Generalized Lindblad master equation for measurement-induced phase transition

The measurement-induced phase transition (MIPT) occurs when the system is evolving under unitary evolution together with local measurements followed by post-selection. We propose a generalized version of the Lindblad master equation as a continuous equa-tion, to describe the dynamics of the second Re ' nyi entropy in the MIPT. This generalized Lindblad equation explicitly takes into account the post-selection in the MIPT, which is realized as the Einstein-Podolsky-Rosen (EPR) state projection in the equation. Also, this generalized Lindblad equation preserves the Hermitian, unit trace, and positive def-initeness of the density matrix. We further use the hard-core Bose-Hubbard model as a concrete example to numerically confirm that our generalized Lindblad equation is applicable to describing the MIPT.