  The question of how fast a quantum state can evolve is considered. Using the
definition of squared speed based on the Euclidean distance given in [Phys.
Rev. Research, {\bf 2}, 033127 (2019)], we present a systematic framework to
obtain the optimal speed of a $d$-dimensional system evolved unitarily under a
time-independent Hamiltonian. Among the set of mixed quantum states having the
same purity, the optimal state is obtained in terms of its purity parameter. We
show that for an arbitrary $d$, the optimal state is represented by a $X$-state
with an additional property of being symmetric with respect to the secondary
diagonal. For sufficiently low purities for which the purity exceeds the purity
of maximally mixed state $\Id/d$ by at most $2/d^2$, the only nonzero
off-diagonal entry of the optimal state is $\varrho_{1d}$, corresponding to the
transition amplitude between two energy eigenstates with minimum and maximum
eigenvalues, respectively. For larger purities, however, whether or not the
other secondary diameter entries $\varrho_{i,d-i+1}$ take nonzero values
depends on their relative energy gaps $|E_{d-i+1}-E_{i}|$. The effects of
coherence and entanglement, with respect to the energy basis, are also examined
and found that for optimal states both resources are monotonic functions of
purity, so they can cause speed up quantum evolution leading to a smaller
quantum speed limit. Our results show that although the coherence of the states
is responsible for the speed of evolution, only the coherence caused by some
off-diagonal entries located on the secondary diagonal play a role in the
fastest states.
