Structures and Transformations for Model Reduction of Linear Quantum Stochastic Systems

The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. It is shown that subsystem truncation preserves the physical realizability property of linear quantum stochastic systems, and that the property of complete passivity is invariant under subsystem truncation. However, generic linear quantum stochastic systems need not have a balanced realization under symplectic transformations. Therefore, alternative notions of balancing, including so-called quasi-balancing, are developed, and necessary and sufficient conditions are derived. A truncation error bound is derived for quasi-balanceable linear quantum stochastic systems and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. An example is provided to illustrate model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities.