[3][4] Quantum communications offer the potential for secure enterprise-scale solutions.
[7] Prototypical quantum communication networks arrived in 2004.
[8] Large scale communication networks tend to have non-trivial topologies and characteristics, such as small world effect, community structure, or scale-free.
[6] In quantum information theory, qubits are analogous to bits in classical systems.
A qubit is a quantum object that, when measured, can be found to be in one of only two states, and that is used to transmit information.
[3] Photon polarization or nuclear spin are examples of binary phenomena that can be used as qubits.
[1][11][12] While models for quantum complex networks are not of identical structure, usually a node represents a set of qubits in the same station (where operations like Bell measurements and entanglement swapping can be applied) and an edge between node
, although those two qubits are in different places and so cannot physically interact.
[1][11] Quantum networks where the links are interaction terms[clarification needed] instead of entanglement are also of interest.[13][which?]
Each node in the network contains a set of qubits in different states.
[1][11] In this notation, two particles are entangled if the joint wave function,
represents the quantum state of the qubit at node i and
represents the quantum state of the qubit at node j.
Another important concept is maximally entangled states.
The four states (the Bell states) that maximize the entropy of entanglement between two qubits can be written as follows:[3][10] The quantum random network model proposed by Perseguers et al. (2009)[1] can be thought of as a quantum version of the Erdős–Rényi model.
The degree of entanglement between a pair of nodes, represented by
in the Erdős–Rényi model in which two nodes form a connection with probability
refers to the probability of converting an entangled pair of qubits to a maximally entangled state using only local operations and classical communication.
[14] Using Dirac notation, a pair of entangled qubits connecting the nodes
, we obtain the maximally entangled state: For intermediate values of
, successfully converted to the maximally entangled state using LOCC operations.
[14] One feature that distinguishes this model from its classical analogue is the fact that, in quantum random networks, links are only truly established after they are measured, and it is possible to exploit this fact to shape the final state of the network.[relevant?]
For an initial quantum complex network with an infinite number of nodes, Perseguers et al.[1] showed that, the right measurements and entanglement swapping, make it possible[how?]
This result is contrary to classical graph theory, where the type of subgraphs contained in a network is bounded by the value of
Entanglement percolation models attempt to determine whether a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find the best strategies to create such connections.
[11][16] Cirac et al. (2007)[16] applied a model to complex networks by Cuquet et al. (2009),[11] in which nodes are distributed in a lattice[16] or in a complex network,[11] and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangled qubit pair with probability
We can think of maximally entangled qubits as the true links between nodes.
a path between two randomly selected nodes exists with a finite probability, and for
[17] A similar phenomenon was found in the model proposed by Cirac et al. (2007),[16] where the probability of forming a maximally entangled state between two randomly selected nodes is zero if
[16] Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower