Operator size growth in Lindbladian SYK

We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with $q$-body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite $q$ and analytically at large $q$. With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large $q$ limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large $q$, leading to classical dynamics of the operator size growth with dissipation.