Quantum systems equilibrate rapidly for most observables

Considering any Hamiltonian, any initial state, and measurements with a small number of possible outcomes compared to the dimension, we show that most measurements are already equilibrated. To investigate nontrivial equilibration, we therefore consider a restricted set of measurements. When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.

Figure 1

$D_{{\rho }_0}({\rho }_t,\omega )$ and its finite-time average for a full period of the harmonic oscillator with level spacing $\nu$ and the initial condition is a pure state spread equally over the first 50 energy levels (irrespective of phases). Notice that, despite the revival (blue solid line), the projector still equilibrates (red dotted line).

Figure 2

Graphic illustration of ${\eta }_{\epsilon }$. The vertical lines display the probability distribution in energy space of a three-dimensional harmonic oscillator with energy levels $E_n=(n+1/2)\nu$, under the Boltzmann distribution (accounting for degeneracies) with temperature $=10\nu$. The blue shaded region represents ${\eta }_{8\nu }$ (the maximum probability that can be found inside an energy interval of size $8\nu$).