I Introduction
Classical information theory has been revitalized into the quantum realm, where it is formally known as quantum Shannon theory or quantum information theory [1]. In both classical and quantum information theory framework, the trajectory traversed by the information carrier is causally welldefined. Interestingly, due to the property of quantum information, the information carrier can traverse quantum channels in a superposition of multiple trajectories, characterized by different causal orders. This implies that the path traversed by the information carrier exhibits an indefinite causal order of quantum channels [2]. Consequently, there is a novel paradigm of quantum information theory involving the existence of indefinite causal order of quantum channels, which leads to a new frontier research field [3].
For instance, let us observe Fig. 1. Assume that in order to transfer the information from the source to the destination, we have to utilize two quantum channels denoted by and . In the conventional framework of quantum information theory, the information carrier can traverse through the quantum channel first and then followed by quantum channel or through the quantum channel first and then followed by quantum channel . In both cases, the causal order of the quantum channels are welldefined, it is either or . In other words, it is said that they have a definite causal order. Furthermore, we may refer to the path traversed by information carrier with a definite causal order as classical trajectory.
Interestingly, the properties of quantum information allow the information carrier to traverse the quantum channels in the superposition of both causal orders as illustrated in Fig. 2. Precisely, in order to transfer the information from the source to the destination, the information carrier can traverse both possible combinations of the classical trajectories of the quantum channels simultaneously, i.e. and . Consequently, the superposition of both classical trajectories traversed by the information carrier exhibits an indefinite causal order. We may refer to the path traversed by an information carrier with an indefinite causal order as quantum trajectory.
Remark.
The term trajectory is used in the relevant literature to denote the path – generally assumed being constituted by a sequence of quantum channels – traversed by the information carrier. If the quantum channels are traversed in a welldefined causal order, the information carrier is said to propagate through a classical trajectory. Conversely, whenever the order of the quantum channels cannot be expressed as a welldefined causal order, the information carrier propagates through a quantum trajectory. In other words, quantum mechanics allows the information carrier to traverse quantum channels placed in a quantum configuration.
As counterintuitive as it seems, the ability of information carrier traversing a superposition of classical trajectories has been experimentally verified using a quantum device called quantum switch [4, 5, 6, 7, 8, 9, 10, 11, 12]. Until recently, the alluring benefits of the indefinite casual order of quantum process have been reported for various aspects of quantum information processing, including quantum computation [13, 14], noiseless quantum teleportation [15], communication complexity [16], quantum resource theory [17, 18], quantum metrology [19], discrimination of quantum process [20], and ultimately for boosting the channel capacities of quantum and classical communications over quantum channels [21, 22, 23, 24].
From a communication engineering point of view, the pivotal question is always how to quantify the advantage that can be obtained from exploiting the indefinite causal order of quantum channels for enhancing the quality of classical and quantum communication. In this treatise, we aim for answering the remaining open questions on how much advantage we can glean from the indefinite causal order of quantum channels to improve the capacities of both entanglementassisted classical and quantum communication. More specifically, within the Quantum Internet framework, multiple quantum devices are interconnected via preshared entanglement for facilitating various applications that require the exchange of classical and quantum information amongst the quantum devices, including quantum communications [25, 26], quantum cryptography [27], quantum sensing [28, 29], distributed quantum computation [30, 31], blind quantum computation [32, 33], quantumsecure directcommunication (QSDC) [34, 35, 36, 37], and quantumsecure secretsharing [38]. Therefore, the preshared entanglement can be viewed as the primary consumable resources for enabling entanglementassisted classical and quantum communications within the Quantum Internet framework [39, 40, 41, 42, 31]. Ultimately, the advantage gleaned from the indefinite causal order of quantum channels for entanglementassisted classical and quantum communication can be immediately extended to the aforementioned applications. Thus, the analysis of entanglementassisted communication over quantum trajectory will provide a critical milestone for the development of Quantum Internet.
In this treatise, we consider quantum superdense coding protocol [43] as our model for entanglementassisted classical communication since a singleletter capacity formulation can be derived for quantum Pauli channel [44]. More specifically, quantum Pauli channel constitutes a set of quantum channel models with various practical applications. Additionally, an entanglementassisted classical communication constituted by quantum superdense coding is known to be the optimal scheme of utilizing a single use of quantum channel and a pair of preshared maximallyentangled quantum state in exchange for two classical bits [43]. Furthermore, the quantum superdense coding versus quantum teleportation tradeoff suggests that the capacity formulation of entanglementassisted quantum communication can be obtained directly from their classical counterparts [45, 46].
Against this background, our contributions can be summarized as follows:

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

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

We present the achievable capacity of entanglementassisted quantum communication over quantum trajectory, which is obtained via quantum superdense versus quantum teleportation tradeoff.
The rest of this treatise is organized as follows. We provide a comprehensive comparison between our work and the stateoftheart in Section II. We present the quantum channel models considered as well as a brief description of quantum superdense coding in Section III. Furthermore, we also provide the tools required for evaluating the capacity of entanglementassisted classical communication based on the description of quantum channel models and quantum superdense coding. It is followed by Section IV where we detail the formal desription of the classical and quantum trajectory. The main results of the entanglementassisted classical and quantum communication capacity are presented in Section V. Finally, we conclude our work in Section VI by also providing several potential directions for future research.
Ii Related Works
The first demonstration of the beneficial capacity gains obtained from the indefinite causal order of quantum channels was presented in [21], which marks the lower bound of quantum communication capacity over quantum trajectory. More specifically, this lower bound is represented by unassisted quantum communication capacity obtained based on the entropy measure of the quantum channels [47, 48]. By contrast, quantum communication capacity is upperbounded by twoway entanglementassisted quantum communication capacity, whose formulation is calculated via relative entropy entanglement of the Choi matrix [49]. Relying on this formulation, the upperbound of quantum communication capacity over quantum trajectory has been recently presented in [23]. Additionally, the lowerbound of quantum communication capacity given in [23] is tighter than that in [21]. Hence, the operating capacity of quantum communication over quantum trajectory has been established. However, a noticeable gap can be observed between these lower and upper bounds, which may cause an inconvenience for determining the exact capacity for specific applications, including those of entanglementassistedbased described in Section I. Thus, to navigate the concept of quantum trajectory closer to practical purposes, in this treatise, we consider the oneway entanglementassisted – both classical and quantum – communication in our investigation, whose capacity conceivably lies between these lower and upper bounds^{1}^{1}1The term unassisted refers to a scenario where the source and the destination does not share a preshared EPR pair, while the term entanglementassisted assumes that the source and the destination have preshared maximallyentangled quantum state such as EPR pairs. The term oneway refers to a scenario where classical communications can be conducted in one direction only, i.e. from the source to the destination (forward direction), while the term twoway refers to a scenario where classical communications can be performed in both directions, i.e. from the source to the destination and vice versa (forward and backward directions). Entanglementassisted communication exhibits higher capacity compared to the unassisted one and the twoway entanglementassisted communication may attain higher capacity than the oneway one..
Since a quantum channel can be used for transferring both classical and quantum information, the investigation related to indefinite causal order of quantum channel is extended to the world of classical communication. However, currently, there is no closedform formula for determining the exact classical communication capacity over a quantum channel, except for the widely known Holevo bound [50, 51], which marks the upperbound of accessible classical information of the unassisted classical communication over a quantum channel. Consequently, it is also generally hard to find a singleletter formula for establishing the classical communication capacity over an indefinite causal order of quantum channels for a wide range channel parameters. However, for very specific cases, such as fullydepolarizing quantum channel and quantum entanglementbreaking channel, the unassisted classical communication capacities over quantum trajectory have been determined [22, 52]. On the other hand, an entanglementassisted classical communication constituted by quantum superdense coding is known to be the optimal scheme of utilizing a single use of quantum channel and a pair of preshared maximallyentangled quantum state in exchange for two classical bits [43]. Therefore, a singleletter formula of entanglementassisted classical communication capacity can be derived for a wide range quantum channel parameters [44]. Relying on these facts, ultimately, we derive closedform expressions for entanglementassisted classical communication capacity over quantum trajectories.
