On Poles and Zeros of Linear Quantum Systems

TL;DR

This paper investigates the structure of zeros and poles in linear quantum systems, revealing fundamental relationships and constraints imposed by quantum mechanics on system dynamics.

Contribution

It establishes the connection between system eigenvalues, invariant zeros, and the physical realizability conditions unique to quantum systems.

Findings

- $-s_0^*$ is a transmission zero if and only if $s_0$ is a pole.

- Generalized relationship between eigenvalues and invariant zeros.

- Analysis of invertibility and tradeoffs in quantum system zeros and poles.

Abstract

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm are manifested by the physical realizability conditions on system matrices. These restrictions endow system matrices with special structure. The purpose of this paper is to study such structure by investigating zeros and poses of linear quantum systems. In particular, we show that $-s_0^\ast$ is a transmission zero if and only if $s_0$ is a pole, and which is further generalized to the relationship between system eigenvalues and invariant zeros. Additionally, we study left-invertibility and fundamental tradeoff for linear quantum systems in terms of their zeros and poles.