On realization theory of quantum linear systems

John E. Gough, Guofeng Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

39 Citations (Scopus)


Abstract The purpose of this paper is to study the realization theory of quantum linear systems. It is shown that for a general quantum linear system its controllability and observability are equivalent and they can be checked by means of a simple matrix rank condition. Based on controllability and observability a specific realization is proposed for general quantum linear systems in which an uncontrollable and unobservable subspace is identified. When restricted to the passive case, it is found that a realization is minimal if and only if it is Hurwitz stable. Computational methods are proposed to find the cardinality of minimal realizations of a quantum linear passive system. It is found that the transfer function G(s) of a quantum linear passive system can be written as a fractional form in terms of a matrix function Σ(s); moreover, G(s) is lossless bounded real if and only if Σ(s) is lossless positive real. A type of realization for multi-input-multi-output quantum linear passive systems is derived, which is related to its controllability and observability decomposition. Two realizations, namely the independent-oscillator realization and the chain-mode realization, are proposed for single-input-single-output quantum linear passive systems, and it is shown that under the assumption of minimal realization, the independent-oscillator realization is unique, and these two realizations are related to the lossless positive real matrix function Σ(s).
Original languageEnglish
Article number6441
Pages (from-to)139-151
Number of pages13
Publication statusPublished - 1 Sept 2015


  • Controllability
  • Observability
  • Quantum linear systems
  • Realization theory

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering


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