Editing Routing Entanglement in the Quantum Internet (section)

==Notation==
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Quantum network with topology described by a graph <math>G(V,E)</math>:
* Each of the <math>N=|V|</math> nodes is equipped with a quantum repeater.
* Each of the <math>M=|E|</math> edges is a lossy optical channel of range <math>L_i</math> (km) and power transmissivity <math>\eta_i \propto e^{-\alpha L_i}</math>, <math>i \in E</math> and <math>\alpha</math> depends on the material of the channel.
* <math>K</math> Source-Destination pairs <math>(A_j, B_j)</math>, <math>1 \leq j \leq k</math> situated at (not necessarily distinct) nodes in <math>V</math>.
Considering this graph<math>G</math> we adopt the following notation for the repeater network.
* Each node <math>v \in V</math> is a repeater. 
* Each edge <math>e \in E</math> is a physical link connecting two repeater nodes.
* <math>S(e) \in \mathbb{Z}^+</math> is an integer edge weight corresponding to the number of parallel channel across edge <math>e</math>.
* <math>\mathcal{N}(v)</math> is the set of neighbor edges of <math>v</math>
* <math>d(v) = |\mathcal{N}(v)|</math> is the degree of node <math>v</math>.
* <math>\sum_{e \in \mathcal{N}(v)} S(e)</math> is the number of memories at node <math>v</math>.
* <math>d_A</math> distance of a node to the node A, using the <math>\mathbb{L}^2</math> norm.
* An ebit represents a maximally entangled qubit.

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