Trace-Distance based End-to-End Entanglement Fidelity with Information Preservation in Quantum Networks

A quantum network, denoted as $G$ = ($\mathcal{V}$, $E$), is composed of a set of nodes $\mathcal{V}$ and a set of edges $E$, which represent quantum entities interconnected by shared quantum channels. For a given quantum network, each node $v$ $\in$ $\mathcal{V}$ is equipped with quantum memory. Each edge $(u,v)\in E$ has a channel capacity $c_{u,v}$, which is defined as the number of EPR pairs shared between the adjacent quantum nodes $u$ and $v$. Let $s_{k}$ and $d_{k}$ be the source node and destination node for the $k^{th}$ source-destination pair $(s_{k},d_{k})$, respectively. Let $P^{SPF}_{minCost}$ be the set of shortest paths, and $P_{i}(s_{k},d_{k})$ be the $i^{th}$ shortest path for the source-destination pair $(s_{k},d_{k})$. In the network, let $m_{u}$ represent the memory size at node $u$, and $(p_{i},q_{i})$ denote a probability distribution of the quantum state. The trace-distance of the quantum state over an edge ($u,v$) is denoted as $D_{u,v}$($\rho$, $\sigma$), and the maximum trace-distance of an edge in the selected path from source $s_{k}$ to destination $d_{k}$ is $D_{s_{k},d_{k}}^{max}$($\rho$, $\sigma$). The closeness centrality of a node $v$ is represented by $C_{v}$. The quantum states of the node are represented by $\left|\psi\right\rangle$ and $\left|\varphi\right\rangle$. The density matrices of quantum states $\left|\psi\right\rangle$ and $\left|\varphi\right\rangle$ are denoted as $\rho$ and $\sigma$, respectively, where $U$ and $V$ are unitary operators that perform quantum operations at the nodes, and $Z$ is the Pauli phase operator. Let $\varepsilon$ be the quantum channel and $\rho_{\varepsilon}$ be the density matrix associated with the quantum channel. The operation of the quantum channel on a state is represented by $\zeta(\left|\psi\right\rangle,\left\langle\psi|\right)$. The fidelity over an edge $(u,v)$ is denoted as $F_{u,v}$, and the minimum fidelity of a quantum channel is $F_{min}\left(\varepsilon\right)$. The fidelity of the quantum state over an edge is $F_{u,v}$($\rho$, $\sigma$). The maximum fidelity of an edge in the selected path from source $s_{k}$ to destination $d_{k}$ is ${\hat{F}}_{s_{k},d_{k}}^{SelEdge}$($\rho$, $\sigma$), and the purified fidelity of the edge with the maximum trace distance in the selected path from source $s_{k}$ to destination $d_{k}$ is $F_{D_{s_{k},d_{k}}^{max}}^{purific}$($\rho$, $\sigma$). The terminologies used in this paper are summarized in Table II.

1) Quantum nodes: Every quantum node serves as a processing unit for quantum information. Each node possesses the capability to generate, store, transmit, and manipulate quantum states. The establishment of link-level entanglement between any two neighboring quantum nodes is facilitated through the process of entanglement generation. Each quantum node is furnished with a finite set of quantum memory units [25] and the requisite hardware for executing quantum operations, such as entanglement swapping. Quantum nodes encompass various entities, primarily quantum end nodes, quantum repeaters, and quantum routers. Quantum end nodes play a pivotal role in teleporting unknown quantum bits and processing quantum information to support diverse quantum applications [26]. Quantum repeaters, on the other hand, focus on extending the reach of entanglement distribution. Acting as networking devices, quantum routers interconnect numerous quantum end nodes and direct each entanglement routing request to the intended destination node. The collaborative efforts of quantum routers and quantum repeaters facilitate the seamless interconnection of multiple quantum end nodes within a quantum network.

2) Quantum repeaters: In quantum networks, establishing connections between non-adjacent nodes is a challenging task that necessitates using quantum repeaters [27] for long-distance entanglement sharing via quantum swapping. Quantum repeaters connect with other repeaters and nodes through the classical internet, enabling information exchange over a channel [28]. Due to their shared functionality, quantum nodes and repeaters are considered network nodes.

3) Quantum links: In quantum networks, a quantum link that connects two neighboring quantum nodes serves as the conduit for the distribution of EPR pairs, essentially functioning as a quantum channel. Quantum links exist in two main forms: optical fiber and free space. Both variants of quantum links inherently suffer from losses and decoherence [29], resulting in the success probability of entanglement distribution between adjacent quantum nodes exponentially diminishing with the physical length of the quantum link [30]. Consequently, two neighboring quantum nodes are compelled to undertake multiple entanglement distribution attempts to establish entanglement over a quantum link. Additionally, quantum memory can assist a quantum link in retaining multiple parallel EPR pairs.

3) Quantum memory: Over the past few decades, there has been extensive exploration into various storage schemes for quantum memory [31]. Advances in quantum memory technologies have led to practical improvements in coherence time, fidelity, and efficiency. Due to the low success probability of entanglement distribution between adjacent quantum nodes and the impressive $99.5$% storage fidelity of quantum memory [32], we have embraced a continuous model. This model entails the sharing of EPR pairs between adjacent quantum nodes before path selection, effectively addressing entanglement routing requests. Furthermore, the design of quantum memory allows for the creation of a composite of several independent and accessible memory units. This feature enables the assignment of a unique identity to each stored entanglement in memory, presented in the form of shared EPR pairs. This distinctive identification ensures the correct establishment of end-to-end entanglement between each S-D pair.

Managing the quantum network entails defining the quantum processor, channel, and repeaters. Classical networks connect quantum nodes, each equipped with computational and storage capabilities. The network operates synchronously, with activities segmented into discrete time slots [14]. At the network layer, a centralized controller oversees network management. Nodes can report and update the data, such as fidelity and trace-distance, within the controller’s records [33]. The entanglement routing process includes: (i) Paired nodes generate entangled pairs, and the controller compiles these routing requests [34] and updates to each node’s routing table. (ii) The controller then identifies routing paths and allocates resources utilizing a routing algorithm. Owing to resource constraints and multiple requests, some requests may be declined. (iii) Following the controller’s directives, nodes carry out purification and swapping actions to establish multi-hop entanglement connections for the requested pairs.

In quantum networks, to preserve information and establish E2E entanglement, we need to consider four unique operations: static, i.e., trace-distance and measurement of the quantum state, entanglement purification, and entanglement swapping. These operations have no direct equivalent in classical networking.

1) Static measurement of quantum state: The static measure of a quantum state, also known as trace-distance, quantifies the distinguishability between two quantum states. It measures quantum information between quantum states and holds significant importance in quantum information theory [35]. A smaller value of trace-distance indicates similarity between quantum states. In comparison, a larger value of trace-distance indicates greater distinguishability, which means the quantum state received at a destination node is different from the source node’s quantum state. In a quantum network, the closeness of quantum states between adjacent nodes is quantified by the trace-distance [36], defined as:

The trace-distance between two quantum states is denoted as $D_{u,v}$, where $\rho$ and $\sigma$ are the density matrices representing the quantum states at nodes $u$ and $v$ in a quantum network respectively. Where $\rho=\sum_{i}{p_{i}\left.\left|\psi_{i}\right.\right\rangle\left\langle\left.\psi_{i}\right|\right.}$ and $\sigma=\sum_{i}{q_{i}\left.\left|\varphi_{i}\right.\right\rangle\left\langle\left.\varphi_{i}\right|\right.}$. In the density matrix, $\left|\psi\right\rangle$ and $\left|\varphi\right\rangle$ represent the quantum state with a probability of $p_{i}$ and $q_{i}$, respectively. $tr$ is the trace operation, which is the sum of the diagonal elements of the matrix. $\left|\rho-\sigma\right|$ refers to the positive semidefinite matrix, which is the absolute value of the difference between $\rho$ and $\sigma$. It is found by taking the matrix difference $\rho-\sigma$, and then computing its eigenvalues. The factor of $\frac{1}{2}$ (see Appendix A) ensures that the trace-distance between the normalized density matrices takes values in the range [0,1], i.e., $0\leq D_{u,v}\leq 1$ [37]. This occurs when the two states $\left|\psi\right\rangle$ and $\left|\varphi\right\rangle$ are orthogonal, meaning they are completely distinguishable. Density matrices allow the description of mixed states, where the system is in a statistical mixture of noisy quantum states.

