The Open Automation and Control Systems Journal




(Discontinued)

ISSN: 1874-4443 ― Volume 11, 2019

Quantum Linear Systems Theory



Ian R. Petersen*
School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia

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-splitters, 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. However, the design of coherent quantum feedback controllers remains a major challenge. 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 occurring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.

Keywords: Coherent control, Linear systems theory, Physical realizability, Quantum control theory, Quantum H-infinity control, Quantum systems+.


Article Information


Identifiers and Pagination:

Year: 2016
Volume: 8
First Page: 67
Last Page: 93
Publisher Id: TOAUTOCJ-8-67
DOI: 10.2174/1874444301608010067

Article History:

Received Date: 06/03/2016
Revision Received Date: 16/07/2016
Acceptance Date: 21/07/2016
Electronic publication date: 31/10/2016
Collection year: 2016

© Ian R. Petersen; Licensee Bentham Open

open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution-Non-Commercial 4.0 International Public License (CC BY-NC 4.0) (https://creativecommons.org/licenses/by-nc/4.0/legalcode), which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.


* Address correspondence to this author at the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia; Tel: +61262688446; Fax: +61262688443; E-mail: i.r.petersen@gmail.com





1. INTRODUCTION

Developments in quantum technology and quantum information provide an important motivation for research in the area of quantum feedback control systems; e.g., see [1V. Belavkin, "On the theory of controlling observable quantum systems", Autom. Remote Control, vol. 42, no. 2, pp. 178-188, 1983.-7D. Dong, and I.R. Petersen, "Quantum control theory and applications: A survey,", IET Control Theory Appl, vol. 4, no. 12, pp. 2651-2671, 2010.
[http://dx.doi.org/10.1049/iet-cta.2009.0508]
]. In particular, in recent years, there has been considerable interest in the feedback control and modeling of linear quantum systems; e.g., see [3A. Doherty, and K. Jacobs, "Feedback-control of quantum systems using continuous state-estimation", Phys. Rev. A, vol. 60, pp. 2700-2711, 1999.
[http://dx.doi.org/10.1103/PhysRevA.60.2700]
, 5M. Yanagisawa, and H. Kimura, "Transfer function approach to quantum control-part I: Dynamics of quantum feedback systems", IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2107-2120, 2003.
[http://dx.doi.org/10.1109/TAC.2003.820063]
, 8S.C. Edwards, and V.P. Belavkin, Optimal quantum feedback control via quantum dynamic programming, University of Nottingham: UK, 2005. quant-ph/0506018-26H. M. Wiseman, and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press: UK, 1994.]. Such linear quantum systems commonly arise in the area of quantum optics; e.g., see [27D.F. Walls, and G.J. Milburn, Quantum Optics., Berlin. Springer-Verlag: New York, 1994.-29H. Bachor, and T. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. Wiley Online Library: USA, 2004.
[http://dx.doi.org/10.1002/9783527619238]
]. Feedback control of quantum optical systems has applications in areas such as quantum communications, quantum teleportation, and gravity wave detection. In particular, linear quantum optics is one of the possible platforms being investigated for future communication systems (see [30B.C. Jacobs, T.B. Pittman, and J.D. Franson, "Quantum relays and noise suppression using linear optics", Phys. Rev. A, vol. 66, p. 052307, 2002.
[http://dx.doi.org/10.1103/PhysRevA.66.052307]
, 31P. Kok, C.P. Williams, and J.P. Dowling, "The construction of a quantum repeater with linear optics", Phys. Rev. A, vol. 68, p. 022301, 2003.
[http://dx.doi.org/10.1103/PhysRevA.68.022301]
]) and quantum computers (see [32E. Knill, R. Laflamme, and G.J. Milburn, "A scheme for efficient quantum computation with linear optics", Nature, vol. 409, no. 6816, pp. 46-52, 2001.
[http://dx.doi.org/10.1038/35051009] [PMID: 11343107]
-34M. Nielsen, and I. Chuang, Quantum Computation and Quantum Information. Cambridge., Cambridge University Press: UK, 2000.]). Feedback control of quantum systems aims to achieve closed loop properties such as stability [35R. Somaraju, and I.R. Petersen, "Lyapunov stability for quantum Markov processes", In: Proceedings of the 2009 American Control Conference St Louis, Mo, 2009.
[http://dx.doi.org/10.1109/ACC.2009.5160264]
, 36R. Somaraju, and I. Petersen, "Feedback interconnection of open quantum systems: A small gain theorem", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5400728]
], robustness [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 37A.J. Shaiju, I.R. Petersen, and M.R. James, "Guaranteed cost LQG control of uncertain linear stochastic quantum systems", In: Proceedings of the 2007 American Control Conference New York, 2007 arXiv:0807.4619
[http://dx.doi.org/10.1109/ACC.2007.4282598]
], entanglement [18N. Yamamoto, H.I. Nurdin, M.R. James, and I.R. Petersen, "Avoiding entanglement sudden-death via feedback control in a quantum network", Phys. Rev. A, vol. 78, no. 4, p. 042339, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.042339]
, 38H.G. Harno, and I.R. Petersen, "Coherent control of linear quantum systems: A differential evolution approach", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531367]
, 39H. Nurdin, I.R. Petersen, and M.R. James, "On the infeasibility of entanglement generation in Gaussian quantum systems via classical control", IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 198-203, 2012.
[http://dx.doi.org/10.1109/TAC.2011.2162888]
].

