Optimal quantum speed for mixed states

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.