Quantum Dissipative Systems and Feedback Control Design by Interconnection

Matthew R. James; John Gough

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Quantum Dissipative Systems and Feedback Control Design by Interconnection

Matthew R. James, John Gough

arXiv:0707.1074v3Jul 7, 2007quant-ph

TL;DR

Quantum technologies need analysis and design methods for dissipative systems. The paper models plant–exosystem interactions as quantum feedback networks and derives an infinitesimal dissipation characterization that generalizes the Positive Real and Bounded Real Lemmas, supporting control by interconnection.

Problem
Quantum technologies need analogous analysis and design methods for dissipative systems, extending a theory rooted in classical physical systems.
Source: §I · p. 2
Method
The paper models open quantum plants interacting with classes of exosystems in quantum feedback networks and characterizes dissipation infinitesimally.
Source: §I · p. 3, §V · p. 29
Results
The infinitesimal characterization generalizes the Positive Real and Bounded Real Lemmas, and essentially all open quantum systems are dissipative for a suitable supply rate.
Source: §I · p. 3, §V · p. 29
Takeaways & Limitations
Quantum network models formulate stabilization, regulation, and robust-control problems, with controllers in principle physically realizable through unitary models.
Source: §I · p. 4

Abstract

from arXiv · show

The purpose of this paper is to extend J.C. Willems' theory of dissipative systems to the quantum domain. This general theory, which combines perspectives from the quantum physics and control engineering communities, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of exosystems in general quantum feedback network terms. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using quantum network models, which implements Willems' `control by interconnection' for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

I. INTRODUCTION

The paper extends dissipative-system analysis to open quantum systems and formulates control design through quantum network interconnections. It uses the classical RLC circuit to connect energy-balance inequalities with the plant–exosystem perspective.

A. The Classical RLC Circuit Revisited

The passive series RLC circuit's energy-balance relation yields a dissipation inequality: stored-energy rate is below supplied rate, with the remainder dissipated.Source: §I · p. 2

I. INTRODUCTION

The optical cavity stores electromagnetic energy as photons and loses them to an external field, analogous to the RLC circuit losing energy as heat.Source: §I · p. 2
The paper extends dissipative-system analysis to open quantum systems and formulates control design through quantum network interconnections.Source: §I · pp. 3–4
An infinitesimal dissipation characterization generalizes the Positive Real and Bounded Real Lemmas and supports analysis of quantum dissipative-system properties.Source: §I · p. 3
Quantum network models formulate stabilization, regulation, and robust-control problems; unitary models mean controllers can in principle be physically realized.Source: §I · p. 4
Stability matters because active network elements can inject energy, through design or unwanted environmental influences.Source: §I · p. 4

A. The Classical RLC Circuit Revisited

Treating each voltage input as generated by an external signal system recasts the RLC inequality as a plant–exosystem interconnection, a bridge to the quantum setting.Source: A. The Classical RLC Circuit Revisited · p. 6

B. Quantum Mechanics

The section introduces quantum observables and open-system interactions, then describes how a plant and exosystem exchange energy. Their interaction Hamiltonian supplies the coupling, while the broader framework also allows field connections.

B. Quantum Mechanics

Quantum physical quantities such as energy, spin, and position are represented by self-adjoint operators on a Hilbert space, while other physical variables may be non-self-adjoint.Source: B. Quantum Mechanics · p. 6
Open quantum systems interact with other systems or an environment and are modeled using a stochastic generalization of the Schrödinger equation involving quantum noise.Source: B. Quantum Mechanics · p. 7

C. A Pair of Interacting Systems

For independent plant and exosystem components, the combined physical variable space is their tensor product, and variables from the two systems commute.Source: C. A Pair of Interacting Systems · p. 7
An interaction Hamiltonian specifies energy exchange between the plant and exosystem; when a plant observable commutes with its isolated Hamiltonian, the plant is lossless and its energy change is delivered by the exosystem.Source: C. A Pair of Interacting Systems · p. 7
The general framework allows open-system plants and exosystems to connect through fields as well as direct couplings.Source: C. A Pair of Interacting Systems · p. 8

D. Definitions

The paper models open quantum systems with parameters that determine their unitary evolution, then builds feedback networks from component interconnections. Concatenation and series connections describe ordinary wiring, while a linear fractional transformation represents feedback loops.

D. Definitions

The field processes include photon annihilation and channel scattering, with canonical inputs specified by their non-vanishing second-order Ito products.System–field coupling is specified using a scattering matrix and a coupling vector.Source: D. Definitions · p. 8
Open-system parameters G = (S, L, H) compactly specify the system and determine its behavior through its unitary evolution.The system's dynamics are described by a Hamiltonian and its coupling to field channels.Source: D. Definitions · pp. 8–9

E. Quantum Feedback Networks

Quantum feedback networks connect open components through directional field channels, and their framework also accommodates nonlinear component dynamics.Components may additionally interact through direct couplings.Source: E. Quantum Feedback Networks · pp. 9–10
Concatenation collects components, series products represent cascade connections, and remaining signal connections are represented by a linear fractional transformation.The feedback transformation is well-defined when I − S22 is invertible.Source: E. Quantum Feedback Networks · pp. 10–11

III. QUANTUM DISSIPATIVE SYSTEMS

The paper defines quantum dissipation by modeling external influences as a class of interconnected exosystems and relating supply to plant storage. It gives an infinitesimal characterization and shows how the framework covers lossless behavior and generalizations of passivity results.

III. QUANTUM DISSIPATIVE SYSTEMS

Quantum dissipation models external influences on a plant as an interconnected exosystem drawn from a specified class.The class lets the definition cover a range of external systems rather than a single signal generator.Source: §III · p. 12

A. Definitions and Differential Characterization

Theorem 3.1 characterizes dissipation if and only if a non-negative system satisfies an infinitesimal condition for all exosystem parameters.The condition involves plant and exosystem operators; the initial vacuum expectation averages out the noise.Source: A. Definitions and Differential… · p. 14
For unrestricted exosystems, a non-negative observable V0 commuting with H yields lossless dissipation with storage function V = V0 and supply rate r0.This result applies to the class of open exosystems that can be connected to the plant without further constraints.Source: A. Definitions and Differential… · p. 14
Theorems 3.1 and 3.3 include special cases of the Bounded and Positive Real Lemmas and fundamental energy-balance properties.The paper also identifies a lossless energy-rate relation that generalizes passivity inequalities.Source: A. Definitions and Differential… · pp. 15–16
The natural supply rate separates quantum-noise damping, signal contributions, and energy flow between the plant and exosystem.A negative damping term represents energy lost from the plant to the field or heat bath.Source: A. Definitions and Differential… · p. 15

B. Stability

The section relates dissipativity to stability through an exponential stability criterion and a result for strictly dissipative systems. For the unperturbed plant, the stated criterion implies convergence of expected storage to zero.

B. Stability

Lemma 3.4 gives a storage-function bound under a differential inequality, and when λ = 0 it implies that expected storage tends to zero.The lemma assumes c > 0 and real λ.Source: B. Stability · p. 16
The proof derives the result by integrating the plant dynamics and applying the vacuum expectation relation.The text states that more general stability results are left for future work.Source: B. Stability · p. 16
For the unperturbed plant, the stated condition with c > 0 implies that expected storage converges to zero for any plant state.The section presents this as a simple exponential stability criterion.Source: B. Stability · p. 16

C. Passivity and Gain

The section gives infinitesimal characterizations of passivity and gain for quantum plants interconnected with exosystems, generalizing the Positive Real and Bounded Real Lemmas. Examples show how these properties apply to quantum systems and classical deterministic systems.

Scope

The passivity and gain results concern series plant–exosystem interconnections and assume commuting exosystems for simplicity; the framework also permits more general results.Source: C. Passivity and Gain · p. 17

Positive Real Lemma

Theorem 3.6 states that a plant P = (I, L, H) is passive with respect to W1 if and only if there exists a non-negative system observable V ∈ AP and a non-negative real number λ.Source: C. Passivity and Gain · p. 17

Bounded Real Lemma

For commuting exosystems in W2, a plant has gain g if and only if a nonnegative observable V and λ satisfy the bounded-real inequality, with a reduced condition when Γ is invertible.The reduced condition is given in (60).Source: C. Passivity and Gain · pp. 17–18

Two-level atom

The two-level atom is lossless and passive, has gain 1 for its stated output and commuting inputs, and is asymptotically stable when D = 0 with zero exosystem input.Source: C. Passivity and Gain · p. 19

Open oscillator

For the open oscillator, passivity holds when |β|^2 − |α|^2 ≤ 0; strict inequality gives stability, while β = 0 additionally makes the expected excitation tend to zero.The β = 0 case also has gain 1 for output √γa + w with commuting inputs.Source: C. Passivity and Gain · p. 19

Classical deterministic systems

Classical deterministic dissipation is a special case of the quantum formulation, obtained by representing the classical system as a commutative subsystem.The text notes that classical open stochastic systems can also be represented, but does not develop that case.Source: C. Passivity and Gain · p. 20

D. Dissipative Networks

The section derives how dissipation properties behave under quantum feedback-network constructions. It applies these rules to feedback and series architectures and shows that component storage functions can yield a storage function for an interconnected network.

Concatenation and series connections

Dissipation can be carried through concatenation and series connections, with combined storage and supply rates determined by the component properties.Under the stated cross-commutation conditions, the supply rate simplifies.Source: D. Dissipative Networks · pp. 20–21

LFT feedback

A well-defined LFT feedback system remains dissipative with the same storage function when its exosystem class and supply rate are transformed as specified.The transformed external exosystem accounts for elimination of the loop's internal signal.Source: D. Dissipative Networks · p. 21

Series architectures

A series connection can shift the supply rate while preserving the storage function and the equivalent dissipation property.Source: D. Dissipative Networks · p. 22

Scattering matrix

The framework also gives equivalent dissipation descriptions when moving a plant's scattering matrix into the supply-rate formulation.Source: D. Dissipative Networks · p. 22

Interconnected network example

For a network of n interconnected dissipative plants, the combined network is dissipative with storage function V = V1 + . . . + Vn under the stated well-definedness and component assumptions.Source: D. Dissipative Networks · p. 22, Figure 9 · p. 23

E. Uncertainty Modeling

The framework models uncertainty by interconnecting a nominal plant with an exosystem; in a damped-oscillator example, parameter uncertainty preserves stability but changes the decay rate.

E. Uncertainty Modeling

Parameter uncertainty is represented by an exosystem connected to a nominal plant, with the plant and exosystem sharing variables.The discussion focuses on parameter uncertainty as a simple, common case; detailed robustness analysis is beyond the paper's scope.Source: E. Uncertainty Modeling · pp. 23–24
For a damped cavity, uncertainty in transmissivity γ and detuning ω is encoded by an uncertain exosystem alongside a nominal tuned cavity.The nominal coupling is √γ0 a; the uncertain plant has coupling (1 + ϵ)√γ0 a and Hamiltonian ωa∗a.Source: E. Uncertainty Modeling · p. 24
The uncertain damped oscillator remains stable for all modeled uncertainty, although its decay rate changes.Detuning also rotates the quadratures, and detailed performance analysis may involve physical variables such as ω.Source: E. Uncertainty Modeling · p. 24

A. Control Design Formulation

The control formulation treats design as choosing a controller to shape the dynamics of an interconnected plant-controller-exosystem network. It accommodates quantum, classical, or mixed controllers and encodes objectives through dissipativity conditions.

A. Control Design Formulation

The framework extends passivity-based, energy-shaping, interconnection-and-damping-assignment, and control-as-interconnection methods to quantum control design.The paper describes these as classical methods its procedure extends.Source: A. Control Design Formulation · p. 25
Choosing a controller shapes the dynamics of the plant-controller-exosystem network, and the controller may be quantum, classical, or mixed.The network parameters and inputs determine the network's dynamical evolution.Source: A. Control Design Formulation · p. 25, Figure 10 · p. 25
The plant-controller boundary may require engineering plant couplings or direct-interaction Hamiltonians to make desired control connections available.For objectives concerning expected network behavior over a range of variables, the authors also discuss working directly with the network generator to determine the controller.Source: A. Control Design Formulation · p. 25
Design objectives are encoded by a non-negative observable, a supply rate, and an exosystem class, then imposed as a dissipativity condition for all exosystem parameters.The observable may represent quantities to be small or to tend to zero, such as regulation errors; exosystems accommodate uncertainty and disturbances.Source: A. Control Design Formulation · pp. 25–26

B. Controller Synthesis

Controller synthesis is framed as finding a controller that makes the closed-loop network dissipative under the chosen objective. The framework gives conditions for exponential stabilization or regulation and for an H∞ gain bound under disturbance inputs.

