The underlying order induced by orthogonality and the quantum speed limit

Abstract

We perform a comprehensive analysis of the set of parameters $\{r_i\}$ that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time $\tau$, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between $\tau$ and the energy spectrum and allowing the classification of $\{r_i\}$ into families organized in a 2-simplex, ${\delta }^2$. Furthermore, the states determined by $\{r_i\}$ are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those $r_i$s in ${\delta }^2$ correspondent to states whose orthogonality time is limited by the Mandelstam--Tamm bound from those restricted by the Margolus--Levitin one. Our results offer a complete characterization of the physical quantities that become relevant in both the preparation and study of the dynamics of three-level states evolving towards orthogonality.

orthogonality time; quantum speed limit; three-level systems; dynamics towards orthogonality