Quantum speed limit theorems

Quantum speed limit theorems are quantum mechanics theorems concerning the orthogonalization interval, the minimum time for a quantum system to evolve between two orthogonal states, also known as the quantum speed limit.

Consider an initial pure quantum state expressed as a superposition of its energy eigenstates

${\displaystyle \left|\psi (0)\right\rangle =\sum _{n}c_{n}\left|E_{n}\right\rangle }$.

If the state ${\displaystyle \left|\psi (0)\right\rangle }$ is let to evolve for an interval ${\displaystyle \delta t}$ by the Schrödinger equation it becomes

${\displaystyle \left|\psi (\delta t)\right\rangle =\sum _{n}c_{n}e^{-i{\frac {E_{n}\delta t}{\hbar }}}\left|E_{n}\right\rangle }$,

where ${\displaystyle \hbar ={\frac {h}{2\pi }}}$ is the reduced Planck constant. If the initial state ${\displaystyle \left|\psi (0)\right\rangle }$ is orthogonal to the evolved state ${\displaystyle \left|\psi (\delta t)\right\rangle }$ then ${\displaystyle \left\langle \psi (0)|\psi (\delta t)\right\rangle =0}$ and the minimum interval ${\displaystyle \delta t_{\perp }}$ required to achieve this condition is called the orthogonalization interval or time.