $q$, respectively. More precisely, we express this
distinction as

$\displaystyle\mathcal{N}_{1}$ $\displaystyle=\mathcal{N}_{1}(p),$ (68)
$\displaystyle\mathcal{N}_{2}$ $\displaystyle=\mathcal{N}_{2}(q).$ (69)

These notations allow us to clearly identify the parameter associated with
each channel, which is essential for analyzing and comparing their effects in
noisy quantum teleportation. Having defined $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$, we can now completely characterize the entangling gate
between Alice and Bob as $\mathcal{E}(p,q)$, colored brown in Fig. 8(b).
Expressed mathematically, the noisy entangling gate $\mathcal{E}(p,q)$ can be
represented by

$\displaystyle\mathcal{E}(p,q)_{AB\to AB}:=\mathcal{N}_{1}(p)_{A\to
A}\otimes\mathcal{N}_{2}(q)_{B\to B}\circ\text{C}_{\text{NOT},AB\to
AB}\circ\text{H}_{A\to A}\otimes\text{id}_{B\to B}.$ (70)

Here, we use the symbol H to refer to the _Hadamard gate_ , and
$\text{C}_{\text{NOT}}$ to denote the controlled NOT gate (also known as the
controlled-X gate). In this scenario, the resultant channel
$\mathcal{F}_{\text{b}}(p,q)$ from system $R$ to $B$ takes the form of

$\displaystyle\mathcal{F}_{\text{b}}(p,q)_{R\to
B}:=\Theta^{\text{post}}_{0,RAB\to B}\circ\mathcal{E}(p,q)_{AB\to
AB}(\outerproduct{0}{0}_{A}\otimes\outerproduct{0}{0}_{B}).$ (71)

The subscript ‘b’ in $\mathcal{F}$ indicates that the channel from $R$ to $B$
is obtained using protocol b. In order to measure and evaluate the performance
of protocol b, we use the metric of average fidelity $f$, which is defined in
Def. I.2. This metric provides a quantitative measure of how accurately the
teleported state $\mathcal{F}_{\text{b}}(p,q)(\psi)$ matches the original
state $\psi$ that was being teleported. In particular, the average fidelity
$f$ of channel $\mathcal{F}_{\text{b}}(p,q)$ is given by

$\displaystyle f(\mathcal{F}_{\text{b}}(p,q))=\int d\psi
F_{\text{U}}\left(\psi,\mathcal{F}_{\text{b}}(p,q)(\psi)\right)=\int
d\psi\Tr[\psi\cdot\left(\mathcal{F}_{\text{b}}(p,q)(\psi)\right)].$ (72)

To simplify notation, we can use the abbreviation $f_{\text{b}}$ to refer to
the performance of protocol b, i.e.,

$\displaystyle f_{\text{b}}:=f(\mathcal{F}_{\text{b}}(p,q)).$ (73)

Using Lem. I.2, we can prove that $f_{b}$ can be expressed as

$\displaystyle f_{\text{b}}=\frac{2F(\mathcal{F}_{\text{b}}(p,q))+1}{3}.$ (74)

Here, $F(\mathcal{F}_{\text{b}}(p,q))$ represents the entanglement fidelity
(defined in Def. I.2) of the channel $\mathcal{F}_{\text{b}}(p,q)$.

  * •

Protocol c: LOCC Teleportation (see Fig. 8(c)). It is worth noting that the
standard teleportation protocol $\Theta^{\text{post}}_{0}$ involves performing
Bell measurements and applying local unitary operations based on the
measurement outcomes. Such an operation can be understood as a specific
instance of a LOCC channel. Given this connection, it is reasonable to
consider whether optimizing over all possible LOCC channels could potentially
improve the performance of teleportation. Formally, if we replace the
standardized operation $\Theta^{\text{post}}_{0}$, as depicted in Fig. 8(b),
with a general LOCC channel $\Theta^{\text{post}}_{1}\in\text{LOCC}(RA:B)$,
the resultant channel $\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)$
from the sender to the receiver can be expressed as

$\displaystyle\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)_{R\to
B}:=\Theta^{\text{post}}_{1,RAB\to B}\circ\mathcal{E}(p,q)_{AB\to
AB}(\outerproduct{0}{0}_{A}\otimes\outerproduct{0}{0}_{B}).$ (75)

While both $\mathcal{F}_{b}(p,q)$ (see Eq. 71) and
$\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)$ (see Eq. 75) are
influenced by parameters $p$ and $q$,
$\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)$ is more complex as it
also takes into account the variable of LOCC channel
$\Theta^{\text{post}}_{1}$. The inclusion of $\Theta^{\text{post}}_{1}$ in
$\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)$ adds an extra layer of
complexity that enhances the performance of teleportation. The expression for
the average fidelity $f$ (see Eq. 11) of
$\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)$ is

$\displaystyle f(\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q))=\int
d\psi
F_{\text{U}}\left(\psi,\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)(\psi)\right)=\int
d\psi\Tr[\psi\cdot\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)(\psi)].$
(76)

Now, if we maximize over all possible
$\Theta^{\text{post}}_{1}\in\text{LOCC}(RA:B)$, we get

$\displaystyle
f_{\text{c}}:=\max_{\Theta^{\text{post}}_{1}\in\text{LOCC}(RA:B)}f(\mathcal{F}_{\text{c}}(\Theta^{\text{post}}_{1},p,q)).$
(77)

Because $\Theta^{\text{post}}_{0}$ is a LOCC channel and we have optimized
over all possible choices of $\Theta^{\text{post}}_{1}$, the resulting
fidelity of $f_{\text{c}}$ is guaranteed to be greater than or equal to that
of $f_{\text{b}}$, i.e.,

$\displaystyle f_{\text{c}}\geqslant f_{\text{b}}.$ (78)

The fidelity $f_{\text{c}}$ represents the maximum achievable fidelity that
can be obtained via conventional teleportation. However, the computation of
the quantity $f_{\text{c}}$ remains elusive to us in general. This difficulty
lies in the fact that the mathematical representation of LOCC operations is
often hard to specify.

