# _EntangleNet:_ Theoretical Reestablishment of Entanglement in Quantum Networks +
## 2 The Idea of Restoring Entanglement
### Our Proposal
Assuming each hop is realized in one unit of time, then [PERSON] and [PERSON] will share entanglement after \(t_{0}\equiv 4k\) units of time. Based on these distribution rules, there are four path segments from \(A\) to \(B\) as the following sets:  
* \(R_{1}=\{1,\ldots,k-1\}\),
* \(R_{2}=\{k+1,\ldots,2k-1\}\),
* \(R_{3}=\{2k+1,\ldots,3k-1\}\),
* \(R_{4}=\{3k+1,\ldots,4k-1\}\). Then, as expressed in (3)-(6), \(R_{F}\equiv R_{1}\cup R_{3}\) and \(R_{B}\equiv R_{2}\cup R_{4}\) contain indices that identify active quantum switches on _forward_ and _backward_ distribution routes, respectively, as shown in Figure 3.  
Considering that a classical switch cannot execute quantum-specific operations, it will not require a particular instruction message, since it can only route the received qubit further until it reaches a quantum switch. The main goal of the protocol is to ensure particles reach nodes \(A^{\prime}\) and \(B^{\prime}\) via both upstream (from \(A\) and \(B\) towards \(O_{A}\)) and downstream (from \(O_{A}\) towards \(A\) and \(B\)) routes. Considering each intermediate quantum switch, i.e., each node from \(\mathcal{S}_{a}\setminus\{A^{\prime},O_{A},B^{\prime}\}\) has successfully received the qubit from its nearest node and has itself sent one from its generated pair, then it performs a BSM on the two qubits from different pairs to accomplish entanglement swapping, thus propagating the entanglement towards middle nodes \(A^{\prime}\) and \(B^{\prime}\). Additional classical information resulting from the measurement is required to be provided to the destination node in order to preserve the original Bell state \(|\beta_{00}\rangle\). This step is completed by applying an appropriate corrective operation \(U\in\{I,c_{x},c_{z},c_{z}c_{x}\}\) to the received qubit. These two nodes \(A^{\prime}\) and \(B^{\prime}\) will be the last ones to possess two qubits received from upstream and downstream. A subsequent BSM performed by both of them on their qubits will then establish entanglement between [PERSON] and [PERSON], therefore achieving the original total of \(N-1=4k-2\) Bell measurements are performed during the entire process, along with the same number of classical messages sent to assist in adjusting the received state to be the correct one. A visual representation of the the distribution process is shown in Figure 4.  
\[S_{a_{2k-1}} \stackrel{{ e_{2k}^{\prime}}}{{\longleftarrow}}O_{A},\]0 Creation of entangled pairs at each node from the subset of \(N\) active quantum switches. Depending on the position relative to middle stations \(A^{\prime}\) and \(B^{\prime}\), a qubit is received from downstream or upstream, while one qubit of the generated pair is further sent to another node. Bell measurements (dotted ellipse around particles) will eventually be performed on qubits from different entangled pairs. \[S_{a_{2k-1}} \stackrel{{ e_{2k}^{\prime}}}{{\longleftarrow}}O_{A},\]1 Tree-like illustration of a configuration of \(N=4k-1\) active quantum switches on route from \(A\) to \(B\) (tilde notation dropped to avoid cluttering). Dotted black lines connect entangled particles, as it is initially seen at Alice. A sequential distribution of entanglement from Alice to Bob implies that one particle of Alice’s entangled pair goes through \(4k\) nodes to eventually reach Bob. This process would take \(t_{0}=4k\) units of time. As previously mentioned, the scheme we have presented takes advantage of a configuration of \(N=4k-1\) quantum switches between Alice and Bob. Thus, \(p-N\) quantum switches will not participate in the protocol. Since each active quantum switch must receive a qubit, be able to generate an entangled pair and then send one qubit to the next quantum switch, the inactive \(p-N\) ones will act as classical switches and only route the qubit further, until another AQS is reached.