2) Closeness centrality: Closeness centrality is a measure used in network analysis to determine the importance of a node based on its proximity to all other nodes in the network. In a quantum network, a node with high closeness centrality using trace-distance can interact with or distribute entanglement to other nodes with minimal quantum state degradation. This centrality facilitates faster entanglement distribution. Routing algorithms can identify paths that minimize overall distance or communication cost by focusing on nodes with higher closeness centrality, leading to more efficient network performance. Additionally, closeness centrality allows for more effective resource allocation in quantum networks with limited resources such as qubits and entangled pairs. Nodes with higher closeness centrality are better positioned to maintain high-fidelity connections, ensuring optimal network operation.

3) Entanglement swapping: To establish long-distance entanglement connections in a distributed quantum network, entanglement swapping is commonly used [27]. It converts one-hop entanglements into direct entanglements between non-adjacent nodes using a quantum repeater, as shown in Figure 1. However, imperfect measurements during swapping can introduce noise and degrade the entanglement quality [38]. Furthermore, the varying fidelity of entangled pairs across different quantum channels requires different routing paths for end-to-end connections after entanglement swapping.

4) Entanglement purification: In practical quantum networks, directly establishing high-fidelity entanglement between adjacent nodes is challenging due to noise. This process improves link quality in quantum networks by employing controlled nodes and heralding stations [39, 40]. Low-fidelity entangled pairs, such as EPR pairs, are purified using entanglement purification, typically involving CNOT gates or beam splitters [41]. The goal is to maximize the fidelity of entangled states by purifying EPR pairs. The fidelity of the resulting purified state is described in Lemma 1.

There are various entanglement purification techniques, including symmetric purification [42], banded purification [43], and entanglement pumping [27], among others. In this paper, we are using the entanglement pumping technique to purify EPR pairs. In this process, initially, the quantum nodes generate low-fidelity entangled EPR pairs. These pairs are probabilistically combined through entanglement purification using (5), producing a higher-fidelity pair. The fidelity of an EPR pair is increased by purifying it with base-level pairs created via the physical entanglement mechanism. This iterative process continues until the desired higher fidelity of the EPR pairs is achieved. An example of the entanglement pumping method is shown in Figure 2. Over multiple rounds, this pumping technique incrementally improves the entanglement quality, allowing the fidelity between quantum states $\left|\psi\right\rangle$ and $\left|\varphi\right\rangle$ to approach the maximum value. The pumping process continues until the desired fidelity threshold¹¹1The target fidelity level set for entangled quantum states, particularly EPR pairs, during quantum communication or quantum computing processes. In this paper, we use a fidelity threshold of $0.80$. is achieved, at which point it stops. This recursive approach ensures that, despite noise, high-fidelity entanglement can be reliably established across quantum networks, enabling robust quantum communication.

6) Dynamic measurement of quantum state: The dynamic measurement of a quantum state involves measuring a quantum system over time while considering the preservation of information. The environmental loss, imperfect swapping, and quantum gate errors cause a loss in fidelity. This is crucial for maintaining coherence and enabling accurate information processing in quantum systems. The fidelity of the quantum channels is calculated based on the initial quantum state $\left|\psi\right\rangle$ of the nodes. Quantum unitary operations are applied to protect the quantum state during communication [35]. The cumulative effect of these losses on the density matrix of the quantum state and fidelity of these states transferred over a quantum channel $\varepsilon$ as:

where $\rho_{\varepsilon}$ it represents the density matrix of quantum channel. The quantum unitary operation minimizes fidelity loss by evaluating all possible initial states $\left|\psi\right\rangle$, thereby aiding information preservation.This minimization is denoted as:

Where $F_{min}\left(\varepsilon\right)$ is the minimum fidelity of the quantum channel and it is designed to mitigate the loss in a quantum channel, ensuring information preservation and E2E entanglement fidelity within a quantum network.

We focused on preserving quantum state information and E2E entanglement fidelity between selected S-D pairs by performing dynamic operations on the quantum links using (1) and (7). The quantum operation is applied in real-time along the selected routing path $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$, as shown in Figure 3(c), when a quantum state with density matrix $\rho$ or $\sigma$ interacts with the adjacent nodes. Environmental noise during quantum communication can cause significant information loss through the quantum link. To address this problem, we perform a quantitative analysis to preserve information and entanglement fidelity. We use Theorem 1 for our analysis showing that dynamically measuring the quantum state can effectively preserve information during communication in the quantum network.