Quantum linear system models have been used in the physics and mathematical physics literature since the 1980's; e.g., see [26H. M. Wiseman, and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press: UK, 1994., 28C. Gardiner, and P. Zoller, Quantum Noise., Berlin Springer: New York, 2000., 40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
-42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992.]. An important class of linear quantum stochastic models describe the Heisenberg evolution of the (canonical) position and momentum, or annihilation and creation operators of several independent open quantum harmonic oscillators that are coupled to external coherent bosonic fields, such as coherent laser beams; e.g., see [8S.C. Edwards, and V.P. Belavkin, Optimal quantum feedback control via quantum dynamic programming, University of Nottingham: UK, 2005. quant-ph/0506018-13J. Gough, R. Gohm, and M. Yanagisawa, "Linear quantum feedback networks", Phys. Rev. A, vol. 78, p. 062104, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.062104]
, 17H. Mabuchi, "Coherent-feedback quantum control with a dynamic compensator", Phys. Rev. A, vol. 78, p. 032323, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.032323]
, 18N. Yamamoto, H.I. Nurdin, M.R. James, and I.R. Petersen, "Avoiding entanglement sudden-death via feedback control in a quantum network", Phys. Rev. A, vol. 78, no. 4, p. 042339, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.042339]
, 22H.I. Nurdin, M.R. James, and A.C. Doherty, "Network synthesis of linear dynamical quantum stochastic systems", SIAM J. Contr. Optim., vol. 48, no. 4, pp. 2686-2718, 2009.
[http://dx.doi.org/10.1137/080728652]
, 25J.E. Gough, M.R. James, and H.I. Nurdin, "Squeezing components in linear quantum feedback networks", Phys. Rev. A, vol. 81, p. 023804, 2010.
[http://dx.doi.org/10.1103/PhysRevA.81.023804]
-28C. Gardiner, and P. Zoller, Quantum Noise., Berlin Springer: New York, 2000., 43H. M. Wiseman, and A. C. Doherty, "Optimal unravellings for feedback control in linear quantum systems", Physics Review Letters, vol. 97, pp. 070 405-1-070 405-1, 2005.
[http://dx.doi.org/10.1103/PhysRevLett.94.070405]
, 44G. Sarma, A. Silberfarb, and H. Mabuchi, "Quantum stochastic calculus approach to modeling double-pass atom-field coupling", Phys. Rev. A, vol. 78, p. 025801, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.025801]
]). These linear stochastic models describe quantum optical devices such as optical cavities [27D.F. Walls, and G.J. Milburn, Quantum Optics., Berlin. Springer-Verlag: New York, 1994., 29H. Bachor, and T. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. Wiley Online Library: USA, 2004.
[http://dx.doi.org/10.1002/9783527619238]
], linear quantum amplifiers [28C. Gardiner, and P. Zoller, Quantum Noise., Berlin Springer: New York, 2000.], and finite bandwidth squeezers [28C. Gardiner, and P. Zoller, Quantum Noise., Berlin Springer: New York, 2000.]. Following [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
, 22H.I. Nurdin, M.R. James, and A.C. Doherty, "Network synthesis of linear dynamical quantum stochastic systems", SIAM J. Contr. Optim., vol. 48, no. 4, pp. 2686-2718, 2009.
[http://dx.doi.org/10.1137/080728652]
], we will refer to this class of models as linear quantum stochastic systems. In particular, we consider linear quantum stochastic differential equations driven by quantum Wiener processes; see [28C. Gardiner, and P. Zoller, Quantum Noise., Berlin Springer: New York, 2000.]. Further details on quantum stochastic differential equations and quantum Wiener processes can be found in [40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 45L. Bouten, R. van Handel, and M. James, "An introduction to quantum filtering", SIAM J. Contr. Optim., vol. 46, no. 6, pp. 2199-2241, 2007.
[http://dx.doi.org/10.1137/060651239]
].

This paper will survey some of the available results on the feedback control of linear quantum systems and related problems. As a survey paper, it does not present new results but rather surveys existing results in a unified framework. An important class of quantum feedback control systems involves the use of measurement devices to obtain classical output signals from the quantum system and no quantum measurements is involved. These classical signals are fed into a classical controller which may be implemented via analog or digital electronics and then the resulting control signal act on the quantum system via an actuator. However, some recent papers on the feedback control of linear quantum systems have considered the case in which the feedback controller itself is also a quantum system. Such feedback control is often referred to as coherent quantum control; e.g., see [5M. Yanagisawa, and H. Kimura, "Transfer function approach to quantum control-part I: Dynamics of quantum feedback systems", IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2107-2120, 2003.
[http://dx.doi.org/10.1109/TAC.2003.820063]
, 6M. Yanagisawa, and H. Kimura, "Transfer function approach to quantum control-part II: Control concepts and applications", IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2121-2132, 2003.
[http://dx.doi.org/10.1109/TAC.2003.820065]
, 11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009.-17H. Mabuchi, "Coherent-feedback quantum control with a dynamic compensator", Phys. Rev. A, vol. 78, p. 032323, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.032323]
, 46H.M. Wiseman, and G.J. Milburn, "All-optical versus electro-optical quantum-limited feedback", Phys. Rev. A, vol. 49, no. 5, pp. 4110-4125, 1994.
[http://dx.doi.org/10.1103/PhysRevA.49.4110] [PMID: 9910711]
-48J.E. Gough, and S. Wildfeuer, "Enhancement of field squeezing using coherent feedback", Phys. Rev. A, vol. 80, p. 042107, 2009.
[http://dx.doi.org/10.1103/PhysRevA.80.042107]
]. Due to the limitations imposed by quantum mechanics on the use of quantum measurement, the use of coherent quantum feedback control may lead to improved control system performance. In addition, in many applications, coherent quantum feedback controllers may be preferable to classical feedback controllers due to considerations of speed and ease of implementation.

One motivation for considering such coherent quantum control problems is that coherent controllers have the potential to achieve improved performance since quantum measurements inherently involve the destruction of quantum information; e.g., see [34M. Nielsen, and I. Chuang, Quantum Computation and Quantum Information. Cambridge., Cambridge University Press: UK, 2000.]. Also, technology is emerging which will enable the implementation of complex coherent quantum controllers (e.g., see [49A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J.L. O’Brien, "Silica-on-silicon waveguide quantum circuits", Science, vol. 320, no. 5876, pp. 646-649, 2008.
[http://dx.doi.org/10.1126/science.1155441] [PMID: 18369104]
]) and the coherent H controllers proposed in [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
] have already been implemented experimentally as described in [17H. Mabuchi, "Coherent-feedback quantum control with a dynamic compensator", Phys. Rev. A, vol. 78, p. 032323, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.032323]
]. Furthermore, coherent controllers implemented using quantum optics have the potential to operate at much higher speeds than classical controllers implemented in analog or digital electronics.

In general, quantum linear stochastic systems represented by linear Quantum Stochastic Differential Equations (QSDEs) with arbitrary constant coefficients need not correspond to physically meaningful systems. In contrast, because classical linear stochastic systems can be implemented at least approximately, using analog or digital electronics, we regard them as always being realizable. Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs. In particular, the laws of quantum mechanics dictate that closed quantum systems evolve unitarily, implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times. Therefore, to characterize physically meaningful systems [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
], has introduced a formal notion of physically realizable quantum linear stochastic systems and derives a pair of necessary and sufficient characterizations for such systems in terms of constraints on their system matrices. However, the design of coherent quantum feedback controllers remains a major challenge. In this paper, we survey some methods which can be applied to this problem.

