The Kalman Decomposition for Linear Quantum Systems

TL;DR

This paper extends the Kalman decomposition to linear quantum systems, revealing unique structural properties and providing a method to transform systems into a canonical form that highlights quantum-specific subsystems and modes.

Contribution

It introduces a construction method for Kalman decomposition in quantum systems, respecting quantum mechanical constraints, and uncovers the structure of controllable, observable, and special quantum modes.

Findings

Decomposition exposes decoherence-free and quantum-nondemolition modes

Passive systems have only controllable/observable and uncontrollable/unobservable subsystems

General systems may have conjugate variable subsystems

Abstract

This paper studies the Kalman decomposition for linear quantum systems. Contrary to the classical case, the coordinate transformation used for the decomposition must belong to a specific class of transformations as a consequence of the laws of quantum mechanics. We propose a construction method for such transformations that put the system in a Kalman canonical form. Furthermore, we uncover an interesting structure for the obtained decomposition. In the case of passive systems, it is shown that there exist only controllable/observable and uncontrollable/unobservable subsystems. In the general case, controllable/unobservable and uncontrollable/observable subsystems may also be present, but their respective system variables must be conjugate variables of each other. This decomposition naturally exposes decoherence-free modes, quantum-nondemolition modes, quantum-mechanics-free subsystems, and back-action evasion measurements in the quantum system, which are useful resources for quantum information processing, and quantum measurements. The theory developed is applied to physical examples.