  We study the operator size growth in open quantum systems with all-to-all
interactions, in which the operator size is defined by counting the number of
non-trivial system operators. We provide a general argument for the existence
of a transition of the operator size dynamics when the system-bath coupling
$\gamma$ is tuned to its critical value $\gamma_c$. We further demonstrate the
transition through the analytical calculation of the operator size distribution
in a solvable Brownian SYK model. Our results show that: (i) For
$\gamma>\gamma_c$, the system is in a dissipative phase where the system
operator size decays with a rate $\sim (\gamma-\gamma_c)$, which indicates the
initial information of the system all dives into the bath eventually. (ii) For
$\gamma<\gamma_c$, the system sustains a scrambling phase, where the average
operator size grows exponentially up to the scrambling time $t_s\sim
(\gamma_c-\gamma)^{-1}\log N$ and saturates to a $O(N)$ value in the long-time
limit. (iii) At the critical point $\gamma=\gamma_c$, which separates the two
phases, the operator size distribution at finite size shows a power-law decay
over time.

Quantum Physics;Quantum Gases;Statistical Mechanics;Strongly Correlated Electrons;High Energy Physics - Theory