[Paper Review] On Realization Theory of Quantum Linear Systems

John Gough, Guofeng Zhang|arXiv (Cornell University)|Nov 6, 2013

Quantum Information and Cryptography27 references4 citations

TL;DR

This paper establishes realization theory for quantum linear systems by proving controllability and observability are equivalent under a simple matrix rank condition, and derives minimal realizations for passive systems. It shows that minimal realizations are Hurwitz stable and provides explicit constructions—such as independent-oscillator and chain-mode realizations—linked to lossless positive real matrix functions, enabling computation of minimal realization cardinality and fractional transfer function representations.

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 of a quantum linear passive system $G$ can be written as a fractional form in terms of a matrix function $Σ$; moreover, $G$ is lossless bounded real if and only if $Σ$ is lossless positive real. A type of realization for multi-input-multi-output quantum linear passive systems is derived, which is closely 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 $Σ$.

Motivation & Objective

• To develop a comprehensive realization theory for general quantum linear systems.
• To establish equivalence between controllability and observability using a matrix rank condition.
• To identify uncontrollable and unobservable subspaces in general quantum linear systems.
• To characterize minimal realizations of quantum linear passive systems via Hurwitz stability.
• To derive computational methods for determining the cardinality of minimal realizations in SISO and MIMO cases.

Proposed method

• Proposes a canonical decomposition of general quantum linear systems by identifying uncontrollable and unobservable subspaces via matrix rank conditions.
• Uses unitary transformations to re-express system variables, enabling a structured realization with explicit Hamiltonian and coupling operators.
• Derives a fractional representation of the transfer function in terms of a matrix function Σ, linking system properties to matrix function theory.
• Introduces two specific realizations—independent-oscillator and chain-mode—for SISO passive systems, both tied to the lossless positive real matrix function Σ.
• Applies the Schur-Feshbach formula and orthogonal polynomial theory to derive inverse matrix expressions and verify unitary transformations.
• Establishes that a passive system is lossless bounded real if and only if the associated matrix function Σ is lossless positive real.

Experimental results

Research questions

• RQ1 Under what conditions are controllability and observability equivalent in general quantum linear systems?
• RQ2 How can the uncontrollable and unobservable subspace be systematically identified in a quantum linear system?
• RQ3 What is the relationship between minimal realization and Hurwitz stability in quantum linear passive systems?
• RQ4 How can the cardinality of minimal realizations be computed for SISO and MIMO quantum linear passive systems?
• RQ5 What is the role of the matrix function Σ in characterizing lossless bounded real transfer functions?

Key findings

• Controllability and observability are equivalent in general quantum linear systems and can be checked via a simple matrix rank condition.
• A canonical realization is constructed that explicitly identifies the uncontrollable and unobservable subspace, generalizing Kalman decomposition to the quantum domain.
• For quantum linear passive systems, minimality is equivalent to Hurwitz stability, providing a clear criterion for minimal realization.
• The cardinality of minimal realizations for SISO systems is given by Proposition 3.7, and for MIMO systems by Proposition 3.8, both derived from spectral properties of the system.
• A quantum linear passive system is lossless bounded real if and only if the associated matrix function Σ is lossless positive real, establishing a fundamental characterization.
• Two distinct realizations—independent-oscillator and chain-mode—are derived for SISO systems, with the independent-oscillator realization proven unique under minimality.