In the paper [21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
], the physical realizability results of [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
] are extended to the most general class of complex linear QSDEs. It is shown that this class of linear quantum systems corresponds to the class of real linear quantum systems considered in [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
] via the use of a suitable state transformation.

The remainder of this paper proceeds as follows. In Section 2, we introduce the class of linear quantum stochastic systems under consideration and consider a number of different representations of these systems. We also introduce a useful special class of linear quantum systems which was considered in [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009.-16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
]. In Section 3, we consider the issue of physical realizability for the class of linear quantum systems under consideration. In Section 4, we will consider the problem of coherent H quantum controller synthesis. In Section 5, we present some conclusions.

2. LINEAR QUANTUM SYSTEM MODELS

In this section, we formulate the class of linear quantum system models under consideration. These linear quantum system models take the form of quantum stochastic differential equations which are derived from the quantum harmonic oscillator.

2.1. Quantum Harmonic Oscillators

We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space e.g., see [25J.E. Gough, M.R. James, and H.I. Nurdin, "Squeezing components in linear quantum feedback networks", Phys. Rev. A, vol. 81, p. 023804, 2010.
[http://dx.doi.org/10.1103/PhysRevA.81.023804]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 50P.A. Meyer, "Quantum Probability for Probabilists", Springer-Verlag: Berlin, 1995.
[http://dx.doi.org/10.1007/BFb0084701]
]. Elements of the Hilbert space H, ψ(x) are the standard complex valued wave functions arising in quantum mechanics where x is a spatial variable. Corresponding to this collection of harmonic oscillators is a vector of annihilation operators

(1)

Each annihilation operator ai is an unbounded linear operator defined on a suitable domain in H by

where ψ Є H is contained in the domain of the operator ai. The adjoint of the operator ai is denoted ai* and is referred to as a creation operator. The operators ai and ai* are such that the following canonical commutation relations are satisfied

(2)

where δij denotes the Kronecker delta multiplied by the identity operator on the Hilbert space H. We also have the commutation relations

(3)

For a general vector of operators

on H, we use the notation

to denote the corresponding vector of adjoint operators. Also, g T denotes the corresponding row vector of operators g T = [g1g2gn], and g = (g #)T. Using this notation, the canonical commutation relations (2), (3) can be written as

(4)

A state on our system of quantum harmonic oscillators is defined by a density operator ρ which is a self-adjoint positive-semidefinite operator on H with tr(ρ) = 1; e.g., see [34M. Nielsen, and I. Chuang, Quantum Computation and Quantum Information. Cambridge., Cambridge University Press: UK, 2000.]. Corresponding to a state ρ and an operator g on H is the quantum expectation

A state on the system is said to be Gaussian with positive-semidefinite covariance matrix and mean vector if given any vector ,

e.g., see [25J.E. Gough, M.R. James, and H.I. Nurdin, "Squeezing components in linear quantum feedback networks", Phys. Rev. A, vol. 81, p. 023804, 2010.
[http://dx.doi.org/10.1103/PhysRevA.81.023804]
, 50P.A. Meyer, "Quantum Probability for Probabilists", Springer-Verlag: Berlin, 1995.
[http://dx.doi.org/10.1007/BFb0084701]
]. Here, u# denotes the complex conjugate of the complex vector u, uT denotes the transpose of the complex vector u, and u denotes the complex conjugate transpose of the complex vector u.

Note that in the zero mean case, α = 0, the covariance matrix Q satisfies

In the special case in which the covariance matrix Q is of the form

and the mean α = 0, the system is said to be in the vacuum state. In the sequel, it will be assumed that the state on the system of harmonic oscillators is a Gaussian vacuum state. The state on the system of harmonic oscillators plays a similar role to the probability distribution of the initial conditions of a classical stochastic system.

The quantum harmonic oscillators described above are assumed to be coupled to m external independent quantum fields modelled by bosonic annihilation field operators A1(t), A2(t), …, Am(t) which are defined on separate Fock spaces Fi defined over for each field operator [39H. Nurdin, I.R. Petersen, and M.R. James, "On the infeasibility of entanglement generation in Gaussian quantum systems via classical control", IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 198-203, 2012.
[http://dx.doi.org/10.1109/TAC.2011.2162888]
, 40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 45L. Bouten, R. van Handel, and M. James, "An introduction to quantum filtering", SIAM J. Contr. Optim., vol. 46, no. 6, pp. 2199-2241, 2007.
[http://dx.doi.org/10.1137/060651239]
]. For each annihilation field operator Aj (t), there is a corresponding creation field operator Aj* (t), which is defined on the same Fock space and is the operator adjoint of Aj (t). The field operators are adapted quantum stochastic processes with forward differentials

and

that have the quantum Itô products [39H. Nurdin, I.R. Petersen, and M.R. James, "On the infeasibility of entanglement generation in Gaussian quantum systems via classical control", IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 198-203, 2012.
[http://dx.doi.org/10.1109/TAC.2011.2162888]
, 40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 45L. Bouten, R. van Handel, and M. James, "An introduction to quantum filtering", SIAM J. Contr. Optim., vol. 46, no. 6, pp. 2199-2241, 2007.
[http://dx.doi.org/10.1137/060651239]
]:

The field annihilation operators are also collected into a vector of operators defined as follows:

For each i, the corresponding system state on the Fock space Fi is assumed to be a Gaussian vacuum state which means that given any complex valued function , then

e.g., see [25J.E. Gough, M.R. James, and H.I. Nurdin, "Squeezing components in linear quantum feedback networks", Phys. Rev. A, vol. 81, p. 023804, 2010.
[http://dx.doi.org/10.1103/PhysRevA.81.023804]
, 40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 45L. Bouten, R. van Handel, and M. James, "An introduction to quantum filtering", SIAM J. Contr. Optim., vol. 46, no. 6, pp. 2199-2241, 2007.
[http://dx.doi.org/10.1137/060651239]
].

