  The quantum speed limit provides a fundamental bound on how fast a quantum
system can evolve between the initial and the final states under any physical
operation. The celebrated Mandelstam-Tamm (MT) bound has been widely studied
for various quantum systems undergoing unitary time evolution. Here, we prove a
new quantum speed limit using the tighter uncertainty relations for pure
quantum systems undergoing arbitrary unitary evolution. We also derive a
tighter uncertainty relation for mixed quantum states and then derive a new
quantum speed limit for mixed quantum states from it such that it reduces to
that of the pure quantum states derived from tighter uncertainty relations. We
show that the MT bound is a special case of the tighter quantum speed limit
derived here. We also show that this bound can be improved when optimized over
many different sets of basis vectors. We illustrate the tighter speed limit for
pure states with examples using random Hamiltonians and show that the new
quantum speed limit outperforms the MT bound.