  * •

Protocol d: PPT Teleportation (see Fig. 8(d)). Expanding the class of
permissible post-processing operations beyond just LOCC channels can lead to
an improvement in establishing an upper bound for the fidelity $f_{\text{c}}$
(see Eq. 77). We can achieve this by incorporating positive partial transpose
(PPT) channels into our analysis. Specifically, we consider the use of a PPT
channel $\Theta^{\text{post}}_{2}\in\text{PPT}(RA:B)$ for post-processing
operations. In this scenario, the resulting channel
$\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)$ from the sender to the
receiver can be fully characterized by

$\displaystyle\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)_{R\to
B}:=\Theta^{\text{post}}_{2,RAB\to B}\circ\mathcal{E}(p,q)_{AB\to
AB}(\outerproduct{0}{0}_{A}\otimes\outerproduct{0}{0}_{B}).$ (79)

The average fidelity $f$ (see Eq. 11) of channel
$\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)$ is now given by

$\displaystyle f(\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q))=\int
d\psi
F_{\text{U}}\left(\psi,\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)(\psi)\right)=\int
d\psi\Tr[\psi\cdot\left(\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)(\psi)\right)].$
(80)

Taking the maximum over all PPT channel $\Theta^{\text{post}}_{2}$, we arrive
at the following fidelity.

$\displaystyle
f_{\text{d}}:=\max_{\Theta^{\text{post}}_{2}\in\text{PPT}(RA:B)}f(\mathcal{F}_{\text{d}}(\Theta^{\text{post}}_{2},p,q)).$
(81)

As LOCC channels represent a subset of PPT channels (as shown in Eq. 63), we
can derive an inequality between the fidelities $f_{\text{c}}$ and
$f_{\text{d}}$ as follows

$\displaystyle f_{\text{d}}\geqslant f_{\text{c}}.$ (82)

The fidelity $f_{\text{d}}$ is the maximum achievable fidelity of
teleportation that can be obtained through post-processing using PPT channels.

  * •

Protocol e: Pre-Processing-Assisted Teleportation (see Fig. 8(e)).
Traditionally, post-processing, such as $\Theta^{\text{post}}_{0}$ in
protocols a (see Fig. 8(a)) and b (see Fig. 8(b)), $\Theta^{\text{post}}_{1}$
in protocol c (see Fig. 8(c)), and $\Theta^{\text{post}}_{2}$ in protocol d
(see Fig. 8(d)), has been the primary focus for mitigating noise in
teleportation protocols. However, incorporating pre-processing techniques
alongside post-processing can further enhance the fidelity and reliability of
teleportation. There are two primary factors that motivate the pre-processing-
assisted quantum teleportation, as outlined below: (i) First, in the realm of
quantum communication theory, particularly in the investigation of quantum
channel capacity, reliable transmission of information requires both encoding
and decoding. Pre-processing serves as a means of encoding the information,
while post-processing functions as a decoding mechanism. Therefore,
incorporating both pre-processing and post-processing techniques is crucial
for achieving efficient and accurate quantum teleportation. (ii) Second, when
viewed through the lens of quantum resource theory [116], specifically the
resource theory of dynamical entanglement [103, 104, 105, 106, 107], the
shared resource between Alice and Bob is the bipartite channel
$\mathcal{E}(p,q)$ (brown boxes in Fig. 8(b)-(f)). As such, it is imperative
to evaluate its maximum performance under all feasible free morphisms, namely
the set of all LOCC superchannels $\mathfrak{F}_{2}(\text{LOCC}(RA:B))$ (for
further information, please refer to Def. II.1 in Subsec. II.1). Additionally,
the preparation of the initial state
$\outerproduct{0}{0}_{A}\otimes\outerproduct{0}{0}_{B}$ is merely a specific
instance of free pre-processing. Therefore, it is not necessary to exclusively
focus on this particular choice, as any free pre-processing can be employed
instead. In this scenario, we examine a specific selection of free pre-
processing $\Theta^{\text{pre}}_{0}$

$\displaystyle\Theta^{\text{pre}}_{0}:=\mathcal{U}_{1}(\outerproduct{0}{0})_{A}\otimes\mathcal{U}_{2}(\outerproduct{0}{0})_{B}$
(83)

and demonstrate that even with this particular choice, the resulting fidelity
cannot be attained by any protocol that employs solely LOCC post-processing.
In Eq. 83, $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are local unitary gates
satisfying $\mathcal{U}_{i}(\cdot)=U_{i}\cdot U_{i}^{{\dagger}}$ with
$i\in\\{1,2\\}$. It is worth noting that any unitary operator $U$ acting on a
qubit can be written in terms of a global phase shift, a rotation operator
about the $y$-axis, and two rotation operators about the $z$-axis [3]. More
precisely, we have

$\displaystyle U=e^{i\alpha}\left[\begin{array}[]{cc}e^{-i\beta/2}&0\\\
0&e^{i\beta/2}\\\
\end{array}\right]\left[\begin{array}[]{cc}\cos\frac{\gamma}{2}&-\sin\frac{\gamma}{2}\\\
\sin\frac{\gamma}{2}&\cos\frac{\gamma}{2}\\\
\end{array}\right]\left[\begin{array}[]{cc}e^{-i\delta/2}&0\\\
0&e^{i\delta/2}\\\ \end{array}\right].$ (90)

Here, $\alpha$, $\beta$, $\gamma$, and $\delta$ are real-valued parameters. To
facilitate our analysis and numerical simulations, we simplify the discussion
by focusing on specific unitary operations, which has the following form:

$\displaystyle
U_{1}(\alpha):=\left[\begin{array}[]{cc}\cos\frac{\alpha}{2}&-\sin\frac{\alpha}{2}\\\
\sin\frac{\alpha}{2}&\cos\frac{\alpha}{2}\\\ \end{array}\right],\quad
U_{2}(\beta):=\left[\begin{array}[]{cc}\cos\frac{\beta}{2}&-\sin\frac{\beta}{2}\\\
\sin\frac{\beta}{2}&\cos\frac{\beta}{2}\\\ \end{array}\right].$ (95)

With the pre-processing step taken into account, the overall effect of the
protocol is to establish a quantum channel, denoted by
$\mathcal{F}_{\text{e}}(\Theta^{\text{pre}}_{0}(\alpha,\beta),p,q)$ or simply
$\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)$, from the sender to the receiver.
This channel can be expressed as follows

