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Title: The Kalman decomposition for linear quantum systems
Authors: Zhang, G 
Grivopoulos, S
Petersen, IR
Gough, JE
Issue Date: Feb-2018
Source: IEEE transactions on automatic control, Feb. 2018, v. 63, no. 2, 7942122, p. 331-346
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.
Keywords: Controllability
Kalman decomposition
Linear quantum systems
Publisher: Institute of Electrical and Electronics Engineers
Journal: IEEE transactions on automatic control 
ISSN: 0018-9286
EISSN: 1558-2523
DOI: 10.1109/TAC.2017.2713343
Rights: © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works
The following publication Zhang, G., Grivopoulos, S., Petersen, I. R., & Gough, J. E. (2017). The Kalman decomposition for linear quantum systems. IEEE Transactions on Automatic Control, 63(2), 331-346 is available at
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