  The attempt to gain a theoretical understanding of the concept of time in
quantum mechanics has triggered significant progress towards the search for
faster and more efficient quantum technologies. One of such advances consists
in the interpretation of the time-energy uncertainty relations as lower bounds
for the minimal evolution time between two distinguishable states of a quantum
system, also known as quantum speed limits. We investigate how the non
uniqueness of a bona fide measure of distinguishability defined on the quantum
state space affects the quantum speed limits and can be exploited in order to
derive improved bounds. Specifically, we establish an infinite family of
quantum speed limits valid for unitary and nonunitary evolutions, based on an
elegant information geometric formalism. Our work unifies and generalizes
existing results on quantum speed limits, and provides instances of novel
bounds which are tighter than any established one based on the conventional
quantum Fisher information. We illustrate our findings with relevant examples,
demonstrating the importance of choosing different information metrics for open
system dynamics, as well as clarifying the roles of classical populations
versus quantum coherences, in the determination and saturation of the speed
limits. Our results can find applications in the optimization and control of
quantum technologies such as quantum computation and metrology, and might
provide new insights in fundamental investigations of quantum thermodynamics.
