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The rate of the trace distance is used to evaluate quantum speed-up forarbitrary mixed states. Compared with some present methods, the approach basedon trace distance can provide an optimal bound to the speed of the evolution.The dynamical transition from no speed-up region to speed-up region takes on inthe spontaneous decay of an two-level atom with detuning. The evolution ischaracteristic of the alternating behavior between quantum speed-up andspeed-down in the strong system-reservoir coupling regime. Under theoff-resonance condition, the dynamical evolution can be accelerated for shortprevious times and then decelerated to a normal process either in the weak orstrong coupling regime. From the time-energy uncertainty relation, wedemonstrate that the potential capacity for quantum speed-up evolution isclosely related to the energy flow-back from the reservoir to the system. Thenegative decay rate for short time intervals leads to the speed-up processwhere the photons previously emitted by the atom are reabsorbed at a latertime. The values of the spontaneous decay rate becomes positive after a longenough time, which results in the normal evolution with no speed-up potential.

Quantum speed-up transition in open system dynamics