| Summary: | We perform a comprehensive analysis of the set of parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>r</mi><mi>i</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> and the energy spectrum and allowing the classification of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>r</mi><mi>i</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> into families organized in a 2-simplex, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>δ</mi><mn>2</mn></msup></semantics></math></inline-formula>. Furthermore, the states determined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>r</mi><mi>i</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>i</mi></msub></semantics></math></inline-formula>s in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>δ</mi><mn>2</mn></msup></semantics></math></inline-formula> 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.
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