Capacity of a continuously distributed quantum network

This paper investigates the capacity of a continuously distributed quantum network. A mathematical model of a multihop quantum communication process is given for capacity analysis. Our model assumes $n$ nodes in a unit square, and both a capacity upper bound for arbitrarily distributed quantum networks and a constructed capacity lower bound for randomly distributed quantum networks are obtained as $n\to \infty$. According to our results, the capacity in arbitrarily distributed networks is proportional to $n$ while capacity in randomly distributed networks is independent of $n$. Moreover, our findings exhibit the bounds on quantum network capacity and could open a door to performance analysis for future quantum networking.

Figure 1

One-hop and multihop teleportation process that Alice teleports her qubit to Bob.

Figure 2

Distribution of 100 nodes in a uniformly distributed quantum network. Here each dot represents a quantum node, and the square area has a unit length.

Figure 3

Structure of a chain in a distributed quantum network. Each node except a source node is responsible for establishing and maintaining elementary links to its previous node.

Figure 4

Sketch map that a node is in three chains at the same time. Different lines represent links in different chains numbered from 1 to 3.

Figure 5

Sketch map of hop-by-hop and modified nested swapping protocol. The arc represents entangled links, and the dark circles in each step represent the nodes in which swapping operations are conducted. Elementary links in initial states are generated by EGC with a success probability $P_g$ in each hop.

Figure 6

Flow chart of modified entanglement swapping protocol. After a sequential state, all links will be divided into pairs sequentially; likewise, the same operation will be done reversely after a reverse state. Note that when $n$ is odd in these two states, there will be a redundant link which remains unchanged during swapping operations. In a waiting state, the chain is waiting for refreshing and resequencing; these operations will discard swapped links, bring in new links generated by swapping operations, and renumber these refreshed links sequentially.

Figure 7

The partition of $A$. The length of each subsquare is $\frac{1}{k}$.

Figure 8

Straight-line routing in a unit square. All dark cells are intersected by a line segment formed by source node $i$ and destination node $j$, and each of them, except those having source and destination nodes, will select one node inside as an intermediate node in the chain.

Figure 9

A four-hop chain with branch links. Source and destination nodes are in Cells 1 and 5. Cell 3 provides two nodes to construct branch links.

Figure 10

Subsquare is in the center of unit square. Circumcircles of these two squares are drawn. Two straight lines consisting angle $\theta$ are both tangent to a circumcircle of a subsquare.

Figure 11

Error convergence curve utilizing the Hooke-Jeeves algorithm. The solution of $I\left(F,p_c\right)$ is found by a enumeration method. The left figure is the direct solution, and the right one is the partition solution of the first subproblem.

Figure 12

The $F\text{−}{p}_c$ area is partitioned into four parts. The lined area is the most common condition.

Figure 13

Number of routes served by each cell. The $x$ coordinate and $y$ coordinate display the positions of subsquares.