Quantum Networks

Quantum networks

In addition, quantum network theory which defines the topology of quantum nodes and the propagation and processing of information via entanglement, provides support for LQG and D-Net.

A quantum network is an ensemble of nodes between which, with a certain probability, is a connection. That is they exhibit a certain degree of entanglement. It is therefore necessary to create efficient protocols that maximise the probability of achieving maximum entanglement between any of the nodes. The protocols resort to the concepts of classical information theory (percolation theory), but they substantially enhance their efficiency by enlisting and utilising quantum phenomena.

One such example is applying repeaters in classical networks to prevent the exponential decay of the signal with the number of nodes. There is no direct analogue to this in quantum information theory, but quantum mechanics affords greater possibilities of manipulating quantum bits in order to obtain the information completely.

The fundamental difference with classical systems is that in quantum networks it is no longer necessary to consider the channels and nodes separately. The network is defined as a single quantum state shared by the nodes, optimised as a global entanglement distribution.

It is also possible under these conditions, for different protocols to lead to very different probabilities of achieving maximum entanglement between different nodes. For some special cases such asone and two-dimensional networks with special regular geometry, protocols are derived that are distinctly superior to classical percolation protocols. For the case of a one-dimensional chain the optimum protocol was found. Even under conditions where the signal would decay exponentially in a classic system, it is possible to achieve zero-loss transmission of quantum information. (Quantum repeaters may thus be regarded as simple quantum networks allowing quantum communication over long distances).

The calculations show that the system passes through a kind of phase transition with respect to the degree of entanglement: below a certain threshold value for the degree of entanglement the percolation. In this case the transmission from A to B is zero. Above this value the percolation assumes a certain fixed value that is now independent of the distance between the nodes.

The entanglement distribution in a quantum network thus defines a framework in which statistical methods and concepts such as classical percolation theory quite naturally find application. This leads to a new kind of critical phenomenon, viz an entanglement phase transition. The appropriate critical parameter is the minimum entanglement necessary to establish a perfect quantum channel over long distances.

Accordingly the percolation probability does not decrease exponentially as the distance or number of nodes. The further development of quantum networks calls for a better understanding of such entanglement and percolation strategies.