Iii Preliminaries
In the classical domain, the information is conveyed by binary digit (bit), which can carry a value of “0” or “1” at a given time. By contrast, the information in quantum domain is carried by the quantum bit (qubit), where it can be used to represent “0” or “1” or even the superposition of both values. Thus, a qubit can also be used to carry classical information if the qubit is wisely encoded. Similar to the classical domain, the transfer of quantum information between the source and the destination is affected by the noise characterized by the quantum channel . However, differently from the classical domain, for a given quantum channel , we may have several different notions of communication capacity – the maximal amount of information may be transferred reliably under a certain encoding and decoding procedure – which include quantum communication capacity as well as classical communication capacity . More precisely, the quantum communication capacity quantifies the maximum amount of quantum information that can be communicated from the source to the destination over many independent uses of a quantum channel . Similarly, the classical communication capacity quantifies the amount of classical information that can be reliably transferred over many independent uses of a quantum channel . However, in this case, the information is encapsulated using a carefully selected classicaltoquantum mapping. We refer the readers to [53] and [54] for an indepth overview about classical and quantum communication capacities. In this treatise, we consider the socalled entanglementassisted communication capacities, which differ from unassisted communication capacities, since maximallyentangled quantum states are preshared between the source and the destination before the communication is commenced. In this section, we provide the fundamental background required for establishing the main results of our paper.
Iiia Quantum Channel
An arbitrary pure quantum state, denoted by , can be formally expressed as a superposition of the orthogonal basis states as follows:
(1) 
The measurement of the quantum state in the orthogonal basis collapses the superposition of the quantum state into one of the basis states, say
, with probability
. Hence, the coefficients are subjected to the normalization condition . A mixed state, i.e., the statistical mixture of multiple pure quantum states , can be described using the socalled density matrix , which is defined as(2) 
with denoting the outer product between two pure quantum states.
Definition 1.
Quantum channel [55]. A quantum channel is a completelypositive tracepreserving (CPTP) map acting on arbitrary quantum states, which can be written in the operatorsum representation as follows:
(3) 
where denotes a set of operators – referred to as Kraus operators – satisfying the following completeness criterion:
(4) 
A special class of the quantum channel representation of (3) is constituted by a quantum channel where the Kraus operators can be expressed in terms of unitary operators as follows:
(5) 
where is a unitary operator satisfying and
is a probability distribution satisfying
. Hence, the relationship between the Kraus operators and the unitary operators is given by .Definition 2.
Let us now consider a joint quantum state shared between two parties and , where quantum channel affects only the quantum state at . Since we consider that the quantum state at undergoes an “errorfree” identity channel, the extended mapping of on the joint quantum state is given by
(7) 
where denotes the set of Kraus operators of the quantum channel and denotes the set of extended Kraus operators.
IiiB Quantum Superdense Coding
The main objective of this study is to evaluate the entanglementassisted classical and quantum communication capacity over quantum trajectories. To this aim, we can determine the entanglementassisted classical communication capacity via quantum superdense coding [43], where a single use of a quantum channel and a pair of preshared maximallyentangled quantum state can be used for transmitting two classical bits. The general schematic of quantum superdense coding over quantum channel is shown in Fig. 3.
The quantum superdense coding protocol is commenced by presharing a maximallyentangled quantum state between Alice and Bob , which is constituted by the following EPR pair:
(8) 
where the subscripts and indicate that the first qubit is held by , while the second is by . Let us assume that the presharing step is errorfree since multiple copies of EPR pairs can be prepared and hence a quantum entanglement distillation protocol can be invoked to eliminate the quantum errors [56].