While transmitting the state $\left|\psi\right\rangle$ from a source to a destination via a quantum channel, the discrete actions of the channel were governed by quantum operations, such as quantum state measurement or unitary operations on the quantum state, as described in (7). It is acknowledged that no quantum channel is completely flawless. In practice, it is possible to measure information in real-time over a quantum communication channel even without knowing the specific details of the quantum state $\left|\psi\right\rangle$. The behavior of the quantum channel could be quantified by E2E fidelity, illustrating that information is maintained during quantum communication. We analyzed an information preservation process on selected S-D pairs in Theorem 1 as follows:

The (8) demonstrates the preservation of information and fidelity within the quantum network, ensuring that the quantum state remains intact following purification on a quantum link. It outlines the conservation of fidelity and information across quantum networks, spanning from E2E within the network. In the worst-case scenario, if the source node attempts to minimize the initial state $\left.|\psi\right\rangle$, the resulting fidelity of the link after this quantum state minimization process will be:

where $F_{min}\left(\varepsilon\right)$ denotes minimum fidelity of quantum channel. As an illustration, consider a quantum network employing a polar channel for entanglement generation across all quantum states $\left.|\psi\right\rangle$. In such a scenario, the minimized fidelity can be expressed as $F_{min}\left(\sqrt{1-\frac{p}{2}}\right)$, with $p$ representing the probability of the quantum state $\left.|\psi\right\rangle$. Now, in the context of a phase-damping channel under similar conditions, the fidelity is given by (13), as follows:

where $p$ denotes a probability, and $Z$ represents a specific operator acting on the quantum state $\left.|\psi\right\rangle$. For a pure quantum state, the right-hand side of the fidelity square term in (13) should always be non-negative. In light of this, we choose the minimum fidelity $F_{min}\left(\varepsilon\right)=\sqrt{p}$ across the quantum link. This minimization strategy aims to protect the information from potential channel noise, ensuring the overall preservation of the pure quantum state in the network using (12). Accordingly, we formulated the information preservation problem to maximize the fidelity of the quantum state as follows:

The objective of (14) is to maximize the fidelity of quantum state to preserve the quantum information and E2E entanglement fidelity, i.e., the E2E entanglement connection established between S-D pairs. The first two constraints, i.e., (15a)-(15b), show that quantum nodes have a finite number of quantum channels and quantum memory, respectively. The next three constraints, i.e., (15c)-(15e) are the flow conservation constraints for the source node, destination node, and any intermediate nodes, respectively, that are held in all routing problems. The inequality in (15f) shows that the number of entanglement link fidelity generated for E2E entanglement paths should not exceed the channel capacity. In (15g) states that after performing purification over on selected path, a quantum state’s fidelity across a quantum edge must be greater than or equal to the previous fidelity value.

In this section, we introduce the TDPP algorithm, which identifies potential paths for E2E entanglement and conducts purification operations on quantum links when necessary. The algorithm, presented in Algorithm 1, systematically explores all potential paths based on the closeness centrality of nodes for each S-D pair in the networks. It updates the path’s trace-distance and fidelity using the routing cost of closeness centrality as shown in Figure 3(a). During each iteration, the algorithm compares the trace-distance of the chosen paths with the fidelity of each edge. When the trace-distance of a selected path exceeds the edge’s fidelity, the algorithm initiates purification to maximize the fidelity and maintain the entanglement connection over that edge.

The TDPP algorithm prioritizes closeness centrality over the number of hops typically favored in classical routing algorithms, such as Dijkstra algorithm [10, 11], to ensure the chosen routing path is the most reliable for information flow in quantum networks. The algorithm takes into account the factors that secure the preservation of information and entanglement fidelity between S-D pairs, ensuring the selected path is optimal for this purpose. This process comprises two specific steps as follows:

1) Initialization:The first step in the TDPP procedure is to measure closeness centrality. It quantifies the average length of the shortest path from a node to all other nodes in the network. Formally, for a given node $v$, the closeness centrality $C_{v}$ [46], measures to which extent a node $v$ is near to all the other nodes along the shortest paths, where $v\in\mathcal{V}$ in a network $G$:

