On the physical realizability of general linear Quantum Stochastic Differential Equations with complex coefficients

Shaiju, A. J.; Petersen, Ian R.

doi:10.1109/cdc.2009.5399947

Cited by 18 publications

(46 citation statements)

References 15 publications

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“…and have the appropriate dimensions. Here q is a vector of the self-adjoint position operators for the system of harmonic oscillators and p is a vector of momentum operators; e.g., see [11], [12], [21], [39]. Also, Q(t) and P(t) are the vectors of position and momentum operators for the quantum noise fields acting on the system of harmonic oscillators.…”

Section: Position and Momentum Operator Linear Quantum Systemsmentioning

confidence: 99%

Quantum Linear Systems Theory

Petersen¹

2016

Open Autom. Control Syst. J.

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106

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Abstract-This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework. Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-spitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems. With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory. Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case. An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability. This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics. The paper will cover recent results relating the question of physical realizability to notions occuring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.

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Section: Position and Momentum Operator Linear Quantum Systemsmentioning

confidence: 99%

Quantum Linear Systems Theory

Petersen¹

2016

Open Autom. Control Syst. J.

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106

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“…Both the plant and the controller are assumed to satisfy the physical realizability (PR) conditions [9], [16], [20] which will be described in Section V. The plant has n dynamic variables x 1 (t), . .…”

Section: Quantum Plantmentioning

confidence: 99%

“…Note that (19), (22) are the conditions for physical realizability (PR) [9], [16], [20] of the quantum plant which describe the preservation of the CCR matrix Θ 1 in (12) and [x, y T ] = 0. Similarly, the relations (20), (23), which describe the preservation of the CCR matrix Θ 2 in (12) and [ξ , η T ] = 0, are the PR conditions for the coherent quantum controller.…”

Section: Quantum Controllermentioning

confidence: 99%

“…Similarly, the relations (20), (23), which describe the preservation of the CCR matrix Θ 2 in (12) and [ξ , η T ] = 0, are the PR conditions for the coherent quantum controller. The PR condition (20) can be regarded as a linear equation with respect to the matrix a, and its general solution is representable as…”

Section: Quantum Controllermentioning

confidence: 99%

“…The resulting PR conditions [9], [20], [21] are organized as quadratic constraints on the coefficients of the QSDEs. The PR constraints for the state-space matrices of a coherent quantum controller complicate the solution of quantum counterparts to the classical H ∞ and Linear Quadratic Gaussian (LQG) control problems.…”

Section: Introductionmentioning

confidence: 99%

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A gradient descent approach to optimal coherent quantum LQG controller design

Sichani

Vladimirov

Petersen

2015

2015 American Control Conference (ACC)

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This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented. A numerical example of a locally optimal CQLQG controller design is provided to demonstrate the algorithm performance.

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Quantum feedback networks and control: A brief survey

Zhang

James

2012

Chin. Sci. Bull.

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The purpose of this paper is to provide a brief review of some recent developments in quantum feedback networks and control. A quantum feedback network (QFN) is an interconnected system consisting of open quantum systems linked by free fields and/or direct physical couplings. Basic network constructs, including series connections as well as feedback loops, are discussed. The quantum feedback network theory provides a natural framework for analysis and design. Basic properties such as dissipation, stability, passivity and gain of open quantum systems are discussed. Control system design is also discussed, primarily in the context of open linear quantum stochastic systems. The issue of physical realizability is discussed, and explicit criteria for stability, positive real lemma, and bounded real lemma are presented. Finally for linear quantum systems, coherent H ∞ and LQG control are described.open quantum systems, quantum feedback networks, physical realizability, H ∞ control, LQG control Citation: Zhang G F, James M R. Quantum feedback networks and control: A brief survey.

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On the physical realizability of general linear Quantum Stochastic Differential Equations with complex coefficients

Cited by 18 publications

References 15 publications

Quantum Linear Systems Theory

Quantum Linear Systems Theory

A gradient descent approach to optimal coherent quantum LQG controller design

Quantum feedback networks and control: A brief survey

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