  We present rigorous performance bounds for the optimal dynamical decoupling
pulse sequence protecting a quantum bit (qubit) against pure dephasing. Our
bounds apply under the assumption of instantaneous pulses and of bounded
perturbing environment and qubit-environment Hamiltonians. We show that if the
total sequence time is fixed the optimal sequence can be used to make the
distance between the protected and unperturbed qubit states arbitrarily small
in the number of applied pulses. If, on the other hand, the minimum pulse
interval is fixed and the total sequence time is allowed to scale with the
number of pulses, then longer sequences need not always be advantageous. The
rigorous bound may serve as testbed for approximate treatments of optimal
decoupling in bounded models of decoherence.