Where $N$ is the total number of nodes in the network and $d\left(v,u\right)$ which is the shortest path length between nodes $v$ and $u$. This metric is particularly useful for identifying influential nodes in terms of information spread or communication efficiency within the network. Consequently, closeness centrality helps to identify the potential paths for entanglement generation in the network. For example, in the given network 3(b), there are two potential paths available $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$, and $s\rightarrow r_{1}\rightarrow d$, receptively. However, given the closeness centrality values shown in Figure 3(b), the closeness is larger in the path $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$ compared to the path $s\rightarrow r_{1}\rightarrow d$. Hence, we select the $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$ path for both communication and E2E entanglement distribution, as depicted in Figure 3(c). We devise an iterative method to determine the most efficient route at the minimum cost possible. The method uses the cost as the basis for each iteration, which helps guide the path search. We use the $K$-shortest path algorithm [38] to identify several equally low-cost routes. After identifying low-cost routes, we find out the trace-distance of quantum states of the selected routes $P_{i}(s_{k},d_{k})$ as shown in Figure 3(c), i.e., $D_{u,v}\left(\rho,\sigma\right)$ for information preservation over the networks. The trace-distance $D_{u,v}\left(\rho,\sigma\right)$ can be calculated as:

where $\left(u,v\right)\in E$ and $\left(\rho,\sigma\right)$ is the density matrix. Assign the trace-distance and fidelity values over an edge to establish an entanglement connection between the S-D pairs. The selected route $P_{i}(s_{k},d_{k})$, i.e., $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$ needs to check the trace-distance and fidelity values over the edge $\left(u,v\right)$. It demonstrates that the degradation of quantum states and the loss of information in the network are reflected by the maximum trace-distance value over the edge, in contrast to fidelity, as illustrated in Figure 3(d). So, the fidelity of the selected path needs to be improved by using extra quantum operations on path $P_{i}\left(s_{k},d_{k}\right)$. We can calculate it using Lemma 1 as:

Where $F_{u,v}\left(\rho,\sigma\right)$ denotes the fidelity over the edge ($u,v$). For example, in Figure 3(d) the selected minimum cost path for entanglement generation is $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$. The trace-distance of the links $\left(s,r_{2}\right)$, $\left(r_{2},r_{3}\right)$ and $\left(r_{3},d\right)$ are 0.60, 0.67, and 0.50, and the corresponding fidelity of the links are 0.80, 0.64, and 0.86, respectively.

2) Path Selection and Purification Process: The TDPP method determines the most efficient routing path by considering the minimal cost associated with a node in quantum networks. This method employs the node’s minimum cost as a metric for each iteration, streamlining both path selection and search processes. This approach uses the K-shortest path algorithm [38] to determine multiple shortest paths with the same minimal closeness distance between quantum nodes.

After the generation of entangled pairs, entanglement switching is necessary to connect the end-to-end entanglement connection. Considering the environmental noise and imperfect measurement of the quantum gate (Pauli-X-gate), the selected path faces where the maximum trace-distance and degradation of fidelity. The purification decision process is designed to ensure the preservation of end-to-end information and fidelity by minimizing the trace-distance. This involves checking the condition using (19) step-by-step to achieve the desired outcome.

To provide all possible purification options and corresponding costs of selected S-D pairs trace-distance, and fidelity over an edge, we calculated the updated cost for edge $\left(u,v\right)\in\ E$ using (20), (21), and (22b) as follows:

where $D^{max}_{s_{k},d_{k}}\left(\rho,\sigma\right)$ is the maximum trace-distance of an edge in the selected path, $\hat{F}_{s_{k},d_{k}}^{SelEdge}\left(\rho,\sigma\right)$ is maximum fidelity of an edge in the selected path and $F_{D_{s_{k},d_{k}}^{max}}^{purific}\left(\rho,\sigma\right)$ is purified fidelity of the edge $(u,v)$ after performing purification operation on a selected path in S-D pairs. The fidelity of edge $(u,v)$, i.e., $F_{s_{k},d_{k}}\left(\rho,\sigma\right)$ is calculated during purification operation as in Lemma 1 to achieve high-fidelity entanglement generation. For example, in the selected path $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$ for entanglement generation, we select the maximum fidelity of the path as threshold fidelity, as shown in Figure 3(e), and the maximum trace-distance of the selected path after an update is $\hat{F}_{s_{k},d_{k}}^{SelEdge}\left(\rho,\sigma\right)$$=0.86$, and $D^{max}_{s_{k},d_{k}}\left(\rho,\sigma\right)$$=0.67$, respectively. This means that along the selected path, the link with the maximum trace distance is precisely the one with the lowest fidelity. Hence, we need to purify the link where the trace-distance is maximum to preserve the information in the quantum network. For this, we use (22a) and (22b) to purify the link fidelity of selected paths $s\rightarrow r_{2}\rightarrow r_{3}\rightarrow d$. Then the resulting fidelity after improvement will be $F_{D_{s_{k},d_{k}}^{max}}^{purifc}\left(\rho,\sigma\right)$$=0.92$, as shown in Figure 3(f). Thus, fidelity improvement is increasing along with decreasing the trace-distance during the purification. Finally, the optimal path for entanglement is shown in Figure 3(f).