$\displaystyle\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)_{R\to B}:=$
$\displaystyle\Theta^{\text{post}}_{0,RAB\to B}\circ\mathcal{E}(p,q)_{AB\to
AB}\circ\Theta^{\text{pre}}_{0,\mathbb{C}\to AB}$ (96) $\displaystyle=$
$\displaystyle\Theta^{\text{post}}_{0,RAB\to B}\circ\mathcal{E}(p,q)_{AB\to
AB}\left(\left(U_{1}(\alpha)\outerproduct{0}{0}U_{1}^{{\dagger}}(\alpha)\right)_{A}\otimes\left(U_{2}(\beta)\outerproduct{0}{0}U_{2}^{{\dagger}}(\beta)\right)_{B}\right),$
(97)

where the unitary operators $U_{1}(\alpha)$ and $U_{2}(\beta)$ are defined in
Eq. 95. Mathematically, the average fidelity $f$ (see Eq. 11) of the resulting
channel $\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)$ is expressed as

$\displaystyle f(\mathcal{F}_{\text{e}}(\alpha,\beta,p,q))=\int d\psi
F_{\text{U}}\left(\psi,\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)(\psi)\right)=\int
d\psi\Tr[\psi\cdot\left(\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)(\psi)\right)].$
(98)

By maximizing over all possible values of $\alpha$ and $\beta$, we arrive at
the following fidelity expression.

$\displaystyle
f_{\text{e}}:=\max_{\alpha,\beta}f(\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)).$
(99)

As $\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)$ (see Eq. 96) equals
$\mathcal{F}_{\text{b}}(p,q)$ (see Eq. 71) when $\alpha=0$ and $\beta=0$ (see
Eq. 95), it follows that

$\displaystyle f_{\text{e}}\geqslant f_{\text{b}},$ (100)

where $f_{\text{b}}$ is defined in Eq. 73. Although protocol e has the ability
to mitigate the effects of noise and improve the fidelity of teleportation,
its interplay with $f_{\text{c}}$ (see Eq. 77) and $f_{\text{d}}$ (see Eq. 81)
is currently unknown.

  * •

Protocol f: Superchannel-Assisted Teleportation (see Fig. 8(f)). A quantum
superchannel is the most comprehensive method for manipulating a quantum
channel, allowing the transformation of a quantum channel into another quantum
channel by enabling the channel to have memory and transmit information across
multiple time steps. This feature makes quantum superchannels a crucial
component in the development of efficient and reliable quantum communication
protocols and quantum computing algorithms. To mitigate the noise that occurs
in $\mathcal{E}(p,q)$, this protocol utilizes a quantum superchannel
$\Theta=\Theta^{\text{post}}\circ\Theta^{\text{pre}}\in\mathfrak{F}_{2}(\text{LOCC}(RA:B))$.
This superchannel consists of both pre-processing $\Theta^{\text{pre}}$ and
post-processing $\Theta^{\text{post}}$ channels, which are both local
operations and classical communication channels. Now, the output channel
$\mathcal{F}_{\text{f}}(\Theta,p,q)$ that transmits information from $R$ to
$B$ reads

$\displaystyle\mathcal{F}_{\text{f}}(\Theta,p,q)_{R\to
B}:=\Theta\left(\mathcal{E}(p,q)\right)=\Theta^{\text{post}}\circ\mathcal{E}(p,q)\circ\Theta^{\text{pre}}.$
(101)

In this case, the average fidelity $f$ (see Eq. 11) of the output channel
$\mathcal{F}_{\text{f}}(\Theta,p,q)$ is calculated as

$\displaystyle f(\mathcal{F}_{\text{f}}(\Theta,p,q))=\int d\psi
F_{\text{U}}\left(\psi,\mathcal{F}_{\text{f}}(\Theta,p,q)(\psi)\right)=\int
d\psi\Tr[\psi\cdot\left(\mathcal{F}_{\text{f}}(\Theta,p,q)(\psi)\right)].$
(102)

Upon maximizing over all LOCC superchannels, we obtain the following
expression for fidelity.

$\displaystyle
f_{\text{f}}:=\max_{\Theta\in\mathfrak{F}_{2}(\text{LOCC}(RA:B))}f(\mathcal{F}_{\text{f}}(\Theta,p,q)).$
(103)

The subscript ‘f’ is used in average fidelity $f$ to signify that the fidelity
is determined using protocol f. Protocol e considers a specific LOCC
superchannel
$\Theta_{0}:=\Theta^{\text{post}}_{0}\circ\Theta^{\text{pre}}_{0}$ (see Fig.
8(e)), which is only a special case of the broader class of LOCC
superchannels. The fidelity $f_{\text{f}}$ is obtained by maximizing over all
LOCC superchannels, and therefore captures the optimal performance of the
teleportation protocol. As a result, we can establish the following
relationship between $f_{\text{e}}$ (see Eq. 99) and $f_{\text{f}}$ (see Eq.
103).

$\displaystyle f_{\text{f}}\geqslant f_{\text{e}}.$ (104)

Additionally, it is worth mentioning that protocol c, as shown in Fig. 8(c),
implements a special form of LOCC superchannel. This further illuminates the
connection between the two protocols, which can be summarized as follows

$\displaystyle f_{\text{f}}\geqslant f_{\text{c}}.$ (105)

The protocol f discussed here provides the most general method for
manipulating the bipartite channel shared between the sender and receiver, and
effectively leverages the resources associated with the noisy entangling gate
$\mathcal{E}(p,q)$. Therefore, if LOCC operations are considered to be free,
the average fidelity achieved by $f_{\text{f}}$ represents the highest
performance achievable in practical scenarios.

To assess the relative performance of each protocol (see Fig. 8), we will
compare their average fidelity (see Eq. 11). From the results presented in Eq.
78 and Eq. 82, it is evident that the fidelity of protocol d is superior to
those of protocols c and b.

$\displaystyle f_{\text{d}}\geqslant f_{\text{c}}\geqslant f_{\text{b}}.$
(106)

Similarly, Eq. 100 and Eq. 104 reveal that the fidelity of protocol f is
higher than that of protocols e and b.