In order to describe the joint evolution of the quantum harmonic oscillators and quantum fields, we first specify the Hamiltonian operator for the quantum system which is a Hermitian operator on H of the form

where is a Hermitian matrix of the form

and M1 = M1, M2 = M2T. Here, M denotes the complex conjugate transpose of the complex matrix M, MT denotes the transpose of the complex matrix M, and M # denotes the complex conjugate of the complex matrix M. Also, we specify the coupling operator for the quantum system to be an operator of the form

where and . Also, we write

In addition, we define a scattering matrix which is a unitary matrix . These quantities then define the joint evolution of the quantum harmonic oscillators and the quantum fields according to a unitary adapted process U(t) (which is an operator valued function of time) satisfying the Hudson-Parthasarathy QSDE [23J. Gough, and M.R. James, "The series product and its application to quantum feedforward and feedback networks", IEEE Trans. Autom. Control, vol. 54, no. 11, pp. 2530-2544, 2009.
[http://dx.doi.org/10.1109/TAC.2009.2031205]
, 40R. Hudson, and K. Parthasarathy, "Quantum Ito's formula and stochastic evolution", Commun. Math. Phys., vol. 93, pp. 301-323, 1984.
[http://dx.doi.org/10.1007/BF01258530]
, 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992., 45L. Bouten, R. van Handel, and M. James, "An introduction to quantum filtering", SIAM J. Contr. Optim., vol. 46, no. 6, pp. 2199-2241, 2007.
[http://dx.doi.org/10.1137/060651239]
]:

where Λ(t) = [Λjk(t)]j,k = 1,…,m. Here, the processes Λjk(t) for j,k = 1,…,m are adapted quantum stochastic processes referred to as gauge processes, and the forward differentials dΛjk(t) = Λjk(t + dt) - Λjk(t) j,k = 1,…,m have the quantum Itô products:

Then, using the Heisenberg picture of quantum mechanics, the harmonic oscillator operators ai(t) evolve with time unitarily according to

for i = 1,2, …, n. Also, the linear quantum system output fields are given by

for i = 1,2, …, m.

We now use the fact that for any adapted processes X(t) and X(t) satisfying a quantum Itô stochastic differential equation, we have the quantum Itô rule

e.g., see [42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992.]. Using the quantum Itô rule and the quantum Itô products given above, as well as exploiting the canonical commutation relations between the operators in a, the following QSDEs describing the linear quantum system can be obtained (e.g., see [25J.E. Gough, M.R. James, and H.I. Nurdin, "Squeezing components in linear quantum feedback networks", Phys. Rev. A, vol. 81, p. 023804, 2010.
[http://dx.doi.org/10.1103/PhysRevA.81.023804]
]):

(5)

where

(6)

From this, we can write

(7)

where

(8)

Also, the equations (6) can be re-written as

(9)

where

Note that matrices of the form (8) occur commonly in the theory of linear quantum systems. It is straightforward to establish the following lemma which characterizes matrices of this form.

Lemma 1 A matrix satisfies

if and only if

where

We now consider the case when the initial condition in the QSDE (5) is no longer the vector of annihilation operators (1) but rather a vector of linear combinations of annihilation operators and creation operators defined by

where

is non-singular. Then, it follows from (4) that

where

(10)

The relationship

(11)

is referred to as a generalized commutation relation [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009.-16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
]. Also, the covariance matrix corresponding to is given by

In terms of the variables , the QSDEs, (7) can be rewritten as

(12)

where

(13)

Now, we can re-write the operators H and L defining the above collection of quantum harmonic oscillators in terms of the variables as

where

(14)

Here,

(15)

Furthermore, equations (9), (13) and (14) can be combined to obtain

(16)

Note that since S is unitary, it follows that

(17)

Also,

(18)

Indeed these two properties characterize all matrices non-singular satisfying Σ = Σ # which are of the form given in (16). Let the non-singular matrix K be such that KΣ = ΣK #, KJK = J, and KK = I. It follows from Lemma 1 that we can write

Also, KK = I implies

and KJK = J implies

The (1,1) block of these two equations imply K1K1 + K2K2 = I and K1K1-K2K2 = I. Hence K2K2 = 0. Therefore, K2 = 0 and K1K1 = I. From this, it follows that the matrix K must be of the form given in (16).

The QSDEs (12), (13), (16) define the general class of linear quantum systems considered in this paper. Such quantum systems can be used to model a large range of devices and networks of devices arising in the area of quantum optics including optical cavities, squeezers, optical parametric amplifiers, cavity Quantum Electrodynamic (cavity QED) systems, beam splitters, and phase shifters; e.g., see [3A. Doherty, and K. Jacobs, "Feedback-control of quantum systems using continuous state-estimation", Phys. Rev. A, vol. 60, pp. 2700-2711, 1999.
[http://dx.doi.org/10.1103/PhysRevA.60.2700]
, 5M. Yanagisawa, and H. Kimura, "Transfer function approach to quantum control-part I: Dynamics of quantum feedback systems", IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2107-2120, 2003.
[http://dx.doi.org/10.1109/TAC.2003.820063]
, 6M. Yanagisawa, and H. Kimura, "Transfer function approach to quantum control-part II: Control concepts and applications", IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2121-2132, 2003.
[http://dx.doi.org/10.1109/TAC.2003.820065]
, 11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 17H. Mabuchi, "Coherent-feedback quantum control with a dynamic compensator", Phys. Rev. A, vol. 78, p. 032323, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.032323]
, 19I.R. Petersen, "Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009., 22H.I. Nurdin, M.R. James, and A.C. Doherty, "Network synthesis of linear dynamical quantum stochastic systems", SIAM J. Contr. Optim., vol. 48, no. 4, pp. 2686-2718, 2009.
[http://dx.doi.org/10.1137/080728652]
, 24B. Roy, and P. Das, "Modelling of quantum networks of feedback QED systems in interacting Fock space", Int. J. Control, vol. 82, no. 12, pp. 2267-2276, 2009.
[http://dx.doi.org/10.1080/00207170903015164]
, 26H. M. Wiseman, and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press: UK, 1994.-29H. Bachor, and T. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. Wiley Online Library: USA, 2004.
[http://dx.doi.org/10.1002/9783527619238]
, 48J.E. Gough, and S. Wildfeuer, "Enhancement of field squeezing using coherent feedback", Phys. Rev. A, vol. 80, p. 042107, 2009.
[http://dx.doi.org/10.1103/PhysRevA.80.042107]
].

