Some quantum states can dynamically evolve to an orthogonal state after a time interval, while some states can never evolve to their orthogonal states. In this paper, we investigate three issues on the orthogonalization properties of quantum states. First, we answer the question as to under what conditions an initial state can evolve to an orthogonal state. Second, we discuss the orthogonalization rate of quantum states, i.e. , the total number of orthogonal states that the evolution can go through in unit time. Finally, we show that the Margolus-Levitin bound for orthogonalization time remains tight for superpositions of more than two eigenstates.
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least $\ensuremath{\pi}\ensuremath{\alpha}(\ensuremath{\epsilon})/2\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$, where $\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\ensuremath{\alpha}(\ensuremath{\epsilon})$ is a function of the fidelity $\ensuremath{\epsilon}$ between the two states. For a long time, only an upper bound ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and a lower bound ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$ are known, although they agree up to at least seven significant figures. Recently, H\"ornedal and S\"onnerborn [arXiv:2301.10063] proved an analytical expression for $\ensuremath{\alpha}(\ensuremath{\epsilon})$, a fully classified system whose evolution time saturates the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ is indeed equal to ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$. More importantly, I point out a numerical stability issue in computing ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and report a simple way to evaluate it efficiently and accurately.
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least
πα(ε)/(2⟨E−E0⟩), where
⟨E−E0⟩ is the expected energy of the system relative to the ground state of the Hamiltonian and
α(ε) is a function of the fidelity
ε between the two state. For a long time, only a upper bound
α>(ε) and lower bound
α<(ε) are known although they agree up to at least seven significant figures. Lately, Hörnedal and Sönnerborn proved an analytical expression for
α(ε), fully classified systems whose evolution times saturate the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that
α>(ε) is indeed equal to
α<(ε). More importantly, I point out a numerical stability issue in computing
α>(ε) and report a simple way to evaluate it efficiently and accurately.
The Margolus-Levitin Theorem is a limitation on the minimal evolving time of a quantum system from its initial state to its orthogonal state. It also supplies a bound on the maximal operations or events can occur within a volume of spacetime. We compare Margolus-Levitin Theorem with other form of limitation on minimal evolving time, and present covariant versions of Margolus-Levitin Theorem in special and general relativity respectively. The relation among entropy bound, maximum information flow and computational limit are discussed, by applying the covariant versions of Margolus-Levitin Theorem.