Robust Control Performance for Open Quantum Systems

TL;DR

This work develops a robust-control framework for open quantum systems subject to structured Hamiltonian/Lindblad uncertainties and decoherence, by modeling performance as a disturbance-to-error transmission $\bm T_{\bm{z},\bm{w}}(s,\delta)$ and applying structured singular-value analysis. A specialized #-inversion lemma resolves the $s=0$ pole arising from trace conservation, enabling continuity of robustness measures and extending classical μ-analysis to quantum dynamics, including cases with multiple poles and symmetry-induced degeneracies. The methodology is validated through analyses of pure dephasing and general dissipative dynamics, and a two-qubit cavity example reveals how entanglement (via concurrence) interacts with log-sensitivity and stability margins under decoherence, highlighting nontrivial performance-robustness trade-offs in quantum control. The results offer a principled path to design and bound robust quantum-control schemes, with implications for quantum technologies where decoherence and imperfect state preparation are unavoidable.

Abstract

Robust performance of control schemes for open quantum systems is investigated under classical uncertainties in the generators of the dynamics and nonclassical uncertainties due to decoherence and initial state preparation errors. A formalism is developed to measure performance based on the transmission of a dynamic perturbation or initial state preparation error to the quantum state error. This makes it possible to apply tools from classical robust control such as structured singular value analysis. A difficulty arising from the singularity of the closed-loop Bloch equations for the quantum state is overcome by introducing the #-inversion lemma, a specialized version of the matrix inversion lemma. Under some conditions, this guarantees continuity of the structured singular value at s = 0. Additional difficulties occur when symmetry gives rise to multiple open-loop poles, which under symmetry-breaking unfold into single eigenvalues. The concepts are applied to systems subject to pure decoherence and a general dissipative system example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. A nonclassical performance index, steady-state entanglement quantified by the concurrence, a nonlinear function of the system state, is introduced. Simulations confirm a conflict between entanglement, its log-sensitivity and stability margin under decoherence.

Figures (4)

• Figure 1: (a) Comparison of the norm of the inverse of $\bm \Phi(s)$ for the #-inverse and the Moore-Penrose pseudo-inverse ($(\cdot)^+$). (b,c) Error gains $\|T_{\bm{z},\bm{w}_u}^{u}(\imath\omega,\delta\bm S_k)\|$ as a function of frequency for the structured uncertainties in Eq. \ref{['eq:S1234567']} and different sizes $\delta\in\{0.1,1\}$. Due to the pairwise similarities $\bm S_1 \sim \bm S_2$, $\bm S_3 \sim \bm S_4$ and $\bm S_6 \sim \bm S_7$, the cases $\bm S_2$, $\bm S_4,$ and $\bm S_6$ are not plotted.
• Figure 2: The maximum gain for the structured uncertainties in Eq. \ref{['eq:S1234567']} suggests that for small $\delta$ ($\delta<0.1$) the system is most sensitive to perturbations of type $\bm S_1$ while for larger $\delta$ sensitivity to $\bm S_3$ dominates.
• Figure 3: (a) Upper bounds on the $\mu_\mathcal{D}$ bounding the error transmission $\bm T_{\bm{z},\bm{w}_u}^u(\imath \omega, \delta \bm S_k)$ for the structured uncertainties in Eq. \ref{['eq:S1234567']} under frequency sweep $s=\imath\omega$. (b) upper and lower bounds for $s \downarrow 0$ along the real axis. Upper and lower bounds in (b) coincide for $\bm S_3$, $\bm S_5$, and $\bm S_7$. $\bm S_1$ displays aberrant behavior under frequency and real axis sweep. (c) Upper bounds on ${\mu}_\mathcal{D}^{(\#)}$ of initial state error transmission $\bar{T}_{\bm{z},\bm{z}_0}^u(s,\delta \bm S_k)$ for the structured uncertainties in Eq. \ref{['eq:S1234567']} for $s\downarrow0$ along the real axis.
• Figure 4: Maximum of the real part of the eigenvalues $\lambda$ of $\overline{\bm A}+\Delta (\overline{\bm S}_3+\overline{\bm S}_4)$, concurrence of steady-state and log-sensitivity of steady-state concurrence as function of detuning $\Delta$ for $\alpha=\gamma=1$. All three figures of merit are concordant, i.e. they decrease with increasing detuning.

Theorems & Definitions (30)

• Example 1
• Theorem 1
• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 2
• proof
• Definition 1
• Lemma 3