A twobit vector
is used for applying a controlled and controlled operations defined by . To elaborate a little further, if the classical bit , a gate is applied to the qubit of the EPR pair on side, otherwise, an identity gate is applied. Similarly, if the classical bit , an gate is applied, otherwise, an identity gate is applied. Consequently, the classicaltoquantum mapping of to EPR pairs is given by(9) 
Next, the qubit of the EPR pair at side is sent through a quantum channel as shown in Fig. 3. Assume that we have a quantum Pauli channel , which means that the quantum channel inflicts the bitflip , phaseflip , as well as the simultaneous bitflip and phaseflip errors with the probability of , , and , respectively. Therefore, the transition probability of every possible combinations of due to , , and errors is depicted in Fig. 4(a). Finally, as both of the qubits of the EPR pair are now at side, a Bellstate measurement is conducted to recover the twobit vector , which is the corrupted version of . With this, the quantum superdense coding protocol over quantum channel is completed.
IiiC The Capacity of EntanglementAssisted Classical Communication
Evaluating the capacity of quantum superdense coding protocol can be reformulated as evaluating the capacity of quaternary discrete classical communication. To obtain a singleletter formula of its capacity, first, we provide the general definition of classical communication capacity.
Definition 3.
Classical communication capacity [57]. The classical communication capacity is defined by
(10) 
where
is the mutual information between random variables
and , is the probability distribution of the source emitting symbol , is the entropy of random variable , is the conditional entropy of random variable conditioned by .Furthermore, the capacity of (10) is also equivalent to
(11) 
Since we assume that we have an equiprobable source for the symbols and is time invariant, the expression of (11) can be further simplified to
(12) 
where is the transition probability of for any characterized by the channel^{3}^{3}3The transition probability of for quantum Pauli channel is known to be symmetric for any ..
In order to glean a clearer idea, let us proceed with an example. Consider the quantum depolarizing channel affecting a singlequbit having the density matrix as follows:
(13) 
where is the depolarizing probability. Now, we have to make the connection between the singlequbit depolarizing channel of (13) and the transition probability of Fig. 4. Utilizing the extended mapping of the quantum channel of (7) and by considering , the effect of a singlequbit depolarizing channel on an EPR pair is given by
(14) 
Based on (14), we obtain the transition probability as follows:
(15) 
where represents the decimal representation of binary vector . Finally, substituting the value and of (15) into (12), we obtain the entanglementassisted classical communication capacity in terms of as follows [44]:
(16) 
For a singlequbit depolarizing channel, the capacity can be derived for both symbolbased and bitbased classical communication. The symbolbased capacity is already given in (16) based on the transition probability of in Fig. 4(a). By contrast, the bitbased capacity can be determined by summing the individual capacity of and based on the transition probability of Fig. 4(b), where we obtain
(17) 
As a benchmark, we provide the unassisted classical communication capacity of a singlequbit depolarizing channel. Luckily, for a singlequbit depolarizing channel, the capacity always correspond to the Von Neumann measurements [58], where in terms of is given by
(18) 
Notice that the unassisted classical capacity is exactly half of the bitbased entanglementassisted classical capacity^{4}^{4}4With reference to the entanglementassisted capacity, this statement is true since the cost of distributing entanglement is not taken into account, as generally assumed in the theoretical quantum information theory framework..
Remark.
The inherent interbit correlation within the symbols makes the symbolbased capacity slightly higher than the sum of individual bitbased capacity. Consequently, the entanglementassisted classical communication capacity is also slightly higher than the sum of individual unassisted classical communication capacity.
We portray the symbolbased capacity of (16), the bitbased capacity of (17), and the unassisted capacity of (18) in Fig. 5. Observe that all the capacities are equal to when . This specific point is associated with the fullydepolarizing quantum channel where we have for any arbitrary . Consequently, there is no classical information can be transmitted, either by employing entanglementassisted or unassisted, through a fullydepolarizing quantum channel.
Finally, following the reasoning we utilized for deriving the entanglementassisted classical communication capacity for singlequbit depolarizing channel, we may also directly derive the entanglementassisted classical communication capacity for the general quantum Pauli channel of (6), which is given by
(19) 
For the rest of treatise, we are going to use the entanglementassisted capacity of (19) for evaluating the bottleneck capacity of the quantum channels arranged in a welldefined causal order, which we will elaborate in the next section.
Iv The Classical and Quantum Trajectory
As we have briefly alluded in Section I, the unique properties of quantum information allow the information carrier to traverse multiple classical trajectories simultaneously. In this section, we provide the formal mathematical description of the classical and quantum trajectory of two quantum channels. More specifically, a classical trajectory of two quantum channels is characterized by a welldefined causal order of either or . By contrast, a quantum trajectory of two quantum channels is characterized by the superposition of two classical trajectories and implying that the trajectory exhibits an indefinite causal order of the quantum channels and .
Iva Classical Trajectory
Consider two quantum channels and as follows:
(20) 
The resultant channel of two quantum channels over a classical trajectory with a definite causal order of is formulated as
(21) 
where the Kraus operators are given by
(22) 
The amount of transferable information over a classical trajectory of two quantum channels and is upperbounded by the socalled bottleneck capacity, which applies for both classical and quantum communications as well as for both unassisted and entanglementassisted communications.
Definition 4.
The bottleneck capacity [59, 23]. The communication capacity over classical trajectory of two quantum channels is upperbounded by
(23) 
where and are the capacities of the individual quantum channels and , respectively. When and represent the entanglementassisted communication capacities of and , respectively, the notation is used for portraying the associated bottleneck capacity.
IvB Quantum Trajectory
Given two quantum channels and , the resultant quantum channel over quantum trajectory is formulated as [21]
(24) 
where is the control qubit and the Kraus operators are given by
(25) 
The formulation of the Kraus operators over quantum trajectory implies that we may create a superposition of classical trajectories using the control qubit . More specifically, when , the information carrier of the quantum state traverses the classical trajectory of . By contrast, when , the information carrier of the quantum state traverses the classical trajectory of . For instance, we may create an equal superposition of both classical trajectories by initializing , which means that the quantum state is traversed through both classical trajectories simultaneously, which results in an indefinite causal order of quantum channels. Thus, throughout this treatise, we assume that the control qubit is always initialized in the quantum state of since it gives us the capability of detecting the superposition of causal orders from anticommuting Kraus operators and ultimately provides us with the highest possible capacity gain [21].
More specifically, let us assume that and are two anticommuting Kraus operators, i.e. . Thus, the Kraus operators of Eq (25) can be expressed as follows:
(26) 
Therefore, the action of Kraus operators of (26) on the initial quantum state of is given by
(27) 
According to this result, when we measure the quantum state on the control qubit, we can infer that a superposition of causal orders from two anticommuting Kraus operators has taken place. Thus, we can utilize the measurement result to our advantage for improving the performance of classical and quantum communication.
Remark.
If the control qubit is initialized in the quantum state of , the superposition of causal orders from two anticommuting Kraus operators transforms the control qubit into .
Based on the formal description of the quantum trajectory in (24), indeed we require an additional auxiliary qubit to control the superposition of the causal orders of the quantum channels and . Intuitively, we have the inclination to make a comparison between the advocated scheme presented in this treatise to another auxiliaryqubits assisted scheme, such as quantum errorcorrection codes [60, 61]. Nevertheless, viewing the control qubit of (24) in the same light as the auxiliary qubits in quantum errorcorrection codes can be very problematic. More specifically, the auxiliary qubits in quantum errorcorrection codes are encoded together with the logical qubits and sent through the quantum channels carrying a certain amount of information. Consequently, the incorporation of auxiliary qubits in quantum errorcorrection codes increases the total number of quantum channel uses. By contrast, the utilization of the auxiliary qubit of (24) does not increase the number of quantum channel uses. In fact, as shown in [18], the encoding operation of (24) must be considered as a nonsidechannel generating operation, since the control qubit does not carry – or embed in any way – the information from the source to the destination. The initial state of the control qubit is fixed as part of the placement and thus, it is deemed inaccessible to the sender for encoding information. Having said that, we refer enthusiastic readers to [18] for the full discourse on quantum resource theories.
V The EntanglementAssisted Capacity over Classical and Quantum Trajectory
We employ the formulation of classical and quantum trajectories of (22) and (25) and incorporate them into (7) for devising the transition probability required for calculating the classical communication capacity of (12). Consider the following quantum Pauli channels and :
(28) 
(29) 
Our results of the entanglementassisted classical communication capacity over classical and quantum trajectories are summarized in the following propositions.
Proposition 1.
The entanglementassisted classical communication capacity of the two arbitrary quantum Pauli channels in (28) and (29) over a classical trajectory is given by
(30) 
where , , , and .
Proof:
Please refer to Appendix A. ∎
Proposition 2.
The entanglementassisted classical communication capacity of the two arbitrary quantum Pauli channels of (28) and (29) over a quantum trajectory is given by
(31) 
where , , , , , , , , and is the binary entropy of defined by .
Proof:
Please refer to Appendix B. ∎
Observe from (30) and (31), the following inequality always holds: . The equality holds when implying , , , , and . It means that the advantage of quantum trajectory can only be observed when the coefficient pair of for noncommuting Pauli matrices is non zero. More specifically, the two classical trajectories of two noncommuting Pauli matrices, for example , transforms the control qubit into , which consequently increases the entanglementassisted classical communication capacity compared to that of classical trajectories.
Remark.
In order to glean the advantage of quantum trajectory for increasing the entanglementassisted classical communication capacity of two arbitrary quantum Pauli channels and , the coefficient pair of for noncommuting Pauli matrices has to be nonzero.
Va Classical Communication Capacity
In the following subsections, we provide several derivative results of Proposition 1 and 2 for various types of quantum Pauli channels, which are widely considered in the practical applications of quantum communications. More specifically, we consider the combination of bitflip and phase flip quantum channels, quantum entanglementbreaking channels, as well as quantum depolarizing channels. Moreover, the experimental implementation of the indefinite causal orders of the aforementioned quantum channels can be found in [8, 10, 11].
VA1 Quantum BitFlip and PhaseFlip Channels
We consider two quantum Pauli channels constituted by bitflip and phaseflip quantum channels. A quantum bitflip channel is defined as follows:
(32) 
where the Kraus operators are given by and . A quantum phaseflip channel is defined as follows:
(33) 
where the Kraus operators are given by and . Based on the Kraus operators of the bitflip and phaseflip quantum channels of (32) and (33), respectively, and based on the formulation of Kraus operators for classical trajectory of (22), we obtain the following corollary.
Corollary 1.
The entanglementassisted classical communication capacity of bitflip and phaseflip quantum channels over classical trajectory is given by
(34) 
Similarly, based on the formulation of Kraus operators for quantum trajectory of (25), we also obtain the following corollary.
Corollary 2.
The entanglementassisted classical communication capacity of bitflip and phaseflip quantum channels over quantum trajectory is given by
(35) 
where .
To demonstrate the advantage of the quantum trajectory compared to the classical trajectory, let us consider a scenario where we have . We plot the entanglementassisted capacity of (34) and (35) in a scenario of in Fig. 6. In addition to these results, we include the bottleneck capacity of the two quantum channels. Observe that the capacity of the resultant channel over classical trajectory indeed cannot violate the stringent bottleneck capacity. However, the capacity over quantum trajectory violates the bottleneck capacity in the region of . This violation of bottleneck capacity is induced by the combination of two causal orders of two noncommuting Pauli matrices from the Kraus operators of the quantum channels, . This combination transforms the control qubit , which is initialized in the quantum state of , into quantum state of . Consequently, this gives us a perfect entanglementassisted classical communication with a probability of since every time we measure on the control qubit, we can apply operation to perfectly recover the quantum state of information qubit . This is very important to highlight that the measurement of the control qubit does not give us any information about which causal order the information carrier has traversed, or in other words it means the measurement does not collapse the superposition of the classical trajectories. Instead, what the measurement tells us is if the quantum channels have inflicted two noncommuting Pauli matrices in two different causal orders, which can only occur when the information carrier has actually traversed both classical trajectories simultaneously.
Remark.
The result in Corollary 2 implies that the indefinite causal order of the quantum channels allows us to violate the bottleneck capacity, which constrains the capacity of classical and quantum communication traversed over a classical trajectory with a definite causal order of quantum channels.
For the sake of demonstration, the results presented in in Fig. 6 is valid for a scenario of . However, the results presented in Corollary 1 and 2 are actually general for a wider range of and . To enrich our discussion and to explore the advantages gleaned by the indefinite causal order of quantum channels, we introduce two additional metrics, namely the capacity gain and the bottleneck violation based on the following definitions.
