The underlying order induced by orthogonality and the quantum speed limit

TL;DR

This paper analyzes the parameter space of pure qutrit states evolving under time-independent Hamiltonians, revealing the relationship between energy distribution, orthogonality time, and quantum speed limits, providing a comprehensive classification.

Contribution

It offers an exact solution to the orthogonality condition for qutrits and classifies states based on their orthogonality time relative to quantum speed limits.

Findings

Orthogonality condition solved exactly for qutrits.

Classification of states based on orthogonality time and quantum speed limits.

Complete characterization of parameters influencing qutrit dynamics.

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.