Editing Routing Entanglement in the Quantum Internet (section)

===Multipath Routing of a Single Entanglement Flow===
'''Input:''' <math>K=1</math>, Source-Destination pair <math>(A_1, B_1)</math>.

'''Output:''' Quantum network with ebits shared between the end points of <math>(A_1, B_1)</math>.

====Protocol for Entanglement Routing with Global Link-state Information====
After the External Phase
# <math>m = 1</math>
# <math>G_m</math> = Subgraph induced by the successful external links and the repeater nodes after the external phase.
# While (True):
## <math>S_m</math> = shortest path in <math>G_m</math> connecting <math>A_1</math> with <math>B_1</math>. 
## If <math>S_m</math> is empty: Break While Loop.
## Else:
### Set <math>L_m</math> as the length of <math>S_m</math>.
### Try connecting all internal links along the nodes of <math>S_m</math> <br />//successfully generating an ebit between <math>A_1</math> and <math>B_1</math> with probability <math>q^{L_m - 1}</math>
## <math>G_{m+1}</math> = <math>G_m -</math> all external and internal links of <math>S_m</math>.
## <math>m = m + 1.</math> 

====Protocol for Entanglement Routing with Local Link-state Information====

Knowledge of success and failure of the External Phase is communicated only to the two repeater nodes connected by the link.
Repeater nodes need to decide on which pair(s) of memories BSMs should be attempted (which internal links to attempt), based only on information about the states of external links adjacent to them.

After the External Phase
# For every repeater except <math>A_1</math> and <math>B_1</math>: <br /> Let <math>u</math> be the node of this iteration.
## If less than one of the neighboring external links is successful: <br />Then no internal links are attempted since this repeater node can not be part of a path from <math>A_1</math> to <math>B_1</math>.
## Else If two or more neighboring external links are successful: <br />Then let <math>v</math> be the node linked to <math>u</math> with the smallest <math>d_{A_1}</math> and <math>w</math> be the node linked to <math>u</math> with the smallest <math>d_{B_1}</math>. <br /> Attempt a BSM on node <math>u</math> on the memories connected to <math>v</math> and <math>w</math>.
### Else If four neighboring external links are successful: <br /> Then attempt a BSM on node <math>u</math> on the remaining memories disconsidering the two memories from the previous step.