Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems

The purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems. A general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented. Fundamental properties of stability, dissipation, passivity, and gain for this class of linear quantum models are presented and characterized using complex Lyapunov equations and linear matrix inequalities (LMIs). Coherent $H^\infty$ and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.

💡 Research Summary

The paper addresses the design of coherent feedback controllers for a class of linear quantum stochastic systems, extending the conventional framework that only accounts for indirect (field‑mediated) couplings to also include direct (Hamiltonian‑mediated) couplings between the plant and auxiliary quantum systems. The authors first construct a comprehensive physical model in which the plant’s state vector evolves according to a quantum stochastic differential equation (QSDE) of the form

dx = (A + A_d) x dt + B dw, y = C x dt + D dw,

where A_d = 2 J K represents the contribution of a direct coupling Hamiltonian term (K is a real symmetric matrix, J is the canonical symplectic matrix). Indirect coupling is captured through the usual SLH parameters (S, L, H) and appears in the matrices B, C, D. The model respects the physical realizability (PR) constraints for linear quantum systems, which enforce a J‑symmetry on the system matrices.

Stability, dissipation, passivity, and gain are analyzed using complex Lyapunov equations and linear matrix inequalities (LMIs). Because direct coupling generally makes the system matrix A non‑Hermitian, the authors employ a complex positive‑definite Lyapunov matrix P (P > 0) and formulate the condition A†P + PA < 0 as an LMI. Dissipativity is expressed through a supply rate S = y†y − w†w ≥ 0, leading to the LMI (A†P + PA + C†C) ≤ 0. They prove that if K is negative semidefinite, the system automatically satisfies a passivity condition, highlighting the energetic role of direct coupling.

For H∞ synthesis, the goal is to minimize the induced L₂ gain γ from the external quantum noise w to the regulated output y. Traditional H∞ design relies on Riccati equations, which may not admit solutions when direct couplings are present. To overcome this, the authors propose a multistep optimization procedure:

1. Step 1 (Indirect‑only design): Solve an LMI feasibility problem to obtain preliminary controller matrices (X₁, Y₁) that guarantee PR and a baseline γ.
2. Step 2 (Incorporate direct coupling): Introduce K as an additional decision variable and reformulate the H∞ LMI using the Schur complement, thereby linearizing the cross‑terms between K and the previously obtained variables.

The optimization iterates between these steps until γ ceases to improve, guaranteeing convergence under standard convexity assumptions. The same multistep scheme is adapted for LQG synthesis, where the quadratic cost J = E