An error bound on balanced truncation of quasi-balanceable linear quantum stochastic systems

Recently, an adaptation of the balanced truncation method for model reduction of classical (non-quantum) linear systems has been proposed for the class of linear quantum stochastic systems that are commonly employed as quantum stochastic Markov models of open quantum oscillators in quantum optics and related fields. In order to preserve the canonical commutation relations of the internal quantum harmonic oscillators, the similarity transformation that can be applied to linear quantum stochastic systems must be sympletic. Due to this restriction, general linear quantum stochastic systems need not have a balanced realization, in contrast to the classical case. However, a necessary and sufficient condition can be derived under which a linear quantum stochastic system can be identified as being quasi-balanceable, meaning that it can be transformed under a symplectic similarity transformation into a realization in which the controllability and observability Gramians are both diagonal but not the same diagonal matrix. In this paper, an error bound is derived for model reduction of quasi-balanceable linear quantum stochastic systems. It is shown that analogous to the classical case, the bound is determined by the Hankel singular values of the linear quantum stochastic system, and a Hankel singular value of the system is in fact the geometric mean of a corresponding pair of symplectic eigenvalues of the controllability and observability Gramians.