  The quantum speed limit is a fundamental concept in quantum mechanics, which
aims at finding the minimum time scale or the maximum dynamical speed for some
fixed targets. In a large number of studies in this field, the construction of
valid bounds for the evolution time is always the core mission, yet the physics
behind it and some fundamental questions like which states can really fulfill
the target, are ignored. Understanding the physics behind the bounds is at
least as important as constructing attainable bounds. Here we provide an
operational approach for the definition of the quantum speed limit, which
utilizes the set of states that can fulfill the target to define the speed
limit. Its performances in various scenarios have been investigated. For
time-independent Hamiltonians, it is inverse-proportional to the difference
between the highest and lowest energies. The fact that its attainability does
not require a zero ground-state energy suggests it can be used as an indicator
of quantum phase transitions. For time-dependent Hamiltonians, it is shown that
contrary to the results given by existing bounds, the true speed limit should
be independent of the time. Moreover, in the case of spontaneous emission, we
find a counterintuitive phenomenon that a lousy purity can benefit the
reduction of the quantum speed limit.
