  The minimal time required for a system to evolve between two different states
is an important notion for developing ultra-speed quantum computer and
communication channel. Here, we introduce a new metric for non-degenerate
density operator evolving along unitary orbit and show that this is
experimentally realizable operation dependent metric on quantum state space.
Using this metric, we obtain the geometric uncertainty relation that leads to a
new quantum speed limit. Furthermore, we argue that this gives a tighter bound
for the evolution time compared to any other bound. We also obtain a Levitin
kind of bound for mixed states. We propose how to measure this new distance and
speed limit in quantum interferometry. Finally, the lower bound for the
evolution time of a quantum system is studied for any completely positive trace
preserving map using this metric.
