The network as a whole may function as a complex transformer of quantum states, perhaps with many degrees of freedom, into desired output states. Examples of quantum networks include quantum cryptography apparatuses, multichannel quantum interferometers and quantum computers. These networks may consist of coherently coupled ions, quantum dots or optical beam splitters and may transform electronic states, nuclear spins or optical states. As quantum engineering improves by leaps and bounds, interest in quantum networks grows, both due to improvements in technology and due to the remarkable properties of quantum networks. We find that Lie algebra and Lie group methods are particularly well-suited to studying quantum networks.
More complicated networks can have more complex Lie symmetries.
Networks may be linear or nonlinear. We treat networks as linear if the transformation for the network, for all its varying parameters, is generated by a linear combination of the Lie algebra obtained from the generators for each device. If, on the other hand, the enveloping Lie algebra is required, then the network is regarded as being nonlinear.