The robustness of quantum control in the presence of uncertainties is
important for practical applications but their quantum nature poses many
challenges for traditional robust control. In addition to uncertainties in the
system and control Hamiltonians and initial state preparation, there is
uncertainty about interactions with the environment leading to decoherence.
This paper investigates the robust performance of control schemes for open
quantum systems subject to such uncertainties. A general formalism is
developed, where performance is measured based on the transmission of a dynamic
perturbation or initial state preparation error to a final density operator
error. This formulation makes it possible to apply tools from classical robust
control, especially structured singular value analysis, to assess robust
performance of controlled, open quantum systems. However, there are additional
difficulties that must be overcome, especially at low frequency (s≈0).
For example, at s=0, the Bloch equations for the density operator are
singular, and this causes lack of continuity of the structured singular value.
We address this issue by analyzing the dynamics on invariant subspaces and
defining a pseudo-inverse that enables us to formulate a specialized version of
the matrix inversion lemma. The concepts are demonstrated with an example of
two qubits in a leaky cavity under laser driving fields and spontaneous
emission. In addition, a new performance index is introduced for this system.
Instead of the tracking or transfer fidelity error, performance is measured by
the steady-steady entanglement generated, which is quantified by a non-linear
function of the system state called concurrence. Simulations show that there is
no conflict between this performance index, its log-sensitivity and stability
margin under decoherence, unlike for conventional control problems [...].Comment: 12 pages, 5 figures, 2 table