# Simulation of Fidelity in Entanglement-Based Networks with Repeater Chains
## 4 Model for Entanglement-Based Networking  
In this work, as the general model for a quantum network, we adopt an abstract weighted directed graph \(G=(V,E,\{\omega_{e}\}_{e\in E})\) where the vertices \(v\in V\) represent either end-nodes or intermediary nodes, the edges \(e\in E\) represent communication links (classical and quantum) between connected nodes, and each edge is labeled with a weight \(\omega_{e}\geq 0\), which in our case will be the average fidelity between the entangled pairs created at \(e\). Hence, we assume that any point-to-point link can be used for creating link-level entanglement between two nodes, a shared state to be used for some specific communication--quantum or entanglement-assisted--or an arbitrary quantum computation task. An end node is just a source or sink of quantum information, i.e., any physical device capable of creating, operating on, or measuring qubits (more generally, \(n\)-dimensional quantum states).  
Intermediary nodes are generic quantum switches (see Figure 1) that support the creation of end-to-end entanglement. A quantum switch is modeled as a device having \(k\) input/output links and \(m\) quantum memory units, where each memory unit can store one qubit.  
The links can connect two adjacent intermediary nodes, or intermediary and end-nodes, and switching the switch generates an extended entanglement between a subset of its connected users.  
This is accomplished by first generating link-level entanglement between the users and intermediary nodes, followed by an entangling measurement on the quantum memories to create the shared entangled state between the intended users (this process is called entanglement swapping (ES) [31]). If a sequence of ESs is conducted on a path on the network graph \(G\), the outcome will be an end-to-end entanglement (EEE) association between a set of end users in the quantum network. This EEE bond will also be referred to as a quantum path. Moreover, by assumption, multiple paths can simultaneously exist between an origin and a destination.  
From this graph model for Entanglement-Based Networks (EBN) where a request from \(s\) to \(d\) can be served by multiple paths, this paper focuses on linear topologies where quantum repeaters (QRs) are connected to the two next nodes and support one single path. The main goal of this paper is to estimate (by simulation) the feasible EEE fidelity on a linear topology of quantum repeaters. As in other works, we suppose that for generating link-level entanglement, entangled qubit (Bell) pairs are continuously generated between the link endpoints, and each part of these bipartite states can be temporarily stored in a single memory unit until spontaneous decoherence destroys the entanglement. Upon an incoming request, the QR selects a pair of its stored active qubits, performs a Bell measurement on these, and generates a 2-qubit entangled state \(|\Phi^{+}\rangle\) between two of its links. Note that, in our model, the memory registers are functionally identical, finite in number, and have a deterministic finite decoherence time beyond which they are useless for EEE. Such a coherence time emerges out of some physical property dependent on the physical realization of the quantum register (see, e.g., [18; 38; 39]), and is a configurable parameter in our setting. In contrast, the fidelity of a generated Bell pair, after a measurement, or after ES is probabilistic. Furthermore, instead of considering that the latter operations succeed with a fixed probability (as in [31]), our switch relies on physical probabilistic models for these key processes (see Section 3). Since, in particular, ES cannot guarantee an increase in the fidelity of the outcome bipartite state, we will allow the switch to use entanglement purification as a mechanism for boosting the fidelity to values close to 1 (maximal entanglement) using some of the known distillation protocols [5].