We introduce the characteristic operator as the generalization of the usual concept of a transfer function of linear input-plant-output systems to arbitrary quantum nonlinear Markovian input-output models. This is intended as a tool in the characterization of quantum feedback control systems that fits in with the general theory of networks. The definition exploits the linearity of noise differentials in both the plant Heisenberg equations of motion and the differential form of the input-output relations. Mathematically, the characteristic operator is a matrix of dimension equal to the number of outputs times the number of inputs (which must coincide), but with entries that are operators of the plant system. In this sense, the characteristic operator retains details of the effective plant dynamical structure and is an essentially quantum object. We illustrate the relevance to model reduction and simplification definition by showing that the convergence of the characteristic operator in adiabatic elimination limit models requires the same conditions and assumptions appearing in the work on limit quantum stochastic differential theorems of Bouten and Silberfarb [Commun. Math. Phys. 283, 491-505 (2008)]. This approach also shows in a natural way that the limit coefficients of the quantum stochastic differential equations in adiabatic elimination problems arise algebraically as Schur complements and amounts to a model reduction where the fast degrees of freedom are decoupled from the slow ones and eliminated.
We introduce the characteristic operator as the generalization of the usual concept of a transfer function of linear input-plant-output systems to arbitrary quantum nonlinear Markovian input-output models. This is intended as a tool in the characterization of quantum feedback control systems that fits in with the general theory of networks. The definition exploits the linearity of noise differentials in both the plant Heisenberg equations of motion and the differential form of the input-output relations. Mathematically, the characteristic operator is a matrix of dimension equal to the number of outputs times the number of inputs (which must coincide), but with entries that are operators of the plant system. In this sense, the characteristic operator retains details of the effective plant dynamical structure and is an essentially quantum object. We illustrate the relevance to model reduction and simplification definition by showing that the convergence of the characteristic operator in adiabatic elimination limit models requires the same conditions and assumptions appearing in the work on limit quantum stochastic differential theorems of Bouten and Silberfarb [Commun. Math. Phys. 283, 491-505 (2008)]. This approach also shows in a natural way that the limit coefficients of the quantum stochastic differential equations in adiabatic elimination problems arise algebraically as Schur complements and amounts to a model reduction where the fast degrees of freedom are decoupled from the slow ones and eliminated.
Based on the stochastic differential equation of quantum mechanical feedback obtained in the first part of this paper, detailed control concepts and applications are discussed for quantum systems interacting with a noncommutative noise source. A feedback system in our framework is purely nonclassical in the sense that feedback control is performed via local operation and quantum communication through a quantum channel. The role of the controller is to alter the quantum dynamic characteristics of the plant through entanglement, shared between the plant and controller by sending quantum states, that is modulated by the Hamiltonian on the controller. The input-output relation of quantum systems provides a natural extension of control theory to the quantum domain. This enables one to present a control theoretical interpretation of some fundamental quantum mechanical notions such as the uncertainty principle, but also to find applications of ideas and tools of control theory. One of the most important applications is the production of squeezed states, which has been an important issue of quantum theory in relation to quantum computation and quantum communication. The method proposed here reduces the application to a conventional noise reduction problem with feedback. The H/sub /spl infin// control then leads to complete squeezing.
This paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators. A quantum optical closed-loop, including a plant and controller, is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view, in order to facilitate the use of control theory in microscopic world described by quantum theory. In particular, an input-output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed. This would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics. This paper is divided into two parts. The first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device, cavity, that can be regarded as a unit of quantum dynamics of bosonic modes. The formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations. The input-output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems. In the follow-up paper, the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented.
We express the rules for forming quantum feedback networks using the Stratonovich form of quantum stochastic calculus rather than the Itō or SLH (J. E. Gough and M. R. James, “Quantum feedback networks: Hamiltonian formulation,” Commun. Math. Phys. 287, 1109 (2009), J. E. Gough and M. R. James, “The Series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control 54, 2530 (2009)) form. Remarkably the feedback reduction rule implies that we obtain the Schur complement of the matrix of Stratonovich coupling operators where we short out the internal input/output coefficients.
Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring D and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-known scattering transformation in Schrödinger picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegle’s matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
The concept of a controlled flow of a dynamical system, especially when the controlling process feeds information back about the system, is of central importance in control engineering. In this paper, we build on the ideas presented by Bouten and van Handel [Quantum Stochastics and Information: Statistics, Filtering and Control (World Scientific, 2008)] and develop a general theory of quantum feedback. We elucidate the relationship between the controlling processes, Z, and the measured processes, Y, and to this end we make a distinction between what we call the input picture and the output picture. We should note that the input-output relations for the noise fields have additional terms not present in the standard theory but that the relationship between the control processes and measured processes themselves is internally consistent—we do this for the two main cases of quadrature measurement and photon-counting measurement. The theory is general enough to include a modulating filter which post-processes the measurement readout Y before returning to the system. This opens up the prospect of applying very general engineering feedback control techniques to open quantum systems in a systematic manner, and we consider a number of specific modulating filter problems. Finally, we give a brief argument as to why most of the rules for making instantaneous feedback connections [J. Gough and M. R. James, Commun. Math. Phys. 287, 1109 (2009)] ought to apply for controlled dynamical networks as well.
We propose a general theoretical framework that is suitable to study a wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we propose definitions of invariant and attractive quantum subsystem, characterize Markovian invariance properties, and provide sufficient conditions for attraction. The general framework and results are illustrated by addressing the potential of output-feedback Markovian control strategies for quantum pure state-stabilization. In particular, constructive results for the synthesis of stabilizing semigroups in arbitrary finite-dimensional Markovian systems are established. I. BACKGROUND AND MOTIVATIONS Stabilization problems are of central relevance for many quantum control applications, ranging from state preparation of quantum-optical and nano-mechanical systems to generation of noise-protected realizations of quantum information in realistic devices [1]. Dynamical systems undergoing Markovian evolution [2], [3] are both widely relevant from a physical standpoint and present distinctive control challenges – preventing, in particular, open-loop quantum-engineering and stabilization methods based on dynamical decoupling to be viable [4], [5]. However, we show here how a wide class of stabilization problems can be effectively treated in a general framework, provided by attractive quantum subsystems. After introducing the main ideas and definitions along with some general results, we shall explore their application to pure-state preparation problems for Markovian output-feedback control. We refer to the forthcoming journal version of the present paper [6] for detailed proofs we shall omit or merely sketch in the following sections. Consider a separable Hilbert space H over the complex field C. Let B(H) represent the set of linear bounded operators on H, H(H) denoting the real subspace of Hermitian operators, with I, O being the identity and the zero operator, respectively. In the standard statistical formulation of quantum mechanics [7], [8], the dimension of the Hilbert space H associated with the quantum system of interest, Q, is determined by the physics of the problem. In what follows, we consider finite-dimensional systems, i.e. dim(H) 0 being described by a TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], [1]. A differential equation for the density operator of I may be derived provided that a forward composition law holds: Definition 1 (QDS): A quantum dynamical semigroup is a one-parameter family of TPCP maps {Tt(·), t ≥ 0} that satisfies: (i) T0 = I, (ii) Tt ◦ Ts = Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) is a continuous function of t, ∀ρ ∈ D(HI), ∀X ∈ B(HI). Due to the trace and positivity preserving assumptions, a QDS is a semigroup of contractions. It has been proved [10], [13] that the Hille-Yoshida generator for a QDS exists and can be cast in the following canonical form: