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Analysis of Lyapunov Method for Control of Quantum Systems

Xiaoting Wang, Sonia G. Schirmer

arXiv:0801.0702v2quant-ph

TL;DR

The paper examines when Lyapunov control can steer finite-dimensional quantum systems to target trajectories, using LaSalle’s invariance principle and stability analysis. It finds that control is effective for most targets under ideal Hamiltonians, but its effectiveness is substantially diminished for non-ideal systems.

  • Problem

    When and for which systems and objectives Lyapunov control is effective for quantum systems remains unresolved.

    Source: §I · p. 2

  • Method

    The paper analyzes convergence using LaSalle’s invariance principle and stability analysis for finite-dimensional quantum systems under ideal and non-ideal Hamiltonians.

    Source: §I · p. 2

  • Results

    Under ideal Hamiltonians, Lyapunov control is effective for most, but not all, target states; for non-ideal systems, most solutions converge to other points on a centre manifold.

    Source: §I · p. 2, §VII · p. 18

  • Takeaways & Limitations

    The analysis supports Lyapunov control as an effective strategy overall for ideal systems, but suggests effectiveness only for a very small subset of controllable quantum systems in realistic settings.

    Source: §VII · pp. 17–18

    The ideal-system conclusion relies on stringent assumptions: a strongly regular drift Hamiltonian H0 and a fully connected control Hamiltonian H1.

    Source: §VII · p. 17

Abstract

from arXiv · show

We present a detailed analysis of the convergence properties of Lyapunov control for finite-dimensional quantum systems based on the application of the LaSalle invariance principle and stability analysis from dynamical systems and control theory. For a certain class of ideal Hamiltonians, convergence results are derived both pure-state and mixed-state control, and the effectiveness of the method for more realistic Hamiltonians is discussed.

I. INTRODUCTION

The paper asks when Lyapunov control effectively steers quantum systems, and analyzes trajectory tracking for finite-dimensional Hamiltonian systems using LaSalle’s principle and stability analysis. It argues that ideal Hamiltonians can yield effective control for most target states, while less ideal Hamiltonians substantially reduce its effectiveness.

I. INTRODUCTION

  • Lyapunov control uses model-based feedback, but when it is effective for which quantum systems and objectives remains unresolved.The paper addresses this question through convergence analysis.Source: §I · pp. 1–2

A. Quantum states and evolution

  • The analysis uses density operators for pure and mixed states and restricts its discussion to Hamiltonian systems.Pure states are rank-one projectors; pseudo-pure states have two eigenvalues, one with multiplicity one, and generic mixed states have distinct eigenvalues.Source: A. Quantum states and evolution · p. 2

B. Control Problem

  • Hamiltonian evolution preserves the state spectrum, so a target is reachable only when initial and target states have the same spectrum.With density-matrix controllability, this condition is sufficient for reachability; the analysis assumes matching spectra.Source: B. Control Problem · p. 2

C. Trajectory vs Orbit Tracking

  • For most targets, control means trajectory tracking rather than orbit tracking, because matching an orbit is weaker and can conflate distinct states.The paper focuses on trajectory convergence, which it describes as the strongest well-defined notion for arbitrary trajectories.Source: C. Trajectory vs Orbit Tracking · p. 3

D. Control Design based on Lyapunov Function

  • Lyapunov feedback makes the chosen function non-increasing, but LaSalle’s principle then guarantees convergence only to an invariant set, not necessarily to the target.The controlled dynamics are formulated as an autonomous system on the joint state space.Source: D. Control Design based on Lyapunov… · pp. 3–4

III. LASALLE INVARIANCE PRINCIPLE AND LASALLE INVARIANT SET

LaSalle’s invariance principle narrows convergence analysis to the invariant set where the Lyapunov function’s derivative is zero. Applied to the joint quantum-state dynamics, it guarantees convergence to that set, whose characterization is the next step.

