Linear quantum systems: poles, zeros, invertibility and sensitivity

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which, in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. This paper aims to study this structure by investigating the zeros and poles of linear quantum systems. Firstly, it is shown that $-s_0$ is a transmission zero if and only if $s_0$ is a pole of the transfer function, and $-s_0$ is an invariant zero if and only if $s_0$ is an eigenvalue of the $A$-matrix, of a linear quantum system. Moreover, $s_0$ is an output-decoupling zero if and only if $-s_0$ is an input-decoupling zero. Secondly, based on these pole-zero correspondences and inspired by a recent work on the stable inversion of classical linear systems \cite{DD2023}, we show that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Two types of stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated; in particular, the fundamental tradeoff between ideal squeezing and system robustness is studied on the basis of system sensitivity analysis.