$\displaystyle f_{\text{f}}\geqslant f_{\text{e}}\geqslant f_{\text{b}}.$
(107)

However, the relationship between the average fidelity of protocol e and those
of protocols d and c remains ambiguous, and requires further examination. The
inequality chains described above, i.e., Eq. 106 and Eq. 107, hold for any
noisy bipartite channel $\mathcal{E}(p,q)$ (brown boxes in Fig. 8(b)-(f)),
meaning that they apply to a broad range of noise models. Therefore, these
bounds provide a general framework for evaluating the performance of quantum
teleportation protocols in the presence of noise. Mathematically, the average
fidelities associated with different protocols form a partially ordered set
(poset), which can be visualized using a _Hasse diagram_ such as Fig. 10. In
this poset, the nodes represent different average fidelities, and there is a
directed edge from node $f_{x}$ to node $f_{y}$ if and only if $f_{y}$ is less
than or equal to $f_{x}$
($x,y\in\\{\text{b},\text{c},\text{d},\text{e},\text{f}\\}$); that is

$\displaystyle f_{x}\to f_{y}\quad\iff\quad f_{x}\geqslant f_{y},\quad
x,y\in\\{\text{b},\text{c},\text{d},\text{e},\text{f}\\}.$ (108)

Figure 10: (Color online) Hasse diagram for the average fidelities. Here the
arrows connecting the nodes indicate the direction of the partial ordering
between the quantities. Specifically, an arrow pointing from node $f_{x}$ to
node $f_{y}$ indicates that the quantity $f_{x}$ is greater than or equal to
$f_{y}$, with $x,y\in\\{\text{b},\text{c},\text{d},\text{e},\text{f}\\}$.
Computing the fidelities, as shown in the blue dashed box, involves optimizing
over a set of allowable operations, making it a difficult problem to solve.

In the upcoming subsection (i.e., Subsec. III.3), we will further demonstrate
that there are specific noise models (in terms of $\mathcal{N}_{1}(p)$ and
$\mathcal{N}_{2}(q)$) for which the fidelity of protocol e is strictly higher
than that of protocol d.

$\displaystyle f_{\text{e}}>f_{\text{d}}.$ (109)

Building on previous analysis, we can now establish the following hierarchy in
this case.

$\displaystyle f_{\text{f}}\geqslant f_{\text{e}}>f_{\text{d}}\geqslant
f_{\text{c}}\geqslant f_{\text{b}}.$ (110)

This analysis highlights the importance of investigating the performance of
quantum teleportation protocols with pre-processing, as the effectiveness of a
simply LOCC pre-processing-assisted protocol like e (see Fig. 8(e)) cannot be
achieved even by allowing all possible PPT post-processing operations (see
Fig. 8(d)). Therefore, these results emphasize the necessity of investigating
the potential benefits of superchannel techniques in the development of robust
quantum communication protocols, such as the protocol f illustrated in Fig.
8(f).

###  III.2 Fidelity Benchmarking: The Role of Temporal Entanglement

To assess the quality of the teleportation protocols, we typically use the
average fidelity, which measures the similarity between the input state and
the teleported state. However, computing the average fidelity can be a
challenging task, especially for high-dimensional states. To simplify the
discussion, we can roughly categorize the fidelities considered in Subsec.
III.1 into two types: (i) The first type corresponds to protocols with
standard teleportation operation $\Theta^{\text{post}}_{0}$, such as
$f_{\text{e}}$ (see Eq. 99) and $f_{\text{b}}$ (see Eq. 73), which are
determined solely by the properties of the quantum channel and the
measurements; (ii) The second type, on the other hand, requires the
maximization of fidelity over a set of allowed operations or protocols, such
as $f_{\text{f}}$ (see Eq. 103), $f_{\text{d}}$ (see Eq. 81), and
$f_{\text{c}}$ (see Eq. 77). This optimization allows us to find the maximum
fidelity achievable using the given set of operations and protocols, and
therefore provides a way to compare the performance of different quantum
teleportation schemes. In this subsection, we will explore the process of
calculating these fidelities and prepare ourselves for upcoming numerical
experiments. Through this discussion, we will gain a deeper understanding of
the mathematical underpinnings of the fidelities and develop the necessary
tools to confidently analyze our experimental results.

Let’s begin our discussion of teleportation protocols b and e. It is worth
noting that both protocols share a common feature (see Fig. 8(b) and (e)):
their post-processing step is based on the standard teleportation operation
$\Theta^{\text{post}}_{0}$. In other words, after the measurements are
performed, the classical information is sent from the receiver to the sender,
who can then apply the appropriate correction operations to reconstruct the
teleported state. This use of a common post-processing operation makes it
easier to compute the performance of the two protocols. To begin, let us refer
to the state of the system $AB$ before post-processing
$\Theta^{\text{post}}_{0}$ as $\rho_{AB}$. It is important to understand the
relationship between this state and the teleportation channel
$\mathcal{F}(\rho)$ that is generated by $\rho_{AB}$. For a visual
representation, consult the accompanying Fig. 11.

Figure 11: (Color online) Teleportation channel $\mathcal{F}(\rho)$ (see Eq.
111), which is generated by first preparing a bipartite state $\rho_{AB}$ and
subsequently implementing a standard teleportation operation
$\Theta^{\text{post}}_{0}$. The resulting channel $\mathcal{F}(\rho)$ can be
used to transmit quantum states over distances.

Here channel $\mathcal{F}(\rho)$ is given by

$\displaystyle\mathcal{F}(\rho)_{R\to B}:=\Theta^{\text{post}}_{0,RAB\to
B}(\rho_{AB}).$ (111)

The average fidelity of the teleportation channel $\mathcal{F}(\rho)$ is a key
metric for assessing its performance in quantum teleportation. However, its
calculation is notoriously challenging. The problem can be simplified by
borrowing the insights from Lemma I.2, which enables us to transform the
calculation of average fidelity $f$ into the calculation of entanglement
fidelity $F$. Although the calculation of entanglement fidelity is still a
formidable task, we present a new lemma that establishes a direct connection
between the entanglement fidelity of $\mathcal{F}(\rho)$ and the fidelity of
the state $\rho$ shared between the sender and receiver. Our proof is based on
a novel diagrammatic approach that offers a fresh perspective on the problem
and provides valuable intuition for understanding the underlying physics.

Lemma III.1:  Connection between Entanglement Fidelity and Uhlmann fidelity
When a state $\rho$ on system $AB$ is subjected to the standard teleportation
operation $\Theta^{\text{post}}_{0}$, as illustrated in Fig. 11, the resulting
entanglement fidelity $F(\mathcal{F}(\rho))$ of the generated channel
$\mathcal{F}(\rho)$ (see Eq. 111) can be expressed as the Uhlmann fidelity
$F_{\text{U}}(\rho,\phi^{+})$ between $\rho$ and the maximally entangled state
$\phi^{+}$. More specifically, this is given by $\displaystyle
F(\mathcal{F}(\rho))=F_{\text{U}}(\rho,\phi^{+})=\Tr[\rho\cdot\phi^{+}].$
(112) Above equation simplifies the calculation of entanglement fidelity $F$
for the teleportation channel $\mathcal{F}(\rho)$. By providing a closed-form
expression for the quantity, our equation allows for efficient and accurate
evaluation of the performance of the channel, which is essential for assessing
its suitability for practical applications in quantum information processing.

###### Proof.

Our analysis is geared towards qubits for the sake of clarity and simplicity.
In order to illustrate our findings, we have opted for a pictorial method.
Specifically, we have depicted the entanglement fidelity
$F(\mathcal{F}(\rho))$ as a diagram, as shown in Fig. 12.

Figure 12: (Color online) A graphical representation of the entanglement
fidelity $F$ (see Def. I.2) of channel $\mathcal{F}(\rho)$ (see Eq. 111).

Given the Bell measurement’s equal probability of all outcomes at $1/4$ in
this scenario, we will narrow our focus to the measurement $\mathbb{1}\otimes
U_{i}\ket{\phi^{+}}$. The mathematical characteristic of the maximally
entangled state permits us to equate Fig. 12(a) to Fig. 12(b) and Fig. 12(c)
to Fig. 12(d). In addition, Fig. 12(b) and Fig. 12(c) are demonstrated to be
identical through the application of the yanking equation. The correspondence
between Fig. 12(d) and the Uhlmann fidelity $F_{\text{U}}(\rho,\phi^{+})$ is
apparent, thereby concluding the proof. ∎

######  Remark III.1.

While our proof in Lemma III.2 focuses on qubits, it should be noted that our
approach is applicable beyond this specific scenario. In fact, one can
substitute the Pauli operators with the Heisenberg-Weyl operators to extend
our method to systems of any finite dimension. Moreover, while we present many
of our results with a pictorial representation in the qubit case, it is
important to emphasize that our findings hold for arbitrary finite-dimensional
systems.

To analyze the properties of the quantum teleportation protocols b and e, let
us denote the state of the system $AB$ before post-processing
$\Theta^{\text{post}}_{0}$ as

$\displaystyle\rho_{\text{b}}$ $\displaystyle:=\mathcal{N}_{1}(p)_{A\to
A}\otimes\mathcal{N}_{2}(q)_{B\to B}\circ\text{C}_{\text{NOT},AB\to
AB}\circ\text{H}_{A\to A}\otimes\text{id}_{B\to
B}(\outerproduct{0}{0}_{A}\otimes\outerproduct{0}{0}_{B}),$ (113)
$\displaystyle\rho_{\text{e}}(\alpha,\beta)$
$\displaystyle:=\mathcal{N}_{1}(p)_{A\to A}\otimes\mathcal{N}_{2}(q)_{B\to
B}\circ\text{C}_{\text{NOT},AB\to AB}\circ\text{H}_{A\to
A}\otimes\text{id}_{B\to B}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left(\left(U_{1}(\alpha)\outerproduct{0}{0}U_{1}^{{\dagger}}(\alpha)\right)_{A}\otimes\left(U_{2}(\beta)\outerproduct{0}{0}U_{2}^{{\dagger}}(\beta)\right)_{B}\right).$
(114)

Lemma III.2 provides a key result for evaluating the entanglement fidelity of
the teleportation channel
$\mathcal{F}(\rho_{\text{b}})=\mathcal{F}_{\text{b}}(p,q)$ (see Eq. 71) and
$\mathcal{F}(\rho_{\text{e}}(\alpha,\beta))=\mathcal{F}_{\text{e}}(\alpha,\beta,p,q)$
(see Eq. 96). Specifically, the entanglement fidelity can be expressed in a
compact form, as shown in the following equation

$\displaystyle F(\mathcal{F}(\rho_{\text{b}}))$
$\displaystyle=\Tr[\rho_{\text{b}}\cdot\phi^{+}],$ (115) $\displaystyle
F(\mathcal{F}(\rho_{\text{e}}(\alpha,\beta)))$
$\displaystyle=\Tr[\rho_{\text{e}}(\alpha,\beta)\cdot\phi^{+}].$ (116)

Drawing upon the insights presented in Lemma I.2, we can obtain a concise
expression for the average fidelity of protocol b. This result is fundamental
for assessing the efficacy of the noisy teleportation protocol in transmitting
quantum states from the sender to the receiver.

Theorem III.2:  Performance of Protocol b The expression for the average
fidelity of protocol b, i.e., $f_{\text{b}}$ (see Eq. 73), is given by
$\displaystyle f_{\text{b}}=\frac{2\Tr[\rho_{\text{b}}\cdot\phi^{+}]+1}{3},$
(117) where $\rho_{\text{b}}$ is defined in Eq. 113.

Protocol e can also be evaluated in terms of average fidelity using Lemma I.2,
similar to protocol b. However, the calculation of the average fidelity for
protocol e is more complex compared to that for protocol b. This is because we
need to maximize over all possible values of the pre-processing parameters
$\alpha$ and $\beta$, which introduces additional computational complexity.

Theorem III.3:  Performance of Protocol e The average fidelity of channel
protocol e, i.e., $f_{\text{e}}$ (see Eq. 99), can be calculated using the
formula $\displaystyle
f_{\text{e}}=\frac{2\left(\max\limits_{\alpha,\beta}\left\\{\Tr[\rho_{\text{e}}(\alpha,\beta)\cdot\phi^{+}]\right\\}\right)+1}{3}.$
(118) Here, $\rho_{\text{e}}(\alpha,\beta)$ is given by Eq. 114.

