# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 2 Results
### Creating a Quantum Channel between Nodes by Manipulating Existing Networks  
In the previous section, we have seen how to optimize the generation of complex clusters when we have multimode squeezed vacuum modes and we perform linear optics transformations. As already said, several experimental setups demonstrated the ability to deterministically generate large clusters following this approach; therefore, they can be used to generate the complex clusters presented above. The generated cluster can then be distributed among different parties, i.e., different optical modes corresponding to different nodes of the network can be sent to different players, which can eventually  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(m_{BA}\) & \(\mu_{12}\) (dB) & \(\mu_{13}\) (dB) & \(\mu\) (dB) \\ \hline
1 & \(-6.51\) & \(-6.51\) & \(-4.61\) \\
5 & \(-6.51\) & \(-6.51\) & \(-5.48\) \\
10 & \(-6.51\) & \(-6.51\) & \(-5.76\) \\
20 & \(-6.51\) & \(-6.51\) & \(-6.10\) \\
47 & \(-6.51\) & \(-6.51\) & \(-6.32\) \\ \hlineuse the entanglement correlations between the shared nodes for quantum communication protocols. According to the given task, it may be necessary to reshape the entanglement correlations among the set of nodes. We consider here the simplest case: the protocol wants to establish a maximally entangled state between two arbitrary nodes of a network in order to use it for quantum teleportation. The two-nodes entangled state is a two mode-squeezed state also called EPR, as it is the approximation of the entangled state used by [PERSON], [PERSON], and [PERSON] in their famous paper in 1935 [PERSON] et al. (1935). The task of generating an entanglement link between chosen nodes corresponds to what is called quantum routing. As already said, the procedure we follow is well suited to CV entangled networks as it is relatively easy to deterministically generate the resources and then reshape them, while in the DV case the generation of optical entangled networks is costly and not deterministic so that the best procedure consists in routing the right entanglement connections at the beginning. In the following, we reshape the resource by allowing only for the easiest operations that preserve the number of nodes, i.e., passive linear optics transformations. Quadrature measurements for cluster state reshaping are also an option, but this would imply the removal of the measured node and subsequently the owner of said mode could be cut off from the communication network. For this reason, we will concentrate on linear optics operations, and we will leave open the possibility, in future works, of adding measurements if no other option is possible. These can be global, when they operate on all the nodes at the same time, or local, when they act on a subset of nodes. The case of a global transformation is somewhat trivial as, if we are provided with a cluster \(A\) implemented with a linear optics transformation \(S_{A}\) acting on a set of squeezed input states, it is always possible to find the transformation that leads us to the cluster \(B\). This transformation is simply \(S=S_{B}\cdot S_{A}^{-1}\), where \(S_{B}\) is the linear optics transformation that we should perform on the same set of input states to build the cluster \(B\). We now consider a more interesting scenario: the modes of the cluster are distributed to two spatially separated parties, such that each party is allowed to perform local linear optics transformations only on its set of nodes, as shown in Figure 5. We then want to check which cluster shapes can be reconfigured in order to get a teleportation channel between two arbitrary nodes. A solution to this problem has already been found if we allow for more general symplectic transformations and weighted graphs [39]. Let us say that \(n\) and \(p\) are the number of nodes of party A ([PERSON]) and B ([PERSON]), respectively. We want to act with a linear optics transformation \(U_{A}\) locally on the \(n\) modes and with a linear optics transformation \(U_{B}\) locally on the \(p\) modes.