The Kalman Decomposition for Linear Quantum Systems

Guofeng Zhang, Symeon Grivopolous, Ian Petersen, John Gough

Research output: Contribution to journalArticlepeer-review

20 Citations (SciVal)
185 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)331-346
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number2
Early online date07 Jun 2017
Publication statusPublished - 28 Feb 2018


  • controllability
  • kalman decomposition
  • linear quantum systems
  • observability


Dive into the research topics of 'The Kalman Decomposition for Linear Quantum Systems'. Together they form a unique fingerprint.

Cite this