Statistical Mechanics of Monitored Dissipative Random Circuits

Dissipation is inevitable in realistic quantum circuits. We examine the effects of dissipation on a class of monitored random circuits that exhibit a measurement-induced entanglement phase transition. This transition was previously understood as an order-to-disorder transition of an effective classical spin model. We extend this mapping to include on-site dephasing and amplitude damping, study the corresponding 2D Ising model with generalized interactions, and develop diagrammatic methods for the exact Boltzmann weights of the bonds in terms of the probability of measurement p, the dissipation rate Γ and the on-site Hilbert space dimension q. The dissipation plays the role of Z2-symmetry-breaking interactions, while small measurement rates reduce the ratio of the symmetry-breaking interactions to the pairwise interactions, conducive to long-range order. We analyze the dynamical regimes of the R ́enyi mutual information and find that the joint action of monitored measurements and dissipation yields short time, intermediate time, and steady-state behavior that can be understood in terms of crossovers between different classical domain wall configurations. The presented analysis applies to monitored open or Lindbladian quantum systems and provides a tool to understand entanglement dynamics in realistic dissipative settings and achievable system sizes.