# _EntangleNet:_ Theoretical Reestablishment of Entanglement in Quantum Networks +
## 2 The Idea of Restoring Entanglement
### Our Proposal
#### 2.2.2 Efficiency  
We continue with emphasizing a feature of our proposal based on a time efficiency analysis. The general scenario involves the existence of these \(N=4k-1\) active quantum switches between [PERSON] and [PERSON]. As depicted in Figure 2, the straightforward approach of [PERSON] preparing an entangled pair and sending the other particle to [PERSON] is done in \(4k\) units of time, an interval we label as \(t_{0}\). When the quantum switch \(O_{A}\) between \(A\) and \(B\) is employed in the protocol, it prepares its entangled pair and sends each particle to them. Thus, the time required for each qubit to reach its destination is reduced to \(t_{1}=t_{0}/2\), assuming the intermediate node is located halfway on the route from [PERSON] to [PERSON] and qubits travel at the same speed.  
The first particular case of our protocol is \(k=1\), for which there are \(N=4k-1=3\) nodes between [PERSON] and [PERSON] that participate in the distribution, i.e., one additional quantum switch found halfway on the segments \(A-O_{A}\) and \(O_{A}-B\). According to the steps of the scheme, qubits must arrive at \(A^{\prime}=S_{a_{1}}\) and \(B^{\prime}=S_{a_{3}}\). A BSM is then performed at these stations, followed by \(N-1=2\) (two) transmissions of classical information as part of entanglement swapping. Considering this classical information takes a time \(t_{ci}\) to reach its destination and the transmissions occur in parallel, we observe that [PERSON] and [PERSON] will have their desired pair in an interval of \(t_{N}=t_{3}=t_{0}/4+t_{ci}\).  
As the number of quantum switches \(N\) increases, we derive the general relation that describes the time resources needed to supply [PERSON] and [PERSON] with entanglement,  
\[t_{N}=\frac{t_{0}}{N+1}+t_{ci}=\frac{t_{0}}{4k}+t_{ci},\quad k\geq 1. \tag{7}\]  
As \(N\) becomes larger, the distance between quantum switches will be short enough to consider \(t_{ci}\) negligible. Therefore, the protocol reduces the distribution time by a factor of \(N+1=4k\) relative to direct transmission of qubits from [PERSON] to [PERSON],  
\[t_{N}=\frac{t_{0}}{N+1}=\frac{t_{0}}{4k},\quad k\gg 1. \tag{8}\]  
We make some important remarks here. Once all AQS receive [PERSON]'s instruction messages, they begin executing their specific actions in parallel. For the overall distribution time, we do not account for the time it takes for AQS to receive their instructions. Given that entanglement swapping is the key principle of operation, the order in which it happens is not important because the entanglement propagates between nodes until it eventually reaches [PERSON] and [PERSON]. Therefore, [PERSON]'s particle from her EPR pair must only go through one hop, unlike standard distribution which requires \(4k\) hops.  
The series of operations we have presented so far are related to the route from [PERSON] to [PERSON]. The protocol is symmetric, i.e., the analogous operations are performed from [PERSON]'s perspective. Hence, [PERSON] and [PERSON] are actually supplied with two ebits in the time interval \(t_{N}\), one ebit from each route.