Coherent feedback $H^\infty$ control of quantum linear systems

1. Problem Statement

The paper addresses the coherent feedback $H_\infty$ control problem for linear quantum systems.

• Context: In quantum control, a "coherent" controller is a quantum system itself that interacts directly with the plant (the system to be controlled) via quantum fields, without converting signals to classical data (which would introduce measurement back-action and decoherence).
• Objective: Design a physically realizable quantum controller $K$ for a quantum plant $P$ such that:
  1. The closed-loop system is internally stable.
  2. The $H_\infty$ norm of the transfer function from disturbance inputs ($w_1$) to performance outputs ($z$) is less than a prescribed attenuation level $\gamma$ (i.e., $\|T_{w_1 \to z}\|_\infty < \gamma$).
• Constraint: The controller must satisfy physical realizability conditions. Unlike classical controllers, a quantum controller must preserve the canonical commutation relations (CCR) of the quantum fields. This imposes strict algebraic constraints on the controller's system matrices.

2. Methodology

The authors utilize the real quadrature operator representation of quantum linear systems, transforming the complex annihilation-creation operator formalism into a real-valued state-space framework.

• System Modeling: The plant and controller are modeled using Itô quantum stochastic differential equations (QSDEs). The system matrices satisfy specific physical realizability conditions (e.g., $A + A^\flat + BB^\flat = 0$).
• Standard Approach vs. Proposed Approach:
  • Standard: Existing literature (e.g., Ref. [8]) solves the problem by finding solutions to two coupled Algebraic Riccati Equations (AREs). This is computationally intensive and requires solving non-linear matrix equations.
  • Proposed: The authors exploit the specific structural properties of quantum linear systems (specifically the relationship between the system matrices $A_x$ and $A_y$) to reformulate the problem.
• Key Mathematical Transformation:
  • The authors define auxiliary matrices $A_x$ and $A_y$ derived from the plant matrices.
  • They demonstrate that for quantum systems, $A_y = -A_x^\sharp$ (where $\sharp$ denotes a specific adjoint operation related to the symplectic structure).
  • By applying Schur decomposition to $A_x$, they separate the system into stable ($A_{x1}$) and anti-stable ($A_{x3}$) subspaces.
  • Instead of solving coupled AREs, they show that the existence of a solution depends on solving linear Lyapunov equations.

3. Key Contributions

A. Simplification of Synthesis (Theorem 7)

The primary contribution is a significant reduction in computational complexity.

• Result: For a general linear quantum plant, a physically realizable $H_\infty$ controller can be constructed by solving at most four Lyapunov equations (linear equations), rather than two coupled non-linear AREs.
• Conditions: The existence of a controller is guaranteed if there exist positive definite solutions $S, T, U, V$ to specific Lyapunov equations satisfying:
  1. $S - T/\gamma^2 > 0$
  2. $U - V/\gamma^2 > 0$
  3. A spectral radius condition: $\sigma_{\min}(S - T/\gamma^2)\sigma_{\min}(U - V/\gamma^2) < \gamma^2$.

B. Symmetric Case Simplification (Corollary 9)

For a subclass of systems where the matrix $A_x$ is symmetric (common in many optical systems), the condition simplifies further. The spectral radius condition becomes a necessary and sufficient condition, and the coordinate transformation matrix becomes the identity, removing the need for complex basis changes.

C. Passive Systems (Theorem 12)

For passive quantum linear systems (where dynamics are governed solely by annihilation operators), the authors derive a necessary and sufficient condition based on two uncoupled pairs of Lyapunov equations. This is a major simplification over the general case, as the coupling terms vanish entirely.

4. Results and Validation

The theoretical results were validated through two detailed examples in quantum optics:

1. Empty Optical Cavity (Passive Case):

   • The authors applied the passive case theory (Theorem 12).
   • They derived an analytical expression for the minimal disturbance attenuation level $\gamma$.
   • Finding: The physical realizability condition was satisfied without adding extra vacuum noise channels, though it required a slightly higher $\gamma$ than the theoretical lower bound, illustrating a trade-off between performance and physical realizability.

2. Degenerate Parametric Amplifier (DPA) (Non-Passive Case):

   • The authors applied the general case theory (Theorem 7) to a non-passive system (active device).
   • They analyzed two scenarios based on the relationship between pump strength and cavity decay rates.
   • Finding: In one case, the controller was physically realizable with a specific analytical $\gamma$. In the other, the condition for physical realizability led to a high-degree polynomial equation, demonstrating the complexity of active systems but confirming the method's applicability.

5. Significance

• Computational Efficiency: The shift from solving coupled non-linear AREs to linear Lyapunov equations drastically reduces the computational burden, making robust control design feasible for larger or more complex quantum systems.
• Theoretical Insight: The paper reveals that the "coupling" in standard AREs for quantum systems is an artifact of the general formulation. The inherent symplectic structure of quantum mechanics allows for decoupling and linearization of the synthesis problem.
• Practical Application: The method provides a direct, algorithmic procedure for designing robust controllers for quantum optical systems (cavities, amplifiers) and optomechanical systems, which are critical for quantum communication, computing, and metrology.
• Physical Realizability: The framework explicitly integrates the constraints of quantum physics (CCR preservation) into the control synthesis, ensuring that the resulting controllers are not just mathematically valid but physically implementable.

In conclusion, this paper establishes a streamlined, computationally efficient framework for coherent feedback $H_\infty$ control, replacing the standard heavy machinery of coupled Riccati equations with a more tractable set of Lyapunov equations, while rigorously ensuring physical realizability.