  • For an autonomous system with a non-increasing Lyapunov function, every bounded solution converges to the invariant set where its derivative is zero.The principle identifies a set containing the positive limiting sets of bounded solutions.Source: A. Invariance Principle for Autonomous… · p. 4
  • The quantum state pair evolves on a compact finite-dimensional manifold, so its trajectories are bounded and LaSalle’s principle applies.The Lyapunov function is zero exactly when the states coincide, although it is not positive definite on the joint space.Source: A. Invariance Principle for Autonomous… · p. 4
  • Under Lyapunov control, every system evolution converges to the set of state pairs where the Lyapunov derivative is zero; for moving targets, analysis uses the joint space M × M.The invariant set must still be characterized to determine whether convergence reaches the desired trajectory.Source: A. Invariance Principle for Autonomous… · p. 4

B. Characterization of the LaSalle Invariant Set

The section characterizes the LaSalle invariant set through commutator conditions. Even under stringent ideal-Hamiltonian assumptions, that set is generally larger than the desired solution, so invariance alone does not establish convergence to the target.

B. Characterization of the LaSalle Invariant Set

  • LaSalle invariance requires the control field to vanish, reducing the characterization to commutator constraints and the system Hamiltonians’ structure.The transition frequencies and control couplings determine whether the generated subspace spans the Cartan subalgebra.Source: B. Characterization of the LaSalle… · p. 5
  • With strongly regular H0 and fully connected H1, the generated subspace equals the Cartan subalgebra; with zero diagonal elements of H1, membership in the invariant set is equivalent to a diagonal commutator.The theorem’s zero-diagonal condition is an additional assumption on H1.Source: B. Characterization of the LaSalle… · pp. 5–6
  • Even under strongly regular H0 and a fully connected transition graph, the invariant set is generally larger than the target, so LaSalle’s principle alone cannot establish target convergence.This conclusion follows after characterizing the invariant set under those ideal conditions.Source: B. Characterization of the LaSalle… · p. 6

IV. CRITICAL POINTS OF THE LYAPUNOV FUNCTION

The section identifies critical points of the Lyapunov function through commutation relations and classifies their stability when the target state has non-degenerate eigenvalues. Most such critical points are saddles rather than extrema.

IV. CRITICAL POINTS OF THE LYAPUNOV FUNCTION

  • For a stationary target, critical points of the Lyapunov function occur when the state and target commute and are simultaneously diagonalizable.Source: §IV · p. 6
  • The critical points of the two-state Lyapunov function commute, and therefore belong to the LaSalle invariant set.Source: §IV · p. 7
  • For a stationary target with non-degenerate eigenvalues, the function is Morse: all critical points except its global maximum and minimum are saddles.The critical points correspond to permutations of the target state’s eigenvalues.Source: §IV · p. 7

V. LYAPUNOV CONTROL UNDER AN IDEAL HAMILTONIAN

The section analyzes Lyapunov control under ideal Hamiltonians, assuming H0 is strongly regular and H1 is off-diagonal and fully connected. It considers pseudo-pure, generic, and degenerate stationary target states.

V. LYAPUNOV CONTROL UNDER AN IDEAL HAMILTONIAN

  • The analysis assumes strongly regular H0 and an off-diagonal, fully connected H1, and distinguishes pseudo-pure targets, generic targets, and degenerate stationary targets.For pseudo-pure and generic targets, the section identifies state-space dimensions of 2n − 2 and n^2 − n, respectively.Source: §V · p. 7

A. Pseudo-pure state control

For pseudo-pure targets, the invariant set and convergence behavior depend on the target state and Hamiltonian setting. Two-level targets outside the equatorial plane yield trajectory convergence except from the antipode; higher-dimensional targets generally converge to the target trajectory, with a stated exception.

2. Pseudo-pure states for n > 2

  • For n > 2, pseudo-pure targets have spectrum {w, u}, with u = (1 − w)/(n − 1), including pure states with spectrum {1, 0}.The analysis first treats two-level systems as a visual aid for the higher-dimensional pure-state discussion.Source: A. Pseudo-pure state control · p. 7

1. Two-level systems

  • Two-level Hamiltonian assumptions reduce to non-degenerate energy levels and a nonzero coupling b12.The passage states these conditions hold in all but trivial cases.Source: §1 · p. 8
  • For two-level targets outside the equatorial plane, every initial state except the antipode converges to the target trajectory.The invariant set consists of the target and its antipode.Source: §1 · p. 8
  • Equatorial two-level targets have an invariant circle, so convergence is only to a trajectory on that circle and need not approach the target trajectory.The distance from the target trajectory can remain fixed and non-zero.Source: §1 · p. 8

2. Pseudo-pure states for n > 2

  • In the exceptional higher-dimensional case, V may approach any value from 0 to Vmax, and the invariant set can include both identical and orthogonal states.The exceptional target has two nonzero components of equal norm.Source: §2 · pp. 9–10
  • For ideal Hamiltonians, solutions with V(ρ(0), ρd(0)) < Vmax converge to pseudo-pure target trajectories, except for targets with a single pair of nonzero off-diagonal entries.For that exceptional target, solutions converge to its orbit, but need not converge to the target trajectory.Source: §2 · p. 10

B. Generic-state Control

The section distinguishes stationary from non-stationary generic targets by whether the target commutes with H0.

B. Generic-state Control

  • A generic target is stationary exactly when it commutes with H0; when H0 has nonzero eigenvalues, this is equivalent to being diagonal in H0’s eigenbasis.Source: B. Generic-state Control · p. 10

1. Generic stationary target state

For generic stationary targets, the analysis characterizes the LaSalle invariant set and classifies the critical points through stability analysis. Under the stated conditions, almost all trajectories outside the invariant set converge to the target.

1. Generic stationary target state

  • The invariant set reduces to the critical points when both the generic target and its commutator with the state are diagonal.In this case the state itself must be diagonal and commute with the target.Source: §1 · p. 10
  • For generic stationary targets, the invariant set consists of exactly n! critical points commuting with the target and sharing its spectrum.The target is the only hyperbolic sink; the other critical points are hyperbolic saddles or sources.Source: §1 · pp. 10–11
  • The dynamics are not a gradient flow, so the paper uses a real Bloch-vector representation to analyze stability near critical points.In that representation, the adjoint action is a real antisymmetric matrix acting on the state vector.Source: §1 · p. 10
  • The stability proof linearizes the dynamics on the state manifold’s tangent space, where the restriction has n^2 − n nonzero eigenvalues.The full real representation also contains an (n − 1)-dimensional Cartan subspace on which the linearization vanishes.Source: §1 · pp. 10–11
  • Under strong regularity of H0, full connectivity of H1, and non-degenerate target eigenvalues, the linearization has no purely imaginary eigenvalues.The proof rules out the remaining possibilities by showing they conflict with full connectivity and non-degeneracy.Source: §1 · pp. 11–12
  • Almost every trajectory outside the invariant set converges to the target because non-target saddle stable manifolds have measure zero.The target is the only sink, while the global maximum is a source.Source: §1 · p. 12

2. Generic non-stationary target state

For generic non-stationary targets, the invariant set contains only commuting pairs under a full-rank condition that holds for almost all target states. Solutions then converge to a permutation of the target, although simulations—not a measure-zero proof—indicate that almost all converge to the target itself.

Generic non-stationary target state

  • For generic targets, the invariant set contains noncommuting pairs if and only if rank ˜A(⃗sd(0)) < n² − n.A full-rank condition ensures that a diagonal commutator must be zero; the determinant condition in the lemma characterizes this case.Source: §2 · p. 13
  • The set of target states whose invariant set contains noncommuting pairs has measure zero, so a randomly chosen generic target has only commuting pairs with probability one.The exceptional condition requires equal diagonal elements or det(˜A1) = 0.Source: §2 · p. 13
  • When the invariant set contains only commuting pairs, every solution converges to one of the n! permutations of the target trajectory.The proof identifies commuting limiting pairs with permutations of the target state.Source: §2 · p. 13
  • In the interaction picture, all permutation fixed points other than the target are unstable.This stability conclusion applies when the invariant set consists only of commuting pairs.Source: §2 · p. 14
  • For non-stationary targets with a commuting-pair invariant set, simulations suggest almost all solutions converge to the target, but the saddle-converging set is not proved to have measure zero.The authors also state that it is difficult to establish stability near a non-stationary solution constructively.Source: §2 · p. 14

C. Other stationary target states

For stationary targets with degenerate eigenvalues, the target remains a locally asymptotically stable sink. The analysis further concludes that most solutions starting outside the LaSalle invariant set converge to it.

C. Other stationary target states

  • For stationary targets with degenerate eigenvalues, the target is a hyperbolic sink and is locally asymptotically stable.The target is also an isolated hyperbolic critical point of the Lyapunov function.Source: C. Other stationary target states · pp. 14–15
  • Other stationary points in the LaSalle invariant set have center manifolds and include directions with positive-real-part eigenvalues.The target is the only isolated sink; the non-target stationary points cannot be local minima of the Lyapunov function.Source: C. Other stationary target states · p. 15
  • Provided the initial state lies outside the LaSalle invariant set, most solutions converge to the target despite the degenerate eigenvalues.Source: C. Other stationary target states · p. 15

VI. CONVERGENCE OF LYAPUNOV CONTROL FOR REALISTIC SYSTEMS

The paper turns from systems with complete regularity and fully connected transitions to the effects of relaxing those requirements. The analysis uses a qutrit example and notes that the argument generalizes to n-level systems.

VI. CONVERGENCE OF LYAPUNOV CONTROL FOR REALISTIC SYSTEMS

  • The preceding convergence analysis assumes complete regularity of H0 and a fully connected transition graph for H1.Source: §VI · p. 15
  • The section examines how relaxing those system requirements changes the invariant set and convergence properties.Source: §VI · p. 15
  • The analysis is presented for a qutrit, with generalization to n-level systems described as straightforward.Source: §VI · p. 15

A. H1 not fully connected

When H1 does not connect every energy level, the qutrit target has a two-dimensional center manifold. Most trajectories then approach other invariant-set trajectories rather than the target or its orbit.

A. H1 not fully connected

  • For the example, the invariant set requires [ρ1, ρ2] ∈ C ⊕ T13 and includes infinitely many trajectories with β13 ≠ 0.Source: A. H1 not fully connected · p. 16
  • The missing 1–3 coupling leaves two purely imaginary eigenvalues and a two-dimensional center manifold at the target.The remaining four eigenvalues have negative real parts in the qutrit case.Source: A. H1 not fully connected · p. 16
  • With H1 not fully connected, most initial states do not converge to the target or another critical point, but approach trajectories in the invariant set outside the target orbit.The authors identify the set of solutions that converge to the target as measure zero.Source: A. H1 not fully connected · p. 16
  • The target is a center on that manifold, with nearby solutions cycling around it.Solutions outside the invariant set converge exponentially to solutions on the center manifold.Source: A. H1 not fully connected · p. 16

B. H0 not strongly regular

The analysis finds that non-ideal systems can have enlarged LaSalle invariant sets and poor convergence, while ideal systems generally converge to the target except for restricted cases. For a non-strongly-regular drift Hamiltonian, the invariant set and local dynamics near a generic stationary target also depart from the ideal picture.

B. H0 not strongly regular

  • With H0 not strongly regular, the invariant set E is a two-dimensional manifold, and most solutions near a generic stationary target do not converge to it.The local analysis identifies the target as a centre on a centre manifold.Source: B. H0 not strongly regular · pp. 16–17

VII. CONCLUSIONS AND FURTHER DISCUSSIONS

  • The authors extend the control problem to an autonomous system on an extended state space, where the LaSalle invariant set contains all critical points of the distance-like Lyapunov function.For ideal systems with strongly regular H0, fully connected H1, and stationary targets, the invariant set contains only those critical points.Source: §VII · p. 17
  • For generic targets, every critical point except the target is unstable; in ideal systems, the target is the only hyperbolic sink.For generic stationary targets, most states converge to the target, although the non-converging initial states can exceed the finite set of antipodal points.Source: §VII · p. 17
  • Strong regularity of H0 and full connectedness of H1 are stringent assumptions that exclude nearest-neighbour-only systems and systems with equally spaced or degenerate energy levels.The passage notes that many such excluded systems can nevertheless be completely controllable as bilinear Hamiltonian systems.Source: §VII · pp. 17–18
  • For non-ideal systems, the target becomes a centre within an enlarged invariant set, and most solutions converge elsewhere, questioning Lyapunov control's effectiveness for realistic systems.The authors conclude that Lyapunov control may be effective only for a very small subset of controllable quantum systems.Source: §VII · pp. 17–18
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