Moving on to more complex problems, we will now examine the performance of
protocols c, d, and f. Our analysis will begin with protocols c and d, which
share the same state of system $AB$ before post-processing (the region
highlighted in pink of Fig. 8), namely $\rho_{\text{b}}$ (see Eq. 113). To
determine the efficacy of protocols c and d, we can use the lemma that
follows, which provides us with a path forward.

Lemma III.4:  Connection between Average Fidelity and Entanglement
Distillation When a bipartite quantum state $\rho$ on system $AB$, with
dimension $\dim A=\dim B=d$, undergoes a post-processing
$\Theta^{\text{post}}_{RAB\to B}\in\mathfrak{F}_{1}(\mathcal{S})$, where
$\mathcal{S}$ is a set of allowed operations that can be one of the following
$\\{\text{LOCC}_{1}(\text{poly}(d)),\text{LOCC}_{k},\text{LOCC}_{\mathds{N}},\text{LOCC},\overline{\text{LOCC}_{\mathds{N}}},\text{SEP},\text{SEPP},\text{PPT}\\}$,
then the maximal average fidelity of the resulting channel
$\Theta^{\text{post}}(\rho)$, as demonstrated in Fig. 13, is completely
characterized by the $d$-fidelity of $\mathcal{S}$ distillation, which is
defined in Def. II.2. In other words, we can determine the maximal average
fidelity as
$\displaystyle\max_{\Theta^{\text{post}}\in\mathfrak{F}_{1}(\mathcal{S})}f(\Theta^{\text{post}}(\rho))=\frac{dF_{d,\mathcal{S}}(\rho)+1}{d+1},$
(119) where $\mathfrak{F}_{1}(\mathcal{S})$ is defined as the collection of
all free channels, as specified in Def. II.1.

Figure 13: (Color online) Teleportation channel $\Theta^{\text{post}}(\rho)$,
which is created by first preparing an initial state $\rho_{AB}$, and then
subjected to post-processing operations $\Theta^{\text{post}}_{RAB\to
B}\in\mathfrak{F}_{1}(\mathcal{S})$. The post-processing operation
$\Theta^{\text{post}}_{RAB\to B}$ is designed to transform the static state
$\rho$ into a dynamical resource, which can be used to teleport quantum
information from message system $R$ to receiver’s system $B$.

######  Remark III.2.

In Eq. 119, $F_{d,\mathcal{S}}(\rho)$, appeared on the right-hand side, is a
specific instance of the more general $d$-fidelity of $\mathcal{S}$
distillation $F_{d,\mathcal{S}}(\mathcal{E})$ (defined in Def. II.2). To
develop a more complete understanding of the expression, we unpack the formula
for $F_{d,\mathcal{S}}(\rho)$.

$\displaystyle
F_{d,\mathcal{S}}(\rho):=\max_{\Omega\in\mathfrak{F}_{1}(\mathcal{S})}F_{\text{U}}(\Omega(\rho),\phi^{+}_{d})=\max_{\Omega\in\mathfrak{F}_{1}(\mathcal{S})}\Tr[\Omega(\rho)\cdot\phi^{+}_{d}].$
(120)

Since the input of $F_{d,\mathcal{S}}$ is a state, the corresponding free
morphism simplifies to the case of a superchannel without pre-processing,
which is equivalent to a quantum channel.

###### Proof.

The proof will start by showing that the left-hand side is smaller than or
equal to the right-hand side. Assuming that the maximum of the left-hand side
is achieved by some quantum channel $\Xi\in\mathfrak{F}_{1}(\mathcal{S})$, we
have

$\displaystyle\max_{\Theta^{\text{post}}\in\mathfrak{F}_{1}(\mathcal{S})}f(\Theta^{\text{post}}(\rho))=f(\Xi(\rho))=\frac{dF(\Xi(\rho))+1}{d+1}=\frac{d\Tr[\Xi(\rho)(\phi^{+}_{d})\cdot\phi^{+}_{d}]+1}{d+1}\leqslant\frac{dF_{d,\mathcal{S}}(\rho)+1}{d+1}.$
(121)

The second equation is derived from Lem. I.2, the third equation is based on
the definition of entanglement fidelity (see Def. I.2), and the inequality is
a direct consequence of Eq. 120. Without loss of generality, we can assume
that both $\rho$ and $\phi^{+}_{d}$ act on the system $AB$. We can then apply
a local operation to transform $\Xi(\rho)(\phi^{+}_{d})$ from system $RB$ to
$AB$. For a visual illustration, see Fig. 14(a). Since local operations (i.e.,
LO) belong to the set of free morphisms $\mathfrak{F}_{1}(\mathcal{S})$, the
overall operation is still a free channel in $\mathfrak{F}_{1}(\mathcal{S})$.

Figure 14: (Color online) Visual demonstration of the proof of Lem. III.2.
Figure (a) illustrates the role of $\Xi$ behind Equation 121, which is a key
element of our proof. On the other hand, Figure (b) depicts the construction
of a teleportation channel based on $\Lambda$ (see Eq. 123). In this diagram,
the systems located above the dashed red line belong to Alice, while the
systems below the dashed line belong to Bob. While we have focused on qubit
cases in this figure for the sake of simplicity, it is worth noting that our
proof holds for systems of arbitrary finite dimension $d$.

Next, we will show the inverse direction of the proof by demonstrating that
the right-hand side is smaller than or equal to the left-hand side. In this
part of the proof, let us assume that the right-hand side value of
$F_{d,\mathcal{S}}(\rho)$ is achieved by a particular quantum channel
$\Lambda\in\mathfrak{F}_{1}(\mathcal{S})$. Then, we have

$\displaystyle
F_{d,\mathcal{S}}(\rho)=\Tr[\Lambda(\rho)\cdot\phi^{+}_{d}]=F(\mathcal{F}(\Lambda(\rho)))=F(\Theta^{\text{post}}_{0}\circ\Lambda(\rho)),$
(122)

where $\Theta^{\text{post}}_{0}$ is the standard teleportation operation (see
the pink box of Fig. 8(a)), this implies

$\displaystyle\frac{dF_{d,\mathcal{S}}(\rho)+1}{d+1}=\frac{dF(\Theta^{\text{post}}_{0}\circ\Lambda(\rho))+1}{d+1}=f(\Theta^{\text{post}}_{0}\circ\Lambda(\rho))\leqslant\max_{\Theta^{\text{post}}\in\mathfrak{F}_{1}(\mathcal{S})}f(\Theta^{\text{post}}(\rho)).$
(123)

Since $\Lambda\in\mathfrak{F}_{1}(\mathcal{S})$ and
$\Theta^{\text{post}}_{0}\in\text{LOCC}_{1}(\text{poly}(d))\subset\mathfrak{F}_{1}(\mathcal{S})$,
we have
$\Theta^{\text{post}}_{0}\circ\Lambda\in\mathfrak{F}_{1}(\mathcal{S})$. An
illustration of the teleportation channel
$\Theta^{\text{post}}_{0}\circ\Lambda(\rho)$ is provided in Fig. 14(b). The
second equation of Eq. 122 is obtained using Lem. III.2, which allows us to
express the $d$-fidelity of $\mathcal{S}$ distillation
$F_{d,\mathcal{S}}(\rho)$ with respect to the entanglement fidelity of channel
$\Theta^{\text{post}}_{0}\circ\Lambda(\rho)$. Combining Eq. 121 with Eq. 123,
we can conclude that the two sides are equal, which completes the proof. ∎

######  Remark III.3.

We make two remarks about Lem. III.2. Firstly, it generalizes the main theorem
presented in Ref. [19], by extending its scope beyond the LOCC operations
considered in that work. Specifically, while LOCC might require an
impractically large number of communication rounds, our lemma applies to a
broader range of operations, including $\text{LOCC}_{k}$, making it more
practical. Secondly, the proof of our lemma demonstrates that our results are
applicable to any set $\mathcal{S}$ that includes
$\text{LOCC}_{1}(\text{poly}(d))$, i.e.,
$\text{LOCC}_{1}(\text{poly}(d))\subset\mathcal{S}$. This means that the
inequality in Eq. 119 still holds for a broader range of operations beyond
those explicitly considered in the lemma. This observation highlights the
generality and versatility of our approach, and its potential applications to
various teleportation scenarios.

Equipped with Lem. III.2, we can now express the performance of protocols c
and d in a more precise and general way. Specifically, we can apply the
inequality in Eq. 119 to obtain the average fidelity of teleportation channel
produced by these protocols. This allows us to better understand the effects
of noise and imperfections on the entanglement, and to optimize the protocols
for practical applications. First, let’s consider the performance of protocol
c. Written out explicitly, we have

Theorem III.5:  Performance of Protocol c To calculate the average fidelity of
protocol c, denoted by $f_{\text{c}}$ (as defined in Eq. 77), we can use the
following expression $\displaystyle
f_{\text{c}}=\frac{2\left(\max\limits_{\Omega\in\text{LOCC}(A:B)}\Tr[\Omega(\rho_{\text{b}})\cdot\phi^{+}]\right)+1}{3},$
(124) where $\rho_{\text{b}}$ is defined in Eq. 113.

Although Eq. 124 provides a formula for the average fidelity of protocol c, it
is still difficult to compute due to the complexity of the mathematical
structure of LOCC. This limitation has motivated the development of protocol
d, which offers a computable solution to the problem.

Theorem III.6:  Performance of Protocol d To determine the average fidelity of
protocol d, which is denoted by $f_{\text{d}}$ as defined in Eq. 81, we can
utilize the following expression $\displaystyle
f_{\text{d}}=\frac{2\left(\max\limits_{\Omega\in\text{PPT}(A:B)}\Tr[\Omega(\rho_{\text{b}})\cdot\phi^{+}]\right)+1}{3},$
(125) where $\rho_{\text{b}}$ is defined in Eq. 113.

Thanks to the mathematical property of positive partial transpose (PPT)
operations, the optimization term in the right-hand side of Eq. 125 now takes
a tractable semi-definite programming (SDP) form [156, 157]. This enables us
to efficiently optimize the fidelity and make performance comparisons between
protocol d and other teleportation protocols. Through a deeper examination of
the aforementioned optimization, we can simplify it to [22]

$\displaystyle\max\limits_{\Omega\in\text{PPT}(A:B)}\Tr[\Omega(\rho_{\text{b}})\cdot\phi^{+}]=$
$\displaystyle\max\,\,\,\Tr[\rho_{\text{b}}\cdot X_{AB}],$
$\displaystyle\,\,\text{s.t.}\,\,\,\,\,\,\,\mathbb{0}\leqslant
X_{AB}\leqslant\mathbb{1},\,\,\,-\frac{1}{2}\mathbb{1}\leqslant
X_{AB}^{\mathbf{T}_{B}}\leqslant\frac{1}{2}\mathbb{1},$ (126)

where $\mathbf{T}_{B}$ denotes the partial transpose with respect to system
$B$. Writing everything out explicitly, we have

Corollary III.7:  Performance of Protocol d Simplified To determine the
average fidelity of protocol d, which is denoted by $f_{\text{d}}$ as defined
in Eq. 81, we can utilize the following expression $\displaystyle
f_{\text{d}}=\frac{2\max\left\\{\Tr[\rho_{\text{b}}\cdot
X_{AB}]\,\bigg{|}\,\mathbb{0}\leqslant
X_{AB}\leqslant\mathbb{1},\,-\frac{1}{2}\mathbb{1}\leqslant
X_{AB}^{\mathbf{T}_{B}}\leqslant\frac{1}{2}\mathbb{1}\right\\}+1}{3},$ (127)
where $\rho_{\text{b}}$ is defined in Eq. 113.

The implementation of a quantum superchannel significantly increases the
complexity of calculating the average fidelity for protocol f (see Fig. 8(f)).
Fortunately, temporal entanglement provides a natural framework for
simplifying this calculation. By exploiting the properties of temporal
entanglement, we can simplify the calculation of the average fidelity and
improve the efficiency of quantum teleportation. Here, we will delve into the
details of how temporal entanglement can be leveraged to enhance the
performance of quantum teleportation and to achieve higher fidelities. Our
starting point will be a generalized version of Lem. III.2.

