# _EntangleNet:_ Theoretical Reestablishment of Entanglement in Quantum Networks +
## 2 The Idea of Restoring Entanglement
### Our Proposal  
Given the already existing network infrastructure and taking into account financial costs, a future large scale quantum network would most likely be mapped onto it. Therefore, optical fiber links will represent quantum channels and most nodes will likely use quantum equipment to achieve various tasks. Our solution is formulated in such a context and has at the very basis the idea described in [9], which we improve and generalize. We now define our general model of the network and the elements that are part of our scheme. We consider a _hybrid_ network (having both classical and quantum nodes) with a graph-like topology. Each component node is identified by an address (as an IP address), it is linked by a quantum channel to at least one other node and has a particular status, i.e., it is tasked with accomplishing a certain operation. An important class of nodes we consider is represented by _quantum switches_ (QS), devices that are able to execute quantum routing protocols, generate entangled pairs of particles and perform various types of measurements on qubits. Our parties [PERSON] and [PERSON] are two nodes separated by an arbitrary distance that frequently exchange quantum information via teleportation. Thus, shared entanglement is a key resource they require to achieve their task. After each completed teleportation, their entangled pair is destroyed and another teleportation is not possible. Considering arbitrary distances between end nodes, they cannot afford to locally create entangled pairs of particles and send half of each to their partner. The idea we investigate is how they can be efficiently supplied with another entangled resource. Before discussing it, we present the assumptions based on which the scheme operates. Assumptions:  
* entanglement generation, Bell measurement (BSM) and other local quantum operations are performed perfectly in a negligible amount of time;
* transmission of entangled qubits between adjacent nodes is realized with high fidelity and does not require purification mechanisms. These quantum operations are treated as theoretical processes that are abstract. We are not concerned with their physical nature; otherwise, we would not be able to model them as perfect and instantaneous in our scheme. Travel time between nodes can be neglected as well; however, that is not the most important aspect for our analysis. We discuss the time efficiency of the protocol in Section 2.2.2 by considering through how many hops entangled particles must pass until shared entanglement is achieved between the two parties. We consider that [PERSON] and [PERSON] initially share several entangled pairs of particles described, for example, by Bell state \(|\beta_{00}\rangle=(|00\rangle+|11\rangle)/\sqrt{2}\). After the last teleportation is performed, [PERSON] and [PERSON] acknowledge their resupply need and synchronize with each other in order to send an appropriate classical message to each other at the same time. We label this special message msg_NE_XY, where \"NE\" stands for \"Need entanglement\", \"X\" is the sender and \"Y\" is the receiver. These two messages may reach their destination by different routes, as depicted in Figure 1. We denote the route taken by each message by the set of nodes it passes through:  
* _[PERSON] to [PERSON]_: \(\{S_{A_{i}}\mid i=1:m\}\) \[A\xrightarrow{\texttt{msg\_NE\_AB}}S_{A_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{A_{m}}\xrightarrow{\texttt{msg\_NE\_AB}}B,\]
* _[PERSON] to [PERSON]_: \(\{S_{B_{j}}\mid j=1:n\}\) \[B\xrightarrow{\texttt{msg\_NE\_BA}}S_{B_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{B_{n}}\xrightarrow{\texttt{msg\_NE\_BA}}A.\]  
These sets of nodes may coincide; however, this particular circumstance will rarely occur when considering that a continuous level of network activity would make some routes congested, therefore requiring a different route to be found. Among each set of nodes, there is a subset consisting entirely of quantum switches.