# _EntangleNet:_ Theoretical Reestablishment of Entanglement in Quantum Networks +
## 2 The Idea of Restoring Entanglement
### Our Proposal
\[\exists\left\{a_{1},a_{2},\ldots,a_{p}\right\}\subseteq\left\{A_{1},A_{2},\ldots, A_{m}\right\},\text{ such that}\]  
\[\mathcal{S}_{a}=\left\{S_{a_{i}}\mid i=1:p\right\}\text{ are QS},\quad p\leq m, \tag{1}\]  
\[\exists\left\{b_{1},b_{2},\ldots,b_{q}\right\}\subseteq\left\{B_{1},B_{2}, \ldots,B_{n}\right\},\text{ such that}\]  
\[\mathcal{S}_{b}=\left\{S_{b_{j}}\mid j=1:q\right\}\text{ are QS},\quad q\leq n. \tag{2}\]  
Each quantum switch is able to interpret the special message msg_NE_XY and subsequently make itself visible to sender \(X\) by confirming its status as a quantum node through a classical message sent back to source, msg_ACK_X:  
\[S_{a_{i}}\xrightarrow{\texttt{msg\_ACK\_A}}A,\]  
\[S_{b_{j}}\xrightarrow{\texttt{msg\_ACK\_B}}B.\]  
Upon receiving confirmation from each quantum switch, [PERSON] and [PERSON] have their own list of IP addresses that describes a route between them consisting exclusively of quantum nodes:  
* _Route from [PERSON] to [PERSON]_ (known by [PERSON]): \[A\to S_{a_{1}}\to S_{a_{2}}\rightarrow\ldots\to S_{a_{p}}\to B,\]
* _Route from [PERSON] to [PERSON]_ (known by [PERSON]): \[A\xrightarrow{\texttt{msg\_NE\_AB}}S_{A_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{A_{m}}\xrightarrow{\texttt{msg\_NE\_AB}}B,\]0  
To begin the distribution process, [PERSON] and [PERSON] each send an encoded classical message, as seen in Table 1, to every node \(S_{a_{i}}\) and \(S_{b_{j}}\), respectively and also to their partner in order to instruct them on how to act further. This classical information is labeled msg_ACT_A and msg_ACT_B to suggest who it is sent from, [PERSON] or [PERSON]:  
\[A\xrightarrow{\texttt{msg\_NE\_AB}}S_{A_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{A_{m}}\xrightarrow{\texttt{msg\_NE\_AB}}B,\]1  
\[A\xrightarrow{\texttt{msg\_NE\_AB}}S_{A_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{A_{m}}\xrightarrow{\texttt{msg\_NE\_AB}}B,\]2  
\[A\xrightarrow{\texttt{msg\_NE\_AB}}S_{A_{1}}\xrightarrow{\cdots}\ldots \xrightarrow{\cdots}S_{A_{m}}\xrightarrow{\texttt{msg\_NE\_AB}}B,\]3 Two possible routes connecting [PERSON] (\(A\)) and [PERSON] (\(B\)). Circular and rectangular nodes represent quantum and classical switches, respectively. Our designed scheme operates by isolating the number of _active_ quantum switches (AQS) on a route between source and destination to \(N=4k-1\), for \(k\geq 1\). The protocol is symmetric; therefore, we will discuss here the case regarding [PERSON]'s perspective. Considering that \(N\leq p\), we may label the selected subset of AQS as \(\hat{\mathcal{S}}_{a}\subseteq\mathcal{S}_{a}\), with \(|\hat{\mathcal{S}}_{a}|~{}=~{}N\). Quantum switches that are not part of this subset will be referred to as being _inactive_ (IQS). An important aspect must be clarified here. IQS are designated by [PERSON] or [PERSON] in order to restrict the number of quantum switches that actively participate during the protocol to \(N=4k-1\leq p\). This is how our scheme is devised and it operates on this symmetry argument given by the number \(N\) of AQS. There can be at most three IQS and they will be sent an instruction message that makes them operate as classical switches (CS). For now, we consider that \(\hat{\mathcal{S}}_{a}=\mathcal{S}_{a}\) for brevity in notation (the case when \(\hat{\mathcal{S}}_{a}\subset\mathcal{S}_{a}\) means that \(\hat{\mathcal{S}}_{a}\) differs from \(\hat{\mathcal{S}}_{a}\) by three maximum IQS, which can be considered as classical ones operating just for qubit routing). According to the choice of \(N\), there will be an odd number of such AQS. Following the labeling convention in Equations (1) and (2) and considering the common notation (see [9]), we will refer to the node situated halfway in the set \(\hat{\mathcal{S}}_{a}\) as \(O_{A}\equiv S_{a_{2k}}\). There are \(\lfloor\frac{N}{2}\rfloor=2k-1\) quantum switches on each of the segments \(A-O_{A}\) and \(O_{A}-B\).