2.2. Annihilation Operator Linear Quantum Systems

An important special case of the linear quantum systems (12), (13), (16) corresponds to the case in which the Hamiltonian operator H and coupling operator L depend only of the vector of annihilation operators a and not on the vector of creation operators a#. This class of linear quantum systems is considered in [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009.-17H. Mabuchi, "Coherent-feedback quantum control with a dynamic compensator", Phys. Rev. A, vol. 78, p. 032323, 2008.
[http://dx.doi.org/10.1103/PhysRevA.78.032323]
, 19I.R. Petersen, "Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009., 20A.I. Maalouf, and I.R. Petersen, "Coherent LQG control for a class of linear complex quantum systems", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009.
[http://dx.doi.org/10.1109/ACC.2009.5159845]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
] and can be used to model “passive” quantum optical devices such as optical cavities, beam splitters, phase shifters and interferometers.

This class of linear quantum systems corresponds to the case in which 2 = 0, 2 = 0, and T2 = 0. In this case, the linear quantum system can be modelled by the QSDEs

(19)

where

(20)

2.3. Position and Momentum Operator Linear Quantum Systems

Note that the matrices in the general QSDEs (12), (13) are in general complex. However, it is possible to apply a particular change of variables to the system (5) so that all of the matrices in the resulting transformed QSDEs are real. This change of variables is defined as follows:

(21)

where the matrices Φ have the form

(22)

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 [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
, 21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
, 39H. Nurdin, I.R. Petersen, and M.R. James, "On the infeasibility of entanglement generation in Gaussian quantum systems via classical control", IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 198-203, 2012.
[http://dx.doi.org/10.1109/TAC.2011.2162888]
]. 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. Furthermore, Qout(t) and Pout(t) are the vectors of position and momentum operators for the output quantum noise fields.

It follows from (22) that

(23)

and hence

(24)

Rather than applying the transformations (21) to the quantum linear system (7) which satisfies the canonical commutation relations (4), corresponding transformations can be applied to the quantum linear system (12) which satisfies the generalized commutation relations (11). These transformations are as follows:

(25)

When these transformations are applied to the quantum linear system (12), this leads to the following real quantum linear system:

(26)

where

(27)

These matrices are all real.

Also, it follows from (10) that

(28)

where

(29)

which is a Hermitian matrix.

Now, we can re-write the operators H and L defining the above collection of quantum harmonic oscillators in terms of the variables and as

where

(30)

Here

(31)

where the matrix R is real but the matrix V may be complex.

However, using (30) and (16), we can write

where

is real as in (27). That is, we can write

Note that the matrix ΦTΦ-1 is real and

(32)

where

Hence, the matrix

must be purely imaginary. Hence, we can define the real skew symmetric matrix

Using this notation, (28) can be written as

In addition, we note that

(33)

Furthermore, equations (16), (27), and (31) can be combined to obtain

(34)

Now from (27), we have D = Φ Φ-1 and DT = Φ-† -†Φ-† and hence,

(35)

using (24), (18) and (23); i.e., D is an orthogonal matrix.

Also, we have

(36)

using (32) and (17); i.e., D is a symplectic matrix.

Conversely suppose a matrix D = Φ Φ-1 satisfies DDT = I and D DT = . It follows in a similar fashion to above that = I and J = J. Hence, as in Section 2.1 the matrix must be of the form in (16). Thus, the matrix D must be of the form in (34).

3. PHYSICAL REALIZABILITY

Not all QSDEs of the form (12), (13) correspond to physical quantum systems. This motivates a notion of physical realizability which has been considered in the papers [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009.-16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
, 19I.R. Petersen, "Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009.-21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
, 38H.G. Harno, and I.R. Petersen, "Coherent control of linear quantum systems: A differential evolution approach", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531367]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
]. This notion is of particular importance in the problem of coherent quantum feedback control in which the controller itself is a quantum system. In this case, if a controller is synthesized using a method such as quantum H control [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
] or quantum linear quadratic Gaussian (LQG) control [12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
, 38H.G. Harno, and I.R. Petersen, "Coherent control of linear quantum systems: A differential evolution approach", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531367]
], it important that the controller can be implemented as a physical quantum system [19I.R. Petersen, "Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009., 22H.I. Nurdin, M.R. James, and A.C. Doherty, "Network synthesis of linear dynamical quantum stochastic systems", SIAM J. Contr. Optim., vol. 48, no. 4, pp. 2686-2718, 2009.
[http://dx.doi.org/10.1137/080728652]
]. We first consider the issue of physical realizability in the case of general linear quantum systems and then we consider the issue of physical realizability for the case of annihilator operator linear quantum system of the form considered in Subsection 2.2.

3.1. Physical Realizability for General Linear Quantum Systems

The formal definition of physically realizable QSDEs requires that they can be realized as a system of quantum harmonic oscillators.

Definition 1 QSDEs of the form (12), (13) are physically realizable if there exist complex matrices Θ = Θ, = , , S such that SS = I, Θ is of the form in (10), is of the form in (15), and (16) is satisfied.

A version of the following theorem was presented in [21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
]; see also [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
] for related results.

Theorem 1 The QSDEs (12), (13) are physically realizable if and only if there exists a complex matrix Θ = Θ of the form in (10) such that

(37)

Proof. If there exist matrices Θ = Θ, = , , S such that SS = I, is of the form in (15), Θ is of the form in (10), and (16) is satisfied, then it follows by straightforward substitution that the first equation in (27) will be satisfied and

(38)

Then, the remaining equations in (37) follow using (17) and (18).

Conversely, suppose there exists a complex matrix Θ = Θ of the form in (10) such that (37) is satisfied. Also, as shown at the end of Section II.A, the conditions J = J and = I imply that there exists a complex matrix S such that SS = I and is of the form

Also, let

It is straightforward to verify that this matrix is Hermitian. Also, it follows from (37) that

as required. Furthermore, using SS = I, it now follows that

Hence, (37) implies

and hence

From this, it follows that

and hence,

as required. Hence, (16) is satisfied.

We now use Lemma 1 to show that is of the form in ((13)). Indeed, we have , and ΣJ = -JΣ. Hence,

Therefore, it follows from Lemma 1 that is of the form in (15) and hence, the QSDEs (12), (13) are physically realizable.

Remark 1 In the canonical case when T = I and Θ = J, the physical realizability equations (37) become

(39)

Following the approach of [21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
], we now relate the physical realizability of the QSDEs (12), (13) to the dual (J, J)-unitary property of the corresponding transfer function matrix

(40)

Definition 2 (See [21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
, 52H. Kimura, "Chain-Scattering Approach to H∞ Control", Birkhäuser: Boston, 1997.].) A transfer function matrix Γ(s) of the form (40) is dual (J, J)-unitary if

for all .

Here, Γ~(s) = Γ(-s*) and denotes the set .

Theorem 2 The transfer function matrix (40) corresponding to the QSDEs (12), (13) is dual (J, J)-unitary if and only if

(41)

and there exists a Hermitian matrix Θ such that

(42)

Theorem 3 (See also [21A.J. Shaiju, and I.R. Petersen, "On the physical realizability of general linear quantum stochastic differential equations with complex coefficients", In: Proceedings of the 48th IEEE Conference on Decision and Control Shanghai, China, 2009.
[http://dx.doi.org/10.1109/CDC.2009.5399947]
]) If the QSDEs (12), (13) are physically realizable, then the corresponding transfer function matrix (40) is dual (J, J)-unitary.

Conversely, suppose the QSDEs (12), (13) satisfy the following conditions:

1. The transfer function matrix (40) corresponding to the QSDEs (12), (13) is dual (J, J)-unitary;

2.

(43)

3. The Hermitian matrix Θ satisfying (42) is of the form in (10).

Then, the QSDEs (12), (13) are physically realizable.

Proof. If the QSDEs (12), (13) are physically realizable, then it follows from Theorem 1 that there exist complex matrices Θ = Θ and S such that SS = I and equations (37) are satisfied. However, J = J and = I imply J = J and hence, it follows from (37) that

That is, the conditions (42) are satisfied and hence it follows from Theorem 2 that the transfer function matrix (40) corresponding to the QSDEs (12), (13) is dual (J, J)-unitary.

Conversely, if the QSDEs (12), (13) satisfy conditions (i) - (iii) of the theorem, then it follows from Theorem 2 that there exists a Hermitian matrix Θ of the form in (10) such that equations (42) are satisfied. Hence,

(44)

Furthermore, we have J = J and = I and therefore J = J . Hence, (44) implies

From this it follows that equations (37) are satisfied. Thus, it follows from Theorem 1 that the QSDEs (12), (13) are physically realizable.

3.2. Physical Realizability for Annihilator Operator Linear Quantum Systems

For annihilator operator linear quantum systems described by QSDEs of the form (19) the corresponding formal definition of physical realizability is as follows.

Definition 3 (See [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
].) The QSDEs of the form (19) are said to be physically realizable if there exist matrices Θ1 = Θ1 > 0, 1 = 1, , and S such that SS = I and (20) is satisfied.

The following theorem from [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
] gives a characterization of physical realizability in this case.

Theorem 4 The QSDEs (19) are physically realizable if and only if there exists a complex matrix Θ1 = Θ1 > 0 such that

(45)

In the case of QSDEs of the form (19), the issue of physical realizability is determined by the lossless bounded real property of the corresponding transfer function matrix

(46)

Definition 4 (See also [53B.D. Anderson, and S. Vongpanitlerd, Network Analysis and Synthesis., Dover Publications: NY, 2006.].) The transfer function matrix (46) corresponding to the QSDEs (19) is said to be lossless bounded real if the following conditions hold:

1. is a Hurwitz matrix; i.e., all of its eigenvalues have strictly negative real parts;

2.

for all

Definition 5 (See also [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
].) The QSDEs (19) are said to define a minimal realization of the transfer function matrix (46) if the following conditions hold:

1. Controllability

2. Observability

The following theorem, which is a complex version of the standard lossless bounded real lemma, gives a state space characterization of the lossless bounded real property.

Theorem 5 (Complex Lossless Bounded Real Lemma; e.g., see [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 53B.D. Anderson, and S. Vongpanitlerd, Network Analysis and Synthesis., Dover Publications: NY, 2006.]). Suppose the QSDEs (19) define a minimal realization of the transfer function matrix (46). Then the transfer function (46) is lossless bounded real if and only if there exists a Hermitian matrix X > 0 such that

(47)

Combining Theorems 4 and 5 leads to the following result which provides a complete characterization of the physical realizability property for minimal QSDEs of the form (19).

Theorem 6 (See [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
].) Suppose the QSDEs (19) define a minimal realization of the transfer function matrix (46). Then, the QSDEs (19) are physically realizable if and only if the transfer function matrix (46) is lossless bounded real.

The following theorem from [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
], is useful in synthesizing coherent quantum controllers using state space methods.

Theorem 7 (See [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
].) Suppose the matrices F,G1,H1 define a minimal realization of the transfer function matrix

Then, there exists matrices G2 and H2 such that the following QSDEs of the form (19)

(48)

are physically realizable if and only if F is Hurwitz and

(49)

3.3. Physical Realizability for Position and Momentum Operator Linear Quantum Systems

It is often convenient to consider the physical realizability of real quantum linear systems of the form (26). This can be achieved by applying the transformations (25) and the equations (34), (32), (33), (36), (35) to obtain the following corollary of Theorem 1.

Corollary 1 (See also [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 12H.I. Nurdin, M.R. James, and I.R. Petersen, "Coherent quantum LQG control", Automatica, vol. 45, no. 8, pp. 1837-1846, 2009.
[http://dx.doi.org/10.1016/j.automatica.2009.04.018]
].) The QSDEs (26) are physically realizable if and only if there exists a real matrix Θ = ΘT such that

(50)

Remark 2 For real QSDEs of the form (26) with corresponding transfer function

It is straightforward using equations (25) to verify that this transfer function is related to the transfer function (40) of the corresponding complex QSDEs (12) according to the relation

(51)

Now if the real QSDEs (26) are physically realizable, it follows that the corresponding complex QSDEs (12), (13) are physically realizable. Hence, using Theorem 3, it follows that the corresponding transfer function matrix (40) is dual (J,J)-unitary; i.e.,

for all . Therefore, it follows from (51) and (32) that

for all . Also, as in the discussion at the end of Section II.C, the conditions (41), (43) are equivalent to the conditions

4. COHERENT QUANTUM H CONTROL

In this section, we formulate a coherent quantum control problem in which a linear quantum system is controlled by a feedback controller which is itself a linear quantum system. The fact that the controller is to be a quantum system means that any controller synthesis method needs to produce controllers which are physically realizable. The problem we consider is the quantum H control problem in which it is desired to design a coherent controller such that the resulting closed loop quantum system is stable and attenuates specified disturbances acting on the system; see [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
]. In the standard quantum H control problem such as considered in [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
], the quantum noises are averaged out and only the external disturbance is considered. Other approaches to coherent quantum controller design include the coherent LQG approach. However the H approach has the advantage that it is more computationally tractable than the coherent LQG approach. Also, as in the case of classical H control, this approach enables the design of control systems which are robust to uncertainties in the plant model.