Definition 5.
The capacity gain is defined as the difference between the entanglementassisted classical communication capacity over quantum trajectory and that of classical trajectory, which depicts the amount of additional classical information that can be sent by exploiting the indefinite causal order of quantum channels. Formally, the capacity gain can be expressed as
(36) 
where is the entanglementassisted classical communication capacity over quantum trajectory presented in (31) of Proposition 2, while is that of over classical trajectory obtained from (30) of Proposition 1.
Definition 6.
The bottleneck violation is defined as the nonnegative gain obtained by entanglementassisted classical communication capacity over quantum trajectory against the bottleneck capacity. The bottleneck violation depicts the amount of capacity gain which cannot be attained through any definite causal order of quantum channels, which is formally expressed as
(37) 
where is the entanglementassisted classical communication capacity over quantum trajectory presented in (31) of Proposition 2, while is the bottleneck capacity of entanglementassisted classical communication based on (23) of Definition 23.
We portray the capacity gain for the bitflip and phaseflip quantum channels over quantum trajectory in Fig. 7(a), while the bottleneck violation in Fig. 7(b). Observe that based on the color map in Fig. 7(a), we obtain the benefit of capacity gain at all range values of . However, the significant capacity gain is more prominent in the region of portrayed by the dark red color area. The maximum gain observed is , which is attained for the values of satisfying for . We are also interested in the region where we can observe the violation of the bottleneck capacity as well as the associated magnitude. In Fig. 7(b), we observe that the violation of bottleneck capacity occurs in the region of for . The maximum violation observed is , which is attained for the values of and . This can be verified directly because the bottleneck capacities for and are . By contrast, for these given values and , we have , which means a perfect entanglementassisted classical communication over quantum trajectory.
VA2 Quantum EntanglementBreaking Channels
Any quantum channel is said to be entanglement breaking if it transforms a maximallyentangled quantum state , for example , into a mixture of two separable quantum states and . Formally, a quantum entanglementbreaking channel is defined as follows [62]:
(38) 
An example of quantum entanglementbreaking channels that can be written as a linear combination of Pauli matrices is given by
(39) 
Now, let us consider two partially entanglementbreaking channels, where the first quantum channel is formulated as
(40) 
where the Kraus operators are given by and , where gives us the quantum entanglementbreaking channel of (39). Similarly, the second quantum channel is formulated as
(41) 
where the Kraus operators are given by and . Based on the Kraus operators of the partially entanglementbreaking channels of (40) and (41) and based on the formulation of Kraus operators for classical trajectory of (22), we have the following corollary.
Corollary 3.
The entanglementassisted classical communication capacity of two partially entanglementbreaking channels over classical trajectory is given by
(42) 
Similarly, based on the formulation of Kraus operators for quantum trajectory of (25), we also have the following corollary.
Corollary 4.
The entanglementassisted classical communication capacity of two partially entanglementbreaking channels over quantum trajectory is given by
(43) 
for every value of and .
To provide a clearer picture, we portray the entanglementassisted capacity of (42) and (43) in a scenario where we have in Fig. 8. Once again, we also include the bottleneck capacity of the two partially entanglementbreaking channels. Here, we observe an interesting phenomenon where we can always achieve a perfect entanglementassisted classical communication over quantum trajectory, which demonstrates a full violation of bottleneck capacity for every value of . Compared to the result presented in Subsection VA1, the Kraus operators of the partially entanglementbreaking channels contain two noncommuting Pauli matrices given by . Therefore, the two different causal orders of two noncommuting Pauli matrices can always be detected by the control qubit , since it transforms the initialized into . By contrast, the remaining combinations of the Pauli matrices given by , leave the control qubit unchanged.