Lemma III.8:  Channel Version of Lem. III.2 Consider a bipartite quantum
channel $\mathcal{E}_{A_{1}B_{1}\to A_{2}B_{2}}$ with dimensions $\dim
A_{1}=\dim A_{2}=\dim B_{1}=\dim B_{2}=d$. Let us apply a superchannel
$\Theta_{RA_{2}B_{2}\to A_{1}B_{1}B}$ from the set
$\mathfrak{F}_{2}(\mathcal{S})$ to the channel $\mathcal{E}$, where
$\mathcal{S}$ is a set of permissible operations from
$\\{\text{LOCC}_{1}(\text{poly}(d)),\text{LOCC}_{k},\text{LOCC}_{\mathds{N}},\text{LOCC},\overline{\text{LOCC}_{\mathds{N}}},\text{SEP},\text{SEPP},\text{PPT}\\}$.
We can then characterize the maximum achievable average fidelity of the
resulting channel $\Theta(\mathcal{E})$ (see Fig. 15) using the $d$-fidelity
of $\mathcal{S}$ distillation, defined in Def. II.2. More precisely, the
connection between the maximum achievable average fidelity under free
superchannels and the temporal entanglement can be formally characterized by
the following equation
$\displaystyle\max_{\Theta\in\mathfrak{F}_{2}(\mathcal{S})}f(\Theta(\mathcal{E}))=\frac{dF_{d,\mathcal{S}}(\mathcal{E})+1}{d+1},$
(128) where $\mathfrak{F}_{2}(\mathcal{S})$ is defined as the collection of
all free superchannels, as specified in Def. II.1.

Figure 15: (Color online) Teleportation channel $\Theta(\mathcal{E})$ used for
transmitting quantum information between distant systems. The channel is
obtained by applying a superchannel $\Theta\in\mathfrak{F}_{2}(\mathcal{S})$
to a bipartite channel $\mathcal{E}$. The bipartite channel plays a crucial
role in generating the essential entanglement required for teleportation,
while the superchannel applies a series of designed operations between the
sender, receiver, and message systems to guarantee the reliable and accurate
transmission of quantum states.

###### Proof.

We will prove the equation by establishing an upper bound for the left-hand
side and a lower bound for the right-hand side. Firstly, we will show that the
left-hand side is bounded above by the right-hand side. Assuming that there
exists a quantum superchannel $\Xi\in\mathfrak{F}_{2}(\mathcal{S})$ that
maximizes the left-hand side of Eq. 128, we see that

$\displaystyle\max_{\Theta\in\mathfrak{F}_{2}(\mathcal{S})}f(\Theta(\mathcal{E}))=f(\Xi(\mathcal{E}))=\frac{dF(\Xi(\mathcal{E}))+1}{d+1}=\frac{d\Tr[\Xi(\mathcal{E})(\phi^{+}_{d})\cdot\phi^{+}_{d}]+1}{d+1}\leqslant\frac{dF_{d,\mathcal{S}}(\mathcal{E})+1}{d+1}.$
(129)

We can derive the second equation of Eq. 129 directly from Lemma I.2. To
obtain the third equation of Eq. 129, we use the definition of entanglement
fidelity for a quantum channel, as given in Def. I.2. We note that the
inequality holds because we can prepare a local $\phi^{+}_{d}$ state on
Alice’s side, followed by the free superchannel
$\Xi\in\mathfrak{F}_{2}(\mathcal{S})$. After the interaction, we can apply the
local noiseless channel $\text{id}_{R\to A}\in\text{LO}$. Therefore, we can
view the overall effect of $\Xi(\cdot)(\phi^{+}_{d})$ as a superchannel that
belongs to $\mathfrak{F}_{2}(\mathcal{S})$. Fig. 16 provides a visual
representation of the action of $\Xi(\cdot)(\phi^{+}_{d})$.

Figure 16: (Color online) Visual demonstration of the proof of Lem. III.2.
Figure (a) shows how the superchannel $\Xi$ supports Eq. 129, which is a
crucial part of our proof. Meanwhile, figure (b) illustrates how the
teleportation channel is constructed using $\Lambda$, as described in Eq. 131.
The dashed red line in the diagram serves as a visual separator between the
systems belonging to Alice, which are positioned above the line, and the
systems belonging to Bob, which are located below the line. Although we have
illustrated qubit cases in this figure for simplicity, it is important to
emphasize that our proof of Lem. III.2 applies to systems of any finite
dimension $d$.

Secondly, we will show that the left-hand side is bounded below by the right-
hand side. Assuming that a particular quantum channel
$\Lambda\in\mathfrak{F}_{2}(\mathcal{S})$ achieves the right-hand side value
of $F_{d,\mathcal{S}}(\mathcal{E})$, then we can express
$F_{d,\mathcal{S}}(\mathcal{E})$ equivalently in the following form

$\displaystyle
F_{d,\mathcal{S}}(\mathcal{E})=\Tr[\Lambda(\mathcal{E})\cdot\phi^{+}_{d}]=F(\mathcal{F}(\Lambda(\mathcal{E})))=F(\Theta^{\text{post}}_{0}\circ\Lambda(\mathcal{E})),$
(130)

where the second equation of Eq. 130 is a direct consequence of Lem. III.2,
and the last equation of Eq. 130 comes from Eq. 111, see also Fig. 11 for an
illustration. It is worth noting that in this context, $\Lambda(\mathcal{E})$
denotes a quantum state that operates on the composite system $AB$, while
$\Theta^{\text{post}}_{0}\circ\Lambda(\mathcal{E})$ represents a quantum
channel that maps the message system $R$ to Bob’s system $B$. This immediately
implies that

$\displaystyle\frac{dF_{d,\mathcal{S}}(\mathcal{E})+1}{d+1}=\frac{dF(\Theta^{\text{post}}_{0}\circ\Lambda(\mathcal{E}))+1}{d+1}=f(\Theta^{\text{post}}_{0}\circ\Lambda(\mathcal{E}))\leqslant\max_{\Theta\in\mathfrak{F}_{2}(\mathcal{S})}f(\Theta(\mathcal{E})).$
(131)

Since the standard teleportation operation $\Theta^{\text{post}}_{0}$ can be
realized by using only local operations and one-way classical communication
with the communication complexity polynomially dependent on the dimension of
the message system, it is an element of the set
$\text{LOCC}_{1}(\text{poly}(d))$. When we compose the superchannel $\Lambda$