4.1. Coherent H Control of General Quantum Linear Systems

In this subsection, we formulate the coherent quantum H control problem for a general class of quantum systems of the form (12), (13).

We consider quantum plants described by linear complex quantum stochastic models of the following form defined in an analogous way to the QSDEs (12), (13):

(52)

where all of the matrices in these QSDEs have a form as in (13). Here, the input

represents a disturbance signal where βw(t) is an adapted process; see [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 42K. Parthasarathy, An Introduction to Quantum Stochastic Calculus., Berlin. Birkhauser, 1992.]. The signal u(t) is a control input of the form

where βu(t) is an adapted process. The quantity

represents any additional quantum noise in the plant. The quantities and are quantum noises of the form described in Section 2.

In the coherent quantum H control problem, we consider controllers which are described by QSDEs of the form (12), (13) as follows:

(53)

where all of the matrices in these QSDEs have a form as in (13). Here the quantities

are controller quantum noises of the form described in Section 2. Also, the outputs du0 and du1 are unused outputs of the controller which have been included so that the controller can satisfy the definition of physical realizability given in Definition 1.

Corresponding to the plant (52) and (53), we form the closed loop quantum system by identifying the output of the plant dy with the input to the controller dy, and identifying the output of the controller du with the input to the plant du. This leads to the following closed-loop QSDEs:

(54)

where

For a given quantum plant of the form (52), the coherent quantum H control problem involves finding a physically realizable quantum controller (53) such that the resulting closed loop system (54) is such that the following conditions are satisfied: [(i)]

1. The matrix

(55)

is Hurwitz;

2. The closed loop transfer function

satisfies

(56)

where

Remark 3 In the paper [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
], a version of the coherent quantum H control problem is solved for linear quantum systems described by real QSDEs which are similar to those in (26). In this case, the problem is solved using a standard two Riccati equation approach such as given in [54J.C. Doyle, K. Glover, P.P. Khargonekar, and B. Francis, "State-space solutions to the standard H2 and H∞ control problems", IEEE Trans. Autom. Control, vol. 34, no. 8, pp. 831-847, 1989.
[http://dx.doi.org/10.1109/9.29425]
, 55K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River., Prentice-Hall: NJ, 1996.]. A result is given in [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
] which shows that any H controller which is synthesized using the two Riccati equation approach can be made physically realizable by adding suitable additional quantum noises. Alternative solutions to the problem could be obtained using linear matrix inequality (LMI) approaches to the H control problem. However, the Riccati equation approach has the advantage that it is numerically less demanding than the LMI approach.

4.2. Coherent H Control of Annihilator Operator Quantum Linear Systems

In this subsection, we consider the special case of coherent quantum H control for annihilation operator quantum linear systems of the form considered in Subsection 2.2 and present the Riccati equation solution to this problem obtained in [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
]. The quantum H control problem being considered is the same as considered in Subsection 4.1 but we restrict attention to annihilation operator plants of the form (19) as follows:

(57)

Also, we restrict attention to annihilation operator controllers of the form (19) as follows:

(58)

The quantum plant (57) is assumed to satisfy the following assumptions:

1. K12K12 = E1 > 0;

2. K21K21 = E2 > 0;

3. The matrix is full rank for all ω ≥ 0;

4. The matrix is full rank for all ω ≥ 0.

The results will be stated in terms of the following pair of complex algebraic Riccati equations:

(59)
(60)

The solutions to these Riccati equations will be required to satisfy the following conditions.

1. The matrix F-G2E1-1 K12H1 + (G1G1-G2E1-1G2)X is Hurwitz; i.e., X is a stabilizing solution to (59).

2. The matrix F-G1K21E2-1H2 + Y(H1H1-H2E2-1H2) is Hurwitz; i.e, Y is a stabilizing solution to (60).

3. The matrices X and Y satisfy

(61)

where ρ(·) denotes the spectral radius.

If the above Riccati equations have suitable solutions, a quantum controller of the form (58) is constructed as follows:

(62)

The following Theorem is presented in [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
].

Theorem 8 Necessity: Consider a quantum plant (57) satisfying the above assumptions. If there exists a quantum controller of the form (58) such that the resulting closed-loop system satisfies the conditions (55), (56), then the Riccati equations (59) and (60) will have stabilizing solutions X ≥ 0 and Y ≥ 0 satisfying (61).

Sufficiency: Suppose the Riccati equations (59) and (60) have stabilizing solutions X ≥ 0 and Y ≥ 0 satisfying (61). If the controller (58) is such that the matrices Fc, Gc, Hc are as defined in (62), then the resulting closed-loop system will satisfy the conditions (55), (56).

Note that this theorem does not guarantee that a controller defined by (58), (62) will be physically realizable. However, if the matrices defined in (62) are such that

then it follows from Theorem 7 that a corresponding physically realizable controller of the form (58) can be constructed.

CONCLUSION

In this paper, we have surveyed some recent results in the area of quantum linear systems theory and the related area of coherent quantum H control. However, a number of other recent results on aspects of quantum linear systems theory have not been covered in this paper. These include results on coherent quantum LQG control (see [11M.R. James, H.I. Nurdin, and I.R. Petersen, "H ∞ control of linear quantum stochastic systems", IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787-1803, 2008.
[http://dx.doi.org/10.1109/TAC.2008.929378]
, 38H.G. Harno, and I.R. Petersen, "Coherent control of linear quantum systems: A differential evolution approach", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531367]
]), and model reduction for quantum linear systems (see [51I.R. Petersen, "Singular perturbation approximations for a class of linear complex quantum systems", In: Proceedings of the 2010 American Control Conference Baltimore, MD, 2010.
[http://dx.doi.org/10.1109/ACC.2010.5531374]
]). These results have not been covered since the emphasis in this paper is on the H approach to coherent quantum control. Also, it is not possible to provide a direct comparison between H control methods and LQG control methods since their performance indices are measuring different quantities.

