Large deviations in quantum dynamics and complexity

Abstract

We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval $t\le t_{\max }$, and (iii) via the distribution of expectation values over $t\le t_{\max }$. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at $t\sim \mathcal{O}(1)$, independently of system size $N$. (ii) and (iii) reach their longtime limit at $t\sim e^N$ and $t\sim \exp (e^N)$, respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with $t_{\max }$. We propose that the evolution of these values with $t_{\max }$ provides a measure of quantum complexity.

17 pages, 6 figures, comments welcome!