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Australian Control Conference

Model reduction of linear quantum stochastic systems by subsystem truncation

Hendra I. Nurdin

Published Nov 1, 2012• 2 citations
OpenAlex

Abstract

The purpose of the paper is to develop new theoretical results on model reduction of linear quantum stochastic systems by truncation of subsystems of such systems. It is shown that a truncated system is guaranteed to be physically realizable (i.e., it represents the dynamics of a valid open Markov quantum system). In addition, it is also shown that the class of completely passive linear quantum stochastic systems is invariant under such subsystem truncation.

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The purpose of the paper is to develop new theoretical results on model reduction of linear quantum stochastic systems by truncation of subsystems of such systems. It is shown that a truncated system is guaranteed to be physically realizable (i.e., it represents the dynamics of a valid open Markov quantum system). In addition, it is also shown that the class of completely passive linear quantum stochastic systems is invariant under such subsystem truncation.
The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. It is shown that subsystem truncation preserves the physical realizability property of linear quantum stochastic systems, and that the property of complete passivity is invariant under subsystem truncation. However, generic linear quantum stochastic systems need not have a balanced realization under symplectic transformations. Therefore, alternative notions of balancing, including so-called quasi-balancing, are developed, and necessary and sufficient conditions are derived. A truncation error bound is derived for quasi-balanceable linear quantum stochastic systems and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. An example is provided to illustrate model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities.
This paper derives conditions under which co-diagonalization of the controllability and observability Gramians can be performed in a physically consistent manner for the class of linear quantum stochastic systems that are common in the fields of quantum optics and related areas like quantum nano- and optomechanical systems. Such a class is essentially a quantum analogue of classical (non-quantum) linear stochastic systems. However, unlike their classical counterparts, the laws of quantum mechanics impose nonlinear equality constraints on the system matrices of linear quantum stochastic systems, the so-called physical realizability constraints. Therefore, a meaningful model reduction procedure must yield a physically realizable reduced model satisfying these constraints. Previous work has shown that model reduction by subsystem truncation preserves physical realizability, but did not propose a method for truncating a subsystem. One candidate approach for this truncation is a quantum adaptation of the well-known balanced truncation method for classical linear time-invariant systems. It is shown in this paper that balanced realization for linear quantum stochastic systems is only possible under a strongly restrictive necessary and sufficient condition. However, the paper also derives less restrictive necessary and sufficient conditions for other realizations with simultaneously diagonal controllability and observability Gramians, and introduces the notion of a quasi-balanced realization. An example of the application of the results is provided to demonstrate quasi-balanced truncation in the linear quantum stochastic setting.
This paper concerns the recently proposed quasi-balanced truncation model reduction method for linear quantum stochastic systems. It has previously been shown that the quasi-balanceable class of systems (i.e. systems that can be truncated via the quasi-balanced method) includes the class of completely passive systems. In this work, we refine the previously established characterization of quasi-balanceable systems and show that the class of quasi-balanceable systems is strictly larger than the class of completely passive systems. In particular, we derive a novel characterization of completely passive linear quantum stochastic systems solely in terms of the controllability Gramian of such systems. Exploiting this result, we prove that all linear quantum stochastic systems with a pure Gaussian steady-state (active systems included) are all quasi-balanceable, and establish a new complete parameterization for this important class of systems. Examples are provided to illustrate our results.
A general approach to obtain reduced models for a wide class of discrete-time quantum systems is proposed. The obtained models not only reproduce exactly the output of a given quantum model, but are also guaranteed to satisfy physical constraints, namely complete positivity and preservation of total probability. A fundamental framework for exact model reduction of quantum systems is constructed leveraging on algebraic methods, as well as novel results on quantum conditional expectations in finite-dimensions. The proposed reduction algorithm is illustrated and tested on prototypical examples, including the quantum walk realizing Grover’s algorithm.
Recently, an adaptation of the balanced truncation method for model reduction of classical (non-quantum) linear systems has been proposed for the class of linear quantum stochastic systems that are commonly employed as quantum stochastic Markov models of open quantum oscillators in quantum optics and related fields. In order to preserve the canonical commutation relations of the internal quantum harmonic oscillators, the similarity transformation that can be applied to linear quantum stochastic systems must be sympletic. Due to this restriction, general linear quantum stochastic systems need not have a balanced realization, in contrast to the classical case. However, a necessary and sufficient condition can be derived under which a linear quantum stochastic system can be identified as being quasi-balanceable, meaning that it can be transformed under a symplectic similarity transformation into a realization in which the controllability and observability Gramians are both diagonal but not the same diagonal matrix. In this paper, an error bound is derived for model reduction of quasi-balanceable linear quantum stochastic systems. It is shown that analogous to the classical case, the bound is determined by the Hankel singular values of the linear quantum stochastic system, and a Hankel singular value of the system is in fact the geometric mean of a corresponding pair of symplectic eigenvalues of the controllability and observability Gramians.
This paper presents a model reduction method for the class of linear quantum stochastic systems often encountered in quantum optics and related fields. The approach is proposed on the basis of an interpolatory projection ensuring that specific input-output responses of the original and the reduced-order systems are matched at multiple selected points (or frequencies). Importantly, the physical realizability property of the original quantum system imposed by the laws of quantum mechanics is preserved under our tangential interpolatory projection. Error bounds are established for the proposed model reduction method and an avenue to select interpolation points is proposed. A passivity preserving model reduction method is also presented. Examples of both active and passive systems are provided to illustrate the merits of our proposed approach.