Measuring Distance in Quantum Many-body Wave Functions

We study quantum chaos by investigating the growing distance of two slightly different initial states under unitary evolution. The distance $d(t)$ is defined to be that of the reduced density matrices, which undergoes a rapid growth from small initial value to the value of independent random states. A clear exponential growing regime is observed in the Floquet spin model when we turn on the non-local power law decay interactions. The operator spreading picture shows that $d(t)$ captures the same physics as the out-of-time-ordered correlator.