B. Controller Synthesis

A controller satisfying the stated dissipation condition makes the expected objective observable decay exponentially for any plant-controller state.The stabilization/regulation setup uses a negative supply rate proportional to the observable and a trivial exosystem.Source: B. Controller Synthesis · p. 26
The nonlinear quantum H∞ result extends earlier linear quantum results and models disturbance inputs as commuting exosystem variables.The theorem applies a slight extension of the Bounded Real Lemma to the plant-controller network.Source: B. Controller Synthesis · p. 26
If the stated inequality holds for all exosystem parameters, the plant-controller network has gain g.The condition includes the disturbance term w and a non-negative λ.Source: B. Controller Synthesis · p. 26

C. Design Examples

The design examples use plant-controller interconnections to illustrate regulation and stabilization, including a classical-style feedback attempt that does not achieve asymptotic stability. The conclusion presents a quantum-network formulation of dissipativity and control by interconnection.

C. Design Examples

For cavity regulation, a series-connected source controller supplies photons to maintain a non-zero steady-state expected photon number.Choosing ν = −α/2 and c = 1/2 makes the dissipation expression equal −Vd/2, so the expected objective tends to zero; the coherent-state drive is also representable as a Hamiltonian term with vacuum input.Source: C. Design Examples · pp. 27–28
A classical-style Hamiltonian stabilization feedback requires Z, which is absent from the available output; adding a field channel makes Z accessible but the resulting loop remains marginally stable.The loop has poles at 0 and −4k; replacing the controller with ˜C = (1, ka, 0) yields strictly passive, asymptotically stable re-engineered systems.Source: C. Design Examples · pp. 28–29

V. CONCLUSIONS

The paper extends Willems' dissipative-systems theory to open quantum models, characterizes dissipation infinitesimally, and generalizes the Positive Real and Bounded Real Lemmas.It also describes external influences through exosystem interactions and implements control by interconnection using quantum network representations.Source: §V · p. 29
The authors identify quantum-network methods as useful for analyzing and designing dissipative quantum systems and networks, including more complex systems contemplated in quantum computing.They describe further development of the network paradigm as merited.Source: §V · p. 29

APPENDIX

The appendix reviews two quantum-system models—the two-level atom and harmonic oscillator—and introduces operator-ordering definitions and implications. It characterizes their state spaces and physical variables, then states operator conclusions under universal conditions on an arbitrary operator.

A. Two Level Atom (Qubit)

The two-level atom models ground and excited states in a two-dimensional Hilbert space, with Pauli matrices spanning its physical variable space and σz eigenvalues representing energy levels.Figure 14(a) depicts raising and lowering operators.Source: A. Two Level Atom (Qubit) · pp. 29–30, Figure 14 · p. 30

B. Quantum Harmonic Oscillator

The quantum harmonic oscillator models optical cavities and boson fields in L2(R, C), with creation and annihilation operators obeying [a, a∗] = 1.Its number states have eigenvalues 0, 1, 2, … for a∗a, while coherent states satisfy a|α⟩ = α|α⟩.Source: B. Quantum Harmonic Oscillator · p. 30

C. Operator Orderings

For self-adjoint operators, A ≥ 0 means ⟨ψ, Aψ⟩ ≥ 0 for every ψ, and A ≥ B means A − B ≥ 0.These definitions set the operator-ordering convention used in the appendix.Source: C. Operator Orderings · p. 31
Under the stated condition holding for all operators w, the appendix concludes A ≤ 0 and verifies the claim by contradiction using w = αI for arbitrary real α.The condition itself is not specified in these passages.Source: C. Operator Orderings · p. 31
A similar universal condition on w is stated to imply C = 0.The condition is not specified in these passages.Source: C. Operator Orderings · p. 31