The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

Abstract

We perform a comprehensive analysis of the set of parameters {ri} that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time τ, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between τ and the energy spectrum and allowing the classification of {ri} into families organized in a 2-simplex, δ2. Furthermore, the states determined by {ri} are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those ris in δ2 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.

Speed of evolution in entangled fermionic systems

“…The couple of equations ( 20a) and ( 20b) is equivalent to the orthogonality condition (14). Clearly it imposes simultaneous restrictions on both the orthogonality time τ = φ/h and the variables (18a)-(18c), which in turn impose conditions on the distribution {p n }.…”

“…In general, pure states of a n-level autonomous system do not even reach an orthogonal state-they do so provided certain conditions are met both by the Hamiltonian and the probability distribution of the energy eigenstates [12]-, and when they do the orthogonality time may approximate, though not reach, the quantum speed limit [13]. The characterization of the initial states that transform into an orthogonal one in a finite time for a given (time-independent) Hamiltonian, and the identification of those that do it faster, is a problem that remains open for arbitrary n (a comprehensive characterization of the states that reach orthogonality has been addressed in [14] for qutrit systems, n = 3), and acquires relevance in the study of the dynamics of multi-level systems, as well as in the preparation of states for specific information tasks.…”