Furthermore, in order to apply synthesis results on coherent quantum feedback controller synthesis, it is necessary to realize a synthesized feedback controller transfer function using physical optical components such as optical cavities, beam-splitters, optical amplifiers, and phase shifters. In a recent paper [22H.I. Nurdin, M.R. James, and A.C. Doherty, "Network synthesis of linear dynamical quantum stochastic systems", SIAM J. Contr. Optim., vol. 48, no. 4, pp. 2686-2718, 2009.
[http://dx.doi.org/10.1137/080728652]
], this issue was addressed for a general class of coherent linear quantum controllers. An alternative approach to this problem is addressed in [19I.R. Petersen, "Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control", In: Proceedings of the 2009 European Control Conference Budapest, Hungary, 2009.] for the class of annihilation operator linear quantum systems considered in Subsection 2.2 and [14A.I. Maalouf, and I.R. Petersen, "Coherent H control for a class of linear complex quantum systems", In: 2009 American Control Conference St Louis, Mo, 2009., 15A.I. Maalouf, and I.R. Petersen, "Bounded real properties for a class of Annihilation-Operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 786-801, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2060970]
, 16A.I. Maalouf, and I.R. Petersen, "Coherent H∞ control for a class of annihilation operator linear quantum systems", IEEE Trans. Autom. Control, vol. 56, no. 2, pp. 309-319, 2011.
[http://dx.doi.org/10.1109/TAC.2010.2052942]
]. For this class of quantum systems, an algorithm is given to realize a physically realizable controller transfer function in terms of a cascade connection of optical cavities and phase shifters.

An important application of both classical and coherent feedback control of quantum systems is in enhancing the property of entanglement for linear quantum systems. Entanglement is an intrinsically quantum mechanical notion which has many applications in the area of quantum computing and quantum communications.

To conclude, we have surveyed some of the important advances in the area of linear quantum control theory. However, many important problems in this area remain open and the area provides a great scope for future research.

CONFLICT OF INTEREST

The author confirms that this article content has no conflict of interest.

ACKNOWLEDGEMENTS

Research supported by the Australian Research Council (ARC). A preliminary version of this paper appeared in the 2010 MTNS Conference.

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Patrice Talaga
(UCB S.A., Belgium)

"Open access journals are a novel concept in the medical literature. They offer accessible information to a wide variety of individuals, including physicians, medical students, clinical investigators, and the general public. They are an outstanding source of medical and scientific information."


Jeffrey M. Weinberg
(St. Luke's-Roosevelt Hospital Center, USA)

"Open access journals are extremely useful for graduate students, investigators and all other interested persons to read important scientific articles and subscribe scientific journals. Indeed, the research articles span a wide range of area and of high quality. This is specially a must for researchers belonging to institutions with limited library facility and funding to subscribe scientific journals."


Debomoy K. Lahiri
(Indiana University School of Medicine, USA)

"Open access journals represent a major break-through in publishing. They provide easy access to the latest research on a wide variety of issues. Relevant and timely articles are made available in a fraction of the time taken by more conventional publishers. Articles are of uniformly high quality and written by the world's leading authorities."


Robert Looney
(Naval Postgraduate School, USA)

"Open access journals have transformed the way scientific data is published and disseminated: particularly, whilst ensuring a high quality standard and transparency in the editorial process, they have increased the access to the scientific literature by those researchers that have limited library support or that are working on small budgets."


Richard Reithinger
(Westat, USA)

"Not only do open access journals greatly improve the access to high quality information for scientists in the developing world, it also provides extra exposure for our papers."


J. Ferwerda
(University of Oxford, UK)

"Open Access 'Chemistry' Journals allow the dissemination of knowledge at your finger tips without paying for the scientific content."


Sean L. Kitson
(Almac Sciences, Northern Ireland)

"In principle, all scientific journals should have open access, as should be science itself. Open access journals are very helpful for students, researchers and the general public including people from institutions which do not have library or cannot afford to subscribe scientific journals. The articles are high standard and cover a wide area."


Hubert Wolterbeek
(Delft University of Technology, The Netherlands)

"The widest possible diffusion of information is critical for the advancement of science. In this perspective, open access journals are instrumental in fostering researches and achievements."


Alessandro Laviano
(Sapienza - University of Rome, Italy)

"Open access journals are very useful for all scientists as they can have quick information in the different fields of science."


Philippe Hernigou
(Paris University, France)

"There are many scientists who can not afford the rather expensive subscriptions to scientific journals. Open access journals offer a good alternative for free access to good quality scientific information."


Fidel Toldrá
(Instituto de Agroquimica y Tecnologia de Alimentos, Spain)

"Open access journals have become a fundamental tool for students, researchers, patients and the general public. Many people from institutions which do not have library or cannot afford to subscribe scientific journals benefit of them on a daily basis. The articles are among the best and cover most scientific areas."


M. Bendandi
(University Clinic of Navarre, Spain)

"These journals provide researchers with a platform for rapid, open access scientific communication. The articles are of high quality and broad scope."


Peter Chiba
(University of Vienna, Austria)

"Open access journals are probably one of the most important contributions to promote and diffuse science worldwide."


Jaime Sampaio
(University of Trás-os-Montes e Alto Douro, Portugal)

"Open access journals make up a new and rather revolutionary way to scientific publication. This option opens several quite interesting possibilities to disseminate openly and freely new knowledge and even to facilitate interpersonal communication among scientists."


Eduardo A. Castro
(INIFTA, Argentina)

"Open access journals are freely available online throughout the world, for you to read, download, copy, distribute, and use. The articles published in the open access journals are high quality and cover a wide range of fields."


Kenji Hashimoto
(Chiba University, Japan)

"Open Access journals offer an innovative and efficient way of publication for academics and professionals in a wide range of disciplines. The papers published are of high quality after rigorous peer review and they are Indexed in: major international databases. I read Open Access journals to keep abreast of the recent development in my field of study."


Daniel Shek
(Chinese University of Hong Kong, Hong Kong)

"It is a modern trend for publishers to establish open access journals. Researchers, faculty members, and students will be greatly benefited by the new journals of Bentham Science Publishers Ltd. in this category."


Jih Ru Hwu
(National Central University, Taiwan)


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