  As shown in previous work, in some cases closed quantum systems exhibit a
non-conventional trade-off in performance and robustness in the sense that
controllers with the highest fidelity can also provide the best robustness to
parameter uncertainty. As the dephasing induced by the interaction of the
system with the environment guides the evolution to a more classically mixed
state, it is worth investigating what effect the introduction of dephasing has
on the relationship between performance and robustness. In this paper we
analyze the robustness of the fidelity error, as measured by the logarithmic
sensitivity function, to dephasing processes. We show that introduction of
dephasing as a perturbation to the nominal unitary dynamics requires a
modification of the log-sensitivity formulation used to measure robustness
about an uncertain parameter with non-zero nominal value used in previous work.
We consider controllers optimized for a number of target objectives ranging
from fidelity under coherent evolution to fidelity under dephasing dynamics to
determine the extent to which optimizing for a specific regime has desirable
effects in terms of robustness. Our analysis is based on two independent
computations of the log-sensitivity: a statistical Monte Carlo approach and an
analytic calculation. We show that despite the different log sensitivity
calculations employed in this study, both demonstrate that the log-sensitivity
of the fidelity error to dephasing results in a conventional trade-off between
performance and robustness.
