Network synthesis of mixed quantum-classical linear stochastic systems

In the last decade, there has been a growing interest in the integration of photonic components into VLSI circuits that in the long term may drive the need for systematic mixed electronic-photonic circuit design methods. On the other hand, theoretical and experimental investigations of coherent-feedback quantum control, the feedback control of a quantum system with another quantum system, have led to the development of a network synthesis theory for linear quantum stochastic models that are commonly employed in (linear) quantum optics, modelling such devices as optical cavities and optical parametric amplifiers. This paper makes theoretical contributions towards connecting these two independent developments, and anticipates potential future interest in systematic design methods for mixed linear electronic-photonic circuits, by establishing a network synthesis theory for linear stochastic systems with mixed quantum and classical degrees of freedom. It is shown how a physically realizable mixed quantum-classical linear stochastic system can be realized as a circuit composed of a feedback interconnection of a fully quantum linear subsystem, that can be implemented by quantum optical devices, and a classical linear sub-system, that can be implemented with standard electrical and electronic devices, together with appropriate interfaces that convert quantum signals to classical signals, and vice-versa. Two feedback architectures are proposed, and a decomposition lemma is derived that shows the structure of linear transformations of bosonic quantum signals into classical signals.