The underlying order induced by orthogonality and the quantum speed limit.

We consider the set of pure three-level states that reach an orthogonal state at a finite time $\tau$, when evolving under an arbitrary, time-independent, Hamiltonian. We perform a comprehensive analysis of the exact solutions of the orthogonality condition, determining the allowed energy probability distributions $\{r_{i}\}$, and their non-trivial interrelation with $\tau$ and the Hamiltonian eigenvalues. The sets $\{r_{i}\}$ are classified and geometrically organized in a 2-simplex, $\delta^{2}$, contained in the probability 2-simplex of $\mathbb{R}^{3}$ that conforms the set of all probability distributions. Furthermore, a map of the quantum speed limit of the states associated to each $\{r_{i}\}$ is constructed on $\delta^{2}$, according to whether the quantum speed limit of the state is given by the Madelstam-Tamm or the Margolus-Levitin bound.