However, the purification constraint $D[u][v]<F[u][v]$ must be satisfied for the routing path. If an edge $(u,v)$ cannot provide entangled pairs that satisfy $D^{max}(s_{k},d_{k})\geq F_{D_{s_{k},d_{k}}^{max}}^{purifc}\left(\rho,\sigma\right)$ even after purification, it must be removed from the graph $G$ to reduce complexity. Once this is done, TDPP creates an updated graph $G^{a}$ to keep track of the purification decisions. Finally, TDPP uses the closeness centrality (16) to find the path in graph $G$, ensuring the minimum possible cost. Throughout the iteration process, TDPP also constructs a priority queue to store potential routing paths $P_{minCost}^{SPF}$ and repeats the process as needed.

We conducted a series of numerical evaluations to implement the proposed routing algorithms in quantum networks. The simulation platform utilized for this purpose featured an AMD Ryzen 7 3700X 3.6GHz CPU with 32GB RAM and ran on a Windows-10 64-bit system. As quantum networks evolve, they are anticipated to become essential infrastructure for secure communications and quantum applications, akin to the current Internet backbone. Therefore, we utilized the US backbone network [29] as the topology for our simulation. We conducted simulations $1000$ times for each parameter configuration and calculated the average results. The initial fidelity of entangled pairs adhered to a normal distribution with a mean of $0.80$ and a standard deviation of $0.1$. The standard qubit lifetime is $1.46$ seconds, so we used a synchronization timestamp of 500ms. Each node is assigned $20$ qubits of quantum memory, and the channel capacity of the links between adjacent nodes ranges from $10$ to $90$ qubits.

As a baseline, we used the routing scheme proposed in [4], which includes a purification operation. The entanglement generation rate, which represents the number of successfully established entangled pairs in the network during a fixed time window, is referred to as the throughput (qubits/slot) of the system in routing terminology. This scheme also adopts a proportional share for resource allocation. In our experiment, we utilized the US backbone network as the network topology. We employed quantum state $\left.\left|\psi\right.\right\rangle=\alpha\left.\left|0\right.\right\rangle+\beta\left.\left|1\right.\right\rangle$ where $\alpha$ and $\beta$ act as probability amplitude of getting state $0$ and state $1$, respectively. For example, $\left.\left|\psi\right.\right\rangle=$0.4$\left.\left|0\right.\right\rangle+$0.5$\left.\left|1\right.\right\rangle$. These probability amplitude coefficients ($\alpha$ and $\beta$) need to be normalized to ensure that they fall within the range of [$0$, $1$]. Therefore, we select the maximal mixed quantum state within the range of (0.4, 0.5, 0.6), which indicates greater mixedness and lower information content. This choice impacts the purity of quantum states used in the network and the efficiency of quantum information processing. To gauge the effectiveness of our algorithm, we compared it against three other existing algorithms, namely REPS [5], EFiRAP [16], and Q-PATH [14]. This comparison was made regarding routing and purification techniques, network throughput, and fidelity.

Figures  4(a)$-$(i) show the fidelity performance of the algorithms under different probability amplitude ranges of $\alpha$ and $\beta$ values of ($0.4$, $0.5$, $0.6$). In all scenarios of $\alpha$ and $\beta$, our proposed TDPP algorithm achieves high-fidelity entangled pairs. This outcome is due to our algorithm performing purification based on trace-distance rather than the fidelity threshold. Upon careful observation of these results, we can categorize them into three cases: case 1. where $\alpha=\beta$ depicted in Figures  4(a), (e), and (i); case 2. $\alpha<\beta$ as in Figures 4(b), (c), and (f); and case 3. $\alpha>\beta$ as in Figure 4(d), (g), and (h).

1) $\alpha=\beta$: When we set the $\alpha=\beta$ = 0.4, 0.5, and 0.6 in Figures 4(a), (e), and (i), respectively, our proposed TDPP algorithm performs better compared to other existing algorithms in all three scenarios. This is because in TDPP, the difference in trace-distance between the nodes becomes minimal, and the fidelity of the link is high. In Figures 4(a) and (e), the fidelity difference between the Q-PATH and EFiRAP algorithms is very minimal. This is because, when the channel capacity is high, multi-round purification operations are needed to meet the end-to-end fidelity requirement. As a result, the number of available entangled pairs on each quantum channel decreases. If all entangled pairs reach their maximum, all the routes are discovered and entangled in solution time-space, leading to a similar routing solution obtained by Q-PATH and EFiRAP algorithms. However, in Figure 4(i), the performance of Q-PATH and EFiRAP is clear in terms of fidelity measurement with a utility factor of 0.6. In all three scenarios, the performance of REPS is the lowest, as it only considers entangled pairs for connection and does not require purification or threshold fidelity values to establish the connection.

2) $\alpha<\beta$: Figures 4(b), (c), and (f) show the scenario where $\alpha<\beta$. We observed that the TDPP algorithm performed better than all other algorithms, like in case-1, i.e., $\alpha=\beta$. However, the fidelity performances are not stable as in case-1. The performance of Q-PATH is relatively better than the other two algorithms. In low channel capacity, EFiRAP performs better than REPS algorithm. With the increase in channel capacity, this gap gradually decreases. This is because in the EFiRAP algorithm, entanglement is generated based on the fidelity threshold, and the REPS algorithm generates an entangled path based on available quantum resources. So, when channel capacity increases, generating a more entangled path in quantum networks is possible, showing the REPS algorithm shows linearly increasing fidelity values.

3) $\alpha>\beta$: Figures 4(d), (g), and (h) depict scenarios where $\alpha>\beta$. Our observations indicate that the performance of the algorithms is quite similar to the case-2 scenario, i.e., $\alpha<\beta$. However, the performance of the TDPP algorithm still surpasses all other algorithms. The performance of the algorithms is somewhat unstable at low channel capacities, but it stabilizes as the capacity increases.

Figures 5(a)$-$(i) show the throughput performance of the algorithms in different ranges of utility factor $\alpha$ and $\beta$ values. As in Figure 4, we can group the results of Figure 5 into three categories as case 1. $\alpha=\beta$ as in Figures 5(a), (e), and (i), case 2. $\alpha<\beta$ as in Figures 5(b), (c), and (f), and case 3. $\alpha>\beta$ as in Figures 5(d), (g), and (h).

1) $\alpha=\beta$: Figures 5(a), (e), and (i) show the TDPP algorithm has better throughput performance as compared to other existing algorithms. This is because the TDPP algorithm works on the closeness centrality, which means that the quantum nodes are close to all other nodes in the network, and it helps to generate high-fidelity qubits. The throughput in other algorithms is also relatively stable in this scenario, i.e., $\alpha=\beta$. With the increased channel capacity, the throughput increases steadily in all cases. Because the solution space of a purifying operation is related to channel capacity, algorithms with higher channel capacity can identify superior solutions.

2) $\alpha<\beta$: When we set the utility factor $\alpha<\beta$ in Figures 5(b), (c), and (f), the TDPP algorithm has better throughput performance than other algorithms. This is because, in our proposed TDPP algorithm, higher link capacity will increase the throughput of the network, since more quantum channels will be available to generate qubits. Because of this, the trace-distance values between adjacent nodes are minimal leading to higher throughput. In this scenario, EFiRAP and REPS algorithms show poor performance in throughput due to the following reasons. In EFiRAP, during entanglement generation, a round of purification operations is performed over links, which requires more qubits to be generated, as well as a larger quantum memory. In the REPS, when channel capacity increases, more entanglement is generated between the adjacent nodes. However, most of it is destroyed due to network congestion and low values of EPR pairs.

3) $\alpha>\beta$: Figures 5(d), (g), and (h) demonstrate that an increase in link capacity leads to higher throughput of quantum networks in the TDPP algorithm. This is because, our proposed TDPP algorithm operates on the principle of minimizing trace-distance values between adjacent nodes, ensuring that the generated qubits are highly entangled and possess high-fidelity values. In contrast, algorithms such as Q-PATH, EFiRAP, and REPS consume more qubits to achieve similar high-fidelity values. However, channel utilization decreases significantly when quantum memory becomes a bottleneck, as fewer qubits are available for entanglement and purification operations, substantially reducing quantum throughput in the network. Increasing the quantum memory hosted by each quantum node leads to higher throughput, but the memory utilization will decrease faster. The performance of REPS is the worst since, in this algorithm, purification occurs before path selection and end-to-end fidelity can barely be assured.