We portray the capacity gain for two partially entanglementbreaking channels over quantum trajectory in Fig. 9(a), while the bottleneck violation in Fig. 9(b). Since we always attain a perfect entanglementassisted classical communication over quantum trajectory for two partially entanglementbreaking channels, the magnitude of capacity gain, which may be observed from the dominantly dark red color area portrayed in Fig. 9(a), is significantly higher than that of bitflip and phaseflip quantum channels. Consequently, the benefit from capacity gain from Fig. 9(a) is directly translated to the capacity violation displayed by the color map in Fig. 9(b). In this case, we have shown that the quantum trajectory of two entanglementbreaking channels is capable of achieving the full violation of bottleneck capacity for every value and , as described in (43) of Corollary 4.
Remark.
The result in Corollary 4 implies that the indefinite causal order of the quantum channels allows us to always enable a perfect entanglementassisted classical communication over two partially entanglementbreaking channels.
VA3 Quantum Depolarizing Channels
Let us now consider two quantum depolarizing channels. For the first quantum channel, we have the following description:
(44) 
where the Kraus operators are given by , , , and . Similarly, for the second quantum channel, we have
(45) 
where the Kraus operators are given by , , , and . Based on the Kraus operators of the quantum depolarizing channels of (44) and (45) and based on the formulation of Kraus operators for classical trajectory of (22), we arrive at the following corollary.
Corollary 5.
The entanglementassisted classical communication capacity of two quantum depolarizing channels over classical trajectory is given by
(46) 
Similarly, based on the the formulation of Kraus operators for quantum trajectory of (25), we also arrive at the following corollary.
Corollary 6.
The entanglementassisted classical communication capacity of two quantum depolarizing channels over quantum trajectory is given by
(47) 
where .
As a special case, we portray the entanglementassisted capacity of (46) and (47) in a scenario where we have in Fig. 10. In Subsection VA1 and VA2, the bottleneck capacities are always strictly positive. Instead, here we have a condition where we have a zerocapacity at . At this point, we have the socalled fullydepolarizing quantum channel, which we have mentioned briefly in Section IIIC. It means that no classical information can be sent by the means of entanglementassisted classical communication through the individual quantum channel. Clearly, this is also the case for the classical trajectory since it cannot violate the bottleneck capacity. Interestingly, observe in Fig. 10 that we have a nonzero capacity for entanglementassisted communication utilizing two fullydepolarizing quantum channels over quantum trajectory. More specifically, we have for . The ability to enable a nonzero capacity communication using two zerocapacity quantum channels over quantum trajectory is referred to as the causal activation [22, 21, 24, 18].
Remark.
The entanglementassisted classical communication capacity over quantum trajectory is always strictly positive, which means that we can always send classical information through quantum channels even when we cannot send any classical information through the individual quantum channel.
We portray the capacity gain for two quantum depolarizing channels over quantum trajectory in Fig. 11(a), while the bottleneck violation in Fig. 11(b). Observe that based on the color map in Fig. 11(a), the capacity gain gleaned at range values of are relatively modest compared to that of bitflip and phaseflip quantum channels. The capacity gain are more prominent in the region of denoted by the light blue to dark red color area. The maximum gain observed is , which is attained for the value of . In Fig. 11(b), we also observe modest violation of bottleneck capacity. Nonetheless, the bottleneck violation can be observed for a wide range of and values. The maximum violation observed is , which is also attained for the value of .
Remark.
The quantum trajectory, which leads to indefinite causal order of quantum channels, enables the violation of bottleneck capacity, implying that the phenomenon inducing this violation cannot be obtained by any process exhibiting a welldefined causal order.
VB Quantum Communication Capacity
In the previous subsections, we have provided a thorough analysis on the entanglementassisted classical communication capacity via classical and quantum trajectories. In this section, we extend the analysis to the entanglementassisted quantum communication. In [44] and [45], the relationship between the entanglementassisted capacity of classical communication and that of quantum capacity is readily given by
(48) 
based on quantum superdense coding versus quantum teleportation tradeoff, which dictates that a pair of preshared maximallyentangled quantum state and a single use of quantum channel can be exchanged for a single qubit or two classical bits. Consequently, by exploiting (48), the results in Proposition 1 and 2 for entanglementassisted classical communication over classical and quantum trajectories can be extended directly to the entanglementassisted quantum communications as shown in the following corollaries.
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